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+---
+title: "Week 5 R tutorial"
+subtitle: "Statistics and statistical programming  \nNorthwestern University  \nMTS 525"
+author: "Aaron Shaw"
+date: "October 13, 3030"
+output:
+  pdf_document:
+    toc: yes
+    toc_depth: '3'
+  html_document:
+    toc: yes
+    number_sections: true
+    toc_depth: 3
+    toc_float:
+      collapsed: false
+      smooth_scroll: true
+    theme: readable
+---
+
+```{r setup, include=FALSE}
+knitr::opts_chunk$set(echo = TRUE)
+```
+
+This week, we'll introduce some ways to do some the discrete math, generate distributions, and use a for-loop to run a small simulation.
+
+# Binomial and factorial functions
+
+In Chapter 3 (and in the previous problem set), you needed to calculate some binomial choice arithmetic and/or factorials. They weren't absolutely necessary for the problem set, but here are the corresponding functions in R.
+
+Let's say we want to calculate how many possible pairs you can draw from a population of ten individuals, a.k.a., $10 \choose 2$ or, instead you wanted to calculate $10!$
+```{r}
+choose(10,2)
+
+factorial(10)
+```
+Note that factorial arithmetic can get quite large quite fast, so consider your processor/memory constraints and options before you try to calculate something truly large like $365!$.
+
+# Distribution functions
+
+R has a number of built-in functions to help you work with distributions in various ways that also started to come up in *OpenIntro* Chapter 3. I will introduce a couple of points about them here, but I also highly recommend you look at the relevant section of the Verzani *Using R Introductory Statistics* book (pp 222-229) for more on this (and, honestly, for more on most of the topics we're covering in R).
+
+The key to using R to analyze distributions is that R has a set of built-in distributions (e.g. uniform, normal, binomial, log-normal, etc.) and a set of functions (`d`, `p`, `q`, and `r`) that can be run for each distribution. In the example that follows, I'll use a uniform distribuition (a distribution between any two values (*min*, *max*) where the values may occur with uniform probability) for my example below. Verzani has others for when you need them.
+
+The `d` function gets you information about the *density function* of the distribution. The `p` function works with the *cumulative probabilities*. The `q` function gets you *quantiles* from the distribution. The `r` function allows you to generate *random samples* from the distribution. As you can see, the letters corresponding to each function *almost* make sense...<*sigh*>. They also each take specific arguments that can vary a bit depending on which kind of distribution you are using them with (as always, the help documentation and the internet can be helpful here).
+
+Onwards to the example code, which uses the different functions to calculate information about a uniform distribution between 0 and 3 (take a moment to think about what that would look like in terms of raw data and/or a plot):
+
+```{r} 
+dunif(x=1, min=0, max=3) # What proportion of the area is the to the left of 1?
+
+punif(q=1, min=0, max=3) # Same as the prior example in this case.
+
+qunif(p=0.5, min=0, max=3) # 50th percentile
+
+runif(n=4, min=0, max=3) # Random values in [0,3]
+```
+Look at the Verzani text for additional examples, including several that can solve binomial probability calculations (e.g., if you flip a fair coin 100 times, what are the odds of observing heads 60 or more times?).
+
+# For-loops and a quick simulation
+
+Beyond proving invaluable for rapid calculations of solutions to problem set questions, the distribution functions are very, very useful for running simulations. We won't really spend a lot of time on simulations in class, but I'll give you an example here that can generalize to more complicated problems. I also use a programming technique we haven't talked about yet called a for-loop to help repeat the sampling process multiple times. For-loops feature prominently in some programming tasks/languages, but I encourage you to minimize your use of them in R for reasons that are sort of beyond the scope of the course. That said, it's still super important to learn how they work!
+
+For my simulation let's say that I want to repeatedly draw random samples from a distribution and examine the distribution of the resulting sample means (this example is going to feature prominently in Chapter 5 of *OpenIntro*). I'll start by generating a vector of 10,000 random data points drawn from a log-normal distribution where the mean and standard deviation of the log-transformed values are 0 and 1 respectively:
+
+```{r}
+d <- rlnorm(10000, meanlog=0, sdlog=1)
+
+head(d)
+mean(d)
+sd(d)
+hist(d)
+```
+
+That sure does look like a logarithmic distribution!
+
+Okay, now, I want to draw 500 samples of 100 observations from this population and take the mean of each sample. Time to write a function! Notice that I require two inputs into my function: the population data and the sample size.
+
+```{r}
+sample.mean <- function(pop, n){
+  s <- sample(pop, n)
+  return(mean(s))
+}
+
+## Run it once to see how it goes:
+sample.mean(d, 100)
+```
+Next step: let's run that 500 times. Here's where the for-loop comes in handy. In somewhat abstract terms, a for-loop allows me to define an index and then repeat an operation for some range of values that can assume a function of that index. The simplest example is to take a range of integers as my index (e.g., 1-10) and then I run my loop ten times (once for each integer). You can also do more sophisticated things such as using the index to substitute and/or subset the objects in your loop along the way.  
+
+We can break that down further through an example. The basic syntax of a for-loop is that you call some operation to occur over some index. Here's a simple example that illustrates how that works. The loop iterates through the integers between 1-10 and prints the square of each value:
+```{r}
+for(x in c(1:10)){
+  print(x^2)
+}
+```
+Notice that the initial statement defines a variable (`x` in this case) within the loop (that's a way you can think about the `for(x in...))` part of the first line of code. The part that comes after `in` defines the index (in this case, the integers between 1 and 10). The part between the curly braces defines what R will do for each iteration of the loop, in which it will substitute the corresponding value from the index for variable. 
+
+Now let's go back to my sample means example. Since I want to store the output of my sample means loop, I will first create an object `s.means` that is a numeric vector with one value (0) that will be replaced as I run through the loop.
+```{r}
+s.means <- 0
+```
+Onwards to the loop itself. In the block of code below, you'll notice that I once again declare an index over which to iterate. That's what happens inside that first set of parentheses where I have `i in c(1:30)`. That's telling R to go through the loop for each value from 1:30 and to assign the current index value to `i` during that iteration of the loop. Then, each time through the loop, the value of `i` advances through  the index (in this case, it just goes up by 1). The result is that each iteration will take the output of my `sample.mean` function and append it as the $i^{th}$ value of `s.means`. The `next` call at the end is optional, but can be important sometimes to help you keep track of what's going on.  
+
+```{r}
+for(i in c(1:500)){
+  s.means[i] <- sample.mean(d, 100)
+  next
+}
+```
+In case you're coming to R from other programming languages that index from 0, you should note that this example very much takes advantage of the fact that R indexes from 1 (i.e., the first value of some vector `v` can be returned by `v[1]`).
+
+Once we've run that loop the `s.means` variable now contains a distribution of sample means. What are the characteristics of the distribution? In other words, how would you summarize the distribution? Well, you already know how to do that.
+
+```{r}
+summary(s.means)
+```
+Let's plot it too:
+```{r}
+library(ggplot2)
+qplot(s.means, geom="density")
+```
+
+That looks pretty "normal." 
+
+Experiment with this example by changing the size of the sample and/or the number of samples we draw.
+
+Now, think back to the original vector `d.` Can you explain what fundamental statistical principle is illustrated in this example? Why do the values in `s.means` fluctuate so much? What is the relationship of `s.means` to `d`?
+
+# Quantile quantile plots
+
+Last, but not least, you might have admired the quantile-quantile plots (a.k.a. "Q-Q plots") presented in some of the examples in the most recent *OpenIntro* chapter. The usual idea with Q-Q- plots is that you want to see how the observed (empirical) quantiles of some data compare against the theoretical quantiles of a normal (or other) distribution. You too can create these plots! 
+
+Here's an example that visualizes the result of our simulation (labeled "sample") against a normal distribution with the same mean and standard deviation (labeled "theoretical"). Notice that to accommodate `ggplot2` I have to turn `s.means` into a data frame first.
+
+```{r}
+s.means <- data.frame(s.means)
+ggplot(s.means, aes(sample=s.means)) + geom_qq() + geom_qq_line(color="red")
+
+```
+
+And/or (finally) we could even standardize the values of `s.means` as z-scores using the `scale()` function:
+
+```{r}
+s.z <- data.frame(scale(s.means)); names(s.z) <- "z"
+ggplot(s.z, aes(sample=z)) + geom_qq() + geom_qq_line(color="red")
+```
+
