jan26 - STA 414/2104 Administration Homework 1 on web page...

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STA 414/2104 Jan 26, 2010 Administration I Homework 1 on web page, due Feb 11 I NSERC summer undergraduate award applications due Feb 5 I Some helpful books 1 / 35
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STA 414/2104 Jan 26, 2010 ... administration 2 / 35
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STA 414/2104 Jan 26, 2010 ... administration I collection of tools for regression and classification I some old (least squares, discriminant analysis) I some new (lasso, support vector machines) I statistical justifications: loss, likelihood, mean squared error, classification error, posterior probabilities... I statistical thinking I framework for analysing new tools 3 / 35
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STA 414/2104 Jan 26, 2010 Linear regression plus I variable selection: forward, backward, stepwise, all possible subsets I comparing models: adjusted R 2 , C p vs. p , AIC 1 and variants, K -fold cross-validation I shrinkage methods: ridge regression, lasso, Least Angle Regression I tuning parameter (amount of shrinkage): validation data, K -fold cross-validation I derived variables: principal components regression, partial least squares 1 C p and AIC can be shown to be estimates of expected prediction error 4 / 35
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STA 414/2104 Jan 26, 2010 Predictions with smoothing regression I on training data: x ’s are centered and scaled when fitting Lasso, LAR, and ridge regression I ˆ β 0 is not included in shrinkage I (3.41) ˆ β ridge = argmin β X ( y i - β 0 - p X j = 1 x ij β j ) 2 + λ p X j = 1 β 2 j I (3.52) ˆ β lasso = argmin β X ( y i - β 0 - p X j = 1 x ij β j ) 2 + λ p X j = 1 | β j | I ˆ β 0 = ¯ y = ¯ y train I for predicting a new y 0 = ˆ β 0 + p j = 1 ( x 0 j - ¯ x j ) ˆ β j : use ˆ β 0 = ¯ y train , and ¯ x j means average for j th feature on the training data I see construction of tx and mm in Table33R.txt this is built into predict.lars 5 / 35
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STA 414/2104 Jan 26, 2010 ...predictions > options(digits=4) > as.vector(predict.lars(pr.lars,newx=test[,1:8],type="fit",s=0.36, mode="fraction")$fit) [1] 2.094 1.443 1.780 2.292 2.695 2.009 2.283 1.731 1.969 1.694 2.588 2.486 [13] 2.712 2.492 2.554 2.345 2.053 2.954 3.189 1.976 2.946 3.035 2.746 2.550 [25] 2.636 2.697 3.135 3.095 3.233 3.698 > as.vector(tx % * % coef(pr.lars,s=0.36,mode="fraction"))+mean(train$lpsa) [1] 2.094 1.443 1.780 2.292 2.695 2.009 2.283 1.731 1.969 1.694 2.588 2.486 [13] 2.712 2.492 2.554 2.345 2.053 2.954 3.189 1.976 2.946 3.035 2.746 2.550 [25] 2.636 2.697 3.135 3.095 3.233 3.698 6 / 35
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STA 414/2104 Jan 26, 2010 Derived features § 3.5 I replace x 1 , . . . x p with linear combinations of columns I principal components from SVD are natural candidates I X = UDV T , U T U = I N , V T V = I p I z m = Xv m , m = 1 , . . . , M < p I z m are orthogonal by construction I ˆ y pcr ( M ) = ¯ y 1 + M X m = 1 ˆ θ m z m I ˆ θ m = h z m , y i h z m , z m i = n i = 1 z mi y i n i = 1 z 2 mi I inputs should be scaled first (mean 0, variance 1) I Angle brackets notation explained at (3.25). Exercise: ¯ z m = 0?? 7 / 35
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STA 414/2104 Jan 26, 2010 ... derived features I closely related method Partial least squares I also constructs derived variables I widely used in chemometrics, where often p > N I see § 3.6 for discussion 8 / 35
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STA 414/2104 Jan 26, 2010 Linear methods for classification (Chapter 4) I inputs X = X 1 , . . . , X p (notation x used on p.101) I output Y takes values in one of K classes I output G is a group label: values 1 , . . . , K I response Y as needed ( Y G ), e.g. Y = 1 , 0 as G = blue , orange Fig 2.1; eq. (2.7) I data ( x i , g i ) , i = 1 , . . . N I goal to learn a model to predict the correct class for a future output, based on inputs code is in ElemStatLearn.pdf 10 / 35
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STA 414/2104 Jan 26, 2010 Linear methods I rule: G = 2
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