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