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