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  1. From n00b to Pro Jack Davis Andrew Henrey

  2. PURPOSE Create 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

  3. 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

  4. 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 • ?ablinegives the help file for the abline() command.

  5. 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.

  6. 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.

  7. 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

  8. 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$coefor model$coefficient will bring up the estimated coefficients • If no such aspect exists, then you’ll get a null response. • Example:Andrew$height

  9. 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

  10. 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)

  11. 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.

  12. break • This slide unintentionally left 98% blank

  13. 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

  14. 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

  15. 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

  16. RUNNING YOU FOR A LOOP – Andrew Henrey • Loop syntax: for (i in 1:n) { #TellVicEverything }

  17. 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

  18. 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.

  19. 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

  20. Function definition – Andrew Henrey • Function syntax • Simple function: F = function() { return (5) } >> F() 5

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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)

  26. 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?

  27. 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.

  28. 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

  29. THE END • Leave plztyvm