1 Logistic Regression
通过学生两门成绩来判别学生能否被录取
1.1 Visualizing the data 数据视觉化
% Find Indices of Positive and Negative Examples
pos = find(y==1); neg = find(y == 0);
% Plot Examples
plot(X(pos, 1), X(pos, 2), 'k+','LineWidth', 2, ...
'MarkerSize', 7);
plot(X(neg, 1), X(neg, 2), 'ko', 'MarkerFaceColor', 'y', ...
'MarkerSize', 7);
1.2 Implementation
1.2.1 Warmup exercise: sigmoid function
function g = sigmoid(z)
%SIGMOID Compute sigmoid function
% g = SIGMOID(z) computes the sigmoid of z.
% You need to return the following variables correctly
g = zeros(size(z));
% ====================== YOUR CODE HERE ======================
% Instructions: Compute the sigmoid of each value of z (z can be a matrix,
% vector or scalar).
g = 1 ./ ( 1 + exp(-z) );
% =============================================================
end
1.2.2 Cost function and gradient
function [J, grad] = costFunction(theta, X, y)
%COSTFUNCTION Compute cost and gradient for logistic regression
% J = COSTFUNCTION(theta, X, y) computes the cost of using theta as the
% parameter for logistic regression and the gradient of the cost
% w.r.t. to the parameters.
% 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 of a particular choice of theta.
% You should set J to the cost.
% Compute the partial derivatives and set grad to the partial
% derivatives of the cost w.r.t. each parameter in theta
%
% Note: grad should have the same dimensions as theta
%
h = sigmoid(X * theta);
J = (-1/m)*sum(y.*log(h)+log(1-h).*(1-y));
grad = sum((1/m)*(h-y).*X);
% =============================================================
end
function p = predict(theta, X)
%PREDICT Predict whether the label is 0 or 1 using learned logistic
%regression parameters theta
% p = PREDICT(theta, X) computes the predictions for X using a
% threshold at 0.5 (i.e., if sigmoid(theta'*x) >= 0.5, predict 1)
m = size(X, 1); % Number of training examples
% You need to return the following variables correctly
p = zeros(m, 1);
% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
% your learned logistic regression parameters.
% You should set p to a vector of 0's and 1's
%
h = sigmoid(X*theta);
for i = 1: m
if h(i) >= 0.5
p(i) = 1;
else
p(i) = 0;
end
% =========================================================================
end
2.3 Cost function and gradient
function [J, grad] = costFunctionReg(theta, X, y, lambda)
%COSTFUNCTIONREG Compute cost and gradient for logistic regression with regularization
% J = COSTFUNCTIONREG(theta, X, y, lambda) computes the cost of using
% theta as the parameter for regularized logistic regression and the
% gradient of the cost w.r.t. to the parameters.
% 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 of a particular choice of theta.
% You should set J to the cost.
% Compute the partial derivatives and set grad to the partial
% derivatives of the cost w.r.t. each parameter in theta
n = length(theta);
h = sigmoid(X * theta);
J = (-1/m)*sum(y.*log(h)+log(1-h).*(1-y))+lambda*(theta'*theta-theta(1)^2)/(2*m);
grad(1) = sum((1/m)*(h-y).*X(:,1));
for j = 2 : n
%以theta参数向量的维度作为循环次数
grad(j) = sum((1/m)*(h-y).*X(:,j))+(lambda/m)*theta(j);
end
% =============================================================
end
2.5 Optional (ungraded) exercises
In this part of the exercise, you will get to try out di erent regularization
parameters for the dataset to understand how regularization prevents over-
tting.
在实验种通过改变lambda的取值来观察正则化如何解决过拟合
lambda取过小:过拟合
lambda取过大:欠拟合