[線代] 線性回歸的一個證明

[left (881 forever)]

H = X(X^tX)^-1X^t
X   : an N by d+1 real matrix
X^t : the transpose of X
X^-1: the inverse of X
e    : an N by 1 error real matrix
w,w^*: an d+1 by 1 real matrix

Suppose the inverse of (X^tX) exists

Consider y = Xw^*+ e and y’= Xw

Show that
||y-y'||^2
=||(I-H)e||^2
=(N-d-1)||e||^2

ps: The following has been proven.
a. H^t=H
b. H^k = H for any positive integer k
c. (I-H)^k = (I-H) for any positive integer k
d. trace(H)=d+1

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※ 編輯: left (61.220.35.20), 08/29/2016 15:36:15
→ kerwinhui   : How are w and w^* related?
→ left        : w^* is given in advance, so we can get a noisy
→ left        : signal y. And, then we want to search a w to get
→ left        : a estimate of y through y'.