lreg - Linear Ridge Regression and Principal Component...

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Linear, Ridge Regression, and Principal Component Analysis Linear, Ridge Regression, and Principal Component Analysis Jia Li Department of Statistics The Pennsylvania State University Email: [email protected] http://www.stat.psu.edu/ jiali Jia Li http://www.stat.psu.edu/ jiali
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Linear, Ridge Regression, and Principal Component Analysis Introduction to Regression Input vector: X = ( X 1 , X 2 , ..., X p ). Output Y is real-valued. Predict Y from X by f ( X ) so that the expected loss function E ( L ( Y , f ( X ))) is minimized. Square loss: L ( Y , f ( X )) = ( Y - f ( X )) 2 . The optimal predictor f * ( X ) = argmin f ( X ) E ( Y - f ( X )) 2 = E ( Y | X ) . The function E ( Y | X ) is the regression function . Jia Li http://www.stat.psu.edu/ jiali
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Linear, Ridge Regression, and Principal Component Analysis Example The number of active physicians in a Standard Metropolitan Statistical Area (SMSA), denoted by Y , is expected to be related to total population ( X 1 , measured in thousands), land area ( X 2 , measured in square miles), and total personal income ( X 3 , measured in millions of dollars). Data are collected for 141 SMSAs, as shown in the following table. i : 1 2 3 ... 139 140 141 X 1 9387 7031 7017 ... 233 232 231 X 2 1348 4069 3719 ... 1011 813 654 X 3 72100 52737 54542 ... 1337 1589 1148 Y 25627 15389 13326 ... 264 371 140 Goal: Predict Y from X 1 , X 2 , and X 3 . Jia Li http://www.stat.psu.edu/ jiali
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Linear, Ridge Regression, and Principal Component Analysis Linear Methods The linear regression model f ( X ) = β 0 + p j =1 X j β j . What if the model is not true? It is a good approximation Because of the lack of training data/or smarter algorithms, it is the most we can extract robustly from the data. Comments on X j : Quantitative inputs Transformations of quantitative inputs, e.g., log( · ), ( · ). Basis expansions: X 2 = X 2 1 , X 3 = X 3 1 , X 3 = X 1 · X 2 . Jia Li http://www.stat.psu.edu/ jiali
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Linear, Ridge Regression, and Principal Component Analysis Jia Li http://www.stat.psu.edu/ jiali
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Linear, Ridge Regression, and Principal Component Analysis Estimation The issue of finding the regression function E ( Y | X ) is converted to estimating β j , j = 0 , 1 , ..., p . Training data: { ( x 1 , y 1 ) , ( x 2 , y 2 ) , ..., ( x N , y N ) } , where x i = ( x i 1 , x i 2 , ..., x ip ) . Denote β = ( β 0 , β 1 , ..., β p ) T . The loss function E ( Y - f ( X )) 2 is approximated by the empirical loss RSS ( β ) / N : RSS ( β ) = N i =1 ( y i - f ( x i )) 2 = N i =1 ( y i - β 0 - p j =1 x ij β j ) 2 . Jia Li http://www.stat.psu.edu/ jiali
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Linear, Ridge Regression, and Principal Component Analysis Notation The input matrix X of dimension N × ( p + 1): 1 x 1 , 1 x 1 , 2 ... x 1 , p 1 x 2 , 1 x 2 , 2 ... x 2 , p ... ... ... ... ... 1 x N , 1 x N , 2 ... x N , p Output vector y : y = y 1 y 2 ... y N Jia Li http://www.stat.psu.edu/ jiali
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Linear, Ridge Regression, and Principal Component Analysis The estimated β is ˆ β . The fitted values at the training inputs: ˆ y i = ˆ β 0 + p j =1 x ij ˆ β j and ˆ y = ˆ y 1 ˆ y 2 ... ˆ y N Jia Li http://www.stat.psu.edu/ jiali
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Linear, Ridge Regression, and Principal Component Analysis Point Estimate The least square estimation of ˆ β is ˆ β = ( X T X ) - 1 X T y The fitted value vector is ˆ y = X ˆ β = X ( X T X ) - 1 X T y Hat matrix: H = X ( X T X ) - 1 X T Jia Li http://www.stat.psu.edu/ jiali
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