chap8 - Shrinkage Methods Revisited (Ch 9 of Faraway) 1...

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Unformatted text preview: Shrinkage Methods Revisited (Ch 9 of Faraway) 1 Shrinkage Methods Principal components regression Partial least squares Ridge regression 2 Principal Components Motivation: Reduce the dimensionality of the data Illustration-2-1 1 2-2-1 1 2 x1 x2 g replacements x 1 x 2 3 Definition 1. Center each variable by its mean, x j- x j X n p . 2. Find u 1 such that var ( Xu 1 ) is maximized subject to u T 1 u 1 = 1. 3. Find u 2 such that var ( Xu 2 ) is maximized subject to u T 2 u 2 = 1 and u T 2 u 1 = 0. 4. And so on. 4 Remarks z j = Xu j are projections of data points on u j . z j = Xu j are called the principal compo- nents . u j are the eigenvectors of X T X . var ( Xu j ) = j , the eigenvalues of X T X . Recommended: scale each variable by its standard deviation beforehand. 5 Principal Components Regression PCR replaces the regression y x with y z Typically only a few eigenvalues will be large so that most variation in X can be represented by a few principal components. Dimension re- duction . 6 Food Analyzer Example Response: fat content Predictors: 100 channel spectrum of ab- sorbances Number of data points: n = 215 Number of predictors: p = 100 7 Prediction Performance Goal : build a model that predicts well on future data. Divide the data into two groups: training sam- ples and testing samples . Build the models us- ing the training samples and evaluate them on the testing samples. 8 Food Analyzer Example Continued > library(faraway) > data(meatspec) > dim(meatspec) [1] 215 101 > ## Training data > tr <- meatspec[1:172,] > ## Test data > te <- meatspec[173:215,] ## Linear model > g1 <- lm(fat ~ ., tr) > ## R2 > summary(g1)$r.squared [1] 0.9970196 > ## Root mean squared error > rmse <- function(x, y) { + sqrt(mean( (x - y)^2 )) + } > rmse(g1$fit, tr$fat) [1] 0.6903167 > ## Prediction > rmse( predict(g1, new=te), te$fat ) [1] 3.814000 ## AIC > g2 <- step(g1) > rmse( g2$fit, tr$fat ) [1] 0.7095069 > rmse( predict(g2, te), te$fat ) 9 [1] 3.590245 > ## Principal components regression > library(mva) > meatpca <- prcomp(tr[,-101]) > ## Square root of the eigenvalues > round(meatpca$sdev, 3) [1] 5.055 0.511 0.282 0.168 0.038 0.025 0.014 [8] 0.011 0.005 0.003 0.002 0.002 0.001 0.001 [15] 0.001 0.000 0.000 0.000 0.000 0.000 0.000 [22] 0.000 0.000 0.000 ... ... > matplot(1:100, meatpca$rot[,1:3], type="l", xlab="Frequency", ylab="") > plot(1:10, meatpca$sdev[1:10], type="l", xlab="PC number", ylab="SD of PC") ## Choose PC number > ## Mean of each variable > mm <- apply( tr[,-101], 2, mean ) > ## Apply it to the test data > tex <- as.matrix( sweep(te[, -101], 2, mm) ) > rmsmeat <- NULL > for (i in 1:50) { + g3 <- lm(fat ~ meatpca$x[, 1:i], tr) + ## Compute the PC for the test data + nx <- tex %*% meatpca$rot[, 1:i] + ## Predicted values + pv <- cbind(1, nx) %*% g3$coef + rmsmeat[i] <- rmse( pv, te$fat ) + } > plot(rmsmeat, xlab="PC number", ylab="Test RMS") > which.min(rmsmeat) [1] 27 > min(rmsmeat) [1] 1.854858 Food Analyzer Example Continued 20 40 60 80 100-0.2-0.1 0.0 0.1 Frequency 10 Food Analyzer Example Continued 2 4 6 8 10 1 2 3 4 5 PC number SD of PC cements C number SD of PC 11 Food Analyzer Example Continued 10 20 30 40 50 2 4 6 8 10 12 PC number Test RMs cements C number T est RMS 12 Cross-Validation...
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chap8 - Shrinkage Methods Revisited (Ch 9 of Faraway) 1...

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