Write a function to implement regularization based training procedure for 1D reg

Question

Write a function to implement regularization based training procedure for 1D regression. The function should have the following form: function a = mytrain(train_x, train_y,n, lambda), where train_x is a column vector containing all the input points in the training, train_y is a column vector containing all the output points in the training, n is the order of the polynomial model, lambda is a non-negative regularization parameter, and a is a column vector containing coefficients of the polynomial model. Write a function to implement testing procedure for 1D regression. The function should have the following form: function test_y = mytest(test_x, a), where test_x is a column vector containing all the input points in the test, a is a column vector containing coefficients of the polynomial model, and test_y is a column vector containing all the output points in the test. Write a script to do the following experiment Create 10 input points for training. They should be evenly distributed between 0 and 1. Create 10 corresponding output points for training as follows: cos(2* pi*x) + 0.3 randn(10, 1) Create 100 input points for testing. They should be evenly distributed between 0 and 1. Generate models for lambda = 0, and n = 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. Generate models for n = 3, and lambda = 1e -8, 1e-5, 1e-2, 1, respectively. Generate models for n = 9, and lambda = 1e-8, 1e-5, 1e-2, 1, respectively. For each model, superimpose the testing results onto cos(2*pi*) Write a report to summarize and discuss your results

Explanation / Answer

function model = linReg(X, t, lambda)

% Fit linear regression model y=w'x+w0

% Input:

%   X: d x n data

%   t: 1 x n response

%   lambda: regularization parameter

% Output:

%   model: trained model structure

if nargin < 3

    lambda = 0;

end

d = size(X,1);

idx = (1:d)';

dg = sub2ind([d,d],idx,idx);

xbar = mean(X,2);

tbar = mean(t,2);

X = bsxfun(@minus,X,xbar);

t = bsxfun(@minus,t,tbar);

XX = X*X';

XX(dg) = XX(dg)+lambda;     % 3.54 XX=inv(S)/beta

% w = XX(X*t');

U = chol(XX);

w = U(U'(X*t')); % 3.15 & 3.28

w0 = tbar-dot(w,xbar); % 3.19

model.w = w;

model.w0 = w0;

model.xbar = xbar;

%% for probability prediction

beta = 1/mean((t-w'*X).^2); % 3.21

% alpha = lambda*beta;           % lambda=a/b P.153 3.55

% model.alpha = alpha;

model.beta = beta;

model.U = U;

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