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Assignment 2 Machine Learning COMS 4771 solved

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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 2x
3
+x
2
+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

1
You are encouraged to use the function in the posted solution for Assignment #1. Using your own function is
allowed, but it is at your own risk.