Question

Problem 4.2.

Please only do 4.2, I only need help with 4.2

. OVERFITTING 4.4 Problems pict the monomials of order i, фі(x-a". As you increase correspond to the intuitive notion of increasing complexity? Prob Problem 4.2 Consider the feature transform z = Lo(2), L1(x), L2(zr and the linear model h(x) w'z. For the hypothesis with w [1,-1, 1 what is h(z) explicitly as a function of z? What is its degree? Problem 4.3 The Legendre Polynomials are a family of orthogonal polynomials which are useful for regression. The first two Legendre Polynomals are Lo (z) = 1, Li (x) = x. The higher order Legendre Polynomials are defined by the recursion: 2k-1 k- 1 de n to

Only 4.2 please

Answer 1

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Legender polynomial L_) (x) = 1 L_1 (x) = x L_2 (x) = 1/ 2 (3x^2 - 1) [Because L_2 (x) = 2 . 2- 1/2 x L_1 (x) - 2 - 1/2 L_0 (x) rightarrow L_2 (x) = 3/ 2 x^2 - 1/ 2] so z = [1, x , 1/ 2 (3x^2 - 1)]^T w = [1, -1, 1]^T Thus h(x) = w^T z = [1, -1, 1] [1 x 1/2 (3 x^2 - 1)] rightarrow h(x) = 1 - x + 3/2 x^2 - 1/ 2 rightarrow h(x) = 3/ 2 x^2 - x - 1/ 2 hence h(x) = 3/ 2 x^2 - x - 1/ 2 degree of h(x) is 2