linearRegCostFunction.m
function [J, grad] = linearRegCostFunction(X, y, theta, lambda)
%LINEARREGCOSTFUNCTION Compute cost and gradient for regularized linear
%regression with multiple variables
% [J, grad] = LINEARREGCOSTFUNCTION(X, y, theta, lambda) computes the
% cost of using theta as the parameter for linear regression to fit the
% data points in X and y. Returns the cost in J and the gradient in grad
% Initialize some useful values
m = length(y); % number of training examples
% You need to return the following variables correctly
J = 0;
grad = zeros(size(theta));
% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost and gradient of regularized linear
% regression for a particular choice of theta.
%
% You should set J to the cost and grad to the gradient.
%
J = 1/(2*m) * sum(((X * theta) – y).^2) + lambda/(2*m) * sum(theta(2:end).^2);
grad = 1/m * (( X * theta) – y)’ * X ;
grad(2:end) = grad(2:end) + lambda/m * theta(2:end)’ ;
grad = grad’ ;
% =========================================================================
grad = grad(:);
end
learningCurve.m
function [error_train, error_val] = …
learningCurve(X, y, Xval, yval, lambda)
%LEARNINGCURVE Generates the train and cross validation set errors needed
%to plot a learning curve
% [error_train, error_val] = …
% LEARNINGCURVE(X, y, Xval, yval, lambda) returns the train and
% cross validation set errors for a learning curve. In particular,
% it returns two vectors of the same length – error_train and
% error_val. Then, error_train(i) contains the training error for
% i examples (and similarly for error_val(i)).
%
% In this function, you will compute the train and test errors for
% dataset sizes from 1 up to m. In practice, when working with larger
% datasets, you might want to do this in larger intervals.
%
% Number of training examples
m = size(X, 1);
% You need to return these values correctly
error_train = zeros(m, 1);
error_val = zeros(m, 1);
mval = size(Xval,1);
% ====================== YOUR CODE HERE ======================
% Instructions: Fill in this function to return training errors in
% error_train and the cross validation errors in error_val.
% i.e., error_train(i) and
% error_val(i) should give you the errors
% obtained after training on i examples.
%
% Note: You should evaluate the training error on the first i training
% examples (i.e., X(1:i, 🙂 and y(1:i)).
%
% For the cross-validation error, you should instead evaluate on
% the _entire_ cross validation set (Xval and yval).
%
% Note: If you are using your cost function (linearRegCostFunction)
% to compute the training and cross validation error, you should
% call the function with the lambda argument set to 0.
% Do note that you will still need to use lambda when running
% the training to obtain the theta parameters.
%
% Hint: You can loop over the examples with the following:
%
% for i = 1:m
% % Compute train/cross validation errors using training examples
% % X(1:i, 🙂 and y(1:i), storing the result in
% % error_train(i) and error_val(i)
% ….
%
% end
%
% ———————- Sample Solution ———————-
for i = 1:m
Xtrain=X(1:i,:);
ytrain=y(1:i,:);
theta = trainLinearReg(Xtrain, ytrain, lambda);
costTrain = 1/(2*i) * sum(((Xtrain * theta) – ytrain).^2);
error_train(i) = costTrain;
costVal = 1/(2*mval) * sum(((Xval * theta) – yval).^2);
error_val(i) = costVal;
end
% ————————————————————-
% =========================================================================
end
polyFeatures.m
function [X_poly] = polyFeatures(X, p)
%POLYFEATURES Maps X (1D vector) into the p-th power
% [X_poly] = POLYFEATURES(X, p) takes a data matrix X (size m x 1) and
% maps each example into its polynomial features where
% X_poly(i, 🙂 = [X(i) X(i).^2 X(i).^3 … X(i).^p];
%
% You need to return the following variables correctly.
X_poly = zeros(numel(X), p);
% ====================== YOUR CODE HERE ======================
% Instructions: Given a vector X, return a matrix X_poly where the p-th
% column of X contains the values of X to the p-th power.
%
%
for r=1:numel(X)
for c=1:p
X_poly(r,c) = X(r,1)^c;
end
end
% =========================================================================
end
validationCurve.m
function [lambda_vec, error_train, error_val] = …
validationCurve(X, y, Xval, yval)
%VALIDATIONCURVE Generate the train and validation errors needed to
%plot a validation curve that we can use to select lambda
% [lambda_vec, error_train, error_val] = …
% VALIDATIONCURVE(X, y, Xval, yval) returns the train
% and validation errors (in error_train, error_val)
% for different values of lambda. You are given the training set (X,
% y) and validation set (Xval, yval).
%
% Selected values of lambda (you should not change this)
lambda_vec = [0 0.001 0.003 0.01 0.03 0.1 0.3 1 3 10]’;
% You need to return these variables correctly.
error_train = zeros(length(lambda_vec), 1);
error_val = zeros(length(lambda_vec), 1);
% ====================== YOUR CODE HERE ======================
% Instructions: Fill in this function to return training errors in
% error_train and the validation errors in error_val. The
% vector lambda_vec contains the different lambda parameters
% to use for each calculation of the errors, i.e,
% error_train(i), and error_val(i) should give
% you the errors obtained after training with
% lambda = lambda_vec(i)
%
% Note: You can loop over lambda_vec with the following:
%
% for i = 1:length(lambda_vec)
% lambda = lambda_vec(i);
% % Compute train / val errors when training linear
% % regression with regularization parameter lambda
% % You should store the result in error_train(i)
% % and error_val(i)
% ….
%
% end
%
%
for i = 1:length(lambda_vec)
lambda = lambda_vec(i);
theta = trainLinearReg(X, y , lambda);
[costTrain, gradTrain] = linearRegCostFunction(X, y , theta, 0);
error_train(i) = costTrain;
[costVal, gradVal] = linearRegCostFunction(Xval, yval, theta, 0);
error_val(i) = costVal;
%Initially I did not pass the test as I pass in the value of current lambda into linearRegCostFunction
%linearRegCostFunction(X,y,theta,lambda)
%Then I realise the Jtrain’s and Jcv’s definition does not include the regularization term and lambda should be 0
end
% =========================================================================
end
ML Study Notes 