909 B
regression
consider the generic problem of fitting a dataset to a linear polynomial
given discrete f: x → y
interpolation: y = a + bx
[[1 x_0] [[y_0] [1 x_1] ⋅ [[a] = [y_1] [1 x_n]] [b]] [y_n]]
consider p ∈ col(A)
then y = p + q for some q ⋅ p = 0
then we can generate n ∈ col(A) by $Az$ and n must be orthogonal to q as well
(Az)^T ⋅ 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 Σn=0^n x_n] [Σn=0^n x_n Σn=0^n x_n^2]]
A^T y = [[Σn=0^n y_n] [Σn=0^n x_n y_n]]
a_11 = n+1 a_12 = Σn=0^n x_n a_21 = a_12 a_22 = Σn=0^n x_n^2 b_1 = Σn=0^n y_n b_2 = Σn=0^n x_n y_n
then apply this with:
log(e(h)) ≤ log(C) + rlog(h)
homework 3:
two columns ⇒ coefficients for linear regression