--- /dev/null
+presentations/ica_hackathon_2022/ica_hackathon_2022_files/*
+*.RData
+*.Rhistory
--- /dev/null
+/auto/
+/vc
+/refs-processed.bib
+/*.aux
+/*.bbl
+/*.blg
+/*.log
+/*.out
+/*.pdf
+/*.fdb_latexmk
+/*.fls
+/*.ttt
--- /dev/null
+#!/usr/bin/make
+
+all: $(patsubst %.tex,%.svg,$(wildcard *.tex)) $(patsubst %.tex,%.png,$(wildcard *.tex))
+
+%.png: %.pdf
+ convert -density 300 -transparent white $< $@
+
+%.svg: %.pdf
+ /usr/bin/inkscape $< --export-plain-svg=$@
+
+%.pdf: %.tex
+ latexmk -f -pdf $<
+
+clean:
+ latexmk -C *.tex
+ rm -f *.tmp
+ rm -f vc
+ rm *.svg
+
+viewpdf: all
+ evince *.pdf
+
+vc:
+ vc-git
+
+pdf: all
+
+.PHONY: clean all
--- /dev/null
+\documentclass[12pt]{standalone}
+
+\usepackage{ucs}
+\usepackage[utf8x]{inputenc}
+
+\usepackage[T1]{fontenc}
+\usepackage{textcomp}
+\renewcommand{\rmdefault}{ugm}
+\renewcommand{\sfdefault}{phv}
+\usepackage[garamond]{mathdesign}
+\usepackage{tikz}
+\usetikzlibrary{positioning, shapes, arrows, shadows, }
+
+\begin{document}
+
+\tikzset{>=latex}
+\begin{tikzpicture}[
+ observed/.style={circle, draw},
+ unobserved/.style={draw, circle, fill=gray!40},
+ legend/.style={rectangle, draw},
+ partly observed/.style 2 args={draw, fill=#2, path picture={
+ \fill[#1, sharp corners] (path picture bounding box.south west) -|
+ (path picture bounding box.north east) -- cycle;},
+ circle}
+ ]
+
+
+ \node[observed] (y) {Y};
+% \node[observed,above=of y, xshift=1cm] (r) {R};
+ \node[partly observed={white}{gray!40}, above=of y] (x) {X};
+ \node[observed, left=of x,xshift=-1] (w) {W};
+ \node[unobserved, above=of w, xshift=1cm] (k) {K};
+
+ \matrix [draw, below, yshift=-0.2cm, font=\small, align=center, column sep=2\pgflinewidth, inner sep=0.6em, outer sep=0em, nodes={align=center, anchor=center}] at (current bounding box.south){
+ \node[observed,label=right:observed] {}; \\
+ \node[unobserved,label=right:unobserved]{}; \\
+ };
+
+ \draw[->] (x) -- (y);
+ \draw[->] (x) -- (w);
+ \draw[-] (k) -- (x);
+ \draw[->] (k) -- (w);
+
+\end{tikzpicture}
+\end{document}
+
--- /dev/null
+/auto/
+/vc
+/refs-processed.bib
+/*.aux
+/*.bbl
+/*.blg
+/*.log
+/*.out
+/*.pdf
+/*.fdb_latexmk
+/*.fls
+/*.ttt
--- /dev/null
+#!/usr/bin/make
+
+all: $(patsubst %.tex,%.svg,$(wildcard *.tex)) $(patsubst %.tex,%.png,$(wildcard *.tex))
+
+%.png: %.pdf
+ convert -density 300 -transparent white $< $@
+
+%.svg: %.pdf
+ /usr/bin/inkscape $< --export-plain-svg=$@
+
+%.pdf: %.tex
+ latexmk -f -pdf $<
+
+clean:
+ latexmk -C *.tex
+ rm -f *.tmp
+ rm -f vc
+ rm *.svg
+
+viewpdf: all
+ evince *.pdf
+
+vc:
+ vc-git
+
+pdf: all
+
+.PHONY: clean all
--- /dev/null
+\documentclass[12pt]{standalone}
+
+\usepackage{ucs}
+\usepackage[utf8x]{inputenc}
+
+\usepackage[T1]{fontenc}
+\usepackage{textcomp}
+\renewcommand{\rmdefault}{ugm}
+\renewcommand{\sfdefault}{phv}
+\usepackage[garamond]{mathdesign}
+\usepackage{tikz}
+\usetikzlibrary{positioning, shapes, arrows, shadows}
+
+\begin{document}
+
+\tikzset{>=latex}
+\begin{tikzpicture}[
+ observed/.style={circle, draw},
+ partly observed/.style 2 args={draw, fill=#2, path picture={
+ \fill[#1, sharp corners] (path picture bounding box.south west) -|
+ (path picture bounding box.north east) -- cycle;},
+ circle},
+ unobserved/.style={draw, circle, fill=gray!40}
+ ]
+
+ \node[observed] (y) {Y};
+ \node[observed,above=of y, xshift=1cm] (r) {R};
+ \node[partly observed={white}{gray!40}, left=of r] (x) {X};
+ \node[observed, left=of x] (w) {W};
+ \node[unobserved, above=of w, xshift=1cm] (k) {K};
+
+ \matrix [draw, below, yshift=-0.2cm, font=\small, align=center, column sep=2\pgflinewidth, inner sep=0.6em, outer sep=0em, nodes={align=center, anchor=center}] at (current bounding box.south){
+ \node[observed,label=right:observed] {}; \\
+ \node[unobserved,label=right:unobserved]{}; \\
+ };
+
+
+ \draw[->] (r) -- (y);
+ \draw[->] (r) -- (x);
+ \draw[->] (x) -- (y);
+ \draw[->] (x) -- (w);
+ \draw[->] (k) -- (x);
+ \draw[->] (k) -- (w);
+
+\end{tikzpicture}
+\end{document}
+
--- /dev/null
+/auto/
+/vc
+/refs-processed.bib
+/*.aux
+/*.bbl
+/*.blg
+/*.log
+/*.out
+/*.pdf
+/*.fdb_latexmk
+/*.fls
+/*.ttt
--- /dev/null
+#!/usr/bin/make
+
+all: $(patsubst %.tex,%.svg,$(wildcard *.tex)) $(patsubst %.tex,%.png,$(wildcard *.tex))
+
+%.png: %.pdf
+ convert -density 300 -transparent white $< $@
+
+%.svg: %.pdf
+ /usr/bin/inkscape $< --export-plain-svg=$@
+
+
+%.pdf: %.tex
+ latexmk -f -pdf $<
+
+clean:
+ latexmk -C *.tex
+ rm -f *.tmp
+ rm -f vc
+ rm *.svg
+
+viewpdf: all
+ evince *.pdf
+
+vc:
+ vc-git
+
+pdf: all
+
+.PHONY: clean all
--- /dev/null
+\documentclass[12pt]{standalone}
+
+\usepackage{ucs}
+\usepackage[utf8x]{inputenc}
+
+\usepackage[T1]{fontenc}
+\usepackage{textcomp}
+\renewcommand{\rmdefault}{ugm}
+\renewcommand{\sfdefault}{phv}
+\usepackage[garamond]{mathdesign}
+\usepackage{tikz}
+\usetikzlibrary{positioning, shapes, arrows, shadows}
+
+\begin{document}
+
+\tikzset{>=latex}
+\begin{tikzpicture}[
+ observed/.style={circle, draw},
+ partly observed/.style 2 args={draw, fill=#2, path picture={
+ \fill[#1, sharp corners] (path picture bounding box.south west) -|
+ (path picture bounding box.north east) -- cycle;},
+ circle},
+ unobserved/.style={draw, circle, fill=gray!40}
+ ]
+
+ \node[observed] (y) {Y};
+% \node[observed,above=of y, xshift=1cm] (r) {R};
+ \node[partly observed={white}{gray!40}, above = of y] (x) {X};
+ \node[observed, left=of x] (w) {W};
+ \node[unobserved, above=of w, xshift=1cm] (k) {K};
+ \node[unobserved, right=of k] (u) {U};
+
+% \draw[->] (r) -- (y);
+% \draw[->] (r) -- (x);
+ \draw[->] (x) -- (y);
+ \draw[->] (x) -- (w);
+ \draw[->] (x) -- (k);
+ \draw[->] (k) -- (w);
+ \draw[->] (u) to [out=270,in=30] (y);
+ \draw[->] (u) -- (k);
+
+
+ \matrix [draw, below, yshift=-0.2cm, inner sep=0.6em, outer sep=0em, nodes={align=center, anchor=center}] at (current bounding box.south){
+ \node[observed,label=right:observed] {}; \\
+ \node[unobserved,label=right:unobserved]{}; \\
+ };
+
+
+\end{tikzpicture}
+\end{document}
+
--- /dev/null
+ls()
+weight
+weight
+lablr
+labelr
+nrow(labelr)
+names(labelr)
+names(labelr$data)
+labelr$data
+labelr
+names(labelr)
+labelr$labelr
+labelr$toxic
+setwd("..")
+q()
+n
+summary(toxicity_calibrated)
+qplot(labelr$toxic,type='hist')
+names(labelr)
+labelr$n
+labelr
+names(labelr)
+fbyg
+gghist(fbyg$weight)
+hist(fbyg$weight)
+hist(log(fbyg$weight))
+fbyg$weight==1
+all(fbyg$weight==1)
+fbyg$weight[fbyg$weight != 1]
+fbyg[fbyg$weight != 1]
+fbyg[,fbyg$weight != 1]
+fbyg[[fbyg$weight != 1]]
+fbyg[fbyg$weight != 1,]
+names(labelr)
+summary(toxicity_calibrated)
+toxicity_calibrated
+val.data
+names(labelr)
+labelr$data
+labelr
+labelr[data]
+labelr[data=='yg']
+labelr[,data=='yg']
+labelr[data=='yg',]
+labelr[labelr$data=='yg']
+labelr[,labelr$data=='yg']
+labelr[labelr$data=='yg',]
+toxicity_calibrated
+summary(toxicity_calibrated)
+yg3
+yg3[,['toxic','toxic_pred']]
+yg3 %>% select('toxic','toxic_pred')
+yg3 |> select('toxic','toxic_pred')
+names(yg3)
+yg3[,c('toxic_pred','toxic')]
+corr(yg3[,c('toxic_pred','toxic')])
+cor(yg3[,c('toxic_pred','toxic')])
+cor(yg3[,c('toxic_pred','toxic')],na.rm=T)
+cor(yg3[,c('toxic_pred','toxic')],rm.na=T)
+?cor(yg3[,c('toxic_pred','toxic')],use=
+?cor
+cor(yg3[,c('toxic_pred','toxic')],use='all.obs')
+?cor
+cor(yg3[,c('toxic_pred','toxic')],use='complete.obs')
+cor(yg3[,c('toxic_pred','toxic')],use='complete.obs',method='spearman')
+?predict
+yg3$toxic_pred
+names(preds)
+preds
+preds
+preds$error
+preds
+preds
+summary(errormod)
+summary(errormod)
+summary(preds)
+names(preds)
+preds
+resids
+qplot(resids)
+resids
+?predict.lm
+dnorm(1)
+dnorm(2)
+dnorm(1)
+pnorm(1)
+preds
+p1 + p2
+p1 + p2
+p1
+p2
+preds
+preds1 <- preds
+preds1$diff - preds$diff
+preds1$diff
+preds1$diff - preds1$diff
+preds1$diff - preds$diff
+preds1$diff - preds$diff
+preds1$diff - preds$diff
+preds1$diff - preds$diff
+preds1
+preds
+dnorm(-1)
+dnorm(1)
+pnorm(1)
+pnorm(-1)
+pnorm(2)
+pnorm(9)
+pnorm(6)
+pnorm(2)
+dnorm(0.95)
+qnorm(0.95)
+qnorm(0.841)
+fulldata_preds
+names(yg3)
+yg3$toxic_feature_1
+yg3$toxic_feature_2
+yg3
+yg3[,.(toxic_pred,toxic_var)]
+yg3[,.(toxic_pred,toxicity_2_pred_sigma,toxicity_1_pred_sigma)]
+yg3[,.(toxic_pred,toxicity_2_pred_sigma,toxicity_1_pred_sigma,cov(toxicity_2_pred,toxicity_1_pred))]
+cov(1,2)
+cov(c(1),c(3))
+cov(c(1),c(3,2))
+cov(c(1,1),c(3,2))
+cov(c(1,2),c(3,2))
+covterm
+covterm
+?cov
+covterm
+yg3
+yg3[,.(toxic_pred,toxicity_2_pred_sigma,toxicity_1_pred_sigma,cov(toxicity_2_pred,toxicity_1_pred))]
+yg3[,.(toxic_pred,toxicity_2_pred_sigma,toxicity_1_pred_sigma,toxic_var)]
+yg3[,.(toxic_pred,toxicity_2_pred_sigma,toxicity_1_pred_sigma,toxic_var,toxic_sd)]
+yg3
+names(yg3)
+print(sg)
+print(sg)
+1+1
+library(stargazer)
+
+stargazer(w1,w2,w3,w4,w5,t1,t2,t3,t4,t5, type="text",
+ keep = c("cond1","meantox","cond1:meantox","Constant"),
+ keep.stat=c("n","adj.rsq"),
+ model.numbers = F,
+ dep.var.labels = c("DV = Willingness to comment","DV = Toxicity of YG respondent comments"),
+ covariate.labels = c("Treatment (top comments shown)",
+ "Average toxicity of top comments",
+ "Treatment $\times$ top comments toxicity",
+ "Constant"),
+ add.lines = list(c("Article fixed effects","No","No","No","Yes","Yes","No","No","No","Yes","Yes")),
+ star.cutoffs = c(0.05,0.01,0.005),
+ notes = "Standard errors are clustered at the respondent level.",
+ column.labels = c("(1)","(2)","(3)","(4)","(5)","(6)","(7)","(8)","(9)","(10)"),
+ style = "apsr")
+
+q()
+n
+yglabels
+labelr
+names(labelr)
+fb
+names(fb
+)
+fb.comment_id
+fb['comment_id']
+fb[,'comment_id']
+labelr[,'comment_id']
+names(fb)
+fb.labeled
+names(fb.labeled)
+names(yg)
+?amelia
+yg
+names(yg)
+names(yg3)
+?rbind
+nrow(yg3)
+nrow(yg)
+yg3[,.(.N),by=.(toxic,fb)]
+yg3.toimpute
+names(yg3.toimpute)
+yg3.toimpute
+names(yg3.toimpute)
+names(labelr)
+nrow(yg3)
+nrow(labelr)
+?merge.data.table
+labelr
+is.data.table(labelr)
+yg3.toimpute
+overimp.grid
+overimp.grid
+?amelia
+q()
+n
+setwd("presentations/ica_hackathon_2022/")
+ls()
+attach(r)
+example_2_B.plot.df
+library(ggplot2)
+example_2_B.plot.df[(variable=='x') && (m < 1000)]
+example_2_B.plot.df[(variable=='x') && (m < 1000)]
+theme_set(theme_default())
+theme_set(theme_minimal())
+theme_set(theme_classic())
+example_2_B.plot.df[(variable=='x') && (m < 1000)]
+example_2_B.plot.df[(variable=='x') && (m < 1000),unique(method)]
+as.factor
+update.packages()
+update.packages()
+update.packages()
+cancel
+plot.df
+example_2_B.plot.df
+plot.df
+example_2_B.plot.df
+example_2_B.plot.df$method %>% unique
+example_2_B.plot.df$method |> unique
+example_2_B.plot.df$method |> uniq
+unique(example_2_B.plot.df$method)
+example_2_B.plot.df$method
+example_2_B.plot.df$method
+example_2_B.plot.df$method
+example_2_B.plot.df$method
+example_2_B.plot.df <- r$example_2_B.plot.df
+q()
+n
+setwd("presentations/ica_hackathon_2022/')
+setwd("presentations/ica_hackathon_2022/")
+example_2_B.plot.df$method
+example_2_B.plot.df$method
+q()
+n
+example_2_B.plot.df$method
+example_2_B.plot.df$method
+q()
+n
+example_2_B.plot.df$method
+example_2_B.plot.df$method
+q()
+n
+q()
+n
+plot.df
+plot.df
+plot.df[,.N,by=.(N,m)]
+plot.df[,.N,by=.(N,m,method)]
+plot.df[variable=='x',.N,by=.(N,m,method)]
+plot.df
+plot.df[(variable=='x') & (m < 1000) & (!is.na(p.true.in.ci))]
+plot.df[(variable=='x') & (m != 1000) & (!is.na(p.true.in.ci))]
+plot.df
+?label_wrap_gen
+install.packages("ggplot2")
+devtools::install_github("tidyverse/ggplot2")
+2
+library(ggplot2)
+ggplot2::version
+sessioninfo()
+sessionInfo()
+q()
+n
+sessionInfo()
+?scale_x_discrete
+?facet_grid
+plot.df
+plot.df
+plot.df[method="2SLS+gmm"]
+plot.df[method=="2SLS+gmm"]
+df <- example_2_B.plot.df
+df
+q()
+n
+plot.df
+plot.df[m=50]
+plot.df[m==50]
+plot.df.example.2[m==50][method=2SLS+gmm]
+plot.df.example.2[m==50][method==2SLS+gmm]
+plot.df.example.2[(m==50) & (method==2SLS+gmm)]
+plot.df.example.2[(m==50) & (method=="2SLS+gmm")]
+plot.df[m==50]
+plot.df.example.3
+plot.df.example.3
+plot.df.example.3[N=25000]
+plot.df.example.3[N==25000]
+plot.df
+plot.df
+plot.df
+q()
+n
--- /dev/null
+#!/usr/bin/make
+all:html pdf
+
+html: $(patsubst %.Rmd,%.html,$(wildcard *.Rmd))
+pdf: $(patsubst %.Rmd,%.pdf,$(wildcard *.Rmd))
+
+remembr.RDS:
+ rsync klone:/gscratch/comdata/users/nathante/ml_measurement_error/mi_simulations/remembr.RDS .
+
+%.pdf: %.html
+ Rscript -e 'xaringan::decktape("$<","$@",docker=FALSE)'
+
+%.html: %.Rmd *.css remembr.RDS
+ Rscript -e 'library(rmarkdown); rmarkdown::render("$<", output_file = "$@")'
+# firefox "$@"
+
+clean:
+ rm *.html
+ rm -r *_files
+ rm -r *_cache
+
+publish: all pdf
+ scp -r *.html groc:/home/nathante/public_html/slides/measurement_error_comdatahack_2022.html
+ scp -r *.pdf groc:/home/nathante/public_html/slides/measurement_error_comdatahack_2022.pdf
+
+.PHONY: clean all
--- /dev/null
+/*!
+ * Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com
+ * License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License)
+ */
+.fa,.fab,.fad,.fal,.far,.fas{-moz-osx-font-smoothing:grayscale;-webkit-font-smoothing:antialiased;display:inline-block;font-style:normal;font-variant:normal;text-rendering:auto;line-height:1}.fa-lg{font-size:1.33333em;line-height:.75em;vertical-align:-.0667em}.fa-xs{font-size:.75em}.fa-sm{font-size:.875em}.fa-1x{font-size:1em}.fa-2x{font-size:2em}.fa-3x{font-size:3em}.fa-4x{font-size:4em}.fa-5x{font-size:5em}.fa-6x{font-size:6em}.fa-7x{font-size:7em}.fa-8x{font-size:8em}.fa-9x{font-size:9em}.fa-10x{font-size:10em}.fa-fw{text-align:center;width:1.25em}.fa-ul{list-style-type:none;margin-left:2.5em;padding-left:0}.fa-ul>li{position:relative}.fa-li{left:-2em;position:absolute;text-align:center;width:2em;line-height:inherit}.fa-border{border:.08em solid #eee;border-radius:.1em;padding:.2em .25em 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\ No newline at end of file
--- /dev/null
+---
+title: "How good of a model do you need? Accounting for classification errors in machine assisted content analysis."
+author: Nathan TeBlunthuis
+date: May 24 2022
+template: "../resources/template.html"
+output:
+ xaringan::moon_reader:
+ lib_dir: libs
+ seal: false
+ nature:
+ highlightStyle: github
+ ratio: 16:9
+ countIncrementalSlides: true
+ slideNumberFormat: |
+ <div class="progress-bar-container">
+ <div class="progress-bar" style="width: calc(%current% / %total% * 100%);">
+ </div>
+ </div>
+ self_contained: false
+ css: [default, my-theme.css, fontawesome.min.css]
+ chakra: libs/remark-latest.min.js
+
+---
+```{r echo=FALSE, warning=FALSE, message=FALSE}
+library(knitr)
+library(ggplot2)
+library(data.table)
+f <- function (x) {formatC(x, format="d", big.mark=',')}
+
+theme_set(theme_bw())
+r <- readRDS('remembr.RDS')
+attach(r)
+
+```
+class: center, middle, narrow
+
+<script type='javascript'>
+window.MathJax = {
+ loader: {load: ['[tex]/xcolor']},
+ tex: {packages: {'[+]': ['xcolor']}}
+};
+</script>
+
+<div class="my-header"></div>
+
+
+### .title-heading[Unlocking the power of big data: The importance of measurement error in machine assisted content analysis]
+## Nathan TeBlunthuis
+
+<img src="images/nu_logo.png" height="170px" style="padding:21px"/> <img src="images/uw_logo.png" height="170px" style="padding:21px"/> <img src="images/cdsc_logo.png" height="170px" style="padding:21px"/>
+
+
+nathan.teblunthuis@northwestern.edu
+
+[https://teblunthuis.cc](https://teblunthuis.cc)
+
+???
+
+This talk will be me presenting my "lab notebook" and not a polished research talk. Maybe it would be a good week of a graduate seminar? In sum, machine assisted content analysis has unique limitations and threats to validity that I wanted to understand better. I've learned how the noise introduced by predictive models can result in misleading statistical inferences, but that a sample of human-labeled validation data can often be used to account for this noise and obtain accurate inferences in the end. Statistical knowledge of this problem and computational tools for addressing are still in development. My goals for this presentation are to start sharing this information with the community and hopeful to stimulate us to work on extending existing approaches or using them in our work.
+
+This is going to be a boring talk about some *very* technical material. If you're not that interested please return to your hackathon. Please interrupt me if I'm going too fast for you or if you don't understand something. I will try to move quickly in the interests of those wishing to wrap up their hackathon projects. I will also ask you to show hands once or twice, if you are already familiar with some concepts that it might be expedient to skip.
+
+---
+
+class:center, middle, inverse
+## Machine assistent content analysis (MACA)
+
+???
+
+I'm going to start by defining a study design that is increasingly common, especially in Communication and Political Science, but also across the social sciences and beyond. I call it *machine assisted content analysis* (MACA).
+
+---
+<div class="my-header"></div>
+
+### .border[Machine assisted content analysis (MACA) uses machine learning for scientific measurement.]
+
+.emph[Content analysis:] Statistical analysis of variables measured by human labeling ("coding") of content. This might be simple categorical labels, or maybe more advanced annotations.
+
+--
+
+*Downside:* Human labeling is *a lot* of work.
+
+--
+
+.emph[Machine assisted content analysis:] Use a *predictive algorithm* (often trained on human-made labels) to measure variables for use in a downstream *primary analysis.*
+
+--
+
+*Downside:* Algorithms can be *biased* and *inaccurate* in ways that could invalidate the statistical analysis.
+
+
+???
+
+A machine assisted content analysis can be part of a more complex or more powerful study design (e.g., an experiment, time series analysis &c).
+
+---
+
+
+<!-- <div class="my-header"></div> -->
+
+<!-- ### .border[Hypothetical Example: Predicting Racial Harassement in Social Media Comments] -->
+
+---
+class:large
+
+<div class="my-header"></div>
+
+### .border[How can MACA go wrong?]
+
+Algorithms can be *biased* and *error prone* (*noisy*).
+
+--
+
+Predictor bias is a potentially difficult problem that requires causal inference methods. I'll focus on *noise* for now.
+
+--
+
+Noise in the predictive model introduces bias in the primary analysis.
+
+--
+
+.indent[We can reduce and sometimes even *eliminate* this bias introduced by noise.]
+
+---
+layout:true
+<div class="my-header"></div>
+
+### .border[Example 1: An unbiased, but noisy classifier]
+
+.large[.left-column[]]
+
+???
+
+Please show hands if you are familiar with causal graphs or baysian networks. Should I explain what this diagram means?
+
+
+---
+
+.right-column[
+$x$ is *partly observed* because we have *validation data* $x^*$.
+]
+
+---
+
+
+.right-column[
+$x$ is *partly observed* because we have *validation data* $x^*$.
+
+$k$ are the *features* used by the *predictive model* $g(k)$.
+
+]
+
+---
+
+.right-column[
+$x$ is *partly observed* because we have *validation data* $x^*$.
+
+$k$ are the *features* used by the *predictive model* $g(k)$.
+
+The predictions $w$ are a *proxy variable* $g(k) = \hat{x} = w$.
+
+]
+
+---
+
+
+.right-column[
+$x$ is *partly observed* because we have *validation data* $x^*$.
+
+$k$ are the *features* used by the *predictive model* $g(k)$.
+
+The predictions $w$ are a *proxy variable* $g(k) = \hat{x} = w$.
+
+$x = w + \xi$ because the predictive model makes errors.
+
+]
+
+---
+
+
+layout:true
+<div class="my-header"></div>
+
+### .border[Noise in a *covariate* creates *attenuation bias*.]
+
+.large[.left-column[]]
+
+---
+.right-column[
+
+We want to estimate, $y = Bx + \varepsilon$, but we estimate $y = Bw + \varepsilon$ instead.
+
+$x = w + \xi$ because the predictive model makes errors.
+
+]
+---
+
+.right-column[
+
+We want to estimate, $y = Bx + \varepsilon$, but we estimate $y = Bw + \varepsilon$ instead.
+
+$x = w + \xi$ because the predictive model makes errors.
+
+
+Assume $g(k)$ is *unbiased* so $E(\xi)=0$. Also assume error is *nondifferential* so $E(\xi y)=0$:
+
+]
+
+---
+
+.right-column[
+
+We want to estimate, $y = Bx + \varepsilon$, but we estimate $y = Bw + \varepsilon$ instead.
+
+$x = w + \xi$ because the predictive model makes errors.
+
+Assume $g(k)$ is *unbiased* so $E(\xi)=0$. Also assume error is *nondifferential* so $E(\xi y)=0$:
+
+$$\widehat{B_w}^{ols}=\frac{\sum^n_{j=j}{(x_j + \xi_j - \overline{(x + \xi)})}(y_j - \bar{y})}{\sum_{j=1}^n{(x_j + \xi_j - \overline{(x+\xi)})^2}} = \frac{\sum^n_{j=j}{(x_j - \bar{x})(y_j -
+ \bar{y})}}{\sum_{j=1}^n{(x_j + \xi_j - \bar{x}){^2}}}$$
+
+]
+
+---
+
+.right-column[
+
+We want to estimate, $y = Bx + \varepsilon$, but we estimate $y = Bw + \varepsilon$ instead.
+
+$x = w + \xi$ because the predictive model makes errors.
+
+Assume $g(k)$ is *unbiased* so $E(\xi)=0$. Also assume error is *nondifferential* so $E(\xi y)=0$:
+
+$$\widehat{B_w}^{ols}=\frac{\sum^n_{j=j}{(x_j + \xi_j - \overline{(x + \xi)})}(y_j - \bar{y})}{\sum_{j=1}^n{(x_j + \xi_j - \overline{(x+\xi)})^2}} = \frac{\sum^n_{j=j}{(x_j - \bar{x})(y_j -
+ \bar{y})}}{\sum_{j=1}^n{(x_j + \color{red}{\xi_j} - \bar{x})\color{red}{^2}}}$$
+
+In this scenario, it's clear that $\widehat{B_w}^{ols} < B_x$.
+
+
+]
+
+
+???
+
+Please raise your hands if you're familiar with attenuation bias. I expect that its covered in some graduate stats classes, but not universally.
+
+---
+class:large
+layout:false
+<div class="my-header"></div>
+
+### .border[Beyond attenuation bias]
+.larger[Measurement error can theaten validity because:]
+
+- Attenuation bias *spreads* (e.g., to marginal effects as illustrated later).
+
+--
+
+- Measurement error can be *differential*— not distributed evenly and possible correlated with $x$, $y$, or $\varepsilon$.
+
+--
+
+- *Bias can be away from 0* in GLMs and nonlinear models or if measurement error is differential.
+
+--
+
+- *Confounding* if the *predictive model is biased* introducing a correlation the measurement error and the residuals $(E[\xi\varepsilon]=0)$.
+
+
+---
+
+class:large
+layout:false
+<div class="my-header"></div>
+
+### .border[Correcting measurement error]
+
+There's a vast literature in statistics on measurement error. Mostly about noise you'd find in sensors. Lots of ideas. No magic bullets.
+
+--
+
+I'm going to briefly cover 3 different approaches: *multiple imputation*, *regression calibration* and *2SLS+GMM*.
+
+--
+
+These all depend on *validation data*. I'm going to ignore where this comes from, but assume it's a random sample of the hypothesis testing dataset.
+
+--
+
+You can *and should* use it to improve your statistical estimates.
+
+---
+
+<div class="my-header"></div>
+
+### .border[Multiple Imputation (MI) treats Measurement Error as a Missing Data Problem]
+
+1. Use validation data to estimate $f(x|w,y)$, a probabilistic model of $x$.
+
+--
+
+2. *Sample* $m$ datasets from $\widehat{f(x|w,y)}$.
+
+--
+
+3. Run your analysis on each of the $m$ datasets.
+
+--
+
+4. Average the results from the $m$ analyses using Rubin's rules.
+
+--
+
+.e[Advantages:] *Very flexible!* Sometimes can work if the predictor $g(k) $ is biased. Good R packages (**`{Amelia}`**, `{mi}`, `{mice}`, `{brms}`).
+
+--
+
+.e[Disadvantages:] Results depend on quality of $\widehat{f(x|w,y)}$; May require more validation data, computationally expensive, statistically inefficient and doesn't seem to benefit much from larger datasets.
+
+---
+
+### .border[Regression calibration directly adjusts for attenuation bias.]
+
+1. Use validation data to estimate the errors $\hat{\xi}$.
+
+--
+
+2. Use $\hat{\xi}$ to correct the OLS estimate.
+
+--
+
+3. Correct the standard errors using MLE or bootstrapping.
+
+--
+
+.e[Advantages:] Simple, fast.
+
+--
+
+.e[Disadvantages:] Limited to OLS models. Requires an unbiased predictor $g(k)$. R support (`{mecor}` R package) is pretty new.
+
+---
+layout:true
+### .border[2SLS+GMM is designed for this specific problem]
+
+.left-column[]
+
+*Regression calibration with a trick.*
+
+---
+.right-column[
+
+1. Estimate $x = w + \xi$ to obtain $\hat{x}$. (First-stage LS).
+
+]
+
+---
+.right-column[
+
+1. Estimate $x = w + \xi$ to obtain $\hat{x}$. (First-stage LS).
+
+2. Estimate $y = B^{2sls}\hat{x} + \varepsilon^{2sls}$. (Second-stage LS / regression calibration).
+
+]
+
+---
+.right-column[
+
+1. Estimate $x = w + \xi$ to obtain $\hat{x}$. (First-stage LS).
+
+2. Estimate $y = B^{2sls}\hat{x} + \varepsilon^{2sls}$. (Second-stage LS / regression calibration).
+
+3. Estimate $y = B^{val}x^* + \varepsilon^{val}$. (Validation dataset model).
+
+]
+
+---
+.right-column[
+
+1. Estimate $x = w + \xi$ to obtain $\hat{x}$. (First-stage LS).
+
+2. Estimate $y = B^{2sls}\hat{x} + \varepsilon^{2sls}$. (Second-stage LS / regression calibration).
+
+3. Estimate $y = B^{val}x^* + \varepsilon^{val}$. (Validation dataset model).
+
+4. Combine $B^{val}$ and $B^{2sls}$ using the generalized method of moments (GMM).
+
+]
+
+---
+.right-column[
+
+1. Estimate $x = w + \xi$ to obtain $\hat{x}$. (First-stage LS).
+
+2. Estimate $y = B^{2sls}\hat{x} + \varepsilon^{2sls}$. (Second-stage LS / regression calibration).
+
+3. Estimate $y = B^{val}x^* + \varepsilon^{val}$. (Validation dataset model).
+
+4. Combine $B^{val}$ and $B^{2sls}$ using the generalized method of moments (GMM).
+
+Advantages: Accurate. Sometimes robust if biased predictor $g(k)$ is biased. In theory, flexible to any models that can be fit using GMM.
+
+]
+
+
+---
+.right-column[
+
+1. Estimate $x = w + \xi$ to obtain $\hat{x}$. (First-stage LS).
+
+2. Estimate $y = B^{2sls}\hat{x} + \varepsilon^{2sls}$. (Second-stage LS / regression calibration).
+
+3. Estimate $y = B^{val}x^* + \varepsilon^{val}$. (Validation dataset model).
+
+4. Combine $B^{val}$ and $B^{2sls}$ using the generalized method of moments (GMM).
+
+Advantages: Accurate. Sometimes robust if biased predictor $g(k)$ is biased. In theory, flexible to any models that can be fit using GMM.
+
+Disadvantages: Implementation (`{predictionError}`) is new. API is cumbersome and only supports linear models. Not robust if $E(w\varepsilon) \ne 0$. GMM may be unfamiliar to audiences.
+
+]
+
+---
+layout:false
+### .border[Testing attention bias correction]
+
+<div class="my-header"></div>
+
+I've run simulations to test these approaches in several scenarios.
+
+The model is not very good: about 70% accurate.
+
+Most plausible scenario:
+
+y is continuous and normal-ish.
+
+--
+
+$x$ is binary (human labels) $P(x)=0.5$.
+
+--
+
+$w$ is the *continuous predictor* (e.g., probability) output of $f(x)$ (not binary predictions).
+
+--
+
+if $w$ is binary, most methods struggle, but regression calibration and 2SLS+GMM can do okay.
+
+---
+layout:false
+
+### .border[Example 1: estimator of the effect of x]
+
+.right-column[
+```{r echo=FALSE, message=FALSE, warning=FALSE, result='asis', dev='svg', fig.width=7.5, fig.asp=.625,cache=F}
+
+#plot.df <-
+plot.df <- plot.df.example.1[,':='(method=factor(method,levels=c("Naive","Multiple imputation", "Multiple imputation (Classifier features unobserved)","Regression Calibration","2SLS+gmm","Feasible"),ordered=T),
+ N=factor(N),
+ m=factor(m))]
+
+plot.df <- plot.df[(variable=='x') & (m != 1000) & (m!=500) & (N!=10000) & !is.na(p.true.in.ci) & (method!="Multiple imputation (Classifier features unobserved)")]
+p <- ggplot(plot.df, aes(y=mean.est, ymax=mean.est + var.est/2, ymin=mean.est-var.est/2, x=method))
+p <- p + geom_hline(aes(yintercept=0.2),linetype=2)
+
+p <- p + geom_pointrange() + facet_grid(m~N,as.table=F) + scale_x_discrete(labels=label_wrap_gen(4))
+
+print(p)
+
+# get gtable object
+
+```
+]
+.left-column[
+
+All methods work in this scenario
+
+Multiple imputation is inefficient.
+
+]
+
+
+---
+### .border[What about bias?]
+
+.left-column[
+.large[]
+]
+
+.right-column[
+A few notes on this scenario.
+
+$B_x = 0.2$, $B_g=-0.2$ and $sd(\varepsilon)=3$. So the signal-to-noise ratio is high.
+
+$r$ can be concieved of as a missing feature in the predictive model $g(k)$ that is also correlated with $y$.
+
+For example $r$ might be the *race* of a commentor, $x$ could be *racial harassment*, $y$ whether the commentor gets banned and $k$ only has textual features but human coders can see user profiles to know $r$.
+
+]
+
+---
+layout:false
+### .border[Example 2: Estimates of the effect of x ]
+
+.center[
+```{r echo=FALSE, message=FALSE, warning=FALSE, result='asis', dev='svg', fig.width=8, fig.asp=.625,cache=F}
+
+#plot.df <-
+plot.df <- plot.df.example.2B[,':='(method=factor(method,levels=c("Naive","Multiple imputation", "Multiple imputation (Classifier features unobserved)","Regression Calibration","2SLS+gmm","Feasible"),ordered=T),
+ N=factor(N),
+ m=factor(m))]
+
+plot.df <- plot.df[(variable=='x') & (m != 1000) & (m!=500) & (N!=10000) & !is.na(p.true.in.ci) & (method!="Multiple imputation (Classifier features unobserved)")]
+p <- ggplot(plot.df, aes(y=mean.est, ymax=mean.est + var.est/2, ymin=mean.est-var.est/2, x=method))
+p <- p + geom_hline(aes(yintercept=0.2),linetype=2)
+
+p <- p + geom_pointrange() + facet_grid(m~N,as.table=F) + scale_x_discrete(labels=label_wrap_gen(4))
+
+print(p)
+
+# get gtable object
+
+```
+]
+---
+layout:false
+
+### .border[Example 2: Estimates of the effect of r]
+
+.center[
+```{r echo=FALSE, message=FALSE, warning=FALSE, result='asis', dev='svg', fig.width=8, fig.asp=.625,cache=F}
+
+#plot.df <-
+plot.df <- plot.df.example.2B[,':='(method=factor(method,levels=c("Naive","Multiple imputation", "Multiple imputation (Classifier features unobserved)","Regression Calibration","2SLS+gmm","Feasible"),ordered=T),
+ N=factor(N),
+ m=factor(m))]
+
+plot.df <- plot.df[(variable=='g') & (m != 1000) & (m!=500) & (N!=10000) & !is.na(p.true.in.ci) & (method!="Multiple imputation (Classifier features unobserved)")]
+p <- ggplot(plot.df, aes(y=mean.est, ymax=mean.est + var.est/2, ymin=mean.est-var.est/2, x=method))
+p <- p + geom_hline(aes(yintercept=-0.2),linetype=2)
+
+p <- p + geom_pointrange() + facet_grid(m~N,as.table=F) + scale_x_discrete(labels=label_wrap_gen(4))
+
+
+print(p)
+```
+]
+---
+
+layout:false
+class:large
+
+###.border[Takeaways from example 2]
+
+Bias in the predictive model creates bias in hypothesis tests.
+
+--
+
+Bias can be corrected *in this case*.
+
+--
+
+The next scenario has bias that's more tricky.
+
+--
+
+Multiple imputation helps, but doesn't fully correct the bias.
+
+---
+
+layout:false
+
+### .border[When will GMM+2SLS fail?]
+
+.large[.left-column[]]
+
+.right-column[The catch with GMM:
+
+.emph[Exclusion restriction:] $E[w \varepsilon] = 0$.
+
+The restriction is violated if a variable $U$ causes both $K$ and $Y$ and $X$ causes $K$ (not visa-versa).
+
+]
+
+???
+
+GMM optimizes a model to a system of equations of which the exclusion restriction is one. So if that assumption isn't true it will biased.
+
+This is a different assumption than that of OLS or GLM models.
+
+---
+
+layout:false
+
+### .border[Example 3: Estimates of the effect of x]
+
+.center[
+```{r echo=FALSE, message=FALSE, warning=FALSE, result='asis', dev='svg', fig.width=8, fig.asp=.625,cache=F}
+
+#plot.df <-
+plot.df <- plot.df.example.3[,':='(method=factor(method,levels=c("Naive","Multiple imputation", "Multiple imputation (Classifier features unobserved)","Regression Calibration","2SLS+gmm","Feasible"),ordered=T),
+ N=factor(N),
+ m=factor(m))]
+
+plot.df <- plot.df[(variable=='x') & (m != 1000) & (m!=500) & (N!=10000) & (method!="Multiple imputation (Classifier features unobserved)")]
+p <- ggplot(plot.df, aes(y=mean.est, ymax=mean.est + var.est/2, ymin=mean.est-var.est/2, x=method))
+p <- p + geom_hline(aes(yintercept=0.2),linetype=2)
+
+p <- p + geom_pointrange() + facet_grid(m~N,as.table=F) + scale_x_discrete(labels=label_wrap_gen(4))
+
+
+print(p)
+```
+]
+
+
+
+---
+
+### .border[Takaways]
+
+- Attenuation bias can be a big problem with noisy predictors—leading to small and biased estimates.
+
+- For more general hypothesis tests or if the predictor is biased, measurement error can lead to false discovery.
+
+- It's fixable with validation data—you may not need that much and you should already be getting it.
+
+- This means it can be okay poor predictors for hypothesis testing.
+
+- The ecosystem is underdeveloped, but a lot of methods have been researched.
+
+- Take advantage of machine learning + big data and get precise estimates when the signal-to-noise ratio is high!
+
+---
+layout:false
+
+### .border[Future work: Noise in the *outcome*]
+
+I've been focusing on noise in *covariates.* What if the predictive algorithm is used to measure the *outcome* $y$?
+
+--
+
+This isn't a problem in the simplest case (linear regression with homoskedastic errors). Noise in $y$ is projected into the error term.
+
+--
+
+Noise in the outcome is still a problem if errors are heteroskedastic and for GLMs / non-linear regression (e.g., logistic regression).
+
+--
+
+Multiple imputation (in theory) could help here. The other method's aren't designed for this case.
+
+--
+
+Solving this problem could be an important methodological contribution with a very broad impact.
+
+---
+# .border[Questions?]
+
+Links to slides:[html](https://teblunthuis.cc/~nathante/slides/ecological_adaptation_ica_2022.html) [pdf](https://teblunthuis.cc/~nathante/slides/ecological_adaptation_ica_2022.pdf)
+
+Link to a messy git repository:
+
+<i class="fa fa-envelope" aria-hidden='true'></i> nathan.teblunthuis@northwestern.edu
+
+<i class="fa fa-twitter" aria-hidden='true'></i> @groceryheist
+
+<i class="fa fa-globe" aria-hidden='true'></i> [https://communitydata.science](https://communitydata.science)
+
+
+
+<!-- ### .border[Multiple imputation struggles with discrete variables] -->
+
+<!-- In my experiments I've found that the 2SLS+GMM method works well with a broader range of data types. -->
+
+<!-- To illustrate, Example 3 is the same as Example 2, but with $x$ and $w$ as discrete variables. -->
+
+<!-- Practicallly speaking, a continuous "score" $w$ is often available, and my opinion is that usually this is better + more informative than model predictions in all cases. Continuous validation data may be more difficult to obtain, but it is often possible using techniques like pairwise comparison. -->
+<!-- layout:false -->
+<!-- ### .border[Example 3: Estimates of the effect of x ] -->
+
+<!-- .center[ -->
+<!-- ```{r echo=FALSE, message=FALSE, warning=FALSE, result='asis', dev='svg', fig.width=8, fig.asp=.625,cache=F} -->
+
+<!-- #plot.df <- -->
+<!-- plot.df <- plot.df.example.2[,':='(method=factor(method,levels=c("Naive","Multiple imputation", "Multiple imputation (Classifier features unobserved)","Regression Calibration","2SLS+gmm","Feasible"),ordered=T), -->
+<!-- N=factor(N), -->
+<!-- m=factor(m))] -->
+
+<!-- plot.df <- plot.df[(variable=='x') & (m != 1000) & (m!=500) & (N!=5000) & (N!=10000) & !is.na(p.true.in.ci) & (method!="Multiple imputation (Classifier features unobserved)")] -->
+<!-- p <- ggplot(plot.df, aes(y=mean.est, ymax=mean.est + var.est/2, ymin=mean.est-var.est/2, x=method)) -->
+<!-- p <- p + geom_hline(aes(yintercept=0.2),linetype=2) -->
+
+<!-- p <- p + geom_pointrange() + facet_grid(m~N,as.table=F) + scale_x_discrete(labels=label_wrap_gen(4)) -->
+
+<!-- print(p) -->
+
+<!-- # get gtable object -->
+
+<!-- .large[.left []] -->
+
+<!-- There are at two general ways using a predictive model can introduce bias: *attenuation*, and *confounding.* -->
+
+<!-- Counfounding can be broken down into 4 types: -->
+
+<!-- .right[Confounding on $X$ by observed variables -->
+
+<!-- Confounding on $Y$ by observed variables -->
+<!-- ] -->
+
+<!-- .left[Confounding on $X$ by *un*observed variables -->
+
+<!-- Confounding on $Y$ by *un*observed variables -->
+<!-- ] -->
+
+<!-- Attenuation and the top-right column can be dealt with relative ease using a few different methods. -->
+
+<!-- The bottom-left column can be addressed, but so far I haven't found a magic bullet. -->
+
+<!-- The left column is pretty much a hopeless situation. -->
--- /dev/null
+<!DOCTYPE html>
+<html lang="" xml:lang="">
+ <head>
+ <title>How good of a model do you need? Accounting for classification errors in machine assisted content analysis.</title>
+ <meta charset="utf-8" />
+ <meta name="author" content="Nathan TeBlunthuis" />
+ <script src="libs/header-attrs-2.14/header-attrs.js"></script>
+ <link href="libs/remark-css-0.0.1/default.css" rel="stylesheet" />
+ <link rel="stylesheet" href="my-theme.css" type="text/css" />
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+ <body>
+ <textarea id="source">
+
+
+class: center, middle, narrow
+
+<script type='javascript'>
+window.MathJax = {
+ loader: {load: ['[tex]/xcolor']},
+ tex: {packages: {'[+]': ['xcolor']}}
+};
+</script>
+
+<div class="my-header"></div>
+
+
+### .title-heading[Unlocking the power of big data: The importance of measurement error in machine assisted content analysis]
+## Nathan TeBlunthuis
+
+<img src="images/nu_logo.png" height="170px" style="padding:21px"/> <img src="images/uw_logo.png" height="170px" style="padding:21px"/> <img src="images/cdsc_logo.png" height="170px" style="padding:21px"/>
+
+
+nathan.teblunthuis@northwestern.edu
+
+[https://teblunthuis.cc](https://teblunthuis.cc)
+
+???
+
+This talk will be me presenting my "lab notebook" and not a polished research talk. Maybe it would be a good week of a graduate seminar? In sum, machine assisted content analysis has unique limitations and threats to validity that I wanted to understand better. I've learned how the noise introduced by predictive models can result in misleading statistical inferences, but that a sample of human-labeled validation data can often be used to account for this noise and obtain accurate inferences in the end. Statistical knowledge of this problem and computational tools for addressing are still in development. My goals for this presentation are to start sharing this information with the community and hopeful to stimulate us to work on extending existing approaches or using them in our work.
+
+This is going to be a boring talk about some *very* technical material. If you're not that interested please return to your hackathon. Please interrupt me if I'm going too fast for you or if you don't understand something. I will try to move quickly in the interests of those wishing to wrap up their hackathon projects. I will also ask you to show hands once or twice, if you are already familiar with some concepts that it might be expedient to skip.
+
+---
+
+class:center, middle, inverse
+## Machine assistent content analysis (MACA)
+
+???
+
+I'm going to start by defining a study design that is increasingly common, especially in Communication and Political Science, but also across the social sciences and beyond. I call it *machine assisted content analysis* (MACA).
+
+---
+<div class="my-header"></div>
+
+### .border[Machine assisted content analysis (MACA) uses machine learning for scientific measurement.]
+
+.emph[Content analysis:] Statistical analysis of variables measured by human labeling ("coding") of content. This might be simple categorical labels, or maybe more advanced annotations.
+
+--
+
+*Downside:* Human labeling is *a lot* of work.
+
+--
+
+.emph[Machine assisted content analysis:] Use a *predictive algorithm* (often trained on human-made labels) to measure variables for use in a downstream *primary analysis.*
+
+--
+
+*Downside:* Algorithms can be *biased* and *inaccurate* in ways that could invalidate the statistical analysis.
+
+
+???
+
+A machine assisted content analysis can be part of a more complex or more powerful study design (e.g., an experiment, time series analysis &c).
+
+---
+
+
+<!-- <div class="my-header"></div> -->
+
+<!-- ### .border[Hypothetical Example: Predicting Racial Harassement in Social Media Comments] -->
+
+---
+class:large
+
+<div class="my-header"></div>
+
+### .border[How can MACA go wrong?]
+
+Algorithms can be *biased* and *error prone* (*noisy*).
+
+--
+
+Predictor bias is a potentially difficult problem that requires causal inference methods. I'll focus on *noise* for now.
+
+--
+
+Noise in the predictive model introduces bias in the primary analysis.
+
+--
+
+.indent[We can reduce and sometimes even *eliminate* this bias introduced by noise.]
+
+---
+layout:true
+<div class="my-header"></div>
+
+### .border[Example 1: An unbiased, but noisy classifier]
+
+.large[.left-column[]]
+
+???
+
+Please show hands if you are familiar with causal graphs or baysian networks. Should I explain what this diagram means?
+
+
+---
+
+.right-column[
+`\(x\)` is *partly observed* because we have *validation data* `\(x^*\)`.
+]
+
+---
+
+
+.right-column[
+`\(x\)` is *partly observed* because we have *validation data* `\(x^*\)`.
+
+`\(k\)` are the *features* used by the *predictive model* `\(g(k)\)`.
+
+]
+
+---
+
+.right-column[
+`\(x\)` is *partly observed* because we have *validation data* `\(x^*\)`.
+
+`\(k\)` are the *features* used by the *predictive model* `\(g(k)\)`.
+
+The predictions `\(w\)` are a *proxy variable* `\(g(k) = \hat{x} = w\)`.
+
+]
+
+---
+
+
+.right-column[
+`\(x\)` is *partly observed* because we have *validation data* `\(x^*\)`.
+
+`\(k\)` are the *features* used by the *predictive model* `\(g(k)\)`.
+
+The predictions `\(w\)` are a *proxy variable* `\(g(k) = \hat{x} = w\)`.
+
+`\(x = w + \xi\)` because the predictive model makes errors.
+
+]
+
+---
+
+
+layout:true
+<div class="my-header"></div>
+
+### .border[Noise in a *covariate* creates *attenuation bias*.]
+
+.large[.left-column[]]
+
+---
+.right-column[
+
+We want to estimate, `\(y = Bx + \varepsilon\)`, but we estimate `\(y = Bw + \varepsilon\)` instead.
+
+`\(x = w + \xi\)` because the predictive model makes errors.
+
+]
+---
+
+.right-column[
+
+We want to estimate, `\(y = Bx + \varepsilon\)`, but we estimate `\(y = Bw + \varepsilon\)` instead.
+
+`\(x = w + \xi\)` because the predictive model makes errors.
+
+
+Assume `\(g(k)\)` is *unbiased* so `\(E(\xi)=0\)`. Also assume error is *nondifferential* so `\(E(\xi y)=0\)`:
+
+]
+
+---
+
+.right-column[
+
+We want to estimate, `\(y = Bx + \varepsilon\)`, but we estimate `\(y = Bw + \varepsilon\)` instead.
+
+`\(x = w + \xi\)` because the predictive model makes errors.
+
+Assume `\(g(k)\)` is *unbiased* so `\(E(\xi)=0\)`. Also assume error is *nondifferential* so `\(E(\xi y)=0\)`:
+
+`$$\widehat{B_w}^{ols}=\frac{\sum^n_{j=j}{(x_j + \xi_j - \overline{(x + \xi)})}(y_j - \bar{y})}{\sum_{j=1}^n{(x_j + \xi_j - \overline{(x+\xi)})^2}} = \frac{\sum^n_{j=j}{(x_j - \bar{x})(y_j -
+ \bar{y})}}{\sum_{j=1}^n{(x_j + \xi_j - \bar{x}){^2}}}$$`
+
+]
+
+---
+
+.right-column[
+
+We want to estimate, `\(y = Bx + \varepsilon\)`, but we estimate `\(y = Bw + \varepsilon\)` instead.
+
+`\(x = w + \xi\)` because the predictive model makes errors.
+
+Assume `\(g(k)\)` is *unbiased* so `\(E(\xi)=0\)`. Also assume error is *nondifferential* so `\(E(\xi y)=0\)`:
+
+`$$\widehat{B_w}^{ols}=\frac{\sum^n_{j=j}{(x_j + \xi_j - \overline{(x + \xi)})}(y_j - \bar{y})}{\sum_{j=1}^n{(x_j + \xi_j - \overline{(x+\xi)})^2}} = \frac{\sum^n_{j=j}{(x_j - \bar{x})(y_j -
+ \bar{y})}}{\sum_{j=1}^n{(x_j + \color{red}{\xi_j} - \bar{x})\color{red}{^2}}}$$`
+
+In this scenario, it's clear that `\(\widehat{B_w}^{ols} < B_x\)`.
+
+
+]
+
+
+???
+
+Please raise your hands if you're familiar with attenuation bias. I expect that its covered in some graduate stats classes, but not universally.
+
+---
+class:large
+layout:false
+<div class="my-header"></div>
+
+### .border[Beyond attenuation bias]
+.larger[Measurement error can theaten validity because:]
+
+- Attenuation bias *spreads* (e.g., to marginal effects as illustrated later).
+
+--
+
+- Measurement error can be *differential*— not distributed evenly and possible correlated with `\(x\)`, `\(y\)`, or `\(\varepsilon\)`.
+
+--
+
+- *Bias can be away from 0* in GLMs and nonlinear models or if measurement error is differential.
+
+--
+
+- *Confounding* if the *predictive model is biased* introducing a correlation the measurement error and the residuals `\((E[\xi\varepsilon]=0)\)`.
+
+
+---
+
+class:large
+layout:false
+<div class="my-header"></div>
+
+### .border[Correcting measurement error]
+
+There's a vast literature in statistics on measurement error. Mostly about noise you'd find in sensors. Lots of ideas. No magic bullets.
+
+--
+
+I'm going to briefly cover 3 different approaches: *multiple imputation*, *regression calibration* and *2SLS+GMM*.
+
+--
+
+These all depend on *validation data*. I'm going to ignore where this comes from, but assume it's a random sample of the hypothesis testing dataset.
+
+--
+
+You can *and should* use it to improve your statistical estimates.
+
+---
+
+<div class="my-header"></div>
+
+### .border[Multiple Imputation (MI) treats Measurement Error as a Missing Data Problem]
+
+1. Use validation data to estimate `\(f(x|w,y)\)`, a probabilistic model of `\(x\)`.
+
+--
+
+2. *Sample* `\(m\)` datasets from `\(\widehat{f(x|w,y)}\)`.
+
+--
+
+3. Run your analysis on each of the `\(m\)` datasets.
+
+--
+
+4. Average the results from the `\(m\)` analyses using Rubin's rules.
+
+--
+
+.e[Advantages:] *Very flexible!* Sometimes can work if the predictor $g(k) $ is biased. Good R packages (**`{Amelia}`**, `{mi}`, `{mice}`, `{brms}`).
+
+--
+
+.e[Disadvantages:] Results depend on quality of `\(\widehat{f(x|w,y)}\)`; May require more validation data, computationally expensive, statistically inefficient and doesn't seem to benefit much from larger datasets.
+
+---
+
+### .border[Regression calibration directly adjusts for attenuation bias.]
+
+1. Use validation data to estimate the errors `\(\hat{\xi}\)`.
+
+--
+
+2. Use `\(\hat{\xi}\)` to correct the OLS estimate.
+
+--
+
+3. Correct the standard errors using MLE or bootstrapping.
+
+--
+
+.e[Advantages:] Simple, fast.
+
+--
+
+.e[Disadvantages:] Limited to OLS models. Requires an unbiased predictor `\(g(k)\)`. R support (`{mecor}` R package) is pretty new.
+
+---
+layout:true
+### .border[2SLS+GMM is designed for this specific problem]
+
+.left-column[]
+
+*Regression calibration with a trick.*
+
+---
+.right-column[
+
+1. Estimate `\(x = w + \xi\)` to obtain `\(\hat{x}\)`. (First-stage LS).
+
+]
+
+---
+.right-column[
+
+1. Estimate `\(x = w + \xi\)` to obtain `\(\hat{x}\)`. (First-stage LS).
+
+2. Estimate `\(y = B^{2sls}\hat{x} + \varepsilon^{2sls}\)`. (Second-stage LS / regression calibration).
+
+]
+
+---
+.right-column[
+
+1. Estimate `\(x = w + \xi\)` to obtain `\(\hat{x}\)`. (First-stage LS).
+
+2. Estimate `\(y = B^{2sls}\hat{x} + \varepsilon^{2sls}\)`. (Second-stage LS / regression calibration).
+
+3. Estimate `\(y = B^{val}x^* + \varepsilon^{val}\)`. (Validation dataset model).
+
+]
+
+---
+.right-column[
+
+1. Estimate `\(x = w + \xi\)` to obtain `\(\hat{x}\)`. (First-stage LS).
+
+2. Estimate `\(y = B^{2sls}\hat{x} + \varepsilon^{2sls}\)`. (Second-stage LS / regression calibration).
+
+3. Estimate `\(y = B^{val}x^* + \varepsilon^{val}\)`. (Validation dataset model).
+
+4. Combine `\(B^{val}\)` and `\(B^{2sls}\)` using the generalized method of moments (GMM).
+
+]
+
+---
+.right-column[
+
+1. Estimate `\(x = w + \xi\)` to obtain `\(\hat{x}\)`. (First-stage LS).
+
+2. Estimate `\(y = B^{2sls}\hat{x} + \varepsilon^{2sls}\)`. (Second-stage LS / regression calibration).
+
+3. Estimate `\(y = B^{val}x^* + \varepsilon^{val}\)`. (Validation dataset model).
+
+4. Combine `\(B^{val}\)` and `\(B^{2sls}\)` using the generalized method of moments (GMM).
+
+Advantages: Accurate. Sometimes robust if biased predictor `\(g(k)\)` is biased. In theory, flexible to any models that can be fit using GMM.
+
+]
+
+
+---
+.right-column[
+
+1. Estimate `\(x = w + \xi\)` to obtain `\(\hat{x}\)`. (First-stage LS).
+
+2. Estimate `\(y = B^{2sls}\hat{x} + \varepsilon^{2sls}\)`. (Second-stage LS / regression calibration).
+
+3. Estimate `\(y = B^{val}x^* + \varepsilon^{val}\)`. (Validation dataset model).
+
+4. Combine `\(B^{val}\)` and `\(B^{2sls}\)` using the generalized method of moments (GMM).
+
+Advantages: Accurate. Sometimes robust if biased predictor `\(g(k)\)` is biased. In theory, flexible to any models that can be fit using GMM.
+
+Disadvantages: Implementation (`{predictionError}`) is new. API is cumbersome and only supports linear models. Not robust if `\(E(w\varepsilon) \ne 0\)`. GMM may be unfamiliar to audiences.
+
+]
+
+---
+layout:false
+### .border[Testing attention bias correction]
+
+<div class="my-header"></div>
+
+I've run simulations to test these approaches in several scenarios.
+
+The model is not very good: about 70% accurate.
+
+Most plausible scenario:
+
+y is continuous and normal-ish.
+
+--
+
+`\(x\)` is binary (human labels) `\(P(x)=0.5\)`.
+
+--
+
+`\(w\)` is the *continuous predictor* (e.g., probability) output of `\(f(x)\)` (not binary predictions).
+
+--
+
+if `\(w\)` is binary, most methods struggle, but regression calibration and 2SLS+GMM can do okay.
+
+---
+layout:false
+
+### .border[Example 1: estimator of the effect of x]
+
+.right-column[
+<!-- -->
+]
+.left-column[
+
+All methods work in this scenario
+
+Multiple imputation is inefficient.
+
+]
+
+
+---
+### .border[What about bias?]
+
+.left-column[
+.large[]
+]
+
+.right-column[
+A few notes on this scenario.
+
+`\(B_x = 0.2\)`, `\(B_g=-0.2\)` and `\(sd(\varepsilon)=3\)`. So the signal-to-noise ratio is high.
+
+`\(r\)` can be concieved of as a missing feature in the predictive model `\(g(k)\)` that is also correlated with `\(y\)`.
+
+For example `\(r\)` might be the *race* of a commentor, `\(x\)` could be *racial harassment*, `\(y\)` whether the commentor gets banned and `\(k\)` only has textual features but human coders can see user profiles to know `\(r\)`.
+
+]
+
+---
+layout:false
+### .border[Example 2: Estimates of the effect of x ]
+
+.center[
+<!-- -->
+]
+---
+layout:false
+
+### .border[Example 2: Estimates of the effect of r]
+
+.center[
+<!-- -->
+]
+---
+
+layout:false
+class:large
+
+###.border[Takeaways from example 2]
+
+Bias in the predictive model creates bias in hypothesis tests.
+
+--
+
+Bias can be corrected *in this case*.
+
+--
+
+The next scenario has bias that's more tricky.
+
+--
+
+Multiple imputation helps, but doesn't fully correct the bias.
+
+---
+
+layout:false
+
+### .border[When will GMM+2SLS fail?]
+
+.large[.left-column[]]
+
+.right-column[The catch with GMM:
+
+.emph[Exclusion restriction:] `\(E[w \varepsilon] = 0\)`.
+
+The restriction is violated if a variable `\(U\)` causes both `\(K\)` and `\(Y\)` and `\(X\)` causes `\(K\)` (not visa-versa).
+
+]
+
+???
+
+GMM optimizes a model to a system of equations of which the exclusion restriction is one. So if that assumption isn't true it will biased.
+
+This is a different assumption than that of OLS or GLM models.
+
+---
+
+layout:false
+
+### .border[Example 3: Estimates of the effect of x]
+
+.center[
+<!-- -->
+]
+
+
+
+---
+
+### .border[Takaways]
+
+- Attenuation bias can be a big problem with noisy predictors—leading to small and biased estimates.
+
+- For more general hypothesis tests or if the predictor is biased, measurement error can lead to false discovery.
+
+- It's fixable with validation data—you may not need that much and you should already be getting it.
+
+- This means it can be okay poor predictors for hypothesis testing.
+
+- The ecosystem is underdeveloped, but a lot of methods have been researched.
+
+- Take advantage of machine learning + big data and get precise estimates when the signal-to-noise ratio is high!
+
+---
+layout:false
+
+### .border[Future work: Noise in the *outcome*]
+
+I've been focusing on noise in *covariates.* What if the predictive algorithm is used to measure the *outcome* `\(y\)`?
+
+--
+
+This isn't a problem in the simplest case (linear regression with homoskedastic errors). Noise in `\(y\)` is projected into the error term.
+
+--
+
+Noise in the outcome is still a problem if errors are heteroskedastic and for GLMs / non-linear regression (e.g., logistic regression).
+
+--
+
+Multiple imputation (in theory) could help here. The other method's aren't designed for this case.
+
+--
+
+Solving this problem could be an important methodological contribution with a very broad impact.
+
+---
+# .border[Questions?]
+
+Links to slides:[html](https://teblunthuis.cc/~nathante/slides/ecological_adaptation_ica_2022.html) [pdf](https://teblunthuis.cc/~nathante/slides/ecological_adaptation_ica_2022.pdf)
+
+Link to a messy git repository:
+
+<i class="fa fa-envelope" aria-hidden='true'></i> nathan.teblunthuis@northwestern.edu
+
+<i class="fa fa-twitter" aria-hidden='true'></i> @groceryheist
+
+<i class="fa fa-globe" aria-hidden='true'></i> [https://communitydata.science](https://communitydata.science)
+
+
+
+<!-- ### .border[Multiple imputation struggles with discrete variables] -->
+
+<!-- In my experiments I've found that the 2SLS+GMM method works well with a broader range of data types. -->
+
+<!-- To illustrate, Example 3 is the same as Example 2, but with `\(x\)` and `\(w\)` as discrete variables. -->
+
+<!-- Practicallly speaking, a continuous "score" `\(w\)` is often available, and my opinion is that usually this is better + more informative than model predictions in all cases. Continuous validation data may be more difficult to obtain, but it is often possible using techniques like pairwise comparison. -->
+<!-- layout:false -->
+<!-- ### .border[Example 3: Estimates of the effect of x ] -->
+
+<!-- .center[ -->
+<!-- ```{r echo=FALSE, message=FALSE, warning=FALSE, result='asis', dev='svg', fig.width=8, fig.asp=.625,cache=F} -->
+
+<!-- #plot.df <- -->
+<!-- plot.df <- plot.df.example.2[,':='(method=factor(method,levels=c("Naive","Multiple imputation", "Multiple imputation (Classifier features unobserved)","Regression Calibration","2SLS+gmm","Feasible"),ordered=T), -->
+<!-- N=factor(N), -->
+<!-- m=factor(m))] -->
+
+<!-- plot.df <- plot.df[(variable=='x') & (m != 1000) & (m!=500) & (N!=5000) & (N!=10000) & !is.na(p.true.in.ci) & (method!="Multiple imputation (Classifier features unobserved)")] -->
+<!-- p <- ggplot(plot.df, aes(y=mean.est, ymax=mean.est + var.est/2, ymin=mean.est-var.est/2, x=method)) -->
+<!-- p <- p + geom_hline(aes(yintercept=0.2),linetype=2) -->
+
+<!-- p <- p + geom_pointrange() + facet_grid(m~N,as.table=F) + scale_x_discrete(labels=label_wrap_gen(4)) -->
+
+<!-- print(p) -->
+
+<!-- # get gtable object -->
+
+<!-- .large[.left []] -->
+
+<!-- There are at two general ways using a predictive model can introduce bias: *attenuation*, and *confounding.* -->
+
+<!-- Counfounding can be broken down into 4 types: -->
+
+<!-- .right[Confounding on `\(X\)` by observed variables -->
+
+<!-- Confounding on `\(Y\)` by observed variables -->
+<!-- ] -->
+
+<!-- .left[Confounding on `\(X\)` by *un*observed variables -->
+
+<!-- Confounding on `\(Y\)` by *un*observed variables -->
+<!-- ] -->
+
+<!-- Attenuation and the top-right column can be dealt with relative ease using a few different methods. -->
+
+<!-- The bottom-left column can be addressed, but so far I haven't found a magic bullet. -->
+
+<!-- The left column is pretty much a hopeless situation. -->
+ </textarea>
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+ );
+ if (!scripts.length) return;
+ for (var i = 0; i < scripts.length; i++) {
+ var s = document.createElement('script');
+ var code = document.createTextNode(scripts[i].textContent);
+ s.appendChild(code);
+ var scriptAttrs = scripts[i].attributes;
+ for (var j = 0; j < scriptAttrs.length; j++) {
+ s.setAttribute(scriptAttrs[j].name, scriptAttrs[j].value);
+ }
+ scripts[i].parentElement.replaceChild(s, scripts[i]);
+ }
+})();
+(function() {
+ var links = document.getElementsByTagName('a');
+ for (var i = 0; i < links.length; i++) {
+ if (/^(https?:)?\/\//.test(links[i].getAttribute('href'))) {
+ links[i].target = '_blank';
+ }
+ }
+})();</script>
+
+<script>
+slideshow._releaseMath = function(el) {
+ var i, text, code, codes = el.getElementsByTagName('code');
+ for (i = 0; i < codes.length;) {
+ code = codes[i];
+ if (code.parentNode.tagName !== 'PRE' && code.childElementCount === 0) {
+ text = code.textContent;
+ if (/^\\\((.|\s)+\\\)$/.test(text) || /^\\\[(.|\s)+\\\]$/.test(text) ||
+ /^\$\$(.|\s)+\$\$$/.test(text) ||
+ /^\\begin\{([^}]+)\}(.|\s)+\\end\{[^}]+\}$/.test(text)) {
+ code.outerHTML = code.innerHTML; // remove <code></code>
+ continue;
+ }
+ }
+ i++;
+ }
+};
+slideshow._releaseMath(document);
+</script>
+<!-- dynamically load mathjax for compatibility with self-contained -->
+<script>
+(function () {
+ var script = document.createElement('script');
+ script.type = 'text/javascript';
+ script.src = 'https://mathjax.rstudio.com/latest/MathJax.js?config=TeX-MML-AM_CHTML';
+ if (location.protocol !== 'file:' && /^https?:/.test(script.src))
+ script.src = script.src.replace(/^https?:/, '');
+ document.getElementsByTagName('head')[0].appendChild(script);
+})();
+</script>
+ </body>
+</html>
--- /dev/null
+../../../charts/example_1_dag/example_1_dag.png
\ No newline at end of file
--- /dev/null
+../../../charts/example_1_dag/example_1_dag.svg
\ No newline at end of file
--- /dev/null
+../../../charts/example_2_dag/example_2_dag.png
\ No newline at end of file
--- /dev/null
+../../../charts/example_2_dag/example_2_dag.svg
\ No newline at end of file
--- /dev/null
+../../../charts/example_3_dag/example_3_dag.png
\ No newline at end of file
--- /dev/null
+../../../charts/example_3_dag/example_3_dag.svg
\ No newline at end of file
--- /dev/null
+// Pandoc 2.9 adds attributes on both header and div. We remove the former (to
+// be compatible with the behavior of Pandoc < 2.8).
+document.addEventListener('DOMContentLoaded', function(e) {
+ var hs = document.querySelectorAll("div.section[class*='level'] > :first-child");
+ var i, h, a;
+ for (i = 0; i < hs.length; i++) {
+ h = hs[i];
+ if (!/^h[1-6]$/i.test(h.tagName)) continue; // it should be a header h1-h6
+ a = h.attributes;
+ while (a.length > 0) h.removeAttribute(a[0].name);
+ }
+});
--- /dev/null
+// Pandoc 2.9 adds attributes on both header and div. We remove the former (to
+// be compatible with the behavior of Pandoc < 2.8).
+document.addEventListener('DOMContentLoaded', function(e) {
+ var hs = document.querySelectorAll("div.section[class*='level'] > :first-child");
+ var i, h, a;
+ for (i = 0; i < hs.length; i++) {
+ h = hs[i];
+ if (!/^h[1-6]$/i.test(h.tagName)) continue; // it should be a header h1-h6
+ a = h.attributes;
+ while (a.length > 0) h.removeAttribute(a[0].name);
+ }
+});
--- /dev/null
+/*
+ * /MathJax/config/TeX-MML-AM_CHTML.js
+ *
+ * Copyright (c) 2010-2017 The MathJax Consortium
+ *
+ * Part of the MathJax library.
+ * See http://www.mathjax.org for details.
+ *
+ * Licensed under the Apache License, Version 2.0;
+ * you may not use this file except in compliance with the License.
+ *
+ * http://www.apache.org/licenses/LICENSE-2.0
+ */
+
+MathJax.Ajax.Preloading(
+ "[MathJax]/jax/input/TeX/config.js",
+ "[MathJax]/jax/input/MathML/config.js",
+ "[MathJax]/jax/input/AsciiMath/config.js",
+ "[MathJax]/jax/output/CommonHTML/config.js",
+ "[MathJax]/jax/output/PreviewHTML/config.js",
+ "[MathJax]/extensions/tex2jax.js",
+ "[MathJax]/extensions/mml2jax.js",
+ "[MathJax]/extensions/asciimath2jax.js",
+ "[MathJax]/extensions/MathEvents.js",
+ "[MathJax]/extensions/MathZoom.js",
+ "[MathJax]/extensions/MathMenu.js",
+ "[MathJax]/jax/element/mml/jax.js",
+ "[MathJax]/extensions/toMathML.js",
+ "[MathJax]/extensions/TeX/noErrors.js",
+ "[MathJax]/extensions/TeX/noUndefined.js",
+ "[MathJax]/jax/input/TeX/jax.js",
+ "[MathJax]/extensions/TeX/AMSmath.js",
+ "[MathJax]/extensions/TeX/AMSsymbols.js",
+ "[MathJax]/jax/input/MathML/jax.js",
+ "[MathJax]/jax/input/AsciiMath/jax.js",
+ "[MathJax]/jax/output/PreviewHTML/jax.js",
+ "[MathJax]/extensions/fast-preview.js",
+ "[MathJax]/extensions/AssistiveMML.js",
+ "[MathJax]/extensions/a11y/accessibility-menu.js"
+);
+
+MathJax.Hub.Config({
+ extensions: ['[a11y]/accessibility-menu.js']
+});
+
+MathJax.InputJax.TeX=MathJax.InputJax({id:"TeX",version:"2.7.2",directory:MathJax.InputJax.directory+"/TeX",extensionDir:MathJax.InputJax.extensionDir+"/TeX",config:{TagSide:"right",TagIndent:"0.8em",MultLineWidth:"85%",equationNumbers:{autoNumber:"none",formatNumber:function(a){return a},formatTag:function(a){return"("+a+")"},formatID:function(a){return"mjx-eqn-"+String(a).replace(/[:"'<>&]/g,"")},formatURL:function(b,a){return a+"#"+escape(b)},useLabelIds:true}},resetEquationNumbers:function(){}});MathJax.InputJax.TeX.Register("math/tex");MathJax.InputJax.TeX.loadComplete("config.js");
+MathJax.InputJax.MathML=MathJax.InputJax({id:"MathML",version:"2.7.2",directory:MathJax.InputJax.directory+"/MathML",extensionDir:MathJax.InputJax.extensionDir+"/MathML",entityDir:MathJax.InputJax.directory+"/MathML/entities",config:{useMathMLspacing:false}});MathJax.InputJax.MathML.Register("math/mml");MathJax.InputJax.MathML.loadComplete("config.js");
+MathJax.InputJax.AsciiMath=MathJax.InputJax({id:"AsciiMath",version:"2.7.2",directory:MathJax.InputJax.directory+"/AsciiMath",extensionDir:MathJax.InputJax.extensionDir+"/AsciiMath",config:{fixphi:true,useMathMLspacing:true,displaystyle:true,decimalsign:"."}});MathJax.InputJax.AsciiMath.Register("math/asciimath");MathJax.InputJax.AsciiMath.loadComplete("config.js");
+MathJax.OutputJax.CommonHTML=MathJax.OutputJax({id:"CommonHTML",version:"2.7.2",directory:MathJax.OutputJax.directory+"/CommonHTML",extensionDir:MathJax.OutputJax.extensionDir+"/CommonHTML",autoloadDir:MathJax.OutputJax.directory+"/CommonHTML/autoload",fontDir:MathJax.OutputJax.directory+"/CommonHTML/fonts",webfontDir:MathJax.OutputJax.fontDir+"/HTML-CSS",config:{matchFontHeight:true,scale:100,minScaleAdjust:50,mtextFontInherit:false,undefinedFamily:"STIXGeneral,'Cambria Math','Arial Unicode MS',serif",EqnChunk:(MathJax.Hub.Browser.isMobile?20:100),EqnChunkFactor:1.5,EqnChunkDelay:100,linebreaks:{automatic:false,width:"container"}}});if(!MathJax.Hub.config.delayJaxRegistration){MathJax.OutputJax.CommonHTML.Register("jax/mml")}MathJax.OutputJax.CommonHTML.loadComplete("config.js");
+MathJax.OutputJax.PreviewHTML=MathJax.OutputJax({id:"PreviewHTML",version:"2.7.2",directory:MathJax.OutputJax.directory+"/PreviewHTML",extensionDir:MathJax.OutputJax.extensionDir+"/PreviewHTML",noFastPreview:true,config:{scale:100,minScaleAdjust:50,mtextFontInherit:false,linebreaks:{automatic:false,width:"container"}}});if(!MathJax.Hub.config.delayJaxRegistration){MathJax.OutputJax.PreviewHTML.Register("jax/mml")}MathJax.OutputJax.PreviewHTML.loadComplete("config.js");
+MathJax.Extension.tex2jax={version:"2.7.2",config:{inlineMath:[["\\(","\\)"]],displayMath:[["$$","$$"],["\\[","\\]"]],balanceBraces:true,skipTags:["script","noscript","style","textarea","pre","code","annotation","annotation-xml"],ignoreClass:"tex2jax_ignore",processClass:"tex2jax_process",processEscapes:false,processEnvironments:true,processRefs:true,preview:"TeX"},ignoreTags:{br:(MathJax.Hub.Browser.isMSIE&&document.documentMode<9?"\n":" "),wbr:"","#comment":""},PreProcess:function(a){if(!this.configured){this.config=MathJax.Hub.CombineConfig("tex2jax",this.config);if(this.config.Augment){MathJax.Hub.Insert(this,this.config.Augment)}if(typeof(this.config.previewTeX)!=="undefined"&&!this.config.previewTeX){this.config.preview="none"}this.configured=true}if(typeof(a)==="string"){a=document.getElementById(a)}if(!a){a=document.body}if(this.createPatterns()){this.scanElement(a,a.nextSibling)}},createPatterns:function(){var d=[],e=[],c,a,b=this.config;this.match={};for(c=0,a=b.inlineMath.length;c<a;c++){d.push(this.patternQuote(b.inlineMath[c][0]));this.match[b.inlineMath[c][0]]={mode:"",end:b.inlineMath[c][1],pattern:this.endPattern(b.inlineMath[c][1])}}for(c=0,a=b.displayMath.length;c<a;c++){d.push(this.patternQuote(b.displayMath[c][0]));this.match[b.displayMath[c][0]]={mode:"; mode=display",end:b.displayMath[c][1],pattern:this.endPattern(b.displayMath[c][1])}}if(d.length){e.push(d.sort(this.sortLength).join("|"))}if(b.processEnvironments){e.push("\\\\begin\\{([^}]*)\\}")}if(b.processEscapes){e.push("\\\\*\\\\\\$")}if(b.processRefs){e.push("\\\\(eq)?ref\\{[^}]*\\}")}this.start=new RegExp(e.join("|"),"g");this.skipTags=new RegExp("^("+b.skipTags.join("|")+")$","i");var f=[];if(MathJax.Hub.config.preRemoveClass){f.push(MathJax.Hub.config.preRemoveClass)}if(b.ignoreClass){f.push(b.ignoreClass)}this.ignoreClass=(f.length?new RegExp("(^| )("+f.join("|")+")( |$)"):/^$/);this.processClass=new RegExp("(^| )("+b.processClass+")( |$)");return(e.length>0)},patternQuote:function(a){return a.replace(/([\^$(){}+*?\-|\[\]\:\\])/g,"\\$1")},endPattern:function(a){return new RegExp(this.patternQuote(a)+"|\\\\.|[{}]","g")},sortLength:function(d,c){if(d.length!==c.length){return c.length-d.length}return(d==c?0:(d<c?-1:1))},scanElement:function(c,b,g){var a,e,d,f;while(c&&c!=b){if(c.nodeName.toLowerCase()==="#text"){if(!g){c=this.scanText(c)}}else{a=(typeof(c.className)==="undefined"?"":c.className);e=(typeof(c.tagName)==="undefined"?"":c.tagName);if(typeof(a)!=="string"){a=String(a)}f=this.processClass.exec(a);if(c.firstChild&&!a.match(/(^| )MathJax/)&&(f||!this.skipTags.exec(e))){d=(g||this.ignoreClass.exec(a))&&!f;this.scanElement(c.firstChild,b,d)}}if(c){c=c.nextSibling}}},scanText:function(b){if(b.nodeValue.replace(/\s+/,"")==""){return b}var a,c;this.search={start:true};this.pattern=this.start;while(b){this.pattern.lastIndex=0;while(b&&b.nodeName.toLowerCase()==="#text"&&(a=this.pattern.exec(b.nodeValue))){if(this.search.start){b=this.startMatch(a,b)}else{b=this.endMatch(a,b)}}if(this.search.matched){b=this.encloseMath(b)}if(b){do{c=b;b=b.nextSibling}while(b&&this.ignoreTags[b.nodeName.toLowerCase()]!=null);if(!b||b.nodeName!=="#text"){return(this.search.close?this.prevEndMatch():c)}}}return b},startMatch:function(a,b){var f=this.match[a[0]];if(f!=null){this.search={end:f.end,mode:f.mode,pcount:0,open:b,olen:a[0].length,opos:this.pattern.lastIndex-a[0].length};this.switchPattern(f.pattern)}else{if(a[0].substr(0,6)==="\\begin"){this.search={end:"\\end{"+a[1]+"}",mode:"; mode=display",pcount:0,open:b,olen:0,opos:this.pattern.lastIndex-a[0].length,isBeginEnd:true};this.switchPattern(this.endPattern(this.search.end))}else{if(a[0].substr(0,4)==="\\ref"||a[0].substr(0,6)==="\\eqref"){this.search={mode:"",end:"",open:b,pcount:0,olen:0,opos:this.pattern.lastIndex-a[0].length};return this.endMatch([""],b)}else{var d=a[0].substr(0,a[0].length-1),g,c;if(d.length%2===0){c=[d.replace(/\\\\/g,"\\")];g=1}else{c=[d.substr(1).replace(/\\\\/g,"\\"),"$"];g=0}c=MathJax.HTML.Element("span",null,c);var e=MathJax.HTML.TextNode(b.nodeValue.substr(0,a.index));b.nodeValue=b.nodeValue.substr(a.index+a[0].length-g);b.parentNode.insertBefore(c,b);b.parentNode.insertBefore(e,c);this.pattern.lastIndex=g}}}return b},endMatch:function(a,c){var b=this.search;if(a[0]==b.end){if(!b.close||b.pcount===0){b.close=c;b.cpos=this.pattern.lastIndex;b.clen=(b.isBeginEnd?0:a[0].length)}if(b.pcount===0){b.matched=true;c=this.encloseMath(c);this.switchPattern(this.start)}}else{if(a[0]==="{"){b.pcount++}else{if(a[0]==="}"&&b.pcount){b.pcount--}}}return c},prevEndMatch:function(){this.search.matched=true;var a=this.encloseMath(this.search.close);this.switchPattern(this.start);return a},switchPattern:function(a){a.lastIndex=this.pattern.lastIndex;this.pattern=a;this.search.start=(a===this.start)},encloseMath:function(b){var a=this.search,g=a.close,f,d,c;if(a.cpos===g.length){g=g.nextSibling}else{g=g.splitText(a.cpos)}if(!g){f=g=MathJax.HTML.addText(a.close.parentNode,"")}a.close=g;d=(a.opos?a.open.splitText(a.opos):a.open);while((c=d.nextSibling)&&c!==g){if(c.nodeValue!==null){if(c.nodeName==="#comment"){d.nodeValue+=c.nodeValue.replace(/^\[CDATA\[((.|\n|\r)*)\]\]$/,"$1")}else{d.nodeValue+=c.nodeValue}}else{var h=this.ignoreTags[c.nodeName.toLowerCase()];d.nodeValue+=(h==null?" 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true}if(!h[l]){return true}return h[l](n,m)},Click:function(m,l){if(this.settings.zoom==="Click"){return this.Zoom(m,l)}},DblClick:function(m,l){if(this.settings.zoom==="Double-Click"||this.settings.zoom==="DoubleClick"){return this.Zoom(m,l)}},Hover:function(m,l){if(this.settings.zoom==="Hover"){this.Zoom(m,l);return true}return false},Zoom:function(o,u){this.Remove();b.ClearHoverTimer();g.ClearSelection();var s=MathJax.OutputJax[u.jaxID];var p=s.getJaxFromMath(u);if(p.hover){b.UnHover(p)}var q=this.findContainer(u);var l=Math.floor(0.85*q.clientWidth),t=Math.max(document.body.clientHeight,document.documentElement.clientHeight);if(this.getOverflow(q)!=="visible"){t=Math.min(q.clientHeight,t)}t=Math.floor(0.85*t);var n=d.Element("span",{id:"MathJax_ZoomFrame"},[["span",{id:"MathJax_ZoomOverlay",onmousedown:this.Remove}],["span",{id:"MathJax_Zoom",onclick:this.Remove,style:{visibility:"hidden",fontSize:this.settings.zscale}},[["span",{style:{display:"inline-block","white-space":"nowrap"}}]]]]);var z=n.lastChild,w=z.firstChild,r=n.firstChild;u.parentNode.insertBefore(n,u);u.parentNode.insertBefore(u,n);if(w.addEventListener){w.addEventListener("mousedown",this.Remove,true)}var m=z.offsetWidth||z.clientWidth;l-=m;t-=m;z.style.maxWidth=l+"px";z.style.maxHeight=t+"px";if(this.msieTrapEventBug){var y=d.Element("span",{id:"MathJax_ZoomEventTrap",onmousedown:this.Remove});n.insertBefore(y,z)}if(this.msieZIndexBug){var v=d.addElement(document.body,"img",{src:"about:blank",id:"MathJax_ZoomTracker",width:0,height:0,style:{width:0,height:0,position:"relative"}});n.style.position="relative";n.style.zIndex=i.styles["#MathJax_ZoomOverlay"]["z-index"];n=v}var x=s.Zoom(p,w,u,l,t);if(this.msiePositionBug){if(this.msieSizeBug){z.style.height=x.zH+"px";z.style.width=x.zW+"px"}if(z.offsetHeight>t){z.style.height=t+"px";z.style.width=(x.zW+this.scrollSize)+"px"}if(z.offsetWidth>l){z.style.width=l+"px";z.style.height=(x.zH+this.scrollSize)+"px"}}if(this.operaPositionBug){z.style.width=Math.min(l,x.zW)+"px"}if(z.offsetWidth>m&&z.offsetWidth-m<l&&z.offsetHeight-m<t){z.style.overflow="visible"}this.Position(z,x);if(this.msieTrapEventBug){y.style.height=z.clientHeight+"px";y.style.width=z.clientWidth+"px";y.style.left=(parseFloat(z.style.left)+z.clientLeft)+"px";y.style.top=(parseFloat(z.style.top)+z.clientTop)+"px"}z.style.visibility="";if(this.settings.zoom==="Hover"){r.onmouseover=this.Remove}if(window.addEventListener){addEventListener("resize",this.Resize,false)}else{if(window.attachEvent){attachEvent("onresize",this.Resize)}else{this.onresize=window.onresize;window.onresize=this.Resize}}a.signal.Post(["math zoomed",p]);return e(o)},Position:function(p,r){p.style.display="none";var q=this.Resize(),m=q.x,s=q.y,l=r.mW;p.style.display="";var o=-l-Math.floor((p.offsetWidth-l)/2),n=r.Y;p.style.left=Math.max(o,10-m)+"px";p.style.top=Math.max(n,10-s)+"px";if(!h.msiePositionBug){h.SetWH()}},Resize:function(m){if(h.onresize){h.onresize(m)}var q=document.getElementById("MathJax_ZoomFrame"),l=document.getElementById("MathJax_ZoomOverlay");var o=h.getXY(q),n=h.findContainer(q);if(h.getOverflow(n)!=="visible"){l.scroll_parent=n;var p=h.getXY(n);o.x-=p.x;o.y-=p.y;p=h.getBorder(n);o.x-=p.x;o.y-=p.y}l.style.left=(-o.x)+"px";l.style.top=(-o.y)+"px";if(h.msiePositionBug){setTimeout(h.SetWH,0)}else{h.SetWH()}return o},SetWH:function(){var l=document.getElementById("MathJax_ZoomOverlay");if(!l){return}l.style.display="none";var m=l.scroll_parent||document.documentElement||document.body;l.style.width=m.scrollWidth+"px";l.style.height=Math.max(m.clientHeight,m.scrollHeight)+"px";l.style.display=""},findContainer:function(l){l=l.parentNode;while(l.parentNode&&l!==document.body&&h.getOverflow(l)==="visible"){l=l.parentNode}return l},getOverflow:(window.getComputedStyle?function(l){return getComputedStyle(l).overflow}:function(l){return(l.currentStyle||{overflow:"visible"}).overflow}),getBorder:function(o){var m={thin:1,medium:2,thick:3};var n=(window.getComputedStyle?getComputedStyle(o):(o.currentStyle||{borderLeftWidth:0,borderTopWidth:0}));var l=n.borderLeftWidth,p=n.borderTopWidth;if(m[l]){l=m[l]}else{l=parseInt(l)}if(m[p]){p=m[p]}else{p=parseInt(p)}return{x:l,y:p}},getXY:function(o){var l=0,n=0,m;m=o;while(m.offsetParent){l+=m.offsetLeft;m=m.offsetParent}if(h.operaPositionBug){o.style.border="1px 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m=(n>=9);h.msiePositionBug=!m;h.msieSizeBug=l.versionAtLeast("7.0")&&(!document.documentMode||n===7||n===8);h.msieZIndexBug=(n<=7);h.msieInlineBlockAlignBug=(n<=7);h.msieTrapEventBug=!window.addEventListener;if(document.compatMode==="BackCompat"){h.scrollSize=52}if(m){delete i.styles["#MathJax_Zoom"].filter}},Opera:function(l){h.operaPositionBug=true;h.operaRefreshBug=true}});h.topImg=(h.msieInlineBlockAlignBug?d.Element("img",{style:{width:0,height:0,position:"relative"},src:"about:blank"}):d.Element("span",{style:{width:0,height:0,display:"inline-block"}}));if(h.operaPositionBug||h.msieTopBug){h.topImg.style.border="1px solid"}MathJax.Callback.Queue(["StartupHook",MathJax.Hub.Register,"Begin Styles",{}],["Styles",f,i.styles],["Post",a.Startup.signal,"MathZoom Ready"],["loadComplete",f,"[MathJax]/extensions/MathZoom.js"])})(MathJax.Hub,MathJax.HTML,MathJax.Ajax,MathJax.OutputJax["HTML-CSS"],MathJax.OutputJax.NativeMML);
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g=MathJax.Menu=h.Subclass({version:p,items:[],posted:false,title:null,margin:5,Init:function(u){this.items=[].slice.call(arguments,0)},With:function(u){if(u){f.Insert(this,u)}return this},Post:function(M,E,B){if(!M){M=window.event||{}}var I=document.getElementById("MathJax_MenuFrame");if(!I){I=g.Background(this);delete c.lastItem;delete c.lastMenu;delete g.skipUp;d.Post(["post",g.jax]);g.isRTL=(MathJax.Localization.fontDirection()==="rtl")}var v=o.Element("div",{onmouseup:g.Mouseup,ondblclick:n,ondragstart:n,onselectstart:n,oncontextmenu:n,menuItem:this,className:"MathJax_Menu",onkeydown:g.Keydown,role:"menu"});if(M.type==="contextmenu"||M.type==="mouseover"){v.className+=" MathJax_ContextMenu"}if(!B){MathJax.Localization.setCSS(v)}for(var N=0,K=this.items.length;N<K;N++){this.items[N].Create(v)}if(g.isMobile){o.addElement(v,"span",{className:"MathJax_MenuClose",menu:E,ontouchstart:g.Close,ontouchend:n,onmousedown:g.Close,onmouseup:n},[["span",{},"\u00D7"]])}I.appendChild(v);this.posted=true;if(v.offsetWidth){v.style.width=(v.offsetWidth+2)+"px"}var H=M.pageX,F=M.pageY;var u=document.body.getBoundingClientRect();var C=(window.getComputedStyle?window.getComputedStyle(document.body):{marginLeft:"0px"});var A=u.right-Math.min(0,u.left)+parseFloat(C.marginLeft);if(!H&&!F&&"clientX" in M){H=M.clientX+document.body.scrollLeft+document.documentElement.scrollLeft;F=M.clientY+document.body.scrollTop+document.documentElement.scrollTop}if(!E){var L=g.CurrentNode()||M.target;if((M.type==="keydown"||(!H&&!F))&&L){var P=window.pageXOffset||document.documentElement.scrollLeft;var O=window.pageYOffset||document.documentElement.scrollTop;var w=L.getBoundingClientRect();H=(w.right+w.left)/2+P;F=(w.bottom+w.top)/2+O}if(H+v.offsetWidth>A-this.margin){H=A-v.offsetWidth-this.margin}if(g.isMobile){H=Math.max(5,H-Math.floor(v.offsetWidth/2));F-=20}g.skipUp=M.isContextMenu}else{var z="left",J=E.offsetWidth;H=(g.isMobile?30:J-2);F=0;while(E&&E!==I){H+=E.offsetLeft;F+=E.offsetTop;E=E.parentNode}if(!g.isMobile){if((g.isRTL&&H-J-v.offsetWidth>this.margin)||(!g.isRTL&&H+v.offsetWidth>A-this.margin)){z="right";H=Math.max(this.margin,H-J-v.offsetWidth+6)}}if(!a){v.style["borderRadiusTop"+z]=0;v.style["WebkitBorderRadiusTop"+z]=0;v.style["MozBorderRadiusTop"+z]=0;v.style["KhtmlBorderRadiusTop"+z]=0}}v.style.left=H+"px";v.style.top=F+"px";if(document.selection&&document.selection.empty){document.selection.empty()}var G=window.pageXOffset||document.documentElement.scrollLeft;var D=window.pageYOffset||document.documentElement.scrollTop;g.Focus(v);if(M.type==="keydown"){g.skipMouseoverFromKey=true;setTimeout(function(){delete 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h=f.merror.prototype.toCommonHTML;f.merror.Augment({toCommonHTML:function(n){if(!this.isError){return h.apply(this,arguments)}n=g.addElement(n,"mjx-noError");var p=this.data[0].data[0].data.join("").split(/\n/);for(var k=0,j=p.length;k<j;k++){i.addText(n,p[k]);if(k!==j-1){g.addElement(n,"br",{isMathJax:true})}}var o=this.CHTML=g.BBOX.zero();o.w=(n.offsetWidth)/g.em;if(j>1){var l=1.2*j/2;o.h=l+0.25;o.d=l-0.25;n.style.verticalAlign=g.Em(0.45-l)}else{o.h=1;o.d=0.2+2/g.em}return n}})});b.Startup.signal.Post("TeX noErrors Ready")})(MathJax.Hub,MathJax.HTML);MathJax.Ajax.loadComplete("[MathJax]/extensions/TeX/noErrors.js");
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m},Top:function(m){if(m==null){m=1}if(this.data.length<m){return null}return this.data[this.data.length-m]},Prev:function(m){var n=this.Top();if(m){return n.data[n.data.length-1]}else{return n.Pop()}},toString:function(){return"stack[\n "+this.data.join("\n ")+"\n]"}});var b=e.Item=MathJax.Object.Subclass({type:"base",endError:["ExtraOpenMissingClose","Extra open brace or missing close brace"],closeError:["ExtraCloseMissingOpen","Extra close brace or missing open brace"],rightError:["MissingLeftExtraRight","Missing \\left or extra \\right"],Init:function(){if(this.isOpen){this.env={}}this.data=[];this.Push.apply(this,arguments)},Push:function(){this.data.push.apply(this.data,arguments)},Pop:function(){return this.data.pop()},mmlData:function(m,n){if(m==null){m=true}if(this.data.length===1&&!n){return this.data[0]}return i.mrow.apply(i,this.data).With((m?{inferred:true}:{}))},checkItem:function(m){if(m.type==="over"&&this.isOpen){m.num=this.mmlData(false);this.data=[]}if(m.type==="cell"&&this.isOpen){if(m.linebreak){return false}d.Error(["Misplaced","Misplaced %1",m.name])}if(m.isClose&&this[m.type+"Error"]){d.Error(this[m.type+"Error"])}if(!m.isNotStack){return true}this.Push(m.data[0]);return false},With:function(m){for(var n in m){if(m.hasOwnProperty(n)){this[n]=m[n]}}return this},toString:function(){return this.type+"["+this.data.join("; ")+"]"}});b.start=b.Subclass({type:"start",isOpen:true,Init:function(m){this.SUPER(arguments).Init.call(this);this.global=m},checkItem:function(m){if(m.type==="stop"){return b.mml(this.mmlData())}return this.SUPER(arguments).checkItem.call(this,m)}});b.stop=b.Subclass({type:"stop",isClose:true});b.open=b.Subclass({type:"open",isOpen:true,stopError:["ExtraOpenMissingClose","Extra open brace or missing close brace"],checkItem:function(n){if(n.type==="close"){var m=this.mmlData();return b.mml(i.TeXAtom(m))}return this.SUPER(arguments).checkItem.call(this,n)}});b.close=b.Subclass({type:"close",isClose:true});b.prime=b.Subclass({type:"prime",checkItem:function(m){if(this.data[0].type!=="msubsup"){return[i.msup(this.data[0],this.data[1]),m]}this.data[0].SetData(this.data[0].sup,this.data[1]);return[this.data[0],m]}});b.subsup=b.Subclass({type:"subsup",stopError:["MissingScript","Missing superscript or subscript argument"],supError:["MissingOpenForSup","Missing open brace for superscript"],subError:["MissingOpenForSub","Missing open brace for subscript"],checkItem:function(m){if(m.type==="open"||m.type==="left"){return true}if(m.type==="mml"){if(this.primes){if(this.position!==2){this.data[0].SetData(2,this.primes)}else{m.data[0]=i.mrow(this.primes.With({variantForm:true}),m.data[0])}}this.data[0].SetData(this.position,m.data[0]);if(this.movesupsub!=null){this.data[0].movesupsub=this.movesupsub}return b.mml(this.data[0])}if(this.SUPER(arguments).checkItem.call(this,m)){d.Error(this[["","subError","supError"][this.position]])}},Pop:function(){}});b.over=b.Subclass({type:"over",isClose:true,name:"\\over",checkItem:function(o,m){if(o.type==="over"){d.Error(["AmbiguousUseOf","Ambiguous use of %1",o.name])}if(o.isClose){var n=i.mfrac(this.num,this.mmlData(false));if(this.thickness!=null){n.linethickness=this.thickness}if(this.open||this.close){n.texWithDelims=true;n=d.fixedFence(this.open,n,this.close)}return[b.mml(n),o]}return this.SUPER(arguments).checkItem.call(this,o)},toString:function(){return"over["+this.num+" / "+this.data.join("; ")+"]"}});b.left=b.Subclass({type:"left",isOpen:true,delim:"(",stopError:["ExtraLeftMissingRight","Extra \\left or missing \\right"],checkItem:function(m){if(m.type==="right"){return b.mml(d.fenced(this.delim,this.mmlData(),m.delim))}return this.SUPER(arguments).checkItem.call(this,m)}});b.right=b.Subclass({type:"right",isClose:true,delim:")"});b.begin=b.Subclass({type:"begin",isOpen:true,checkItem:function(m){if(m.type==="end"){if(m.name!==this.name){d.Error(["EnvBadEnd","\\begin{%1} ended with \\end{%2}",this.name,m.name])}if(!this.end){return b.mml(this.mmlData())}return this.parse[this.end].call(this.parse,this,this.data)}if(m.type==="stop"){d.Error(["EnvMissingEnd","Missing \\end{%1}",this.name])}return this.SUPER(arguments).checkItem.call(this,m)}});b.end=b.Subclass({type:"end",isClose:true});b.style=b.Subclass({type:"style",checkItem:function(n){if(!n.isClose){return this.SUPER(arguments).checkItem.call(this,n)}var m=i.mstyle.apply(i,this.data).With(this.styles);return[b.mml(m),n]}});b.position=b.Subclass({type:"position",checkItem:function(n){if(n.isClose){d.Error(["MissingBoxFor","Missing box for %1",this.name])}if(n.isNotStack){var m=n.mmlData();switch(this.move){case"vertical":m=i.mpadded(m).With({height:this.dh,depth:this.dd,voffset:this.dh});return[b.mml(m)];case"horizontal":return[b.mml(this.left),n,b.mml(this.right)]}}return this.SUPER(arguments).checkItem.call(this,n)}});b.array=b.Subclass({type:"array",isOpen:true,copyEnv:false,arraydef:{},Init:function(){this.table=[];this.row=[];this.frame=[];this.hfill=[];this.SUPER(arguments).Init.apply(this,arguments)},checkItem:function(n){if(n.isClose&&n.type!=="over"){if(n.isEntry){this.EndEntry();this.clearEnv();return false}if(n.isCR){this.EndEntry();this.EndRow();this.clearEnv();return false}this.EndTable();this.clearEnv();var o=this.arraydef.scriptlevel;delete this.arraydef.scriptlevel;var m=i.mtable.apply(i,this.table).With(this.arraydef);if(this.frame.length===4){m.frame=(this.frame.dashed?"dashed":"solid")}else{if(this.frame.length){m.hasFrame=true;if(this.arraydef.rowlines){this.arraydef.rowlines=this.arraydef.rowlines.replace(/none( none)+$/,"none")}m=i.menclose(m).With({notation:this.frame.join(" "),isFrame:true});if((this.arraydef.columnlines||"none")!="none"||(this.arraydef.rowlines||"none")!="none"){m.padding=0}}}if(o){m=i.mstyle(m).With({scriptlevel:o})}if(this.open||this.close){m=d.fenced(this.open,m,this.close)}m=b.mml(m);if(this.requireClose){if(n.type==="close"){return m}d.Error(["MissingCloseBrace","Missing close brace"])}return[m,n]}return this.SUPER(arguments).checkItem.call(this,n)},EndEntry:function(){var m=i.mtd.apply(i,this.data);if(this.hfill.length){if(this.hfill[0]===0){m.columnalign="right"}if(this.hfill[this.hfill.length-1]===this.data.length){m.columnalign=(m.columnalign?"center":"left")}}this.row.push(m);this.data=[];this.hfill=[]},EndRow:function(){var m=i.mtr;if(this.isNumbered&&this.row.length===3){this.row.unshift(this.row.pop());m=i.mlabeledtr}this.table.push(m.apply(i,this.row));this.row=[]},EndTable:function(){if(this.data.length||this.row.length){this.EndEntry();this.EndRow()}this.checkLines()},checkLines:function(){if(this.arraydef.rowlines){var m=this.arraydef.rowlines.split(/ /);if(m.length===this.table.length){this.frame.push("bottom");m.pop();this.arraydef.rowlines=m.join(" ")}else{if(m.length<this.table.length-1){this.arraydef.rowlines+=" none"}}}if(this.rowspacing){var n=this.arraydef.rowspacing.split(/ /);while(n.length<this.table.length){n.push(this.rowspacing+"em")}this.arraydef.rowspacing=n.join(" ")}},clearEnv:function(){for(var m in this.env){if(this.env.hasOwnProperty(m)){delete this.env[m]}}}});b.cell=b.Subclass({type:"cell",isClose:true});b.mml=b.Subclass({type:"mml",isNotStack:true,Add:function(){this.data.push.apply(this.data,arguments);return this}});b.fn=b.Subclass({type:"fn",checkItem:function(n){if(this.data[0]){if(n.isOpen){return true}if(n.type!=="fn"){if(n.type!=="mml"||!n.data[0]){return[this.data[0],n]}if(n.data[0].isa(i.mspace)){return[this.data[0],n]}var m=n.data[0];if(m.isEmbellished()){m=m.CoreMO()}if([0,0,1,1,0,1,1,0,0,0][m.Get("texClass")]){return[this.data[0],n]}}return[this.data[0],i.mo(i.entity("#x2061")).With({texClass:i.TEXCLASS.NONE}),n]}return this.SUPER(arguments).checkItem.apply(this,arguments)}});b.not=b.Subclass({type:"not",checkItem:function(n){var m,o;if(n.type==="open"||n.type==="left"){return true}if(n.type==="mml"&&n.data[0].type.match(/^(mo|mi|mtext)$/)){m=n.data[0],o=m.data.join("");if(o.length===1&&!m.movesupsub&&m.data.length===1){o=b.not.remap[o.charCodeAt(0)];if(o){m.SetData(0,i.chars(String.fromCharCode(o)))}else{m.Append(i.chars("\u0338"))}return 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n=o.movesupsub,m=o.sub;if((o.type==="msubsup"&&o.data[o.sub])||(o.type==="munderover"&&o.data[o.under]&&!o.subsupOK)){d.Error(["DoubleSubscripts","Double subscripts: use braces to clarify"])}if(o.type!=="msubsup"){if(n){if(o.type!=="munderover"||o.data[o.under]){if(o.movablelimits&&o.isa(i.mi)){o=this.mi2mo(o)}o=i.munderover(o,null,null).With({movesupsub:true})}m=o.under}else{o=i.msubsup(o,null,null);m=o.sub}}this.Push(b.subsup(o).With({position:m,primes:q,movesupsub:n}))},PRIME:"\u2032",SMARTQUOTE:"\u2019",Prime:function(o){var n=this.stack.Prev();if(!n){n=i.mi()}if(n.type==="msubsup"&&n.data[n.sup]){d.Error(["DoubleExponentPrime","Prime causes double exponent: use braces to clarify"])}var m="";this.i--;do{m+=this.PRIME;this.i++,o=this.GetNext()}while(o==="'"||o===this.SMARTQUOTE);m=["","\u2032","\u2033","\u2034","\u2057"][m.length]||m;this.Push(b.prime(n,this.mmlToken(i.mo(m))))},mi2mo:function(m){var n=i.mo();n.Append.apply(n,m.data);var o;for(o in n.defaults){if(n.defaults.hasOwnProperty(o)&&m[o]!=null){n[o]=m[o]}}for(o in i.copyAttributes){if(i.copyAttributes.hasOwnProperty(o)&&m[o]!=null){n[o]=m[o]}}n.lspace=n.rspace="0";n.useMMLspacing&=~(n.SPACE_ATTR.lspace|n.SPACE_ATTR.rspace);return n},Comment:function(m){while(this.i<this.string.length&&this.string.charAt(this.i)!="\n"){this.i++}},Hash:function(m){d.Error(["CantUseHash1","You can't use 'macro parameter character #' in math mode"])},Other:function(o){var n,m;if(this.stack.env.font){n={mathvariant:this.stack.env.font}}if(g.remap[o]){o=g.remap[o];if(f(o)){n=o[1];o=o[0]}m=i.mo(i.entity("#x"+o)).With(n)}else{m=i.mo(o).With(n)}if(m.autoDefault("stretchy",true)){m.stretchy=false}if(m.autoDefault("texClass",true)==""){m=i.TeXAtom(m)}this.Push(this.mmlToken(m))},SetFont:function(n,m){this.stack.env.font=m},SetStyle:function(n,m,o,p){this.stack.env.style=m;this.stack.env.level=p;this.Push(b.style().With({styles:{displaystyle:o,scriptlevel:p}}))},SetSize:function(m,n){this.stack.env.size=n;this.Push(b.style().With({styles:{mathsize:n+"em"}}))},Color:function(o){var n=this.GetArgument(o);var m=this.stack.env.color;this.stack.env.color=n;var p=this.ParseArg(o);if(m){this.stack.env.color}else{delete this.stack.env.color}this.Push(i.mstyle(p).With({mathcolor:n}))},Spacer:function(m,n){this.Push(i.mspace().With({width:n,mathsize:i.SIZE.NORMAL,scriptlevel:0}))},LeftRight:function(m){this.Push(b[m.substr(1)]().With({delim:this.GetDelimiter(m)}))},Middle:function(m){var n=this.GetDelimiter(m);this.Push(i.TeXAtom().With({texClass:i.TEXCLASS.CLOSE}));if(this.stack.Top().type!=="left"){d.Error(["MisplacedMiddle","%1 must be within \\left and \\right",m])}this.Push(i.mo(n).With({stretchy:true}));this.Push(i.TeXAtom().With({texClass:i.TEXCLASS.OPEN}))},NamedFn:function(n,o){if(!o){o=n.substr(1)}var m=i.mi(o).With({texClass:i.TEXCLASS.OP});this.Push(b.fn(this.mmlToken(m)))},NamedOp:function(n,o){if(!o){o=n.substr(1)}o=o.replace(/ /,"\u2006");var m=i.mo(o).With({movablelimits:true,movesupsub:true,form:i.FORM.PREFIX,texClass:i.TEXCLASS.OP});m.useMMLspacing&=~m.SPACE_ATTR.form;this.Push(this.mmlToken(m))},Limits:function(n,m){var p=this.stack.Prev("nopop");if(!p||(p.Get("texClass")!==i.TEXCLASS.OP&&p.movesupsub==null)){d.Error(["MisplacedLimits","%1 is allowed only on operators",n])}var o=this.stack.Top();if(p.type==="munderover"&&!m){p=o.data[o.data.length-1]=i.msubsup.apply(i.subsup,p.data)}else{if(p.type==="msubsup"&&m){p=o.data[o.data.length-1]=i.munderover.apply(i.underover,p.data)}}p.movesupsub=(m?true:false);p.Core().movablelimits=false;if(p.movablelimits){p.movablelimits=false}},Over:function(o,n,p){var m=b.over().With({name:o});if(n||p){m.open=n;m.close=p}else{if(o.match(/withdelims$/)){m.open=this.GetDelimiter(o);m.close=this.GetDelimiter(o)}}if(o.match(/^\\above/)){m.thickness=this.GetDimen(o)}else{if(o.match(/^\\atop/)||n||p){m.thickness=0}}this.Push(m)},Frac:function(n){var m=this.ParseArg(n);var o=this.ParseArg(n);this.Push(i.mfrac(m,o))},Sqrt:function(p){var q=this.GetBrackets(p),m=this.GetArgument(p);if(m==="\\frac"){m+="{"+this.GetArgument(m)+"}{"+this.GetArgument(m)+"}"}var o=d.Parse(m,this.stack.env).mml();if(!q){o=i.msqrt.apply(i,o.array())}else{o=i.mroot(o,this.parseRoot(q))}this.Push(o)},Root:function(o){var p=this.GetUpTo(o,"\\of");var m=this.ParseArg(o);this.Push(i.mroot(m,this.parseRoot(p)))},parseRoot:function(r){var o=this.stack.env,m=o.inRoot;o.inRoot=true;var q=d.Parse(r,o);r=q.mml();var p=q.stack.global;if(p.leftRoot||p.upRoot){r=i.mpadded(r);if(p.leftRoot){r.width=p.leftRoot}if(p.upRoot){r.voffset=p.upRoot;r.height=p.upRoot}}o.inRoot=m;return r},MoveRoot:function(m,p){if(!this.stack.env.inRoot){d.Error(["MisplacedMoveRoot","%1 can appear only within a root",m])}if(this.stack.global[p]){d.Error(["MultipleMoveRoot","Multiple use of %1",m])}var o=this.GetArgument(m);if(!o.match(/-?[0-9]+/)){d.Error(["IntegerArg","The argument to %1 must be an integer",m])}o=(o/15)+"em";if(o.substr(0,1)!=="-"){o="+"+o}this.stack.global[p]=o},Accent:function(o,m,s){var r=this.ParseArg(o);var q={accent:true};if(this.stack.env.font){q.mathvariant=this.stack.env.font}var n=this.mmlToken(i.mo(i.entity("#x"+m)).With(q));n.stretchy=(s?true:false);var p=(r.isEmbellished()?r.CoreMO():r);if(p.isa(i.mo)){p.movablelimits=false}this.Push(i.TeXAtom(i.munderover(r,null,n).With({accent:true})))},UnderOver:function(o,s,m,q){var r={o:"over",u:"under"}[o.charAt(1)];var p=this.ParseArg(o);if(p.Get("movablelimits")){p.movablelimits=false}if(p.isa(i.munderover)&&p.isEmbellished()){p.Core().With({lspace:0,rspace:0});p=i.mrow(i.mo().With({rspace:0}),p)}var n=i.munderover(p,null,null);n.SetData(n[r],this.mmlToken(i.mo(i.entity("#x"+s)).With({stretchy:true,accent:!q})));if(m){n=i.TeXAtom(n).With({texClass:i.TEXCLASS.OP,movesupsub:true})}this.Push(n.With({subsupOK:true}))},Overset:function(m){var o=this.ParseArg(m),n=this.ParseArg(m);if(n.movablelimits){n.movablelimits=false}this.Push(i.mover(n,o))},Underset:function(m){var o=this.ParseArg(m),n=this.ParseArg(m);if(n.movablelimits){n.movablelimits=false}this.Push(i.munder(n,o))},TeXAtom:function(p,r){var q={texClass:r},o;if(r==i.TEXCLASS.OP){q.movesupsub=q.movablelimits=true;var m=this.GetArgument(p);var n=m.match(/^\s*\\rm\s+([a-zA-Z0-9 ]+)$/);if(n){q.mathvariant=i.VARIANT.NORMAL;o=b.fn(this.mmlToken(i.mi(n[1]).With(q)))}else{o=b.fn(i.TeXAtom(d.Parse(m,this.stack.env).mml()).With(q))}}else{o=i.TeXAtom(this.ParseArg(p)).With(q)}this.Push(o)},MmlToken:function(o){var p=this.GetArgument(o),m=this.GetBrackets(o,"").replace(/^\s+/,""),s=this.GetArgument(o),r={attrNames:[]},n;if(!i[p]||!i[p].prototype.isToken){d.Error(["NotMathMLToken","%1 is not a token element",p])}while(m!==""){n=m.match(/^([a-z]+)\s*=\s*('[^']*'|"[^"]*"|[^ ,]*)\s*,?\s*/i);if(!n){d.Error(["InvalidMathMLAttr","Invalid MathML attribute: %1",m])}if(i[p].prototype.defaults[n[1]]==null&&!this.MmlTokenAllow[n[1]]){d.Error(["UnknownAttrForElement","%1 is not a recognized attribute for %2",n[1],p])}var q=this.MmlFilterAttribute(n[1],n[2].replace(/^(['"])(.*)\1$/,"$2"));if(q){if(q.toLowerCase()==="true"){q=true}else{if(q.toLowerCase()==="false"){q=false}}r[n[1]]=q;r.attrNames.push(n[1])}m=m.substr(n[0].length)}this.Push(this.mmlToken(i[p](s).With(r)))},MmlFilterAttribute:function(m,n){return n},MmlTokenAllow:{fontfamily:1,fontsize:1,fontweight:1,fontstyle:1,color:1,background:1,id:1,"class":1,href:1,style:1},Strut:function(m){this.Push(i.mpadded(i.mrow()).With({height:"8.6pt",depth:"3pt",width:0}))},Phantom:function(n,m,o){var p=i.mphantom(this.ParseArg(n));if(m||o){p=i.mpadded(p);if(o){p.height=p.depth=0}if(m){p.width=0}}this.Push(i.TeXAtom(p))},Smash:function(o){var n=this.trimSpaces(this.GetBrackets(o,""));var m=i.mpadded(this.ParseArg(o));switch(n){case"b":m.depth=0;break;case"t":m.height=0;break;default:m.height=m.depth=0}this.Push(i.TeXAtom(m))},Lap:function(n){var m=i.mpadded(this.ParseArg(n)).With({width:0});if(n==="\\llap"){m.lspace="-1width"}this.Push(i.TeXAtom(m))},RaiseLower:function(m){var n=this.GetDimen(m);var o=b.position().With({name:m,move:"vertical"});if(n.charAt(0)==="-"){n=n.slice(1);m={raise:"\\lower",lower:"\\raise"}[m.substr(1)]}if(m==="\\lower"){o.dh="-"+n;o.dd="+"+n}else{o.dh="+"+n;o.dd="-"+n}this.Push(o)},MoveLeftRight:function(m){var p=this.GetDimen(m);var o=(p.charAt(0)==="-"?p.slice(1):"-"+p);if(m==="\\moveleft"){var n=p;p=o;o=n}this.Push(b.position().With({name:m,move:"horizontal",left:i.mspace().With({width:p,mathsize:i.SIZE.NORMAL}),right:i.mspace().With({width:o,mathsize:i.SIZE.NORMAL})}))},Hskip:function(m){this.Push(i.mspace().With({width:this.GetDimen(m),mathsize:i.SIZE.NORMAL}))},Rule:function(n,p){var m=this.GetDimen(n),o=this.GetDimen(n),r=this.GetDimen(n);var q={width:m,height:o,depth:r};if(p!=="blank"){q.mathbackground=(this.stack.env.color||"black")}this.Push(i.mspace().With(q))},rule:function(p){var n=this.GetBrackets(p),m=this.GetDimen(p),q=this.GetDimen(p);var o=i.mspace().With({width:m,height:q,mathbackground:(this.stack.env.color||"black")});if(n){o=i.mpadded(o).With({voffset:n});if(n.match(/^\-/)){o.height=n;o.depth="+"+n.substr(1)}else{o.height="+"+n}}this.Push(o)},MakeBig:function(m,p,n){n*=g.p_height;n=String(n).replace(/(\.\d\d\d).+/,"$1")+"em";var o=this.GetDelimiter(m,true);this.Push(i.TeXAtom(i.mo(o).With({minsize:n,maxsize:n,fence:true,stretchy:true,symmetric:true})).With({texClass:p}))},BuildRel:function(m){var n=this.ParseUpTo(m,"\\over");var o=this.ParseArg(m);this.Push(i.TeXAtom(i.munderover(o,null,n)).With({texClass:i.TEXCLASS.REL}))},HBox:function(m,n){this.Push.apply(this,this.InternalMath(this.GetArgument(m),n))},FBox:function(m){this.Push(i.menclose.apply(i,this.InternalMath(this.GetArgument(m))).With({notation:"box"}))},Not:function(m){this.Push(b.not())},Dots:function(m){this.Push(b.dots().With({ldots:this.mmlToken(i.mo(i.entity("#x2026")).With({stretchy:false})),cdots:this.mmlToken(i.mo(i.entity("#x22EF")).With({stretchy:false}))}))},Require:function(m){var n=this.GetArgument(m).replace(/.*\//,"").replace(/[^a-z0-9_.-]/ig,"");this.Extension(null,n)},Extension:function(m,n,o){if(m&&!typeof(m)==="string"){m=m.name}n=d.extensionDir+"/"+n;if(!n.match(/\.js$/)){n+=".js"}if(!j.loaded[j.fileURL(n)]){if(m!=null){delete g[o||"macros"][m.replace(/^\\/,"")]}c.RestartAfter(j.Require(n))}},Macro:function(o,r,q,s){if(q){var n=[];if(s!=null){var m=this.GetBrackets(o);n.push(m==null?s:m)}for(var p=n.length;p<q;p++){n.push(this.GetArgument(o))}r=this.SubstituteArgs(n,r)}this.string=this.AddArgs(r,this.string.slice(this.i));this.i=0;if(++this.macroCount>d.config.MAXMACROS){d.Error(["MaxMacroSub1","MathJax maximum macro substitution count exceeded; is there a recursive macro call?"])}},Matrix:function(n,p,v,r,u,o,m,w,t){var s=this.GetNext();if(s===""){d.Error(["MissingArgFor","Missing argument for %1",n])}if(s==="{"){this.i++}else{this.string=s+"}"+this.string.slice(this.i+1);this.i=0}var q=b.array().With({requireClose:true,arraydef:{rowspacing:(o||"4pt"),columnspacing:(u||"1em")}});if(w){q.isCases=true}if(t){q.isNumbered=true;q.arraydef.side=t}if(p||v){q.open=p;q.close=v}if(m==="D"){q.arraydef.displaystyle=true}if(r!=null){q.arraydef.columnalign=r}this.Push(q)},Entry:function(p){this.Push(b.cell().With({isEntry:true,name:p}));if(this.stack.Top().isCases){var o=this.string;var t=0,s=-1,q=this.i,n=o.length;while(q<n){var u=o.charAt(q);if(u==="{"){t++;q++}else{if(u==="}"){if(t===0){n=0}else{t--;if(t===0&&s<0){s=q-this.i}q++}}else{if(u==="&"&&t===0){d.Error(["ExtraAlignTab","Extra alignment tab in \\cases text"])}else{if(u==="\\"){if(o.substr(q).match(/^((\\cr)[^a-zA-Z]|\\\\)/)){n=0}else{q+=2}}else{q++}}}}}var r=o.substr(this.i,q-this.i);if(!r.match(/^\s*\\text[^a-zA-Z]/)||s!==r.replace(/\s+$/,"").length-1){this.Push.apply(this,this.InternalMath(r,0));this.i=q}}},Cr:function(m){this.Push(b.cell().With({isCR:true,name:m}))},CrLaTeX:function(m){var q;if(this.string.charAt(this.i)==="["){q=this.GetBrackets(m,"").replace(/ /g,"").replace(/,/,".");if(q&&!this.matchDimen(q)){d.Error(["BracketMustBeDimension","Bracket argument to %1 must be a dimension",m])}}this.Push(b.cell().With({isCR:true,name:m,linebreak:true}));var p=this.stack.Top();if(p.isa(b.array)){if(q&&p.arraydef.rowspacing){var o=p.arraydef.rowspacing.split(/ /);if(!p.rowspacing){p.rowspacing=this.dimen2em(o[0])}while(o.length<p.table.length){o.push(this.Em(p.rowspacing))}o[p.table.length-1]=this.Em(Math.max(0,p.rowspacing+this.dimen2em(q)));p.arraydef.rowspacing=o.join(" ")}}else{if(q){this.Push(i.mspace().With({depth:q}))}this.Push(i.mspace().With({linebreak:i.LINEBREAK.NEWLINE}))}},emPerInch:7.2,pxPerInch:72,matchDimen:function(m){return m.match(/^(-?(?:\.\d+|\d+(?:\.\d*)?))(px|pt|em|ex|mu|pc|in|mm|cm)$/)},dimen2em:function(q){var o=this.matchDimen(q);var n=parseFloat(o[1]||"1"),p=o[2];if(p==="em"){return n}if(p==="ex"){return n*0.43}if(p==="pt"){return n/10}if(p==="pc"){return n*1.2}if(p==="px"){return n*this.emPerInch/this.pxPerInch}if(p==="in"){return n*this.emPerInch}if(p==="cm"){return n*this.emPerInch/2.54}if(p==="mm"){return n*this.emPerInch/25.4}if(p==="mu"){return n/18}return 0},Em:function(n){if(Math.abs(n)<0.0006){return"0em"}return n.toFixed(3).replace(/\.?0+$/,"")+"em"},HLine:function(n,o){if(o==null){o="solid"}var p=this.stack.Top();if(!p.isa(b.array)||p.data.length){d.Error(["Misplaced","Misplaced %1",n])}if(p.table.length==0){p.frame.push("top")}else{var m=(p.arraydef.rowlines?p.arraydef.rowlines.split(/ /):[]);while(m.length<p.table.length){m.push("none")}m[p.table.length-1]=o;p.arraydef.rowlines=m.join(" ")}},HFill:function(m){var n=this.stack.Top();if(n.isa(b.array)){n.hfill.push(n.data.length)}else{d.Error(["UnsupportedHFill","Unsupported use of %1",m])}},BeginEnd:function(o){var p=this.GetArgument(o),r=false;if(p.match(/^\\end\\/)){r=true;p=p.substr(5)}if(p.match(/\\/i)){d.Error(["InvalidEnv","Invalid environment name '%1'",p])}var q=this.envFindName(p);if(!q){d.Error(["UnknownEnv","Unknown environment '%1'",p])}if(!f(q)){q=[q]}var m=(f(q[1])?q[1][0]:q[1]);var 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q=0,n=0,t=s.len,r=-s.len,p=0;if(v.width!==""){s=this.PHTMLdimen(v.width,"w",0);if(s.pm){r+=s.len}else{u.style.width=g.Em(s.len)}}if(v.height!==""){s=this.PHTMLdimen(v.height,"h",0);if(!s.pm){q+=-this.PHTMLbboxFor(0).h}q+=s.len}if(v.depth!==""){s=this.PHTMLdimen(v.depth,"d",0);if(!s.pm){n+=-this.PHTMLbboxFor(0).d;p+=-s.len}n+=s.len}if(v.voffset!==""){s=this.PHTMLdimen(v.voffset);q-=s.len;n+=s.len;p+=s.len}if(q){o.style.marginTop=g.Em(q)}if(n){o.style.marginBottom=g.Em(n)}if(t){o.style.marginLeft=g.Em(t)}if(r){o.style.marginRight=g.Em(r)}if(p){u.style.verticalAlign=g.Em(p)}return u},PHTMLdimen:function(q,r,n){if(n==null){n=-k}q=String(q);var o=q.match(/width|height|depth/);var p=(o?this.PHTML[o[0].charAt(0)]:(r?this.PHTML[r]:0));return{len:g.length2em(q,p)||0,pm:!!q.match(/^[-+]/)}}});h.munderover.Augment({toPreviewHTML:function(r){var t=this.getValues("displaystyle","accent","accentunder","align");var 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r=e.Element("span",{className:"MJXp-script-box",style:{height:g.Em(o+n.PHTML.h*0.8+u+p.PHTML.d*0.8),"vertical-align":g.Em(-u-p.PHTML.d*0.8)}},[["span",{},[["span",{},[["span",{style:{"margin-bottom":g.Em(-(n.PHTML.d-0.05))}}]]]]],["span",{},[["span",{},[["span",{style:{"margin-top":g.Em(-(n.PHTML.h-0.05))}}]]]]]]);p.PHTMLhandleScriptlevel(r.firstChild);n.PHTMLhandleScriptlevel(r.lastChild);r.firstChild.firstChild.firstChild.appendChild(q.lastChild);r.lastChild.firstChild.firstChild.appendChild(q.lastChild);q.appendChild(r);t.h=Math.max(s.PHTML.h,n.PHTML.h*0.8+o);t.d=Math.max(s.PHTML.d,p.PHTML.d*0.8+u);t.w=s.PHTML.w+Math.max(n.PHTML.w,p.PHTML.w)+0.07}else{if(n){q.lastChild.style.verticalAlign=g.Em(o);n.PHTMLhandleScriptlevel(q.lastChild);t.h=Math.max(s.PHTML.h,n.PHTML.h*0.8+o);t.d=Math.max(s.PHTML.d,n.PHTML.d*0.8-o);t.w=s.PHTML.w+n.PHTML.w+0.07}else{if(p){q.lastChild.style.verticalAlign=g.Em(-u);p.PHTMLhandleScriptlevel(q.lastChild);t.h=Math.max(s.PHTML.h,p.PHTML.h*0.8-u);t.d=Math.max(s.PHTML.d,p.PHTML.d*0.8+u);t.w=s.PHTML.w+p.PHTML.w+0.07}}}return q}});h.mfrac.Augment({toPreviewHTML:function(r){r=this.PHTMLdefaultSpan(r,{childSpans:true,className:"MJXp-box",forceChild:true,noBBox:true});var o=this.getValues("linethickness","displaystyle");if(!o.displaystyle){if(this.data[0]){this.data[0].PHTMLhandleScriptlevel(r.firstChild)}if(this.data[1]){this.data[1].PHTMLhandleScriptlevel(r.lastChild)}}var n=e.Element("span",{className:"MJXp-box"},[["span",{className:"MJXp-denom"},[["span",{},[["span",{className:"MJXp-rule",style:{height:"1em"}}]]],["span"]]]]);n.firstChild.lastChild.appendChild(r.lastChild);r.appendChild(n);var s=this.PHTMLbboxFor(0),p=this.PHTMLbboxFor(1),v=this.PHTML;v.w=Math.max(s.w,p.w)*0.8;v.h=s.h+s.d+0.1+0.25;v.d=p.h+p.d-0.25;v.l=v.r=0.125;o.linethickness=Math.max(0,g.length2em(o.linethickness||"0",0));if(o.linethickness){var u=n.firstChild.firstChild.firstChild;var q=g.Em(o.linethickness);u.style.borderTop="none";u.style.borderBottom=(o.linethickness<0.15?"1px":q)+" solid";u.style.margin=q+" 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+}
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+}
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+}
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+}
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+}
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+}
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+}
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ChartElementFunction ChartElements ChartLabels ChartLayout ChartLegends ChartStyle Chebyshev1FilterModel Chebyshev2FilterModel ChebyshevDistance ChebyshevT ChebyshevU Check CheckAbort CheckAll Checkbox CheckboxBar CheckboxBox CheckboxBoxOptions ChemicalData ChessboardDistance ChiDistribution ChineseRemainder ChiSquareDistribution ChoiceButtons ChoiceDialog CholeskyDecomposition Chop Circle CircleBox CircleDot CircleMinus CirclePlus CircleTimes CirculantGraph CityData Clear ClearAll ClearAttributes ClearSystemCache ClebschGordan ClickPane Clip ClipboardNotebook ClipFill ClippingStyle ClipPlanes ClipRange Clock ClockGauge ClockwiseContourIntegral Close Closed CloseKernels ClosenessCentrality Closing ClosingAutoSave ClosingEvent ClusteringComponents CMYKColor Coarse Coefficient CoefficientArrays CoefficientDomain CoefficientList CoefficientRules CoifletWavelet Collect Colon ColonForm ColorCombine ColorConvert ColorData ColorDataFunction ColorFunction ColorFunctionScaling Colorize ColorNegate ColorOutput ColorProfileData ColorQuantize ColorReplace ColorRules ColorSelectorSettings ColorSeparate ColorSetter ColorSetterBox ColorSetterBoxOptions ColorSlider ColorSpace Column ColumnAlignments ColumnBackgrounds ColumnForm ColumnLines ColumnsEqual ColumnSpacings ColumnWidths CommonDefaultFormatTypes Commonest CommonestFilter CommonUnits CommunityBoundaryStyle CommunityGraphPlot CommunityLabels CommunityRegionStyle CompatibleUnitQ CompilationOptions CompilationTarget Compile Compiled CompiledFunction Complement CompleteGraph CompleteGraphQ CompleteKaryTree CompletionsListPacket Complex Complexes ComplexExpand ComplexInfinity ComplexityFunction ComponentMeasurements ComponentwiseContextMenu Compose ComposeList ComposeSeries Composition CompoundExpression CompoundPoissonDistribution CompoundPoissonProcess CompoundRenewalProcess Compress CompressedData Condition ConditionalExpression Conditioned Cone ConeBox ConfidenceLevel ConfidenceRange ConfidenceTransform ConfigurationPath Congruent Conjugate ConjugateTranspose Conjunction Connect ConnectedComponents ConnectedGraphQ ConnesWindow ConoverTest ConsoleMessage ConsoleMessagePacket ConsolePrint Constant ConstantArray Constants ConstrainedMax ConstrainedMin ContentPadding ContentsBoundingBox ContentSelectable ContentSize Context ContextMenu Contexts ContextToFilename ContextToFileName Continuation Continue ContinuedFraction ContinuedFractionK ContinuousAction ContinuousMarkovProcess ContinuousTimeModelQ ContinuousWaveletData ContinuousWaveletTransform ContourDetect ContourGraphics ContourIntegral ContourLabels ContourLines ContourPlot ContourPlot3D Contours ContourShading ContourSmoothing ContourStyle ContraharmonicMean Control ControlActive ControlAlignment ControllabilityGramian ControllabilityMatrix ControllableDecomposition ControllableModelQ ControllerDuration ControllerInformation ControllerInformationData ControllerLinking ControllerManipulate ControllerMethod ControllerPath ControllerState ControlPlacement ControlsRendering ControlType Convergents ConversionOptions ConversionRules ConvertToBitmapPacket ConvertToPostScript ConvertToPostScriptPacket Convolve ConwayGroupCo1 ConwayGroupCo2 ConwayGroupCo3 CoordinateChartData CoordinatesToolOptions CoordinateTransform CoordinateTransformData CoprimeQ Coproduct CopulaDistribution Copyable CopyDirectory CopyFile CopyTag CopyToClipboard CornerFilter CornerNeighbors Correlation CorrelationDistance CorrelationFunction CorrelationTest Cos Cosh CoshIntegral CosineDistance CosineWindow CosIntegral Cot Coth Count CounterAssignments CounterBox CounterBoxOptions CounterClockwiseContourIntegral CounterEvaluator CounterFunction CounterIncrements CounterStyle CounterStyleMenuListing CountRoots CountryData Covariance CovarianceEstimatorFunction CovarianceFunction CoxianDistribution CoxIngersollRossProcess CoxModel CoxModelFit CramerVonMisesTest CreateArchive CreateDialog CreateDirectory CreateDocument CreateIntermediateDirectories CreatePalette CreatePalettePacket CreateScheduledTask CreateTemporary CreateWindow CriticalityFailureImportance CriticalitySuccessImportance CriticalSection Cross CrossingDetect CrossMatrix Csc Csch CubeRoot Cubics Cuboid CuboidBox Cumulant CumulantGeneratingFunction Cup CupCap Curl CurlyDoubleQuote CurlyQuote CurrentImage CurrentlySpeakingPacket CurrentValue CurvatureFlowFilter CurveClosed Cyan CycleGraph CycleIndexPolynomial Cycles CyclicGroup Cyclotomic Cylinder CylinderBox CylindricalDecomposition D DagumDistribution DamerauLevenshteinDistance DampingFactor Darker Dashed Dashing DataCompression DataDistribution DataRange DataReversed Date DateDelimiters DateDifference DateFunction DateList DateListLogPlot DateListPlot DatePattern DatePlus DateRange DateString DateTicksFormat DaubechiesWavelet DavisDistribution DawsonF DayCount DayCountConvention DayMatchQ DayName DayPlus DayRange DayRound DeBruijnGraph Debug DebugTag Decimal DeclareKnownSymbols DeclarePackage Decompose Decrement DedekindEta Default DefaultAxesStyle DefaultBaseStyle DefaultBoxStyle DefaultButton DefaultColor DefaultControlPlacement DefaultDuplicateCellStyle DefaultDuration DefaultElement DefaultFaceGridsStyle DefaultFieldHintStyle DefaultFont DefaultFontProperties DefaultFormatType DefaultFormatTypeForStyle DefaultFrameStyle DefaultFrameTicksStyle DefaultGridLinesStyle DefaultInlineFormatType DefaultInputFormatType DefaultLabelStyle DefaultMenuStyle DefaultNaturalLanguage DefaultNewCellStyle DefaultNewInlineCellStyle DefaultNotebook DefaultOptions DefaultOutputFormatType DefaultStyle DefaultStyleDefinitions DefaultTextFormatType DefaultTextInlineFormatType DefaultTicksStyle DefaultTooltipStyle DefaultValues Defer DefineExternal DefineInputStreamMethod DefineOutputStreamMethod Definition Degree DegreeCentrality DegreeGraphDistribution DegreeLexicographic DegreeReverseLexicographic Deinitialization Del Deletable Delete DeleteBorderComponents DeleteCases DeleteContents DeleteDirectory DeleteDuplicates DeleteFile DeleteSmallComponents DeleteWithContents DeletionWarning Delimiter DelimiterFlashTime DelimiterMatching Delimiters Denominator DensityGraphics DensityHistogram DensityPlot DependentVariables Deploy Deployed Depth DepthFirstScan Derivative DerivativeFilter DescriptorStateSpace DesignMatrix Det DGaussianWavelet DiacriticalPositioning Diagonal DiagonalMatrix Dialog DialogIndent DialogInput DialogLevel DialogNotebook DialogProlog DialogReturn DialogSymbols Diamond DiamondMatrix DiceDissimilarity DictionaryLookup DifferenceDelta DifferenceOrder DifferenceRoot DifferenceRootReduce Differences DifferentialD DifferentialRoot DifferentialRootReduce DifferentiatorFilter DigitBlock DigitBlockMinimum DigitCharacter DigitCount DigitQ DihedralGroup Dilation Dimensions DiracComb DiracDelta DirectedEdge DirectedEdges DirectedGraph DirectedGraphQ DirectedInfinity Direction Directive Directory DirectoryName DirectoryQ DirectoryStack DirichletCharacter DirichletConvolve DirichletDistribution DirichletL DirichletTransform DirichletWindow DisableConsolePrintPacket DiscreteChirpZTransform DiscreteConvolve DiscreteDelta DiscreteHadamardTransform DiscreteIndicator DiscreteLQEstimatorGains DiscreteLQRegulatorGains DiscreteLyapunovSolve DiscreteMarkovProcess DiscretePlot DiscretePlot3D DiscreteRatio DiscreteRiccatiSolve DiscreteShift DiscreteTimeModelQ DiscreteUniformDistribution DiscreteVariables DiscreteWaveletData DiscreteWaveletPacketTransform DiscreteWaveletTransform Discriminant Disjunction Disk DiskBox DiskMatrix Dispatch DispersionEstimatorFunction Display DisplayAllSteps DisplayEndPacket DisplayFlushImagePacket DisplayForm DisplayFunction DisplayPacket DisplayRules DisplaySetSizePacket DisplayString DisplayTemporary DisplayWith DisplayWithRef DisplayWithVariable DistanceFunction DistanceTransform Distribute Distributed DistributedContexts DistributeDefinitions DistributionChart DistributionDomain DistributionFitTest DistributionParameterAssumptions DistributionParameterQ Dithering Div Divergence Divide DivideBy Dividers Divisible Divisors DivisorSigma DivisorSum DMSList DMSString Do DockedCells DocumentNotebook DominantColors DOSTextFormat Dot DotDashed DotEqual Dotted DoubleBracketingBar DoubleContourIntegral DoubleDownArrow DoubleLeftArrow DoubleLeftRightArrow DoubleLeftTee DoubleLongLeftArrow DoubleLongLeftRightArrow DoubleLongRightArrow DoubleRightArrow DoubleRightTee DoubleUpArrow DoubleUpDownArrow DoubleVerticalBar DoublyInfinite Down DownArrow DownArrowBar DownArrowUpArrow DownLeftRightVector DownLeftTeeVector DownLeftVector DownLeftVectorBar DownRightTeeVector DownRightVector DownRightVectorBar Downsample DownTee DownTeeArrow DownValues DragAndDrop DrawEdges DrawFrontFaces DrawHighlighted Drop DSolve Dt DualLinearProgramming DualSystemsModel DumpGet DumpSave DuplicateFreeQ Dynamic DynamicBox DynamicBoxOptions DynamicEvaluationTimeout DynamicLocation DynamicModule DynamicModuleBox DynamicModuleBoxOptions DynamicModuleParent DynamicModuleValues DynamicName DynamicNamespace DynamicReference DynamicSetting DynamicUpdating DynamicWrapper DynamicWrapperBox DynamicWrapperBoxOptions E EccentricityCentrality EdgeAdd EdgeBetweennessCentrality EdgeCapacity EdgeCapForm EdgeColor EdgeConnectivity EdgeCost EdgeCount EdgeCoverQ EdgeDashing EdgeDelete EdgeDetect EdgeForm EdgeIndex EdgeJoinForm EdgeLabeling EdgeLabels EdgeLabelStyle EdgeList EdgeOpacity EdgeQ EdgeRenderingFunction EdgeRules EdgeShapeFunction EdgeStyle EdgeThickness EdgeWeight Editable EditButtonSettings EditCellTagsSettings EditDistance EffectiveInterest Eigensystem Eigenvalues EigenvectorCentrality Eigenvectors Element ElementData Eliminate EliminationOrder EllipticE EllipticExp EllipticExpPrime EllipticF EllipticFilterModel EllipticK EllipticLog EllipticNomeQ EllipticPi EllipticReducedHalfPeriods EllipticTheta EllipticThetaPrime EmitSound EmphasizeSyntaxErrors EmpiricalDistribution Empty EmptyGraphQ EnableConsolePrintPacket Enabled Encode End EndAdd EndDialogPacket EndFrontEndInteractionPacket EndOfFile EndOfLine EndOfString EndPackage EngineeringForm Enter EnterExpressionPacket EnterTextPacket Entropy EntropyFilter Environment Epilog Equal EqualColumns EqualRows EqualTilde EquatedTo Equilibrium EquirippleFilterKernel Equivalent Erf Erfc Erfi ErlangB ErlangC ErlangDistribution Erosion ErrorBox ErrorBoxOptions ErrorNorm ErrorPacket ErrorsDialogSettings EstimatedDistribution EstimatedProcess EstimatorGains EstimatorRegulator EuclideanDistance EulerE EulerGamma EulerianGraphQ EulerPhi Evaluatable Evaluate Evaluated EvaluatePacket EvaluationCell EvaluationCompletionAction EvaluationElements EvaluationMode EvaluationMonitor EvaluationNotebook EvaluationObject EvaluationOrder Evaluator EvaluatorNames EvenQ EventData EventEvaluator EventHandler EventHandlerTag EventLabels ExactBlackmanWindow ExactNumberQ ExactRootIsolation ExampleData Except ExcludedForms ExcludePods Exclusions ExclusionsStyle Exists Exit ExitDialog Exp Expand ExpandAll ExpandDenominator ExpandFileName ExpandNumerator Expectation ExpectationE ExpectedValue ExpGammaDistribution ExpIntegralE ExpIntegralEi Exponent ExponentFunction ExponentialDistribution ExponentialFamily ExponentialGeneratingFunction ExponentialMovingAverage ExponentialPowerDistribution ExponentPosition ExponentStep Export ExportAutoReplacements ExportPacket ExportString Expression ExpressionCell ExpressionPacket ExpToTrig ExtendedGCD Extension ExtentElementFunction ExtentMarkers ExtentSize ExternalCall ExternalDataCharacterEncoding Extract ExtractArchive ExtremeValueDistribution FaceForm FaceGrids FaceGridsStyle Factor FactorComplete Factorial Factorial2 FactorialMoment FactorialMomentGeneratingFunction FactorialPower FactorInteger FactorList FactorSquareFree FactorSquareFreeList FactorTerms FactorTermsList Fail FailureDistribution False FARIMAProcess FEDisableConsolePrintPacket FeedbackSector FeedbackSectorStyle FeedbackType FEEnableConsolePrintPacket Fibonacci FieldHint FieldHintStyle FieldMasked FieldSize File FileBaseName FileByteCount FileDate FileExistsQ FileExtension FileFormat FileHash FileInformation FileName FileNameDepth FileNameDialogSettings FileNameDrop FileNameJoin FileNames FileNameSetter FileNameSplit FileNameTake FilePrint FileType FilledCurve FilledCurveBox Filling FillingStyle FillingTransform FilterRules FinancialBond FinancialData FinancialDerivative FinancialIndicator Find FindArgMax FindArgMin FindClique FindClusters FindCurvePath FindDistributionParameters FindDivisions FindEdgeCover FindEdgeCut FindEulerianCycle FindFaces FindFile FindFit FindGeneratingFunction FindGeoLocation FindGeometricTransform FindGraphCommunities FindGraphIsomorphism FindGraphPartition FindHamiltonianCycle FindIndependentEdgeSet FindIndependentVertexSet FindInstance FindIntegerNullVector FindKClan FindKClique FindKClub FindKPlex FindLibrary FindLinearRecurrence FindList FindMaximum FindMaximumFlow FindMaxValue FindMinimum FindMinimumCostFlow FindMinimumCut FindMinValue FindPermutation FindPostmanTour FindProcessParameters FindRoot FindSequenceFunction FindSettings FindShortestPath FindShortestTour FindThreshold FindVertexCover FindVertexCut Fine FinishDynamic FiniteAbelianGroupCount FiniteGroupCount FiniteGroupData First FirstPassageTimeDistribution FischerGroupFi22 FischerGroupFi23 FischerGroupFi24Prime FisherHypergeometricDistribution FisherRatioTest FisherZDistribution Fit FitAll FittedModel FixedPoint FixedPointList FlashSelection Flat Flatten FlattenAt FlatTopWindow FlipView Floor FlushPrintOutputPacket Fold FoldList Font FontColor FontFamily FontForm FontName FontOpacity FontPostScriptName FontProperties FontReencoding FontSize FontSlant FontSubstitutions FontTracking FontVariations FontWeight For ForAll Format FormatRules FormatType FormatTypeAutoConvert FormatValues FormBox FormBoxOptions FortranForm Forward ForwardBackward Fourier FourierCoefficient FourierCosCoefficient FourierCosSeries FourierCosTransform FourierDCT FourierDCTFilter FourierDCTMatrix FourierDST FourierDSTMatrix FourierMatrix FourierParameters FourierSequenceTransform FourierSeries FourierSinCoefficient FourierSinSeries FourierSinTransform FourierTransform FourierTrigSeries FractionalBrownianMotionProcess FractionalPart FractionBox FractionBoxOptions FractionLine Frame FrameBox FrameBoxOptions Framed FrameInset FrameLabel Frameless FrameMargins FrameStyle FrameTicks FrameTicksStyle FRatioDistribution FrechetDistribution FreeQ FrequencySamplingFilterKernel FresnelC FresnelS Friday FrobeniusNumber FrobeniusSolve FromCharacterCode FromCoefficientRules FromContinuedFraction FromDate FromDigits FromDMS Front FrontEndDynamicExpression FrontEndEventActions FrontEndExecute FrontEndObject FrontEndResource FrontEndResourceString FrontEndStackSize FrontEndToken FrontEndTokenExecute FrontEndValueCache FrontEndVersion FrontFaceColor FrontFaceOpacity Full FullAxes FullDefinition FullForm FullGraphics FullOptions FullSimplify Function FunctionExpand FunctionInterpolation FunctionSpace FussellVeselyImportance GaborFilter GaborMatrix GaborWavelet GainMargins GainPhaseMargins Gamma GammaDistribution GammaRegularized GapPenalty Gather GatherBy GaugeFaceElementFunction GaugeFaceStyle GaugeFrameElementFunction GaugeFrameSize GaugeFrameStyle GaugeLabels GaugeMarkers GaugeStyle GaussianFilter GaussianIntegers GaussianMatrix GaussianWindow GCD GegenbauerC General GeneralizedLinearModelFit GenerateConditions GeneratedCell GeneratedParameters GeneratingFunction Generic GenericCylindricalDecomposition GenomeData GenomeLookup GeodesicClosing GeodesicDilation GeodesicErosion GeodesicOpening GeoDestination GeodesyData GeoDirection GeoDistance GeoGridPosition GeometricBrownianMotionProcess GeometricDistribution GeometricMean GeometricMeanFilter GeometricTransformation GeometricTransformation3DBox GeometricTransformation3DBoxOptions GeometricTransformationBox GeometricTransformationBoxOptions GeoPosition GeoPositionENU GeoPositionXYZ GeoProjectionData GestureHandler GestureHandlerTag Get GetBoundingBoxSizePacket GetContext GetEnvironment GetFileName GetFrontEndOptionsDataPacket GetLinebreakInformationPacket GetMenusPacket GetPageBreakInformationPacket Glaisher GlobalClusteringCoefficient GlobalPreferences GlobalSession Glow GoldenRatio GompertzMakehamDistribution GoodmanKruskalGamma GoodmanKruskalGammaTest Goto Grad Gradient GradientFilter GradientOrientationFilter Graph GraphAssortativity GraphCenter GraphComplement GraphData GraphDensity GraphDiameter GraphDifference GraphDisjointUnion GraphDistance GraphDistanceMatrix GraphElementData GraphEmbedding GraphHighlight GraphHighlightStyle GraphHub Graphics Graphics3D Graphics3DBox Graphics3DBoxOptions GraphicsArray GraphicsBaseline GraphicsBox GraphicsBoxOptions GraphicsColor GraphicsColumn GraphicsComplex GraphicsComplex3DBox GraphicsComplex3DBoxOptions GraphicsComplexBox GraphicsComplexBoxOptions GraphicsContents GraphicsData GraphicsGrid GraphicsGridBox GraphicsGroup GraphicsGroup3DBox GraphicsGroup3DBoxOptions GraphicsGroupBox GraphicsGroupBoxOptions GraphicsGrouping GraphicsHighlightColor GraphicsRow GraphicsSpacing GraphicsStyle GraphIntersection GraphLayout GraphLinkEfficiency GraphPeriphery GraphPlot GraphPlot3D GraphPower GraphPropertyDistribution GraphQ GraphRadius GraphReciprocity GraphRoot GraphStyle GraphUnion Gray GrayLevel GreatCircleDistance Greater GreaterEqual GreaterEqualLess GreaterFullEqual GreaterGreater GreaterLess GreaterSlantEqual GreaterTilde Green Grid GridBaseline GridBox GridBoxAlignment GridBoxBackground GridBoxDividers GridBoxFrame GridBoxItemSize GridBoxItemStyle GridBoxOptions GridBoxSpacings GridCreationSettings GridDefaultElement GridElementStyleOptions GridFrame GridFrameMargins GridGraph GridLines GridLinesStyle GroebnerBasis GroupActionBase GroupCentralizer GroupElementFromWord GroupElementPosition GroupElementQ GroupElements GroupElementToWord GroupGenerators GroupMultiplicationTable GroupOrbits GroupOrder GroupPageBreakWithin GroupSetwiseStabilizer GroupStabilizer GroupStabilizerChain Gudermannian GumbelDistribution HaarWavelet HadamardMatrix HalfNormalDistribution HamiltonianGraphQ HammingDistance HammingWindow HankelH1 HankelH2 HankelMatrix HannPoissonWindow HannWindow HaradaNortonGroupHN HararyGraph HarmonicMean HarmonicMeanFilter HarmonicNumber Hash HashTable Haversine HazardFunction Head HeadCompose Heads HeavisideLambda HeavisidePi HeavisideTheta HeldGroupHe HeldPart HelpBrowserLookup HelpBrowserNotebook HelpBrowserSettings HermiteDecomposition HermiteH HermitianMatrixQ HessenbergDecomposition Hessian HexadecimalCharacter Hexahedron HexahedronBox HexahedronBoxOptions HiddenSurface HighlightGraph HighlightImage HighpassFilter HigmanSimsGroupHS HilbertFilter HilbertMatrix Histogram Histogram3D HistogramDistribution HistogramList HistogramTransform HistogramTransformInterpolation HitMissTransform HITSCentrality HodgeDual HoeffdingD HoeffdingDTest Hold HoldAll HoldAllComplete HoldComplete HoldFirst HoldForm HoldPattern HoldRest HolidayCalendar HomeDirectory HomePage Horizontal HorizontalForm HorizontalGauge HorizontalScrollPosition HornerForm HotellingTSquareDistribution HoytDistribution HTMLSave Hue HumpDownHump HumpEqual HurwitzLerchPhi HurwitzZeta HyperbolicDistribution HypercubeGraph HyperexponentialDistribution Hyperfactorial Hypergeometric0F1 Hypergeometric0F1Regularized Hypergeometric1F1 Hypergeometric1F1Regularized Hypergeometric2F1 Hypergeometric2F1Regularized HypergeometricDistribution HypergeometricPFQ HypergeometricPFQRegularized HypergeometricU Hyperlink HyperlinkCreationSettings Hyphenation HyphenationOptions HypoexponentialDistribution HypothesisTestData I Identity IdentityMatrix If IgnoreCase Im Image Image3D Image3DSlices ImageAccumulate ImageAdd ImageAdjust ImageAlign ImageApply ImageAspectRatio ImageAssemble ImageCache ImageCacheValid ImageCapture ImageChannels ImageClip ImageColorSpace ImageCompose ImageConvolve ImageCooccurrence ImageCorners ImageCorrelate ImageCorrespondingPoints ImageCrop ImageData ImageDataPacket ImageDeconvolve ImageDemosaic ImageDifference ImageDimensions ImageDistance ImageEffect ImageFeatureTrack ImageFileApply ImageFileFilter ImageFileScan ImageFilter ImageForestingComponents ImageForwardTransformation ImageHistogram ImageKeypoints ImageLevels ImageLines ImageMargins ImageMarkers ImageMeasurements ImageMultiply ImageOffset ImagePad ImagePadding ImagePartition ImagePeriodogram ImagePerspectiveTransformation ImageQ ImageRangeCache ImageReflect ImageRegion ImageResize ImageResolution ImageRotate ImageRotated ImageScaled ImageScan ImageSize ImageSizeAction ImageSizeCache ImageSizeMultipliers ImageSizeRaw ImageSubtract ImageTake ImageTransformation ImageTrim ImageType ImageValue ImageValuePositions Implies Import ImportAutoReplacements ImportString ImprovementImportance In IncidenceGraph IncidenceList IncidenceMatrix IncludeConstantBasis IncludeFileExtension IncludePods IncludeSingularTerm Increment Indent IndentingNewlineSpacings IndentMaxFraction IndependenceTest IndependentEdgeSetQ IndependentUnit IndependentVertexSetQ Indeterminate IndexCreationOptions Indexed IndexGraph IndexTag Inequality InexactNumberQ InexactNumbers Infinity Infix Information Inherited InheritScope Initialization InitializationCell InitializationCellEvaluation InitializationCellWarning InlineCounterAssignments InlineCounterIncrements InlineRules Inner Inpaint Input InputAliases InputAssumptions InputAutoReplacements InputField InputFieldBox InputFieldBoxOptions InputForm InputGrouping InputNamePacket InputNotebook InputPacket InputSettings InputStream InputString InputStringPacket InputToBoxFormPacket Insert InsertionPointObject InsertResults Inset Inset3DBox Inset3DBoxOptions InsetBox InsetBoxOptions Install InstallService InString Integer IntegerDigits IntegerExponent IntegerLength IntegerPart IntegerPartitions IntegerQ Integers IntegerString Integral Integrate Interactive InteractiveTradingChart Interlaced Interleaving InternallyBalancedDecomposition InterpolatingFunction InterpolatingPolynomial Interpolation InterpolationOrder InterpolationPoints InterpolationPrecision Interpretation InterpretationBox InterpretationBoxOptions InterpretationFunction InterpretTemplate InterquartileRange Interrupt InterruptSettings Intersection Interval IntervalIntersection IntervalMemberQ IntervalUnion Inverse InverseBetaRegularized InverseCDF InverseChiSquareDistribution InverseContinuousWaveletTransform InverseDistanceTransform InverseEllipticNomeQ InverseErf InverseErfc InverseFourier InverseFourierCosTransform InverseFourierSequenceTransform InverseFourierSinTransform InverseFourierTransform InverseFunction InverseFunctions InverseGammaDistribution InverseGammaRegularized InverseGaussianDistribution InverseGudermannian InverseHaversine InverseJacobiCD InverseJacobiCN InverseJacobiCS InverseJacobiDC InverseJacobiDN InverseJacobiDS InverseJacobiNC InverseJacobiND InverseJacobiNS InverseJacobiSC InverseJacobiSD InverseJacobiSN InverseLaplaceTransform InversePermutation InverseRadon InverseSeries InverseSurvivalFunction InverseWaveletTransform InverseWeierstrassP InverseZTransform Invisible InvisibleApplication InvisibleTimes IrreduciblePolynomialQ IsolatingInterval IsomorphicGraphQ IsotopeData Italic Item ItemBox ItemBoxOptions ItemSize ItemStyle ItoProcess JaccardDissimilarity JacobiAmplitude Jacobian JacobiCD JacobiCN JacobiCS JacobiDC JacobiDN JacobiDS JacobiNC JacobiND JacobiNS JacobiP JacobiSC JacobiSD JacobiSN JacobiSymbol JacobiZeta JankoGroupJ1 JankoGroupJ2 JankoGroupJ3 JankoGroupJ4 JarqueBeraALMTest JohnsonDistribution Join Joined JoinedCurve JoinedCurveBox JoinForm JordanDecomposition JordanModelDecomposition K KagiChart KaiserBesselWindow KaiserWindow KalmanEstimator KalmanFilter KarhunenLoeveDecomposition KaryTree KatzCentrality KCoreComponents KDistribution KelvinBei KelvinBer KelvinKei KelvinKer KendallTau KendallTauTest KernelExecute KernelMixtureDistribution KernelObject Kernels Ket Khinchin KirchhoffGraph KirchhoffMatrix KleinInvariantJ KnightTourGraph KnotData KnownUnitQ KolmogorovSmirnovTest KroneckerDelta KroneckerModelDecomposition KroneckerProduct KroneckerSymbol KuiperTest KumaraswamyDistribution Kurtosis KuwaharaFilter Label Labeled LabeledSlider LabelingFunction LabelStyle LaguerreL LambdaComponents LambertW LanczosWindow LandauDistribution Language LanguageCategory LaplaceDistribution LaplaceTransform Laplacian LaplacianFilter LaplacianGaussianFilter Large Larger Last Latitude LatitudeLongitude LatticeData LatticeReduce Launch LaunchKernels LayeredGraphPlot LayerSizeFunction LayoutInformation LCM LeafCount LeapYearQ LeastSquares LeastSquaresFilterKernel Left LeftArrow LeftArrowBar LeftArrowRightArrow LeftDownTeeVector LeftDownVector LeftDownVectorBar LeftRightArrow LeftRightVector LeftTee LeftTeeArrow LeftTeeVector LeftTriangle LeftTriangleBar LeftTriangleEqual LeftUpDownVector LeftUpTeeVector LeftUpVector LeftUpVectorBar LeftVector LeftVectorBar LegendAppearance Legended LegendFunction LegendLabel LegendLayout LegendMargins LegendMarkers LegendMarkerSize LegendreP LegendreQ LegendreType Length LengthWhile LerchPhi Less LessEqual LessEqualGreater LessFullEqual LessGreater LessLess LessSlantEqual LessTilde LetterCharacter LetterQ Level LeveneTest LeviCivitaTensor LevyDistribution Lexicographic LibraryFunction LibraryFunctionError LibraryFunctionInformation LibraryFunctionLoad LibraryFunctionUnload LibraryLoad LibraryUnload LicenseID LiftingFilterData LiftingWaveletTransform LightBlue LightBrown LightCyan Lighter LightGray LightGreen Lighting LightingAngle LightMagenta LightOrange LightPink LightPurple LightRed LightSources LightYellow Likelihood Limit LimitsPositioning LimitsPositioningTokens LindleyDistribution Line Line3DBox LinearFilter LinearFractionalTransform LinearModelFit LinearOffsetFunction LinearProgramming LinearRecurrence LinearSolve LinearSolveFunction LineBox LineBreak LinebreakAdjustments LineBreakChart LineBreakWithin LineColor LineForm LineGraph LineIndent LineIndentMaxFraction LineIntegralConvolutionPlot LineIntegralConvolutionScale LineLegend LineOpacity LineSpacing LineWrapParts LinkActivate LinkClose LinkConnect LinkConnectedQ LinkCreate LinkError LinkFlush LinkFunction LinkHost LinkInterrupt LinkLaunch LinkMode LinkObject LinkOpen LinkOptions LinkPatterns LinkProtocol LinkRead LinkReadHeld LinkReadyQ Links LinkWrite LinkWriteHeld LiouvilleLambda List Listable ListAnimate ListContourPlot ListContourPlot3D ListConvolve ListCorrelate ListCurvePathPlot ListDeconvolve ListDensityPlot Listen ListFourierSequenceTransform ListInterpolation ListLineIntegralConvolutionPlot ListLinePlot ListLogLinearPlot ListLogLogPlot ListLogPlot ListPicker ListPickerBox ListPickerBoxBackground ListPickerBoxOptions ListPlay ListPlot ListPlot3D ListPointPlot3D ListPolarPlot ListQ ListStreamDensityPlot ListStreamPlot ListSurfacePlot3D ListVectorDensityPlot ListVectorPlot ListVectorPlot3D ListZTransform Literal LiteralSearch LocalClusteringCoefficient LocalizeVariables LocationEquivalenceTest LocationTest Locator LocatorAutoCreate LocatorBox LocatorBoxOptions LocatorCentering LocatorPane LocatorPaneBox LocatorPaneBoxOptions LocatorRegion Locked Log Log10 Log2 LogBarnesG LogGamma LogGammaDistribution LogicalExpand LogIntegral LogisticDistribution LogitModelFit LogLikelihood LogLinearPlot LogLogisticDistribution LogLogPlot LogMultinormalDistribution LogNormalDistribution LogPlot LogRankTest LogSeriesDistribution LongEqual Longest LongestAscendingSequence LongestCommonSequence LongestCommonSequencePositions LongestCommonSubsequence LongestCommonSubsequencePositions LongestMatch LongForm Longitude LongLeftArrow LongLeftRightArrow LongRightArrow Loopback LoopFreeGraphQ LowerCaseQ LowerLeftArrow LowerRightArrow LowerTriangularize LowpassFilter LQEstimatorGains LQGRegulator LQOutputRegulatorGains LQRegulatorGains LUBackSubstitution LucasL LuccioSamiComponents LUDecomposition LyapunovSolve LyonsGroupLy MachineID MachineName MachineNumberQ MachinePrecision MacintoshSystemPageSetup Magenta Magnification Magnify MainSolve MaintainDynamicCaches Majority MakeBoxes MakeExpression MakeRules MangoldtLambda ManhattanDistance Manipulate Manipulator MannWhitneyTest MantissaExponent Manual Map MapAll MapAt MapIndexed MAProcess MapThread MarcumQ MardiaCombinedTest MardiaKurtosisTest MardiaSkewnessTest MarginalDistribution MarkovProcessProperties Masking MatchingDissimilarity MatchLocalNameQ MatchLocalNames MatchQ Material MathematicaNotation MathieuC MathieuCharacteristicA MathieuCharacteristicB MathieuCharacteristicExponent MathieuCPrime MathieuGroupM11 MathieuGroupM12 MathieuGroupM22 MathieuGroupM23 MathieuGroupM24 MathieuS MathieuSPrime MathMLForm MathMLText Matrices MatrixExp MatrixForm MatrixFunction MatrixLog MatrixPlot MatrixPower MatrixQ MatrixRank Max MaxBend MaxDetect MaxExtraBandwidths MaxExtraConditions MaxFeatures MaxFilter Maximize MaxIterations MaxMemoryUsed MaxMixtureKernels MaxPlotPoints MaxPoints MaxRecursion MaxStableDistribution MaxStepFraction MaxSteps MaxStepSize MaxValue MaxwellDistribution McLaughlinGroupMcL Mean MeanClusteringCoefficient MeanDegreeConnectivity MeanDeviation MeanFilter MeanGraphDistance MeanNeighborDegree MeanShift MeanShiftFilter Median MedianDeviation MedianFilter Medium MeijerG MeixnerDistribution MemberQ MemoryConstrained MemoryInUse Menu MenuAppearance MenuCommandKey MenuEvaluator MenuItem MenuPacket MenuSortingValue MenuStyle MenuView MergeDifferences Mesh MeshFunctions MeshRange MeshShading MeshStyle Message MessageDialog MessageList MessageName MessageOptions MessagePacket Messages MessagesNotebook MetaCharacters MetaInformation Method MethodOptions MexicanHatWavelet MeyerWavelet Min MinDetect MinFilter MinimalPolynomial MinimalStateSpaceModel Minimize Minors MinRecursion MinSize MinStableDistribution Minus MinusPlus MinValue Missing MissingDataMethod MittagLefflerE MixedRadix MixedRadixQuantity MixtureDistribution Mod Modal Mode Modular ModularLambda Module Modulus MoebiusMu Moment Momentary MomentConvert MomentEvaluate MomentGeneratingFunction Monday Monitor MonomialList MonomialOrder MonsterGroupM MorletWavelet MorphologicalBinarize MorphologicalBranchPoints MorphologicalComponents MorphologicalEulerNumber MorphologicalGraph MorphologicalPerimeter MorphologicalTransform Most MouseAnnotation MouseAppearance MouseAppearanceTag MouseButtons Mouseover MousePointerNote MousePosition MovingAverage MovingMedian MoyalDistribution MultiedgeStyle MultilaunchWarning MultiLetterItalics MultiLetterStyle MultilineFunction Multinomial MultinomialDistribution MultinormalDistribution MultiplicativeOrder Multiplicity Multiselection MultivariateHypergeometricDistribution MultivariatePoissonDistribution MultivariateTDistribution N NakagamiDistribution NameQ Names NamespaceBox Nand NArgMax NArgMin NBernoulliB NCache NDSolve NDSolveValue Nearest NearestFunction NeedCurrentFrontEndPackagePacket NeedCurrentFrontEndSymbolsPacket NeedlemanWunschSimilarity Needs Negative NegativeBinomialDistribution NegativeMultinomialDistribution NeighborhoodGraph Nest NestedGreaterGreater NestedLessLess NestedScriptRules NestList NestWhile NestWhileList NevilleThetaC NevilleThetaD NevilleThetaN NevilleThetaS NewPrimitiveStyle NExpectation Next NextPrime NHoldAll NHoldFirst NHoldRest NicholsGridLines NicholsPlot NIntegrate NMaximize NMaxValue NMinimize NMinValue NominalVariables NonAssociative NoncentralBetaDistribution NoncentralChiSquareDistribution NoncentralFRatioDistribution NoncentralStudentTDistribution NonCommutativeMultiply NonConstants None NonlinearModelFit NonlocalMeansFilter NonNegative NonPositive Nor NorlundB Norm Normal NormalDistribution NormalGrouping Normalize NormalizedSquaredEuclideanDistance NormalsFunction NormFunction Not NotCongruent NotCupCap NotDoubleVerticalBar Notebook NotebookApply NotebookAutoSave NotebookClose NotebookConvertSettings NotebookCreate NotebookCreateReturnObject NotebookDefault NotebookDelete NotebookDirectory NotebookDynamicExpression NotebookEvaluate NotebookEventActions NotebookFileName NotebookFind NotebookFindReturnObject NotebookGet NotebookGetLayoutInformationPacket NotebookGetMisspellingsPacket NotebookInformation NotebookInterfaceObject NotebookLocate NotebookObject NotebookOpen NotebookOpenReturnObject NotebookPath NotebookPrint NotebookPut NotebookPutReturnObject NotebookRead NotebookResetGeneratedCells Notebooks NotebookSave NotebookSaveAs NotebookSelection NotebookSetupLayoutInformationPacket NotebooksMenu NotebookWrite NotElement NotEqualTilde NotExists NotGreater NotGreaterEqual NotGreaterFullEqual NotGreaterGreater NotGreaterLess NotGreaterSlantEqual NotGreaterTilde NotHumpDownHump NotHumpEqual NotLeftTriangle NotLeftTriangleBar NotLeftTriangleEqual NotLess NotLessEqual NotLessFullEqual NotLessGreater NotLessLess NotLessSlantEqual NotLessTilde NotNestedGreaterGreater NotNestedLessLess NotPrecedes NotPrecedesEqual NotPrecedesSlantEqual NotPrecedesTilde NotReverseElement NotRightTriangle NotRightTriangleBar NotRightTriangleEqual NotSquareSubset NotSquareSubsetEqual NotSquareSuperset NotSquareSupersetEqual NotSubset NotSubsetEqual NotSucceeds NotSucceedsEqual NotSucceedsSlantEqual NotSucceedsTilde NotSuperset NotSupersetEqual NotTilde NotTildeEqual NotTildeFullEqual NotTildeTilde NotVerticalBar NProbability NProduct NProductFactors NRoots NSolve NSum NSumTerms Null NullRecords NullSpace NullWords Number NumberFieldClassNumber NumberFieldDiscriminant NumberFieldFundamentalUnits NumberFieldIntegralBasis NumberFieldNormRepresentatives NumberFieldRegulator NumberFieldRootsOfUnity NumberFieldSignature NumberForm NumberFormat NumberMarks NumberMultiplier NumberPadding NumberPoint NumberQ NumberSeparator NumberSigns NumberString Numerator NumericFunction NumericQ NuttallWindow NValues NyquistGridLines NyquistPlot O ObservabilityGramian ObservabilityMatrix ObservableDecomposition ObservableModelQ OddQ Off Offset OLEData On ONanGroupON OneIdentity Opacity Open OpenAppend Opener OpenerBox OpenerBoxOptions OpenerView OpenFunctionInspectorPacket Opening OpenRead OpenSpecialOptions OpenTemporary OpenWrite Operate OperatingSystem OptimumFlowData Optional OptionInspectorSettings OptionQ Options OptionsPacket OptionsPattern OptionValue OptionValueBox OptionValueBoxOptions Or Orange Order OrderDistribution OrderedQ Ordering Orderless OrnsteinUhlenbeckProcess Orthogonalize Out Outer OutputAutoOverwrite OutputControllabilityMatrix OutputControllableModelQ OutputForm OutputFormData OutputGrouping OutputMathEditExpression OutputNamePacket OutputResponse OutputSizeLimit OutputStream Over OverBar OverDot Overflow OverHat Overlaps Overlay OverlayBox OverlayBoxOptions Overscript OverscriptBox OverscriptBoxOptions OverTilde OverVector OwenT OwnValues PackingMethod PaddedForm Padding PadeApproximant PadLeft PadRight PageBreakAbove PageBreakBelow PageBreakWithin PageFooterLines PageFooters PageHeaderLines PageHeaders PageHeight PageRankCentrality PageWidth PairedBarChart PairedHistogram PairedSmoothHistogram PairedTTest PairedZTest PaletteNotebook PalettePath Pane PaneBox PaneBoxOptions Panel PanelBox PanelBoxOptions Paneled PaneSelector PaneSelectorBox PaneSelectorBoxOptions PaperWidth ParabolicCylinderD ParagraphIndent ParagraphSpacing ParallelArray ParallelCombine ParallelDo ParallelEvaluate Parallelization Parallelize ParallelMap ParallelNeeds ParallelProduct ParallelSubmit ParallelSum ParallelTable ParallelTry Parameter ParameterEstimator ParameterMixtureDistribution ParameterVariables ParametricFunction ParametricNDSolve ParametricNDSolveValue ParametricPlot ParametricPlot3D ParentConnect ParentDirectory ParentForm Parenthesize ParentList ParetoDistribution Part PartialCorrelationFunction PartialD ParticleData Partition 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RangeSpecification RankedMax RankedMin Raster Raster3D Raster3DBox Raster3DBoxOptions RasterArray RasterBox RasterBoxOptions Rasterize RasterSize Rational RationalFunctions Rationalize Rationals Ratios Raw RawArray RawBoxes RawData RawMedium RayleighDistribution Re Read ReadList ReadProtected Real RealBlockDiagonalForm RealDigits RealExponent Reals Reap Record RecordLists RecordSeparators Rectangle RectangleBox RectangleBoxOptions RectangleChart RectangleChart3D RecurrenceFilter RecurrenceTable RecurringDigitsForm Red Reduce RefBox ReferenceLineStyle ReferenceMarkers ReferenceMarkerStyle Refine ReflectionMatrix ReflectionTransform Refresh RefreshRate RegionBinarize RegionFunction RegionPlot RegionPlot3D RegularExpression Regularization Reinstall Release ReleaseHold ReliabilityDistribution ReliefImage ReliefPlot Remove RemoveAlphaChannel RemoveAsynchronousTask Removed RemoveInputStreamMethod RemoveOutputStreamMethod RemoveProperty RemoveScheduledTask RenameDirectory RenameFile RenderAll RenderingOptions RenewalProcess RenkoChart Repeated RepeatedNull RepeatedString Replace ReplaceAll ReplaceHeldPart ReplaceImageValue ReplaceList ReplacePart ReplacePixelValue ReplaceRepeated Resampling Rescale RescalingTransform ResetDirectory ResetMenusPacket ResetScheduledTask Residue Resolve Rest Resultant ResumePacket Return ReturnExpressionPacket ReturnInputFormPacket ReturnPacket ReturnTextPacket Reverse ReverseBiorthogonalSplineWavelet ReverseElement ReverseEquilibrium ReverseGraph ReverseUpEquilibrium RevolutionAxis RevolutionPlot3D RGBColor RiccatiSolve RiceDistribution RidgeFilter RiemannR RiemannSiegelTheta RiemannSiegelZ Riffle Right RightArrow RightArrowBar RightArrowLeftArrow RightCosetRepresentative RightDownTeeVector RightDownVector RightDownVectorBar RightTee RightTeeArrow RightTeeVector RightTriangle RightTriangleBar RightTriangleEqual RightUpDownVector RightUpTeeVector RightUpVector RightUpVectorBar RightVector RightVectorBar RiskAchievementImportance RiskReductionImportance RogersTanimotoDissimilarity Root RootApproximant RootIntervals RootLocusPlot RootMeanSquare RootOfUnityQ RootReduce Roots RootSum Rotate RotateLabel RotateLeft RotateRight RotationAction RotationBox RotationBoxOptions RotationMatrix RotationTransform Round RoundImplies RoundingRadius Row RowAlignments RowBackgrounds RowBox RowHeights RowLines RowMinHeight RowReduce RowsEqual RowSpacings RSolve RudvalisGroupRu Rule RuleCondition RuleDelayed RuleForm RulerUnits Run RunScheduledTask RunThrough RuntimeAttributes RuntimeOptions RussellRaoDissimilarity SameQ SameTest SampleDepth SampledSoundFunction SampledSoundList SampleRate SamplingPeriod SARIMAProcess SARMAProcess SatisfiabilityCount SatisfiabilityInstances SatisfiableQ Saturday Save Saveable SaveAutoDelete SaveDefinitions SawtoothWave Scale Scaled ScaleDivisions ScaledMousePosition ScaleOrigin ScalePadding ScaleRanges ScaleRangeStyle ScalingFunctions ScalingMatrix ScalingTransform Scan ScheduledTaskActiveQ 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eliminate_using ellipse elliptic_e elliptic_ec elliptic_eu elliptic_f elliptic_kc elliptic_pi ematrix empty_graph emptyp endcons entermatrix entertensor entier equal equalp equiv_classes erf erfc erf_generalized erfi errcatch error errormsg errors euler ev eval_string evenp every evolution evolution2d evundiff example exp expand expandwrt expandwrt_factored expint expintegral_chi expintegral_ci expintegral_e expintegral_e1 expintegral_ei expintegral_e_simplify expintegral_li expintegral_shi expintegral_si explicit explose exponentialize express expt exsec extdiff extract_linear_equations extremal_subset ezgcd %f f90 facsum factcomb factor factorfacsum factorial factorout factorsum facts fast_central_elements fast_linsolve fasttimes featurep fernfale fft fib fibtophi fifth filename_merge file_search file_type fillarray findde find_root find_root_abs find_root_error find_root_rel first fix flatten flength float floatnump floor flower_snark flush flush1deriv flushd flushnd flush_output fmin_cobyla forget fortran fourcos fourexpand fourier fourier_elim fourint fourintcos fourintsin foursimp foursin fourth fposition frame_bracket freeof freshline fresnel_c fresnel_s from_adjacency_matrix frucht_graph full_listify fullmap fullmapl fullratsimp fullratsubst fullsetify funcsolve fundamental_dimensions fundamental_units fundef funmake funp fv g0 g1 gamma gamma_greek gamma_incomplete gamma_incomplete_generalized gamma_incomplete_regularized gauss gauss_a gauss_b gaussprob gcd gcdex gcdivide gcfac gcfactor gd generalized_lambert_w genfact gen_laguerre genmatrix gensym geo_amortization geo_annuity_fv geo_annuity_pv geomap geometric geometric_mean geosum get getcurrentdirectory get_edge_weight getenv get_lu_factors get_output_stream_string get_pixel get_plot_option get_tex_environment get_tex_environment_default get_vertex_label gfactor gfactorsum ggf girth global_variances gn gnuplot_close gnuplot_replot gnuplot_reset gnuplot_restart gnuplot_start go Gosper GosperSum gr2d gr3d gradef gramschmidt graph6_decode graph6_encode graph6_export graph6_import graph_center graph_charpoly graph_eigenvalues graph_flow graph_order graph_periphery graph_product graph_size graph_union great_rhombicosidodecahedron_graph great_rhombicuboctahedron_graph grid_graph grind grobner_basis grotzch_graph hamilton_cycle hamilton_path hankel hankel_1 hankel_2 harmonic harmonic_mean hav heawood_graph hermite hessian hgfred hilbertmap hilbert_matrix hipow histogram histogram_description hodge horner hypergeometric i0 i1 %ibes ic1 ic2 ic_convert ichr1 ichr2 icosahedron_graph icosidodecahedron_graph icurvature ident identfor identity idiff idim idummy ieqn %if ifactors iframes ifs igcdex igeodesic_coords ilt image imagpart imetric implicit implicit_derivative implicit_plot indexed_tensor indices induced_subgraph inferencep inference_result infix info_display init_atensor init_ctensor in_neighbors innerproduct inpart inprod inrt integerp integer_partitions integrate intersect intersection intervalp intopois intosum invariant1 invariant2 inverse_fft inverse_jacobi_cd inverse_jacobi_cn inverse_jacobi_cs inverse_jacobi_dc inverse_jacobi_dn inverse_jacobi_ds inverse_jacobi_nc inverse_jacobi_nd inverse_jacobi_ns inverse_jacobi_sc inverse_jacobi_sd inverse_jacobi_sn invert invert_by_adjoint invert_by_lu inv_mod irr is is_biconnected is_bipartite is_connected is_digraph is_edge_in_graph is_graph is_graph_or_digraph ishow is_isomorphic isolate isomorphism is_planar isqrt isreal_p is_sconnected is_tree is_vertex_in_graph items_inference %j j0 j1 jacobi jacobian jacobi_cd jacobi_cn jacobi_cs jacobi_dc jacobi_dn jacobi_ds jacobi_nc jacobi_nd jacobi_ns jacobi_p jacobi_sc jacobi_sd jacobi_sn JF jn join jordan julia julia_set julia_sin %k kdels kdelta kill killcontext kostka kron_delta kronecker_product kummer_m kummer_u kurtosis kurtosis_bernoulli kurtosis_beta kurtosis_binomial kurtosis_chi2 kurtosis_continuous_uniform kurtosis_discrete_uniform kurtosis_exp kurtosis_f kurtosis_gamma kurtosis_general_finite_discrete kurtosis_geometric kurtosis_gumbel kurtosis_hypergeometric kurtosis_laplace kurtosis_logistic kurtosis_lognormal kurtosis_negative_binomial kurtosis_noncentral_chi2 kurtosis_noncentral_student_t kurtosis_normal kurtosis_pareto kurtosis_poisson kurtosis_rayleigh kurtosis_student_t kurtosis_weibull label labels lagrange laguerre lambda lambert_w laplace laplacian_matrix last lbfgs lc2kdt lcharp lc_l lcm lc_u ldefint ldisp ldisplay legendre_p legendre_q leinstein length let letrules letsimp levi_civita lfreeof lgtreillis lhs li liediff limit Lindstedt linear linearinterpol linear_program linear_regression line_graph linsolve listarray list_correlations listify list_matrix_entries list_nc_monomials listoftens listofvars listp lmax lmin load loadfile local locate_matrix_entry log logcontract log_gamma lopow lorentz_gauge lowercasep lpart lratsubst lreduce lriemann lsquares_estimates lsquares_estimates_approximate lsquares_estimates_exact lsquares_mse lsquares_residual_mse lsquares_residuals lsum ltreillis lu_backsub lucas lu_factor %m macroexpand macroexpand1 make_array makebox makefact makegamma make_graph make_level_picture makelist makeOrders make_poly_continent make_poly_country make_polygon make_random_state make_rgb_picture makeset make_string_input_stream make_string_output_stream make_transform mandelbrot mandelbrot_set map mapatom maplist matchdeclare matchfix mat_cond mat_fullunblocker mat_function mathml_display mat_norm matrix matrixmap matrixp matrix_size mattrace mat_trace mat_unblocker max max_clique max_degree max_flow maximize_lp max_independent_set max_matching maybe md5sum mean mean_bernoulli mean_beta mean_binomial mean_chi2 mean_continuous_uniform mean_deviation mean_discrete_uniform mean_exp mean_f mean_gamma mean_general_finite_discrete mean_geometric mean_gumbel mean_hypergeometric mean_laplace mean_logistic mean_lognormal mean_negative_binomial mean_noncentral_chi2 mean_noncentral_student_t mean_normal mean_pareto mean_poisson mean_rayleigh mean_student_t mean_weibull median median_deviation member mesh metricexpandall mgf1_sha1 min min_degree min_edge_cut minfactorial minimalPoly minimize_lp minimum_spanning_tree minor minpack_lsquares minpack_solve min_vertex_cover min_vertex_cut mkdir mnewton mod mode_declare mode_identity ModeMatrix moebius mon2schur mono monomial_dimensions multibernstein_poly multi_display_for_texinfo multi_elem multinomial multinomial_coeff multi_orbit multiplot_mode multi_pui multsym multthru mycielski_graph nary natural_unit nc_degree ncexpt ncharpoly negative_picture neighbors new newcontext newdet new_graph newline newton new_variable next_prime nicedummies niceindices ninth nofix nonarray noncentral_moment nonmetricity nonnegintegerp nonscalarp nonzeroandfreeof notequal nounify nptetrad npv nroots nterms ntermst nthroot nullity nullspace num numbered_boundaries numberp number_to_octets num_distinct_partitions numerval numfactor num_partitions nusum nzeta nzetai nzetar octets_to_number octets_to_oid odd_girth oddp ode2 ode_check odelin oid_to_octets op opena opena_binary openr openr_binary openw openw_binary operatorp opsubst optimize %or orbit orbits ordergreat ordergreatp orderless orderlessp orthogonal_complement orthopoly_recur orthopoly_weight outermap out_neighbors outofpois pade parabolic_cylinder_d parametric parametric_surface parg parGosper parse_string parse_timedate part part2cont partfrac partition partition_set partpol path_digraph path_graph pathname_directory pathname_name pathname_type pdf_bernoulli pdf_beta pdf_binomial pdf_cauchy pdf_chi2 pdf_continuous_uniform pdf_discrete_uniform pdf_exp pdf_f pdf_gamma pdf_general_finite_discrete pdf_geometric pdf_gumbel pdf_hypergeometric pdf_laplace pdf_logistic pdf_lognormal pdf_negative_binomial pdf_noncentral_chi2 pdf_noncentral_student_t pdf_normal pdf_pareto pdf_poisson pdf_rank_sum pdf_rayleigh pdf_signed_rank pdf_student_t pdf_weibull pearson_skewness permanent permut permutation permutations petersen_graph petrov pickapart picture_equalp picturep piechart piechart_description planar_embedding playback plog plot2d plot3d plotdf ploteq plsquares pochhammer points poisdiff poisexpt poisint poismap poisplus poissimp poissubst poistimes poistrim polar polarform polartorect polar_to_xy poly_add poly_buchberger poly_buchberger_criterion poly_colon_ideal poly_content polydecomp poly_depends_p poly_elimination_ideal poly_exact_divide poly_expand poly_expt poly_gcd polygon poly_grobner poly_grobner_equal poly_grobner_member poly_grobner_subsetp poly_ideal_intersection poly_ideal_polysaturation poly_ideal_polysaturation1 poly_ideal_saturation poly_ideal_saturation1 poly_lcm poly_minimization polymod poly_multiply polynome2ele polynomialp poly_normal_form poly_normalize poly_normalize_list poly_polysaturation_extension poly_primitive_part poly_pseudo_divide poly_reduced_grobner poly_reduction poly_saturation_extension poly_s_polynomial poly_subtract polytocompanion pop postfix potential power_mod powerseries powerset prefix prev_prime primep primes principal_components print printf printfile print_graph printpois printprops prodrac product properties propvars psi psubst ptriangularize pui pui2comp pui2ele pui2polynome pui_direct puireduc push put pv qput qrange qty quad_control quad_qag quad_qagi quad_qagp quad_qags quad_qawc quad_qawf quad_qawo quad_qaws quadrilateral quantile quantile_bernoulli quantile_beta quantile_binomial quantile_cauchy quantile_chi2 quantile_continuous_uniform quantile_discrete_uniform quantile_exp quantile_f quantile_gamma quantile_general_finite_discrete quantile_geometric quantile_gumbel quantile_hypergeometric quantile_laplace quantile_logistic quantile_lognormal quantile_negative_binomial quantile_noncentral_chi2 quantile_noncentral_student_t quantile_normal quantile_pareto quantile_poisson quantile_rayleigh quantile_student_t quantile_weibull quartile_skewness quit qunit quotient racah_v racah_w radcan radius random random_bernoulli random_beta random_binomial random_bipartite_graph random_cauchy random_chi2 random_continuous_uniform random_digraph random_discrete_uniform random_exp random_f random_gamma random_general_finite_discrete random_geometric random_graph random_graph1 random_gumbel random_hypergeometric random_laplace random_logistic random_lognormal random_negative_binomial random_network random_noncentral_chi2 random_noncentral_student_t random_normal random_pareto random_permutation random_poisson random_rayleigh random_regular_graph random_student_t random_tournament random_tree random_weibull range rank rat ratcoef ratdenom ratdiff ratdisrep ratexpand ratinterpol rational rationalize ratnumer ratnump ratp ratsimp ratsubst ratvars ratweight read read_array read_binary_array read_binary_list read_binary_matrix readbyte readchar read_hashed_array readline read_list read_matrix read_nested_list readonly read_xpm real_imagpart_to_conjugate realpart realroots rearray rectangle rectform rectform_log_if_constant recttopolar rediff reduce_consts reduce_order region region_boundaries region_boundaries_plus rem remainder remarray rembox remcomps remcon remcoord remfun remfunction remlet remove remove_constvalue remove_dimensions remove_edge remove_fundamental_dimensions remove_fundamental_units remove_plot_option remove_vertex rempart remrule remsym remvalue rename rename_file reset reset_displays residue resolvante resolvante_alternee1 resolvante_bipartite resolvante_diedrale resolvante_klein resolvante_klein3 resolvante_produit_sym resolvante_unitaire resolvante_vierer rest resultant return reveal reverse revert revert2 rgb2level rhs ricci riemann rinvariant risch rk rmdir rncombine romberg room rootscontract round row rowop rowswap rreduce run_testsuite %s save saving scalarp scaled_bessel_i scaled_bessel_i0 scaled_bessel_i1 scalefactors scanmap scatterplot scatterplot_description scene schur2comp sconcat scopy scsimp scurvature sdowncase sec sech second sequal sequalignore set_alt_display setdifference set_draw_defaults set_edge_weight setelmx setequalp setify setp set_partitions set_plot_option set_prompt set_random_state set_tex_environment set_tex_environment_default setunits setup_autoload set_up_dot_simplifications set_vertex_label seventh sexplode sf sha1sum sha256sum shortest_path shortest_weighted_path show showcomps showratvars sierpinskiale sierpinskimap sign signum similaritytransform simp_inequality simplify_sum simplode simpmetderiv simtran sin sinh sinsert sinvertcase sixth skewness skewness_bernoulli skewness_beta skewness_binomial skewness_chi2 skewness_continuous_uniform skewness_discrete_uniform skewness_exp skewness_f skewness_gamma skewness_general_finite_discrete skewness_geometric skewness_gumbel skewness_hypergeometric skewness_laplace skewness_logistic skewness_lognormal skewness_negative_binomial skewness_noncentral_chi2 skewness_noncentral_student_t skewness_normal skewness_pareto skewness_poisson skewness_rayleigh skewness_student_t skewness_weibull slength smake small_rhombicosidodecahedron_graph small_rhombicuboctahedron_graph smax smin smismatch snowmap snub_cube_graph snub_dodecahedron_graph solve solve_rec solve_rec_rat some somrac sort sparse6_decode sparse6_encode sparse6_export sparse6_import specint spherical spherical_bessel_j spherical_bessel_y spherical_hankel1 spherical_hankel2 spherical_harmonic spherical_to_xyz splice split sposition sprint sqfr sqrt sqrtdenest sremove sremovefirst sreverse ssearch ssort sstatus ssubst ssubstfirst staircase standardize standardize_inverse_trig starplot starplot_description status std std1 std_bernoulli std_beta std_binomial std_chi2 std_continuous_uniform std_discrete_uniform std_exp std_f std_gamma std_general_finite_discrete std_geometric std_gumbel std_hypergeometric std_laplace std_logistic std_lognormal std_negative_binomial std_noncentral_chi2 std_noncentral_student_t std_normal std_pareto std_poisson std_rayleigh std_student_t std_weibull stemplot stirling stirling1 stirling2 strim striml strimr string stringout stringp strong_components struve_h struve_l sublis sublist sublist_indices submatrix subsample subset subsetp subst substinpart subst_parallel substpart substring subvar subvarp sum sumcontract summand_to_rec supcase supcontext symbolp symmdifference symmetricp system take_channel take_inference tan tanh taylor taylorinfo taylorp taylor_simplifier taytorat tcl_output tcontract tellrat tellsimp tellsimpafter tentex tenth test_mean test_means_difference test_normality test_proportion test_proportions_difference test_rank_sum test_sign test_signed_rank test_variance test_variance_ratio tex tex1 tex_display texput %th third throw time timedate timer timer_info tldefint tlimit todd_coxeter toeplitz tokens to_lisp topological_sort to_poly to_poly_solve totaldisrep totalfourier totient tpartpol trace tracematrix trace_options transform_sample translate translate_file transpose treefale tree_reduce treillis treinat triangle triangularize trigexpand trigrat trigreduce trigsimp trunc truncate truncated_cube_graph truncated_dodecahedron_graph truncated_icosahedron_graph truncated_tetrahedron_graph tr_warnings_get tube tutte_graph ueivects uforget ultraspherical underlying_graph undiff union unique uniteigenvectors unitp units unit_step unitvector unorder unsum untellrat untimer untrace uppercasep uricci uriemann uvect vandermonde_matrix var var1 var_bernoulli var_beta var_binomial var_chi2 var_continuous_uniform var_discrete_uniform var_exp var_f var_gamma var_general_finite_discrete var_geometric var_gumbel var_hypergeometric var_laplace var_logistic var_lognormal var_negative_binomial var_noncentral_chi2 var_noncentral_student_t var_normal var_pareto var_poisson var_rayleigh var_student_t var_weibull vector vectorpotential vectorsimp verbify vers vertex_coloring vertex_connectivity vertex_degree vertex_distance vertex_eccentricity vertex_in_degree vertex_out_degree vertices vertices_to_cycle vertices_to_path %w weyl wheel_graph wiener_index wigner_3j wigner_6j wigner_9j with_stdout write_binary_data writebyte write_data writefile wronskian xreduce xthru %y Zeilberger zeroequiv zerofor zeromatrix zeromatrixp zeta zgeev zheev zlange zn_add_table zn_carmichael_lambda zn_characteristic_factors zn_determinant zn_factor_generators zn_invert_by_lu zn_log zn_mult_table absboxchar activecontexts adapt_depth additive adim aform algebraic algepsilon algexact aliases allbut all_dotsimp_denoms allocation allsym alphabetic animation antisymmetric arrays askexp assume_pos assume_pos_pred assumescalar asymbol atomgrad atrig1 axes axis_3d axis_bottom axis_left axis_right axis_top azimuth background background_color backsubst berlefact bernstein_explicit besselexpand beta_args_sum_to_integer beta_expand bftorat bftrunc bindtest border boundaries_array box boxchar breakup %c capping cauchysum cbrange cbtics center cflength cframe_flag cnonmet_flag color color_bar color_bar_tics colorbox columns commutative complex cone context contexts contour contour_levels cosnpiflag ctaypov ctaypt ctayswitch ctayvar ct_coords ctorsion_flag ctrgsimp cube current_let_rule_package cylinder data_file_name debugmode decreasing default_let_rule_package delay dependencies derivabbrev derivsubst detout diagmetric diff dim dimensions dispflag display2d|10 display_format_internal distribute_over doallmxops domain domxexpt domxmxops domxnctimes dontfactor doscmxops doscmxplus dot0nscsimp dot0simp dot1simp dotassoc dotconstrules dotdistrib dotexptsimp dotident dotscrules draw_graph_program draw_realpart edge_color edge_coloring edge_partition edge_type edge_width %edispflag elevation %emode endphi endtheta engineering_format_floats enhanced3d %enumer epsilon_lp erfflag erf_representation errormsg error_size error_syms error_type %e_to_numlog eval even evenfun evflag evfun ev_point expandwrt_denom expintexpand expintrep expon expop exptdispflag exptisolate exptsubst facexpand facsum_combine factlim factorflag factorial_expand factors_only fb feature features file_name file_output_append file_search_demo file_search_lisp file_search_maxima|10 file_search_tests file_search_usage file_type_lisp file_type_maxima|10 fill_color fill_density filled_func fixed_vertices flipflag float2bf font font_size fortindent fortspaces fpprec fpprintprec functions gamma_expand gammalim gdet genindex gensumnum GGFCFMAX GGFINFINITY globalsolve gnuplot_command gnuplot_curve_styles gnuplot_curve_titles gnuplot_default_term_command gnuplot_dumb_term_command gnuplot_file_args gnuplot_file_name gnuplot_out_file gnuplot_pdf_term_command gnuplot_pm3d gnuplot_png_term_command gnuplot_postamble gnuplot_preamble gnuplot_ps_term_command gnuplot_svg_term_command gnuplot_term gnuplot_view_args Gosper_in_Zeilberger gradefs grid grid2d grind halfangles head_angle head_both head_length head_type height hypergeometric_representation %iargs ibase icc1 icc2 icounter idummyx ieqnprint ifb ifc1 ifc2 ifg ifgi ifr iframe_bracket_form ifri igeowedge_flag ikt1 ikt2 imaginary inchar increasing infeval infinity inflag infolists inm inmc1 inmc2 intanalysis integer integervalued integrate_use_rootsof integration_constant integration_constant_counter interpolate_color intfaclim ip_grid ip_grid_in irrational isolate_wrt_times iterations itr julia_parameter %k1 %k2 keepfloat key key_pos kinvariant kt label label_alignment label_orientation labels lassociative lbfgs_ncorrections lbfgs_nfeval_max leftjust legend letrat let_rule_packages lfg lg lhospitallim limsubst linear linear_solver linechar linel|10 linenum line_type linewidth line_width linsolve_params linsolvewarn lispdisp listarith listconstvars listdummyvars lmxchar load_pathname loadprint logabs logarc logcb logconcoeffp logexpand lognegint logsimp logx logx_secondary logy logy_secondary logz lriem m1pbranch macroexpansion macros mainvar manual_demo maperror mapprint matrix_element_add matrix_element_mult 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simplified_output simplify_products simpproduct simpsum sinnpiflag solvedecomposes solveexplicit solvefactors solvenullwarn solveradcan solvetrigwarn space sparse sphere spring_embedding_depth sqrtdispflag stardisp startphi starttheta stats_numer stringdisp structures style sublis_apply_lambda subnumsimp sumexpand sumsplitfact surface surface_hide svg_file symmetric tab taylordepth taylor_logexpand taylor_order_coefficients taylor_truncate_polynomials tensorkill terminal testsuite_files thetaresolution timer_devalue title tlimswitch tr track transcompile transform transform_xy translate_fast_arrays transparent transrun tr_array_as_ref tr_bound_function_applyp tr_file_tty_messagesp tr_float_can_branch_complex tr_function_call_default trigexpandplus trigexpandtimes triginverses trigsign trivial_solutions tr_numer tr_optimize_max_loop tr_semicompile tr_state_vars tr_warn_bad_function_calls tr_warn_fexpr tr_warn_meval tr_warn_mode tr_warn_undeclared tr_warn_undefined_variable tstep ttyoff 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attackEnabled backpack backpackCargo backpackContainer backpackItems backpackMagazines backpackSpaceFor behaviour benchmark binocular blufor boundingBox boundingBoxReal boundingCenter breakOut breakTo briefingName buildingExit buildingPos buttonAction buttonSetAction cadetMode call callExtension camCommand camCommit camCommitPrepared camCommitted camConstuctionSetParams camCreate camDestroy cameraEffect cameraEffectEnableHUD cameraInterest cameraOn cameraView campaignConfigFile camPreload camPreloaded camPrepareBank camPrepareDir camPrepareDive camPrepareFocus camPrepareFov camPrepareFovRange camPreparePos camPrepareRelPos camPrepareTarget camSetBank camSetDir camSetDive camSetFocus camSetFov camSetFovRange camSetPos camSetRelPos camSetTarget camTarget camUseNVG canAdd canAddItemToBackpack canAddItemToUniform canAddItemToVest cancelSimpleTaskDestination canFire canMove canSlingLoad canStand canUnloadInCombat captive captiveNum cbChecked cbSetChecked ceil cheatsEnabled checkAIFeature civilian className clearAllItemsFromBackpack clearBackpackCargo clearBackpackCargoGlobal clearGroupIcons clearItemCargo clearItemCargoGlobal clearItemPool clearMagazineCargo clearMagazineCargoGlobal clearMagazinePool clearOverlay clearRadio clearWeaponCargo clearWeaponCargoGlobal clearWeaponPool closeDialog closeDisplay closeOverlay collapseObjectTree combatMode commandArtilleryFire commandChat commander commandFire commandFollow commandFSM commandGetOut commandingMenu commandMove commandRadio commandStop commandTarget commandWatch comment commitOverlay compile compileFinal completedFSM composeText configClasses configFile configHierarchy configName configProperties configSourceMod configSourceModList connectTerminalToUAV controlNull controlsGroupCtrl copyFromClipboard copyToClipboard copyWaypoints cos count countEnemy countFriendly countSide countType countUnknown createAgent createCenter createDialog createDiaryLink createDiaryRecord createDiarySubject createDisplay createGearDialog createGroup createGuardedPoint createLocation createMarker createMarkerLocal createMenu createMine createMissionDisplay createSimpleTask createSite createSoundSource createTask createTeam createTrigger createUnit createUnit array createVehicle createVehicle array createVehicleCrew createVehicleLocal crew ctrlActivate ctrlAddEventHandler ctrlAutoScrollDelay ctrlAutoScrollRewind ctrlAutoScrollSpeed ctrlChecked ctrlClassName ctrlCommit ctrlCommitted ctrlCreate ctrlDelete ctrlEnable ctrlEnabled ctrlFade ctrlHTMLLoaded ctrlIDC ctrlIDD ctrlMapAnimAdd ctrlMapAnimClear ctrlMapAnimCommit ctrlMapAnimDone ctrlMapCursor ctrlMapMouseOver ctrlMapScale ctrlMapScreenToWorld ctrlMapWorldToScreen ctrlModel ctrlModelDirAndUp ctrlModelScale ctrlParent ctrlPosition ctrlRemoveAllEventHandlers ctrlRemoveEventHandler ctrlScale ctrlSetActiveColor ctrlSetAutoScrollDelay ctrlSetAutoScrollRewind ctrlSetAutoScrollSpeed ctrlSetBackgroundColor ctrlSetChecked ctrlSetEventHandler ctrlSetFade ctrlSetFocus ctrlSetFont ctrlSetFontH1 ctrlSetFontH1B ctrlSetFontH2 ctrlSetFontH2B ctrlSetFontH3 ctrlSetFontH3B ctrlSetFontH4 ctrlSetFontH4B ctrlSetFontH5 ctrlSetFontH5B ctrlSetFontH6 ctrlSetFontH6B ctrlSetFontHeight ctrlSetFontHeightH1 ctrlSetFontHeightH2 ctrlSetFontHeightH3 ctrlSetFontHeightH4 ctrlSetFontHeightH5 ctrlSetFontHeightH6 ctrlSetFontP ctrlSetFontPB ctrlSetForegroundColor ctrlSetModel ctrlSetModelDirAndUp ctrlSetModelScale ctrlSetPosition ctrlSetScale ctrlSetStructuredText ctrlSetText ctrlSetTextColor ctrlSetTooltip ctrlSetTooltipColorBox ctrlSetTooltipColorShade ctrlSetTooltipColorText ctrlShow ctrlShown ctrlText ctrlTextHeight ctrlType ctrlVisible curatorAddons curatorCamera curatorCameraArea curatorCameraAreaCeiling curatorCoef curatorEditableObjects curatorEditingArea curatorEditingAreaType curatorMouseOver curatorPoints curatorRegisteredObjects curatorSelected curatorWaypointCost currentChannel currentCommand currentMagazine currentMagazineDetail currentMagazineDetailTurret currentMagazineTurret currentMuzzle currentNamespace currentTask currentTasks currentThrowable currentVisionMode currentWaypoint currentWeapon currentWeaponMode currentWeaponTurret currentZeroing cursorTarget customChat customRadio cutFadeOut cutObj cutRsc cutText damage date dateToNumber daytime deActivateKey debriefingText debugFSM debugLog deg deleteAt deleteCenter deleteCollection deleteEditorObject deleteGroup deleteIdentity deleteLocation deleteMarker deleteMarkerLocal deleteRange deleteResources deleteSite deleteStatus deleteTeam deleteVehicle deleteVehicleCrew deleteWaypoint detach detectedMines diag activeMissionFSMs diag activeSQFScripts diag activeSQSScripts diag captureFrame diag captureSlowFrame diag fps diag fpsMin diag frameNo diag log diag logSlowFrame diag tickTime dialog diarySubjectExists didJIP didJIPOwner difficulty difficultyEnabled difficultyEnabledRTD direction directSay disableAI disableCollisionWith disableConversation disableDebriefingStats disableSerialization disableTIEquipment disableUAVConnectability disableUserInput displayAddEventHandler displayCtrl displayNull displayRemoveAllEventHandlers displayRemoveEventHandler displaySetEventHandler dissolveTeam distance distance2D distanceSqr distributionRegion doArtilleryFire doFire doFollow doFSM doGetOut doMove doorPhase doStop doTarget doWatch drawArrow drawEllipse drawIcon drawIcon3D drawLine drawLine3D drawLink drawLocation drawRectangle driver drop east echo editObject editorSetEventHandler effectiveCommander emptyPositions enableAI enableAIFeature enableAttack enableCamShake enableCaustics enableCollisionWith enableCopilot enableDebriefingStats enableDiagLegend enableEndDialog enableEngineArtillery enableEnvironment enableFatigue enableGunLights enableIRLasers enableMimics enablePersonTurret enableRadio enableReload enableRopeAttach enableSatNormalOnDetail enableSaving enableSentences enableSimulation enableSimulationGlobal enableTeamSwitch enableUAVConnectability enableUAVWaypoints endLoadingScreen endMission engineOn enginesIsOnRTD enginesRpmRTD enginesTorqueRTD entities estimatedEndServerTime estimatedTimeLeft evalObjectArgument everyBackpack everyContainer exec execEditorScript execFSM execVM exp expectedDestination eyeDirection eyePos face faction fadeMusic fadeRadio fadeSound fadeSpeech failMission fillWeaponsFromPool find findCover findDisplay findEditorObject findEmptyPosition findEmptyPositionReady findNearestEnemy finishMissionInit finite fire fireAtTarget firstBackpack flag flagOwner fleeing floor flyInHeight fog fogForecast fogParams forceAddUniform forceEnd forceMap forceRespawn forceSpeed forceWalk forceWeaponFire forceWeatherChange forEachMember forEachMemberAgent forEachMemberTeam format formation formationDirection formationLeader formationMembers formationPosition formationTask formatText formLeader freeLook fromEditor fuel fullCrew gearSlotAmmoCount gearSlotData getAllHitPointsDamage getAmmoCargo getArray getArtilleryAmmo getArtilleryComputerSettings getArtilleryETA getAssignedCuratorLogic getAssignedCuratorUnit getBackpackCargo getBleedingRemaining getBurningValue getCargoIndex getCenterOfMass getClientState getConnectedUAV getDammage getDescription getDir getDirVisual getDLCs getEditorCamera getEditorMode getEditorObjectScope getElevationOffset getFatigue getFriend getFSMVariable getFuelCargo getGroupIcon getGroupIconParams getGroupIcons getHideFrom getHit getHitIndex getHitPointDamage getItemCargo getMagazineCargo getMarkerColor getMarkerPos getMarkerSize getMarkerType getMass getModelInfo getNumber getObjectArgument getObjectChildren getObjectDLC getObjectMaterials getObjectProxy getObjectTextures getObjectType getObjectViewDistance getOxygenRemaining getPersonUsedDLCs getPlayerChannel getPlayerUID getPos getPosASL getPosASLVisual getPosASLW getPosATL getPosATLVisual getPosVisual getPosWorld getRepairCargo getResolution getShadowDistance getSlingLoad getSpeed getSuppression getTerrainHeightASL getText getVariable getWeaponCargo getWPPos glanceAt globalChat globalRadio goggles goto group groupChat groupFromNetId groupIconSelectable groupIconsVisible groupId groupOwner groupRadio groupSelectedUnits groupSelectUnit grpNull gunner gusts halt handgunItems handgunMagazine handgunWeapon handsHit hasInterface hasWeapon hcAllGroups hcGroupParams hcLeader hcRemoveAllGroups hcRemoveGroup hcSelected hcSelectGroup hcSetGroup hcShowBar hcShownBar headgear hideBody hideObject hideObjectGlobal hint hintC hintCadet hintSilent hmd hostMission htmlLoad HUDMovementLevels humidity image importAllGroups importance in incapacitatedState independent inflame inflamed inGameUISetEventHandler inheritsFrom initAmbientLife inputAction inRangeOfArtillery insertEditorObject intersect isAbleToBreathe isAgent isArray isAutoHoverOn isAutonomous isAutotest isBleeding isBurning isClass isCollisionLightOn isCopilotEnabled isDedicated isDLCAvailable isEngineOn isEqualTo isFlashlightOn isFlatEmpty isForcedWalk 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lbSetValue lbSize lbSort lbSortByValue lbText lbValue leader leaderboardDeInit leaderboardGetRows leaderboardInit leaveVehicle libraryCredits libraryDisclaimers lifeState lightAttachObject lightDetachObject lightIsOn lightnings limitSpeed linearConversion lineBreak lineIntersects lineIntersectsObjs lineIntersectsSurfaces lineIntersectsWith linkItem list listObjects ln lnbAddArray lnbAddColumn lnbAddRow lnbClear lnbColor lnbCurSelRow lnbData lnbDeleteColumn lnbDeleteRow lnbGetColumnsPosition lnbPicture lnbSetColor lnbSetColumnsPos lnbSetCurSelRow lnbSetData lnbSetPicture lnbSetText lnbSetValue lnbSize lnbText lnbValue load loadAbs loadBackpack loadFile loadGame loadIdentity loadMagazine loadOverlay loadStatus loadUniform loadVest local localize locationNull locationPosition lock lockCameraTo lockCargo lockDriver locked lockedCargo lockedDriver lockedTurret lockTurret lockWP log logEntities lookAt lookAtPos magazineCargo magazines magazinesAllTurrets magazinesAmmo magazinesAmmoCargo magazinesAmmoFull magazinesDetail magazinesDetailBackpack magazinesDetailUniform magazinesDetailVest magazinesTurret magazineTurretAmmo mapAnimAdd mapAnimClear mapAnimCommit mapAnimDone mapCenterOnCamera mapGridPosition markAsFinishedOnSteam markerAlpha markerBrush markerColor markerDir markerPos markerShape markerSize markerText markerType max members min mineActive mineDetectedBy missionConfigFile missionName missionNamespace missionStart mod modelToWorld modelToWorldVisual moonIntensity morale move moveInAny moveInCargo moveInCommander moveInDriver moveInGunner moveInTurret moveObjectToEnd moveOut moveTime moveTo moveToCompleted moveToFailed musicVolume name name location nameSound nearEntities nearestBuilding nearestLocation nearestLocations nearestLocationWithDubbing nearestObject nearestObjects nearObjects nearObjectsReady nearRoads nearSupplies nearTargets needReload netId netObjNull newOverlay nextMenuItemIndex nextWeatherChange nMenuItems not numberToDate objectCurators objectFromNetId objectParent objNull objStatus onBriefingGroup onBriefingNotes onBriefingPlan onBriefingTeamSwitch onCommandModeChanged onDoubleClick onEachFrame onGroupIconClick onGroupIconOverEnter onGroupIconOverLeave onHCGroupSelectionChanged onMapSingleClick onPlayerConnected onPlayerDisconnected onPreloadFinished onPreloadStarted onShowNewObject onTeamSwitch openCuratorInterface openMap openYoutubeVideo opfor or orderGetIn overcast overcastForecast owner param params parseNumber parseText parsingNamespace particlesQuality pi pickWeaponPool pitch playableSlotsNumber playableUnits playAction playActionNow player playerRespawnTime playerSide playersNumber playGesture playMission playMove playMoveNow playMusic playScriptedMission playSound playSound3D position positionCameraToWorld posScreenToWorld posWorldToScreen ppEffectAdjust ppEffectCommit ppEffectCommitted ppEffectCreate ppEffectDestroy ppEffectEnable ppEffectForceInNVG precision preloadCamera preloadObject preloadSound preloadTitleObj preloadTitleRsc preprocessFile preprocessFileLineNumbers primaryWeapon primaryWeaponItems primaryWeaponMagazine priority private processDiaryLink productVersion profileName profileNamespace profileNameSteam progressLoadingScreen progressPosition progressSetPosition publicVariable publicVariableClient publicVariableServer pushBack putWeaponPool queryItemsPool queryMagazinePool queryWeaponPool rad radioChannelAdd radioChannelCreate radioChannelRemove radioChannelSetCallSign radioChannelSetLabel radioVolume rain rainbow random rank rankId rating rectangular registeredTasks registerTask reload reloadEnabled remoteControl remoteExec remoteExecCall removeAction removeAllActions removeAllAssignedItems removeAllContainers removeAllCuratorAddons removeAllCuratorCameraAreas removeAllCuratorEditingAreas removeAllEventHandlers removeAllHandgunItems removeAllItems removeAllItemsWithMagazines removeAllMissionEventHandlers removeAllMPEventHandlers removeAllMusicEventHandlers removeAllPrimaryWeaponItems removeAllWeapons removeBackpack removeBackpackGlobal removeCuratorAddons removeCuratorCameraArea removeCuratorEditableObjects removeCuratorEditingArea removeDrawIcon removeDrawLinks removeEventHandler removeFromRemainsCollector removeGoggles removeGroupIcon removeHandgunItem removeHeadgear removeItem removeItemFromBackpack removeItemFromUniform removeItemFromVest removeItems removeMagazine removeMagazineGlobal removeMagazines removeMagazinesTurret removeMagazineTurret removeMenuItem removeMissionEventHandler removeMPEventHandler removeMusicEventHandler removePrimaryWeaponItem removeSecondaryWeaponItem removeSimpleTask removeSwitchableUnit removeTeamMember removeUniform removeVest removeWeapon removeWeaponGlobal removeWeaponTurret requiredVersion resetCamShake resetSubgroupDirection resistance resize resources respawnVehicle restartEditorCamera reveal revealMine reverse reversedMouseY roadsConnectedTo roleDescription ropeAttachedObjects ropeAttachedTo ropeAttachEnabled ropeAttachTo ropeCreate ropeCut ropeEndPosition ropeLength ropes ropeUnwind ropeUnwound rotorsForcesRTD rotorsRpmRTD round runInitScript safeZoneH safeZoneW safeZoneWAbs safeZoneX safeZoneXAbs safeZoneY saveGame saveIdentity saveJoysticks saveOverlay saveProfileNamespace saveStatus saveVar savingEnabled say say2D say3D scopeName score scoreSide screenToWorld scriptDone scriptName scriptNull scudState secondaryWeapon secondaryWeaponItems secondaryWeaponMagazine select selectBestPlaces selectDiarySubject selectedEditorObjects selectEditorObject selectionPosition selectLeader selectNoPlayer selectPlayer selectWeapon selectWeaponTurret sendAUMessage sendSimpleCommand sendTask sendTaskResult sendUDPMessage serverCommand serverCommandAvailable serverCommandExecutable serverName serverTime set setAccTime setAirportSide setAmmo setAmmoCargo setAperture setApertureNew setArmoryPoints setAttributes setAutonomous setBehaviour setBleedingRemaining setCameraInterest setCamShakeDefParams setCamShakeParams setCamUseTi setCaptive setCenterOfMass setCollisionLight setCombatMode setCompassOscillation setCuratorCameraAreaCeiling setCuratorCoef setCuratorEditingAreaType setCuratorWaypointCost setCurrentChannel setCurrentTask setCurrentWaypoint setDamage setDammage setDate setDebriefingText setDefaultCamera setDestination setDetailMapBlendPars setDir setDirection setDrawIcon setDropInterval setEditorMode setEditorObjectScope setEffectCondition setFace setFaceAnimation setFatigue setFlagOwner setFlagSide setFlagTexture setFog setFog array setFormation setFormationTask setFormDir setFriend setFromEditor setFSMVariable setFuel setFuelCargo setGroupIcon setGroupIconParams setGroupIconsSelectable setGroupIconsVisible setGroupId setGroupIdGlobal setGroupOwner setGusts setHideBehind setHit setHitIndex setHitPointDamage setHorizonParallaxCoef setHUDMovementLevels setIdentity setImportance setLeader setLightAmbient setLightAttenuation setLightBrightness setLightColor setLightDayLight setLightFlareMaxDistance setLightFlareSize setLightIntensity setLightnings setLightUseFlare setLocalWindParams setMagazineTurretAmmo setMarkerAlpha setMarkerAlphaLocal setMarkerBrush setMarkerBrushLocal setMarkerColor setMarkerColorLocal setMarkerDir setMarkerDirLocal setMarkerPos setMarkerPosLocal setMarkerShape setMarkerShapeLocal setMarkerSize setMarkerSizeLocal setMarkerText setMarkerTextLocal setMarkerType setMarkerTypeLocal setMass setMimic setMousePosition setMusicEffect setMusicEventHandler setName setNameSound setObjectArguments setObjectMaterial setObjectProxy setObjectTexture setObjectTextureGlobal setObjectViewDistance setOvercast setOwner setOxygenRemaining setParticleCircle setParticleClass setParticleFire setParticleParams setParticleRandom setPilotLight setPiPEffect setPitch setPlayable setPlayerRespawnTime setPos setPosASL setPosASL2 setPosASLW setPosATL setPosition setPosWorld setRadioMsg setRain setRainbow setRandomLip setRank setRectangular setRepairCargo setShadowDistance setSide setSimpleTaskDescription setSimpleTaskDestination setSimpleTaskTarget setSimulWeatherLayers setSize setSkill setSkill array setSlingLoad setSoundEffect setSpeaker setSpeech setSpeedMode setStatValue setSuppression setSystemOfUnits setTargetAge setTaskResult setTaskState setTerrainGrid setText setTimeMultiplier setTitleEffect setTriggerActivation setTriggerArea setTriggerStatements setTriggerText setTriggerTimeout setTriggerType setType setUnconscious setUnitAbility setUnitPos setUnitPosWeak setUnitRank setUnitRecoilCoefficient setUnloadInCombat setUserActionText setVariable setVectorDir setVectorDirAndUp setVectorUp setVehicleAmmo setVehicleAmmoDef setVehicleArmor setVehicleId setVehicleLock setVehiclePosition setVehicleTiPars setVehicleVarName setVelocity setVelocityTransformation setViewDistance setVisibleIfTreeCollapsed setWaves setWaypointBehaviour setWaypointCombatMode setWaypointCompletionRadius setWaypointDescription setWaypointFormation setWaypointHousePosition setWaypointLoiterRadius setWaypointLoiterType setWaypointName setWaypointPosition setWaypointScript setWaypointSpeed setWaypointStatements setWaypointTimeout setWaypointType setWaypointVisible setWeaponReloadingTime setWind setWindDir setWindForce setWindStr setWPPos show3DIcons showChat showCinemaBorder showCommandingMenu showCompass showCuratorCompass showGPS showHUD showLegend showMap shownArtilleryComputer shownChat shownCompass shownCuratorCompass showNewEditorObject shownGPS shownHUD shownMap shownPad shownRadio shownUAVFeed shownWarrant shownWatch showPad showRadio showSubtitles showUAVFeed showWarrant showWatch showWaypoint side sideChat sideEnemy sideFriendly sideLogic sideRadio sideUnknown simpleTasks simulationEnabled simulCloudDensity simulCloudOcclusion simulInClouds simulWeatherSync sin size sizeOf skill skillFinal skipTime sleep sliderPosition sliderRange sliderSetPosition sliderSetRange sliderSetSpeed sliderSpeed slingLoadAssistantShown soldierMagazines someAmmo sort soundVolume spawn speaker speed speedMode splitString sqrt squadParams stance startLoadingScreen step stop stopped str sunOrMoon supportInfo suppressFor surfaceIsWater surfaceNormal surfaceType swimInDepth switchableUnits switchAction switchCamera switchGesture switchLight switchMove synchronizedObjects synchronizedTriggers synchronizedWaypoints synchronizeObjectsAdd synchronizeObjectsRemove synchronizeTrigger synchronizeWaypoint synchronizeWaypoint trigger systemChat systemOfUnits tan targetKnowledge targetsAggregate targetsQuery taskChildren taskCompleted taskDescription taskDestination taskHint taskNull taskParent taskResult taskState teamMember teamMemberNull teamName teams teamSwitch teamSwitchEnabled teamType terminate terrainIntersect terrainIntersectASL text text location textLog textLogFormat tg time timeMultiplier titleCut titleFadeOut titleObj titleRsc titleText toArray toLower toString toUpper triggerActivated triggerActivation triggerArea triggerAttachedVehicle triggerAttachObject triggerAttachVehicle triggerStatements triggerText triggerTimeout triggerTimeoutCurrent triggerType turretLocal turretOwner turretUnit tvAdd tvClear tvCollapse tvCount tvCurSel tvData tvDelete tvExpand tvPicture tvSetCurSel tvSetData tvSetPicture tvSetPictureColor tvSetTooltip tvSetValue tvSort tvSortByValue tvText tvValue type typeName typeOf UAVControl uiNamespace uiSleep unassignCurator unassignItem unassignTeam unassignVehicle underwater uniform uniformContainer uniformItems uniformMagazines unitAddons unitBackpack unitPos unitReady unitRecoilCoefficient units unitsBelowHeight unlinkItem unlockAchievement unregisterTask updateDrawIcon updateMenuItem updateObjectTree useAudioTimeForMoves vectorAdd vectorCos vectorCrossProduct vectorDiff vectorDir vectorDirVisual vectorDistance vectorDistanceSqr vectorDotProduct vectorFromTo vectorMagnitude vectorMagnitudeSqr vectorMultiply vectorNormalized vectorUp vectorUpVisual vehicle vehicleChat vehicleRadio vehicles vehicleVarName velocity velocityModelSpace verifySignature vest vestContainer vestItems vestMagazines viewDistance visibleCompass visibleGPS visibleMap visiblePosition visiblePositionASL visibleWatch waitUntil waves waypointAttachedObject waypointAttachedVehicle waypointAttachObject waypointAttachVehicle waypointBehaviour waypointCombatMode waypointCompletionRadius waypointDescription waypointFormation waypointHousePosition waypointLoiterRadius waypointLoiterType waypointName waypointPosition waypoints waypointScript waypointsEnabledUAV waypointShow waypointSpeed waypointStatements waypointTimeout waypointTimeoutCurrent waypointType waypointVisible weaponAccessories weaponCargo weaponDirection weaponLowered weapons weaponsItems weaponsItemsCargo weaponState weaponsTurret weightRTD west WFSideText wind windDir windStr wingsForcesRTD worldName worldSize worldToModel worldToModelVisual worldToScreen _forEachIndex _this _x",
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decompose decrement decrypt deduplicate def defa defau defaul default defaults deferred defi defin define degrees delayed delegate delete delete_all delimited demand dense_rank depth dequeue des_decrypt des_encrypt des_key_file desc descr descri describ describe descriptor deterministic diagnostics difference dimension direct_load directory disable disable_all disallow disassociate discardfile disconnect diskgroup distinct distinctrow distribute distributed div do document domain dotnet double downgrade drop dumpfile duplicate duration each edition editionable editions element ellipsis else elsif elt empty enable enable_all enclosed encode encoding encrypt end end-exec endian enforced engine engines enqueue enterprise entityescaping eomonth error errors escaped evalname evaluate event eventdata events except exception exceptions exchange exclude excluding execu execut execute exempt exists exit exp expire explain export export_set extended extent external external_1 external_2 externally extract failed failed_login_attempts failover failure far fast feature_set feature_value fetch field fields file file_name_convert filesystem_like_logging final finish first first_value fixed flash_cache flashback floor flush following follows for forall force form forma format found found_rows freelist freelists freepools fresh from from_base64 from_days ftp full function general generated get get_format get_lock getdate getutcdate global global_name globally go goto grant grants greatest group group_concat group_id grouping grouping_id groups gtid_subtract guarantee guard handler hash hashkeys having hea head headi headin heading heap help hex hierarchy high high_priority hosts hour http id ident_current ident_incr ident_seed identified identity idle_time if ifnull ignore iif ilike ilm immediate import in include including increment index indexes indexing indextype indicator indices inet6_aton inet6_ntoa inet_aton inet_ntoa infile initial initialized initially initrans 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merge microsecond mid migration min minextents minimum mining minus minute minvalue missing mod mode model modification modify module monitoring month months mount move movement multiset mutex name name_const names nan national native natural nav nchar nclob nested never new newline next nextval no no_write_to_binlog noarchivelog noaudit nobadfile nocheck nocompress nocopy nocycle nodelay nodiscardfile noentityescaping noguarantee nokeep nologfile nomapping nomaxvalue nominimize nominvalue nomonitoring none noneditionable nonschema noorder nopr nopro noprom nopromp noprompt norely noresetlogs noreverse normal norowdependencies noschemacheck noswitch not nothing notice notrim novalidate now nowait nth_value nullif nulls num numb numbe nvarchar nvarchar2 object ocicoll ocidate ocidatetime ociduration ociinterval ociloblocator ocinumber ociref ocirefcursor ocirowid ocistring ocitype oct octet_length of off offline offset oid oidindex old on online only opaque open operations operator 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publishingservername purge quarter query quick quiesce quota quotename radians raise rand range rank raw read reads readsize rebuild record records recover recovery recursive recycle redo reduced ref reference referenced references referencing refresh regexp_like register regr_avgx regr_avgy regr_count regr_intercept regr_r2 regr_slope regr_sxx regr_sxy reject rekey relational relative relaylog release release_lock relies_on relocate rely rem remainder rename repair repeat replace replicate replication required reset resetlogs resize resource respect restore restricted result result_cache resumable resume retention return returning returns reuse reverse revoke right rlike role roles rollback rolling rollup round row row_count rowdependencies rowid rownum rows rtrim rules safe salt sample save savepoint sb1 sb2 sb4 scan schema schemacheck scn scope scroll sdo_georaster sdo_topo_geometry search sec_to_time second section securefile security seed segment select self sequence sequential serializable server servererror session session_user sessions_per_user set sets settings sha sha1 sha2 share shared shared_pool short show shrink shutdown si_averagecolor si_colorhistogram si_featurelist si_positionalcolor si_stillimage si_texture siblings sid sign sin size size_t sizes skip slave sleep smalldatetimefromparts smallfile snapshot some soname sort soundex source space sparse spfile split sql sql_big_result sql_buffer_result sql_cache sql_calc_found_rows sql_small_result sql_variant_property sqlcode sqldata sqlerror sqlname sqlstate sqrt square standalone standby start starting startup statement static statistics stats_binomial_test stats_crosstab stats_ks_test stats_mode stats_mw_test stats_one_way_anova stats_t_test_ stats_t_test_indep stats_t_test_one stats_t_test_paired stats_wsr_test status std stddev stddev_pop stddev_samp stdev stop storage store stored str str_to_date straight_join strcmp strict string struct stuff style subdate subpartition subpartitions substitutable substr substring subtime subtring_index subtype success sum suspend switch switchoffset switchover sync synchronous synonym sys sys_xmlagg sysasm sysaux sysdate sysdatetimeoffset sysdba sysoper system system_user sysutcdatetime table tables tablespace tan tdo template temporary terminated tertiary_weights test than then thread through tier ties time time_format time_zone timediff timefromparts timeout timestamp timestampadd timestampdiff timezone_abbr timezone_minute timezone_region to to_base64 to_date to_days to_seconds todatetimeoffset trace tracking transaction transactional translate translation treat trigger trigger_nestlevel triggers trim truncate try_cast try_convert try_parse type ub1 ub2 ub4 ucase unarchived unbounded uncompress under undo unhex unicode uniform uninstall union unique unix_timestamp unknown unlimited unlock unpivot unrecoverable unsafe unsigned until untrusted unusable unused update updated upgrade upped upper upsert url urowid usable usage use use_stored_outlines user user_data user_resources users using utc_date utc_timestamp uuid uuid_short validate validate_password_strength validation valist value values var var_samp varcharc vari varia variab variabl variable variables variance varp varraw varrawc varray verify version versions view virtual visible void wait wallet warning warnings week weekday weekofyear wellformed when whene whenev wheneve whenever where while whitespace with within without work wrapped xdb xml xmlagg xmlattributes xmlcast xmlcolattval xmlelement xmlexists xmlforest xmlindex xmlnamespaces xmlpi xmlquery xmlroot xmlschema xmlserialize xmltable xmltype xor year year_to_month years yearweek",literal:"true false null",built_in:"array bigint binary bit blob boolean char character date dec decimal float int int8 integer interval number numeric real record serial serial8 smallint text varchar varying 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checksum chelp ci cii cl class classutil clear cli clis clist clo clog clog_lf clog_p clogi clogi_sw clogit clogit_lf clogit_p clogitp clogl_sw cloglog clonevar clslistarray cluster cluster_measures cluster_stop cluster_tree cluster_tree_8 clustermat cmdlog cnr cnre cnreg cnreg_p cnreg_sw cnsreg codebook collaps4 collapse colormult_nb colormult_nw compare compress conf confi confir confirm conren cons const constr constra constrai constrain constraint continue contract copy copyright copysource cor corc corr corr2data corr_anti corr_kmo corr_smc corre correl correla correlat correlate corrgram cou coun count cox cox_p cox_sw coxbase coxhaz coxvar cprplot cprplot_7 crc cret cretu cretur creturn cross cs cscript cscript_log csi ct ct_is ctset ctst_5 ctst_st cttost cumsp cumsp_7 cumul cusum cusum_7 cutil d|0 datasig datasign datasigna datasignat datasignatu datasignatur datasignature datetof db dbeta de dec deco decod decode deff des desc descr descri describ describe destring dfbeta dfgls dfuller di di_g dir dirstats dis discard disp disp_res disp_s displ displa display distinct do doe doed doedi doedit dotplot dotplot_7 dprobit drawnorm drop ds ds_util dstdize duplicates durbina dwstat dydx e|0 ed edi edit egen eivreg emdef en enc enco encod encode eq erase ereg ereg_lf ereg_p ereg_sw ereghet ereghet_glf ereghet_glf_sh ereghet_gp ereghet_ilf ereghet_ilf_sh ereghet_ip eret eretu eretur ereturn err erro error est est_cfexist est_cfname est_clickable est_expand est_hold est_table est_unhold est_unholdok estat estat_default estat_summ estat_vce_only esti estimates etodow etof etomdy ex exi exit expand expandcl fac fact facto factor factor_estat factor_p factor_pca_rotated factor_rotate factormat fcast fcast_compute fcast_graph fdades fdadesc fdadescr fdadescri fdadescrib fdadescribe fdasav fdasave fdause fh_st file open file read file close file filefilter fillin find_hlp_file findfile findit findit_7 fit fl fli flis flist for5_0 form forma format fpredict frac_154 frac_adj frac_chk frac_cox frac_ddp frac_dis frac_dv frac_in frac_mun frac_pp frac_pq frac_pv frac_wgt frac_xo fracgen fracplot fracplot_7 fracpoly fracpred fron_ex fron_hn fron_p fron_tn fron_tn2 frontier ftodate ftoe ftomdy ftowdate g|0 gamhet_glf gamhet_gp gamhet_ilf gamhet_ip gamma gamma_d2 gamma_p gamma_sw gammahet gdi_hexagon gdi_spokes ge gen gene gener genera generat generate genrank genstd genvmean gettoken gl gladder gladder_7 glim_l01 glim_l02 glim_l03 glim_l04 glim_l05 glim_l06 glim_l07 glim_l08 glim_l09 glim_l10 glim_l11 glim_l12 glim_lf glim_mu glim_nw1 glim_nw2 glim_nw3 glim_p glim_v1 glim_v2 glim_v3 glim_v4 glim_v5 glim_v6 glim_v7 glm glm_6 glm_p glm_sw glmpred glo glob globa global glogit glogit_8 glogit_p gmeans gnbre_lf gnbreg gnbreg_5 gnbreg_p gomp_lf gompe_sw gomper_p gompertz gompertzhet gomphet_glf gomphet_glf_sh gomphet_gp gomphet_ilf gomphet_ilf_sh gomphet_ip gphdot gphpen gphprint gprefs gprobi_p gprobit gprobit_8 gr gr7 gr_copy gr_current gr_db gr_describe gr_dir gr_draw gr_draw_replay gr_drop gr_edit gr_editviewopts gr_example gr_example2 gr_export gr_print gr_qscheme gr_query gr_read gr_rename gr_replay gr_save gr_set gr_setscheme gr_table gr_undo gr_use graph graph7 grebar greigen greigen_7 greigen_8 grmeanby grmeanby_7 gs_fileinfo gs_filetype gs_graphinfo gs_stat gsort gwood h|0 hadimvo hareg hausman haver he heck_d2 heckma_p heckman heckp_lf heckpr_p heckprob hel help hereg hetpr_lf hetpr_p hetprob hettest hexdump hilite hist hist_7 histogram hlogit hlu hmeans hotel hotelling hprobit hreg hsearch icd9 icd9_ff icd9p iis impute imtest inbase include inf infi infil infile infix inp inpu input ins insheet insp inspe inspec inspect integ inten intreg intreg_7 intreg_p intrg2_ll intrg_ll intrg_ll2 ipolate iqreg ir irf irf_create irfm iri is_svy is_svysum isid istdize ivprob_1_lf ivprob_lf ivprobit ivprobit_p ivreg ivreg_footnote ivtob_1_lf ivtob_lf ivtobit ivtobit_p jackknife jacknife jknife jknife_6 jknife_8 jkstat joinby kalarma1 kap kap_3 kapmeier kappa kapwgt kdensity kdensity_7 keep ksm ksmirnov ktau kwallis l|0 la lab labe label labelbook ladder levels levelsof leverage lfit lfit_p li lincom line linktest lis list lloghet_glf lloghet_glf_sh lloghet_gp lloghet_ilf lloghet_ilf_sh lloghet_ip llogi_sw llogis_p llogist llogistic llogistichet lnorm_lf lnorm_sw lnorma_p lnormal lnormalhet lnormhet_glf lnormhet_glf_sh lnormhet_gp lnormhet_ilf lnormhet_ilf_sh lnormhet_ip lnskew0 loadingplot loc loca local log logi logis_lf logistic logistic_p logit logit_estat logit_p loglogs logrank loneway lookfor lookup lowess lowess_7 lpredict lrecomp lroc lroc_7 lrtest ls lsens lsens_7 lsens_x lstat ltable ltable_7 ltriang lv lvr2plot lvr2plot_7 m|0 ma mac macr macro makecns man manova manova_estat manova_p manovatest mantel mark markin markout marksample mat mat_capp mat_order mat_put_rr mat_rapp mata mata_clear mata_describe mata_drop mata_matdescribe mata_matsave mata_matuse mata_memory mata_mlib mata_mosave mata_rename mata_which matalabel matcproc matlist matname matr matri matrix matrix_input__dlg matstrik mcc mcci md0_ md1_ md1debug_ md2_ md2debug_ mds mds_estat mds_p mdsconfig mdslong mdsmat mdsshepard mdytoe mdytof me_derd mean means median memory memsize meqparse mer merg merge mfp mfx mhelp mhodds minbound mixed_ll mixed_ll_reparm mkassert mkdir mkmat mkspline ml ml_5 ml_adjs ml_bhhhs ml_c_d ml_check ml_clear ml_cnt ml_debug ml_defd ml_e0 ml_e0_bfgs ml_e0_cycle ml_e0_dfp ml_e0i ml_e1 ml_e1_bfgs ml_e1_bhhh ml_e1_cycle ml_e1_dfp ml_e2 ml_e2_cycle ml_ebfg0 ml_ebfr0 ml_ebfr1 ml_ebh0q ml_ebhh0 ml_ebhr0 ml_ebr0i ml_ecr0i ml_edfp0 ml_edfr0 ml_edfr1 ml_edr0i ml_eds ml_eer0i ml_egr0i ml_elf ml_elf_bfgs ml_elf_bhhh ml_elf_cycle ml_elf_dfp ml_elfi ml_elfs ml_enr0i ml_enrr0 ml_erdu0 ml_erdu0_bfgs ml_erdu0_bhhh ml_erdu0_bhhhq ml_erdu0_cycle ml_erdu0_dfp ml_erdu0_nrbfgs ml_exde ml_footnote ml_geqnr ml_grad0 ml_graph ml_hbhhh ml_hd0 ml_hold ml_init ml_inv ml_log ml_max ml_mlout ml_mlout_8 ml_model ml_nb0 ml_opt ml_p ml_plot ml_query ml_rdgrd ml_repor ml_s_e ml_score ml_searc ml_technique ml_unhold mleval mlf_ mlmatbysum mlmatsum mlog mlogi mlogit mlogit_footnote mlogit_p mlopts mlsum mlvecsum mnl0_ mor more mov move mprobit mprobit_lf mprobit_p mrdu0_ mrdu1_ mvdecode mvencode mvreg mvreg_estat n|0 nbreg nbreg_al nbreg_lf nbreg_p nbreg_sw nestreg net newey newey_7 newey_p news nl nl_7 nl_9 nl_9_p nl_p nl_p_7 nlcom nlcom_p nlexp2 nlexp2_7 nlexp2a nlexp2a_7 nlexp3 nlexp3_7 nlgom3 nlgom3_7 nlgom4 nlgom4_7 nlinit nllog3 nllog3_7 nllog4 nllog4_7 nlog_rd nlogit nlogit_p nlogitgen nlogittree nlpred no nobreak noi nois noisi noisil noisily note notes notes_dlg nptrend numlabel numlist odbc old_ver olo olog ologi ologi_sw ologit ologit_p ologitp on one onew onewa oneway op_colnm op_comp op_diff op_inv op_str opr opro oprob oprob_sw oprobi oprobi_p oprobit oprobitp opts_exclusive order orthog orthpoly ou out outf outfi outfil outfile outs outsh outshe outshee outsheet ovtest pac pac_7 palette parse parse_dissim pause pca pca_8 pca_display pca_estat pca_p pca_rotate pcamat pchart pchart_7 pchi pchi_7 pcorr pctile pentium pergram pergram_7 permute permute_8 personal peto_st pkcollapse pkcross pkequiv pkexamine pkexamine_7 pkshape pksumm pksumm_7 pl plo plot plugin pnorm pnorm_7 poisgof poiss_lf poiss_sw poisso_p poisson poisson_estat post postclose postfile postutil pperron pr prais prais_e prais_e2 prais_p predict predictnl preserve print pro prob probi probit probit_estat probit_p proc_time procoverlay procrustes procrustes_estat procrustes_p profiler prog progr progra program prop proportion prtest prtesti pwcorr pwd q\\s qby qbys qchi qchi_7 qladder qladder_7 qnorm qnorm_7 qqplot qqplot_7 qreg qreg_c qreg_p qreg_sw qu quadchk quantile quantile_7 que quer query range ranksum ratio rchart rchart_7 rcof recast reclink recode reg reg3 reg3_p regdw regr regre regre_p2 regres regres_p regress regress_estat regriv_p remap ren rena renam rename renpfix repeat replace report reshape restore ret retu retur return rm rmdir robvar roccomp roccomp_7 roccomp_8 rocf_lf rocfit rocfit_8 rocgold rocplot rocplot_7 roctab roctab_7 rolling rologit rologit_p rot rota rotat rotate rotatemat rreg rreg_p ru run runtest rvfplot rvfplot_7 rvpplot rvpplot_7 sa safesum sample sampsi sav save savedresults saveold sc sca scal scala scalar scatter scm_mine sco scob_lf scob_p scobi_sw scobit scor score scoreplot scoreplot_help scree screeplot screeplot_help sdtest sdtesti se search separate seperate serrbar serrbar_7 serset set set_defaults sfrancia sh she shel shell shewhart shewhart_7 signestimationsample signrank signtest simul simul_7 simulate simulate_8 sktest sleep slogit slogit_d2 slogit_p smooth snapspan so sor sort spearman spikeplot spikeplot_7 spikeplt spline_x split sqreg sqreg_p sret sretu sretur sreturn ssc st st_ct st_hc st_hcd st_hcd_sh st_is st_issys st_note st_promo st_set st_show st_smpl st_subid stack statsby statsby_8 stbase stci stci_7 stcox stcox_estat stcox_fr stcox_fr_ll stcox_p stcox_sw stcoxkm stcoxkm_7 stcstat stcurv stcurve stcurve_7 stdes stem stepwise stereg stfill stgen stir stjoin stmc stmh stphplot stphplot_7 stphtest stphtest_7 stptime strate strate_7 streg streg_sw streset sts sts_7 stset stsplit stsum sttocc sttoct stvary stweib su suest suest_8 sum summ summa summar summari summariz summarize sunflower sureg survcurv survsum svar svar_p svmat svy svy_disp svy_dreg svy_est svy_est_7 svy_estat svy_get svy_gnbreg_p svy_head svy_header svy_heckman_p svy_heckprob_p svy_intreg_p svy_ivreg_p svy_logistic_p svy_logit_p svy_mlogit_p svy_nbreg_p svy_ologit_p svy_oprobit_p svy_poisson_p svy_probit_p svy_regress_p svy_sub svy_sub_7 svy_x svy_x_7 svy_x_p svydes svydes_8 svygen svygnbreg svyheckman svyheckprob svyintreg svyintreg_7 svyintrg svyivreg svylc svylog_p svylogit svymarkout svymarkout_8 svymean svymlog svymlogit svynbreg svyolog svyologit svyoprob svyoprobit svyopts svypois svypois_7 svypoisson svyprobit svyprobt svyprop 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+
+ <script>
+
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+ 8: function() { RevealChalkboard.reset() }, // reset chalkboard data on current slide when 'BACKSPACE' is pressed
+ 68: function() { RevealChalkboard.download() }, // downlad recorded chalkboard drawing when 'd' is pressed
+ },
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+
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+ dependencies: [
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+ { src: '$revealjs-url$/plugin/zoom-js/zoom.js', async: true },
+$endif$
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+ { src: '$revealjs-url$/plugin/chalkboard/chalkboard.js', async: true },
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+$endif$
+ ]
+ });
+ </script>
+$if(mathjax-url)$
+ <!-- dynamically load mathjax for compatibility with self-contained -->
+ <script>
+ (function () {
+ var script = document.createElement("script");
+ script.type = "text/javascript";
+ script.src = "$mathjax-url$";
+ document.getElementsByTagName("head")[0].appendChild(script);
+ })();
+ </script>
+$endif$
+
+<script>
+ (function() {
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+ Reveal.addEventListener( 'slidechanged', function(event) {
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+ window.jQuery(event.currentSlide).trigger('shown');
+ });
+ }
+ })();
+</script>
+
+$for(include-after)$
+$include-after$
+$endfor$
+
+ </body>
+</html>
--- /dev/null
+ls()
+weight
+weight
+lablr
+labelr
+nrow(labelr)
+names(labelr)
+names(labelr$data)
+labelr$data
+labelr
+names(labelr)
+labelr$labelr
+labelr$toxic
+setwd("..")
+q()
+n
+summary(w2)
+summary(w2)
+q()
+n
+summary(fit1)
+0.5*(dat$x1 + sapply(dat$sdx, function(sd) rnorm(1,0,sd)))
+summary(fit1)
+summary(fit2)
+summary(fit2)
+conditional_effects(fit2,resp='y')
+plot(conditional_effects(fit2,resp='y'))
+stancode(fit2)
+stancode(fit1)
+sessionInfo()
+q()
+y
+p.y
+range(p.y)
+rbinom
+df2
+df2
+df2
+brms.corrected.logit
+q()
+n
+summary(brms.corrected.logit)
+summary(brms.corrected.logit)
+p.y
+q()
+n
+mw
+summary(mw)
+)
+summary(true.model)
+true.model
+true.model$R
+true.model$null.deviance
+true.model$deviance
+getwd()
+setwd("../../)
+setwd("../../)
+
+setwd("../../partitioning_reddit")
+ls
+getwd()
+list.files()
+install.packages("filelock")
+q()
+n
--- /dev/null
+### EXAMPLE 2: demonstrates how measurement error can lead to a type sign error in a covariate
+### Even when you have a good predictor, if it's biased against a covariate you can get the wrong sign.
+### Even when you include the proxy variable in the regression.
+### But with some ground truth and multiple imputation, you can fix it.
+
+library(argparser)
+library(mecor)
+library(ggplot2)
+library(data.table)
+library(filelock)
+library(arrow)
+library(Amelia)
+library(Zelig)
+
+options(amelia.parallel="multicore",
+ amelia.ncpus=40)
+
+## SETUP:
+### we want to estimate g -> y and x -> y; g is observed, x is MAR
+### we have k -> x; g -> x; g->k; k is used to predict x via the model w.
+### we have k -> w; x -> w; w is observed.
+### for illustration, g is binary (e.g., gender==male).
+### A realistic scenario is that we have an NLP model predicting something like "racial harassment" in social media comments
+### Whether a comment is "racial harassment" depends on context, like the kind of person (i.e.,) the race of the person making the comment
+### e.g., a Black person saying "n-word" is less likely to be racial harassement than if a white person does it.
+### Say we have a language model that predicts "racial harassment," but it doesn't know the race of the writer.
+### Our content analyzers can see signals of the writer's race (e.g., a profile or avatar). So our "ground truth" takes this into accont.
+### Our goal is to predict an outcome (say that someone gets banned from the platform) as a function of whether they made a racial harassing comment and of their race.
+
+### simulation:
+#### how much power do we get from the model in the first place? (sweeping N and m)
+####
+logistic <- function(x) {1/(1+exp(-1*x))}
+
+simulate_latent_cocause <- function(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed){
+ set.seed(seed)
+
+ ## the true value of x
+
+ g <- rbinom(N, 1, 0.5)
+ k <- rnorm(N, 0, 1)
+ xprime <- Bkx*k + Bgx * g
+ x <- rbinom(N, 1, logistic(xprime - mean(xprime)))
+ w.model <- glm(x ~ k,family='binomial')
+ w <- as.integer(predict(w.model,data.frame(k=k),type='response') > 0.5)
+ ## y = B0 + B1x + e
+ y <- Bxy * x + Bgy * g + rnorm(N, 0, 1) + B0
+ df <- data.table(x=x,k=k,y=y,w=w,g=g)
+ if( m < N){
+ df <- df[sample(nrow(df), m), x.obs := x]
+ } else {
+ df <- df[, x.obs := x]
+ }
+
+ return(df)
+}
+
+
+run_simulation <- function(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed){
+ result <- list()
+ df <- simulate_latent_cocause(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed)
+
+ result <- append(result, list(N=N,
+ m=m,
+ B0=B0,
+ Bxy=Bxy,
+ Bgy=Bgy,
+ Bkx=Bkx,
+ seed=seed))
+
+ accuracy <- df[,.(mean(w==x))]$V1
+ result <- append(result, list(accuracy=accuracy))
+
+ model.true <- lm(y ~ x + g, data=df)
+ true.ci.Bxy <- confint(model.true)['x',]
+ true.ci.Bgy <- confint(model.true)['g',]
+
+ result <- append(result, list(Bxy.est.true=coef(model.true)['x'],
+ Bgy.est.true=coef(model.true)['g'],
+ Bxy.ci.upper.true = true.ci.Bxy[2],
+ Bxy.ci.lower.true = true.ci.Bxy[1],
+ Bgy.ci.upper.true = true.ci.Bgy[2],
+ Bgy.ci.lower.true = true.ci.Bgy[1]))
+
+
+ model.naive <- lm(y~w+g, data=df)
+
+ naive.ci.Bxy <- confint(model.naive)['w',]
+ naive.ci.Bgy <- confint(model.naive)['g',]
+
+ result <- append(result, list(Bxy.est.naive=coef(model.naive)['w'],
+ Bgy.est.naive=coef(model.naive)['g'],
+ Bxy.ci.upper.naive = naive.ci.Bxy[2],
+ Bxy.ci.lower.naive = naive.ci.Bxy[1],
+ Bgy.ci.upper.naive = naive.ci.Bgy[2],
+ Bgy.ci.lower.naive = naive.ci.Bgy[1]))
+
+
+ ## multiple imputation when k is observed
+ amelia.out.k <- amelia(df, m=200, p2s=0, idvars=c('x'),noms=c("x.obs","w","g"))
+ mod.amelia.k <- zelig(y~x.obs+g, model='ls', data=amelia.out.k$imputations, cite=FALSE)
+ coefse <- combine_coef_se(mod.amelia.k, messages=FALSE)
+
+ est.x.mi <- coefse['x.obs','Estimate']
+ est.x.se <- coefse['x.obs','Std.Error']
+ result <- append(result,
+ list(Bxy.est.amelia.full = est.x.mi,
+ Bxy.ci.upper.amelia.full = est.x.mi + 1.96 * est.x.se,
+ Bxy.ci.lower.amelia.full = est.x.mi - 1.96 * est.x.se
+ ))
+
+ est.g.mi <- coefse['g','Estimate']
+ est.g.se <- coefse['g','Std.Error']
+
+ result <- append(result,
+ list(Bgy.est.amelia.full = est.g.mi,
+ Bgy.ci.upper.amelia.full = est.g.mi + 1.96 * est.g.se,
+ Bgy.ci.lower.amelia.full = est.g.mi - 1.96 * est.g.se
+ ))
+
+ ## What if we can't observe k -- most realistic scenario. We can't include all the ML features in a model.
+ amelia.out.nok <- amelia(df, m=200, p2s=0, idvars=c("x","k"), noms=c("x.obs","w",'g'))
+ mod.amelia.nok <- zelig(y~x.obs+g, model='ls', data=amelia.out.nok$imputations, cite=FALSE)
+ coefse <- combine_coef_se(mod.amelia.nok, messages=FALSE)
+
+ est.x.mi <- coefse['x.obs','Estimate']
+ est.x.se <- coefse['x.obs','Std.Error']
+ result <- append(result,
+ list(Bxy.est.amelia.nok = est.x.mi,
+ Bxy.ci.upper.amelia.nok = est.x.mi + 1.96 * est.x.se,
+ Bxy.ci.lower.amelia.nok = est.x.mi - 1.96 * est.x.se
+ ))
+
+ est.g.mi <- coefse['g','Estimate']
+ est.g.se <- coefse['g','Std.Error']
+
+ result <- append(result,
+ list(Bgy.est.amelia.nok = est.g.mi,
+ Bgy.ci.upper.amelia.nok = est.g.mi + 1.96 * est.g.se,
+ Bgy.ci.lower.amelia.nok = est.g.mi - 1.96 * est.g.se
+ ))
+
+ return(result)
+}
+
+Ns <- c(100, 200, 300, 400, 500, 1000, 2500, 5000, 7500)
+ms <- c(30, 50, 100, 200, 300, 500)
+B0 <- 0
+Bxy <- 1
+Bgy <- 0.3
+Bkx <- 3
+Bgx <- -4
+seeds <- 1:100
+
+rows <- list()
+
+for(N in Ns){
+ print(N)
+ for(m in ms){
+ if(N>m){
+ for(seed in seeds){
+ rows <- append(rows, list(run_simulation(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed)))
+ }
+ }
+ }
+}
+
+result <- rbindlist(rows)
+write_feather(result, "example_2_simulation.feather")
--- /dev/null
+** Next simulations to run
+- Focus on the BKH *multiple imputation* approach --- We don't actually need overimputation when we have some gold standard data.
+- The idea of this paper is basically to say "Supervised learning fits well into this framework. So use it!"
+- Important findings will involve:
+ - How well does this work when the learner is biased?
+ Probably works great as long as you can account for the bias in imputation + regression
+ - How well does this work when features are unavailable?
+ - if the
+ - How well does this work when featues are available?
+ - How well does this work when measurement error is correlated with the independent and dependent variable?
+ - How well does this work when there's a very large number of missing values?
+
+
+
+** How well does supervised ML measurement fit into BHK?
+*** Unbiased proxy assumption: If data is missing but there's a proxy variable that's unbiased we can use m=1 techniques instead of m=2 techniques.
+
+M=1 techniques are simpler / more powerful, but maybe m=2 techniques are more robust given we can't assume supervised learners are unbiased.
+
+That said, if we have access to the features, then we can use the predictions and the features as our proxy variable. Since any bias in w will be correlated with the features, including the features in the likelihood will reduce the bias.
+
+*** IMMA assumption: distribution of the mismeasurement indicator, m,is the same no matter the value of the missing data.
+
+This isn't a problem if ground truth is randomly sampled.
+It is a problem if ground truth is based on a stratified sample.
+
+*** Measurement error distribution assumption: The distribution of the proxy variable (conditional on missing and observed data and its distributional parameters) known up to its parameters. The parameters are either known or a consistent estimator is available.
+
+This isn't a problem if we have ground truth because we can use the ground truth to estimate the parameters.
+
+If we don't have ground truth, we'll have to guess.
projects / ml_measurement_error_public.git / commitdiff
