Thus far, we only considered regression with scalar-valued responses. In some applications, the response is itself...

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Thus far, we only considered regression with scalar-valued responses. In some applications, the response is itself a vector: y = Rm1. We posit the relationship between the fea- tures/predictors (x; = Rdx1) and the vector-valued response y; is linear: = xB*+ error, for i = 1, n where B* Rdxm is a matrix of regression coefficients. Here note that for the linear regression model in class, the dimension of response variable y is m = 1. (a) Express the sum of squared residuals (also called residual sum of squares, RSS) in matrix notation (i.e. without using any summations). Similarly to the linear regression model, the RSS is defined as n RSS(B) = (y-x{B)(y{ x{B)T. i=1 Hint: work out how to express the RSS in terms of the data matrices X = = T T Rnxm Rnxd, Y = y n Also note that for a matrix A = (aij)nm with its ith row vector denoted by a;, we have tr(AAT) = a = inaj. 'n (b) Find the matrix of regression coefficients that minimizes the RSS. Thus far, we only considered regression with scalar-valued responses. In some applications, the response is itself a vector: y = Rm1. We posit the relationship between the fea- tures/predictors (x; = Rdx1) and the vector-valued response y; is linear: = xB*+ error, for i = 1, n where B* Rdxm is a matrix of regression coefficients. Here note that for the linear regression model in class, the dimension of response variable y is m = 1. (a) Express the sum of squared residuals (also called residual sum of squares, RSS) in matrix notation (i.e. without using any summations). Similarly to the linear regression model, the RSS is defined as n RSS(B) = (y-x{B)(y{ x{B)T. i=1 Hint: work out how to express the RSS in terms of the data matrices X = = T T Rnxm Rnxd, Y = y n Also note that for a matrix A = (aij)nm with its ith row vector denoted by a;, we have tr(AAT) = a = inaj. 'n (b) Find the matrix of regression coefficients that minimizes the RSS.