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 Examplesplot(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) ); % =============================================================end1.2.2 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
1.2.4 Evaluating logistic regression
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: mif h(i) >= 0.5p(i) = 1;else p(i) = 0;end % =========================================================================end2 Regularized logistic regression
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 % =============================================================end2.5 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