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* regression
consider the generic problem of fitting a dataset to a linear polynomial
given discrete f: x \rightarrow y
interpolation: y = a + bx
[[1 x_0] [[y_0]
[1 x_1] \cdot [[a] = [y_1]
[1 x_n]] [b]] [y_n]]
consider p \in col(A)
then y = p + q for some q \cdot p = 0
then we can generate n \in col(A) by $Az$ and n must be orthogonal to q as well
(Az)^T \cdot q = 0 = (Az)^T (y - p)
0 = (z^T A^T)(y - Ax)
= z^T (A^T y - A^T A x)
= A^T Ax
= A^T y
A^T A = [[n+1 \Sigma_{n=0}^n x_n]
[\Sigma_{n=0}^n x_n \Sigma_{n=0}^n x_n^2]]
A^T y = [[\Sigma_{n=0}^n y_n]
[\Sigma_{n=0}^n x_n y_n]]
a_11 = n+1
a_12 = \Sigma_{n=0}^n x_n
a_21 = a_12
a_22 = \Sigma_{n=0}^n x_n^2
b_1 = \Sigma_{n=0}^n y_n
b_2 = \Sigma_{n=0}^n x_n y_n
then apply this with:
log(e(h)) \leq log(C) + rlog(h)
* homework 3:
two columns \Rightarrow coefficients for linear regression
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