Thus far, we only considered regression with scalar-valued outcomes. In some applications, the outcome is itself...
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Thus far, we only considered regression with scalar-valued outcomes. In some applications, the outcome is itself a vector: y₁ ERK. We posit the relationship between the features and the vector-valued outcome is linear: yf xf B, ≈ for some matrix of regression coefficients BERPXK 1. The sum of squared residuals (SSR) here is X = n K 1 SSR(B) = (yi,k — xTbk.)², 2 i=1 k=1 where by ERP is the k-th column of B. Express SSR(B) in matrix notation (i.e. without using any explicit summations). Hint: work out how to express the SSR in terms of B, € RXP, and y= [1] ER"XK 2. Find a closed-form expression for the matrix of regression coefficients that minimizes the SSR; i.e. find a (closed-form) expression for Be arg min BERPXK SSR(B). 3. Instead of minimizing the SSR, we break up the problem into K separate regression problems with scalar-valued responses. That is, we fit K linear models of the form ~xTBk, Yi,k~ where yi,k is k-th outcome of the i-th sample (i.e. the k-th entry of y;) and 3 ERP are the regression coefficients of the k-th linear model. How are the fitted coefficients from the K separate regressions B₁,...,K related to the matrix of regression coefficients that minimizes the SSR B? Thus far, we only considered regression with scalar-valued outcomes. In some applications, the outcome is itself a vector: y₁ ERK. We posit the relationship between the features and the vector-valued outcome is linear: yf xf B, ≈ for some matrix of regression coefficients BERPXK 1. The sum of squared residuals (SSR) here is X = n K 1 SSR(B) = (yi,k — xTbk.)², 2 i=1 k=1 where by ERP is the k-th column of B. Express SSR(B) in matrix notation (i.e. without using any explicit summations). Hint: work out how to express the SSR in terms of B, € RXP, and y= [1] ER"XK 2. Find a closed-form expression for the matrix of regression coefficients that minimizes the SSR; i.e. find a (closed-form) expression for Be arg min BERPXK SSR(B). 3. Instead of minimizing the SSR, we break up the problem into K separate regression problems with scalar-valued responses. That is, we fit K linear models of the form ~xTBk, Yi,k~ where yi,k is k-th outcome of the i-th sample (i.e. the k-th entry of y;) and 3 ERP are the regression coefficients of the k-th linear model. How are the fitted coefficients from the K separate regressions B₁,...,K related to the matrix of regression coefficients that minimizes the SSR B?
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Related Book For
Applied Regression Analysis and Other Multivariable Methods
ISBN: 978-1285051086
5th edition
Authors: David G. Kleinbaum, Lawrence L. Kupper, Azhar Nizam, Eli S. Rosenberg
Posted Date:
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