Exercise 3.4(a)

squidsforlunch · #1 10-08-2014, 06:33 PM

For Exercise 3.4 (a) I did

$y = w*^{T}X+E$
$\hat{y} = Hy$
$\hat{y} = H(w*^{T}X+E)$
$\hat{y} = Hw*^{T}x+HE$
$\hat{y} = (X(X^{T}X)^{-1}X^{T})w*^{T}X+HE$
$\hat{y} = (XX^{-1}(X^{T})^{-1}X^{T})w*^{T}X+HE$
$\hat{y} = (I(X^{T})^{-1}X^{T})w*^{T}X+HE$
$\hat{y} = (I(I))w*^{T}X+HE$
$\hat{y} = w*^{T}X+HE$

but
$w*^{T}X$
is not equal to

What am I missing?

mddsangster · #2 10-09-2014, 10:53 AM

Everything is right, however;
$y = w*^{T}X+E$

the signal
$y_n = w*^{T} x_n+ \epsilon$
over the entire domain of x becomes

$y = Xw^* + \epsilon$

from pg 84 :
the nth component of the vector

is exactly $w^{T}x_n-y_n$

Quote:

Originally Posted by squidsforlunch

For Exercise 3.4 (a) I did

$y = w*^{T}X+E$
$\hat{y} = Hy$
$\hat{y} = H(w*^{T}X+E)$
$\hat{y} = Hw*^{T}x+HE$
$\hat{y} = (X(X^{T}X)^{-1}X^{T})w*^{T}X+HE$
$\hat{y} = (XX^{-1}(X^{T})^{-1}X^{T})w*^{T}X+HE$
$\hat{y} = (I(X^{T})^{-1}X^{T})w*^{T}X+HE$
$\hat{y} = (I(I))w*^{T}X+HE$
$\hat{y} = w*^{T}X+HE$

but
$w*^{T}X$
is not equal to

What am I missing?

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