LECTURE 11: LINEAR MODEL SELECTION PT. 2. October 18, 2017 SDS 293: Machine Learning
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1 LECTURE 11: LINEAR MODEL SELECTION PT. 2 October 18, 2017 SDS 293: Machine Learning
2 Announcements 1/2 CS Internship Lunch Presentations Come hear where Computer Science majors interned in Summer 2017! Employers range from companies in the tech industry to research labs. All are welcome! Pizza lunch provided. Thursday, October 26th 12:10-1 pm Ford Hall 241 ***Extra credit opportunity*** Want to drop a missing lab? Attend and post to #talks!
3 Announcements 2/2 Presentation of the CS Major & Minors lunch Ford 240 FREE FOOD!
4 Correcting a fuzzy statement Low bias CV (n) = avg( MSE i ) No variance w.r.t. which test case has been left out variance of the estimated MSE may actually be higher than k-fold CV
5 Outline Model selection: alternatives to least-squares Subset selection - Best subset - Stepwise selection (forward and backward) - Estimating error using cross-validation Shrinkage methods - Ridge regression and the Lasso - Dimension reduction Labs for each part
6 Flashback: subset selection Big idea: if having too many predictors is the problem maybe we can get rid of some Three methods: - Best subset: try all possible combinations of predictors - Forward: start with no predictors, greedily add one at a time - Backward: start with all predictors, greedily remove one at a time greedy = Add/remove whichever predictor improves your model right now
7 Flashback: comparing methods Best Subset Selection Forward Selection Backward Selection How many models get compared? 2 p 1+ p ( p +1 ) 2 1+ p ( p +1 ) 2 Benefits? Provably optimal Inexpensive Inexpensive; doesn t ignore interaction Drawbacks? Exhaustive search is expensive Not guaranteed to be optimal; ignores interaction Not guaranteed to be optimal; breaks when p>n
8 Flashback: choosing the optimal model We know measures of training error (RSS and R 2 ) aren t good predictors of test error (what we actually care about) Two options: - We can indirectly estimate test error by making an adjustment to the training error to account for the bias: 2 R adj C p AIC BIC Pros: inexpensive to compute Cons: makes additional assumptions about the model - We can directly estimate the test error, using either a validation set approach or a cross-validation approach
9 Validation set: how would this work? Recipe for: Best Subset Selection First divide the data into training and test sets Preheat the null model M 0 with no predictors. From the kitchen of: Grandma SDS * on the training set 1. For k = 1, 2,, p: a. Fit all the models that contain exactly k predictors. b. Keep only the model with the smallest training error. Call it M k. 2. Estimate the error, and select a single best model from among M 0 M p ^ Calculate the error rate on the test set In my day, we just eye-balled the error as best we could
10 Discussion: potential problems? Only training on a subset of the data means our model is less accurate Recipe for: Best Subset Selection First divide the data into training and test sets Preheat the null model M 0 with no predictors. From the kitchen of: Grandma SDS * on the training set 1. For k = 1, 2,, p: a. Fit all the models that contain exactly k predictors. b. Keep only the model with the smallest training error. Call it M k. 2. Estimate the error, and select a single best model from among M 0 M p ^ Calculate the error rate on the test set Kids these days, wastin data all willy-nilly like it grows on trees!
11 Cross-validation: how would this work? Recipe for: Best Subset Selection Preheat the null model M 0 with no predictors. From the kitchen of: Grandma SDS 1. For k = 1, 2,, p: a. Fit all the models that contain exactly k predictors. b. Keep only the model with the smallest training error. Call it M k. 2. Estimate the error, and select a single best model from among M 0 M p ^ Use k-fold cross-validation to calculate the CV error Good grief, child! I m never going to make it to bingo!
12 Time to get our hands dirty
13 Lab: subset selection using validation To do today s lab in R: <nothing new> To do today s lab in python: <nothing new> Instructions and code for part 1: Full version can be found beginning on p. 248 of ISLR For part 2: - Apply these techniques to a dataset of your choice - You re welcome (encouraged?) to work in teams!
14 Coming up Reminder: A4 due tonight by 11:59pm Monday: shrinkage methods - ridge regression - the lasso
LECTURE 12: LINEAR MODEL SELECTION PT. 3. October 23, 2017 SDS 293: Machine Learning
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