Description
1. Use SimHousingPrices1 to simulate data that is a polynomial with normally-distributed noise. The function SimPoly should receive as input: RealThetas: A real vector of D+1 coefficients for a D-degree polynomial P(x) StdDev: A non-negative scalar that denotes the scale of fluctuation of the output around the polynomial value x: A real vector of input datapoints The function should provide as output: y: The outputs. Each output yi is P(xi)+ei where ei is a simulated value of a normally-distributed random variable, with mean zero and variance 2 . The function should be in a submitted folder called Assignment02.Problem01 [20 points]
2. Define a cubic polynomial with based on the digits in your UNI (mine would be 2×3+x2+6x+9 as my uni is ip2169). Use SimPoly to simulate outputs with this polynomial and =0.1. Simulate outputs for N training inputs and M testing inputs that are uniformly distributed in [1,1]. Perform polynomial curve fitting of degrees 0 to 8 by defining the relevant pseudoinverse for the relevant matrices, and compare empirical risks on training and testing data by plotting them along the degree axis. Do all this three times: run #1 with N=10, M=10; run #2 with N=100, M=10; run #3 with N=10, M=100. Your code should save files with the following information (as columns of numbers): x.train.[R].txt – for R =1,2,3: 3 training inputs for the corresponding run x.test.[R].txt – for R =1,2,3: 3 testing inputs for the corresponding run y.train.[R].txt – for R =1,2,3: 3 training outputs for the corresponding run y.test.[R].txt – for R =1,2,3: 3 testing outputs for the corresponding run ThetaStar.[R].[D].txt – for R =1,2,3, and D =0, … ,8 : 3×11 files, each with the fit coefficients for for the corresponding run and corresponding degree polynomial. Risk.train.[R].txt – for R =1,2,3: 3 training empirical risk values for the corresponding run Risk.test.[R].txt – for R =1,2,3: 3 testing outputs for the corresponding run The function to do all of this should be called should be called FitCubic() in a submitted folder called Assignment02.Problem02 [60 points]
3. Simulate data for logistic regression. Use SimHousingPrices1 to simulate classification data that is drawn with probability that is logistically dependent on a linear combination of inputs, plus normally-distributed noise. The function SimLogistic should receive as input: RealThetas: A real vector of D+1 linear coefficients x: A real matrix with D columns of input datapoints (row vectors) The function should provide as output: y: The binary outputs. Each output yi is randomly chosen with probability Pr(yi =1) = 1/(1+exp(-zi)) where zi is a -defined linear combination of the coordinates of the i-th input vector The function should be in a submitted folder called Assignment02.Problem03 [20 points]
Good luck!1
