[線代] 線性回歸的一個證明
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
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