FROM N00B TO PRO
Jack Davis
Andrew Henrey
PURPOSE
Create a simulator from scratch that:
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Generates data from a variety of distributions
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Makes a response variable from a known function of the data (plus an error term)
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Constructs a linear model that estimates the coefficients of the function
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Repeats generation and modeling many times to compare the average estimates of the
linear model to the known parameters.
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Package the whole thing nicely into a function that we can call in a single line in later
work.
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If you’re experienced, the commands themselves may seem trivial
OUTLINE
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1) Learning how to learn
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2) Randomly Generating Data
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3) Data Frames and Manipulation
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4) Linear Models
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BREAK – Quality of presenter improves
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5) Running loops
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6) Function Definition
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7) More advanced function topics
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8) Using functions
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9) A short simulation study
LEARNING HOW TO LEARN – JACK DAVIS
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Google CRAN Packages to get the package list
• From here you can get a description of every command in a package.
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??<term> searches for commands related to <term>
• ??plot will find commands related to plot
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?<command> calls up the help file for that command
• ?abline gives the help file for the abline() command.
LEARNING HOW TO LEARN – JACK DAVIS
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Exercises:
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Name one function in the darts game package.
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What is the e-mail of the author of the Texas Holdem simulation package?
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(Bonus) Tell the author about your day via e-mail; s/he likes hearing from fans.
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Find a function to make a histogram
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Find some example code on the heatmap() command.
RANDOMLY GENERATING DATA – JACK DAVIS
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The r<dist> commands randomly generate data from a distribution
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rnorm( n , mean, sd)
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rexp( n, rate)
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rbinom( n, size, prob)
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rt( n, df) From Student’s T. (Mean is zero, so setting a mean is up to you)
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set.seed() Allows you to generate the same data every time, so you or others can
verify work.
Generates from normal distribution (default N(0,1))
RANDOMLY GENERATING DATA – JACK DAVIS
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Set a random seed
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Generate a vector of 50 values from the Normal (mean=10,sd=4) distribution, name
the vector x1.
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Do the same with
• Poisson ( lambda = 5), named x2,
• Exponential (rate = 1/7) named x3,
• Student’s t distribution (df =5), with a mean of 5, named x4,
• Normal (mean=0, sd=20), named err
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Make a new variable y, let it be 3 + 20x1 + 15x2 – 12x3 – 10x4 + err
DATA FRAMES – JACK DAVIS
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data.frame() makes a dataframe object of the vectors listed in the ()
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The advantage of having a data frame is that it can be treated as a single object..
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Data frames, models, and even matrix decompositions can be objects in R.
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You can call parts of objects by name using $
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model$coef or model$coefficient will bring up the estimated coefficients
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If no such aspect exists, then you’ll get a null response.
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Example: Andrew$height
DATA FRAMES – JACK DAVIS
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Exercises:
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Make a data.frame() of x1,x2,x3,x4, and y Name it dat
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(if you’re stuck from the last part, run “Q3-dataframethis.txt” first)
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Use index indicators like dat[4,3], dat [2:7,3], dat [4,], and dat [4,-1] to get
• The 3rd row, 5th entry of dat
• The 2nd – 7th values of the 5 th column
• The entire 3rd row
• The 3rd row without the 1st entry
LINEAR MODELS – JACK DAVIS
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The results of the lm() function are an object.
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Example: mod = lm(y ~ x1 + I(x2^2) + x1:x2, data=dat)
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Useful aspects
• mod$fitted
• mod$residuals
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Useful functions
• summary(mod)
• predict(mod, newdata)
LINEAR MODELS – JACK DAVIS
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Use the lm command to create a linear model of y as a function of x1,x2,x3, and x4
additively using dat data, name it mod. (No interactions or transformations)
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Get the summary of mod
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Display the estimated coefficients with no other values.
BREAK
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This slide unintentionally left 98% blank
OUTLINE
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BREAK – Quality of presenter improves deteriorates
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5) Running loops
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6) Function Definition
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7) More advanced function topics
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8) Using functions
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9) A short simulation study
RUNNING YOU FOR A LOOP – ANDREW HENREY
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Similar to other programming languages, loops in R allow you to repeat the same block of
code several times
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Unlike other programming languages, large loops in R are exceedingly slow
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Any loop of less than about 100,000 total iterations is not going to give you much trouble
in terms of time
RUNNING YOU FOR A LOOP – ANDREW HENREY
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An R loop that executes a million commands takes about a second. Conditions vary wildly
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Generating 100,000 data sets of size 50,000 and looping through the dataset to calculate
a mean for each one would take longer to run than Jack Davis heading up Burnaby
Mountain (ouch)
Sup d00dz late for tutorial 
Jack
RUNNING YOU FOR A LOOP – ANDREW HENREY
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Loop syntax:
for (i in 1:n)
{
#TellVicEverything
}
RUNNING YOU FOR A LOOP – ANDREW HENREY
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No need to run from 1:K
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Can use an arbitrary vector instead
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Runs for length(vect) iterations
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Takes on the i th value of the vector each iteration
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e.g.
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V = c(1,5,3,-6)
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for (count in V) {print(count);}
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## 1, 5 , 3 , -6
RUNNING YOU FOR A LOOP – ANDREW HENREY
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Exercises:
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A) Define a variable runs to be the number 10,000
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B) Define a matrix() called mat with 5 columns and runs rows (10,000 rows)
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C) Put a for() loop around the code found in q5-loopthis.txt . Loop from 1:runs.
Use index indicators like a[k,] to save the estimated coefficients of the model in a
new row of mat.
OR, if you think you are a total coding BOSS, then put the loop around your code in
parts 2-4 that generates data and finds the linear model estimates of the betas.
FUNCTION DEFINITION – ANDREW HENREY
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Functions are a slightly abstract concept
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Mathematics: f(x) = x 2+4x-16
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Computing: mean(x) = sum(x)/length(x) – All 3 are functions!
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Functions map INPUTS to OUTPUT
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Possibly no inputs
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In one way or another, always some form of output
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Example functions:
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SORT, MEDIAN, OPTIM / NLM, LM/GLM
FUNCTION DEFINITION – ANDREW HENREY
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Function syntax
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Simple function:
F = function()
{
return (5)
}
>> F()
5
FUNCTION DEFINITION – ANDREW HENREY
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Exercises:
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Make a function out of the code you wrote in part 5. The syntax should be similar to the
previous slide. The function:
• Should be called simulate.lm
• Should include everything needed to generate the data several times, find a linear
model, and extract the coefficients
• Does NOT take any inputs
• Should return the matrix of 10,000 runs of coefficients
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Use the function and save the results to a matrix called test
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If nothing is working (), you can use the example code in q6 – function this.txt
ADVANCED FUNCTIONS – ANDREW HENREY
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A more complicated example:
MSE = function(X=c(0,3,11),Y)
{
return (mean((X-Y)^2))
}
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Observe that X has default values
>>MSE(Y=c(4,5,6))
15
ADVANCED FUNCTIONS – ANDREW HENREY
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If an input argument to a function has default values, you don’t have to specify them when
calling the function
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If an input argument has no default values, running the function without specifying them
gives you an error
ADVANCED FUNCTIONS – ANDREW HENREY
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Exercises:
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Modify simulate.lm() by adding input parameters. Include:
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nruns, the number of runs in the simulation, with no default
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seed, the random initial seed of the simulation, defaulting to 1337
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verbose, a Boolean true/false to report progress, defaulting to FALSE (caps matter)
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Set runs to nruns at the beginning of your function
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Use set.seed(seed) in your function
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Add code that prints out how far along you are in the loop, but only when verbose is true
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Run this new function to overwrite the old one
USING FUNCTIONS – ANDREW HENREY
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Exercises:
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A) Run your simulate.lm() with 25000 runs. Store these results as a variable
called betas
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B) Use hist() on the first column of betas to see the sampling distribution of the
intercept
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C) Use summary(), mean() , and sd() on this column as well
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D) Use par(mfrow=c(2,2)) and then some hist() commands to display
histograms of the other four sampling distributions in a 2x2 grid
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E) Compare your results to the known values (The means of the sampling distributions to
the true values, and the standard deviations to the estimated values in Q4)
SIMULATION STUDY – ANDREW HENREY
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Idea: You have a binomial experiment with 9 successes and 3 failures.
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You would like to construct a 95% CI for the true proportion of successes
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You DON’T know whether the normal approximation is appropriate
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How can we find out whether or not it’s OK?
SIMULATION STUDY – ANDREW HENREY
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Overall procedure:
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Construct a LOT of samples from a population with 12 trials and p=0.75
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For each sample, calculate the 95% CI using the normal approximation
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For each sample, see whether the CI overlaps with 0.75
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Count the number of samples for which the CI overlaps with 0.75
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The proportion of the samples that have a CI that overlaps is called the “true coverage
probability”
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If the true coverage probability is close to 95% , the normal approximation to the sampling
distribution of p is a good one.
SIMULATION STUDY – ANDREW HENREY
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Steps:
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Generate 100,000 samples of binomial(12,0.75) data using rbinom()
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For each sample, calculate the usual estimate of p
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For each sample, calculate SE = sqrt(p*(1-p)/12)
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For each sample, calculate the lower and upper bounds of the 95% CI
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Find out how many intervals actually contain 0.75
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Optional: Look at hist(x) to gain intuition of why the normal approximation isn’t
perfect
THE END
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Leave plz tyvm