已经学习吴恩达的机器学习四周，但对编程还是不够熟练，所以想重新总结一下自己的编程作业，加强巩固。
在写代码之前一定要搞清楚X、y、theta是几乘几的矩阵。
一元线性回归，步骤：
1、设置代价函数
2、梯度下降，对代价函数求θ的偏导，更新θ的值，迭代更新。
% 代价函数
function J = computeCost(X, y, theta)
% Initialize some useful values
m = length(y); % number of training examples
% You need to return the following variables correctly
J = 0;
% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost of a particular choice of theta
% You should set J to the cost.
% ------ my first try ----------
% 使用迭代的方法计算
% for i = 1 : m
% J = J + ((theta(1) + X(i, 2) * theta(2)) - y(i))^2;
% endfor
% J = J / 2 / m;
% ------------------------------
% ------ my second try ---------
% 使用向量化计算
J = sum((X * theta - y) .^ 2) / 2 / m;
% ------------------------------
% =========================================================================
end
% 梯度下降
function [theta, J_history] = gradientDescent(X, y, theta, alpha, num_iters)
%GRADIENTDESCENT Performs gradient descent to learn theta
% theta = GRADIENTDESCENT(X, y, theta, alpha, num_iters) updates theta by
% taking num_iters gradient steps with learning rate alpha
% Initialize some useful values
m = length(y); % number of training examples
J_history = zeros(num_iters, 1);
for iter = 1:num_iters
% ====================== YOUR CODE HERE ======================
% Instructions: Perform a single gradient step on the parameter vector
% theta.
%
% Hint: While debugging, it can be useful to print out the values
% of the cost function (computeCost) and gradient here.
%
% ------------------------------------------------------------
% 使用迭代的方法计算，速度很慢
% d1 = 0;
% d2 = 0;
% for i = 1 : m
% h = theta(1) + theta(2) * X(i, 2) - y(i);
% d1 = d1 + h * X(i, 1);
% d2 = d2 + h * X(i, 2);
% endfor
% theta(1) = theta(1) - alpha / m * d1;
% theta(2) = theta(2) - alpha / m * d2;
% ------------------------------------------------------------
% 向量化计算
theta = theta - alpha / m * (X' * (X * theta - y));
% ============================================================
% Save the cost J in every iteration
J_history(iter) = computeCost(X, y, theta);
end
end
多元线性回归
1、特征缩放
2、代价函数
3、梯度下降
4、正规方程
如果使用向量化计算的话，多元线性回归的代价函数和梯度下降算法和一元线性回归一样。
% 特征缩放
function [X_norm, mu, sigma] = featureNormalize(X)
%FEATURENORMALIZE Normalizes the features in X
% FEATURENORMALIZE(X) returns a normalized version of X where
% the mean value of each feature is 0 and the standard deviation
% is 1. This is often a good preprocessing step to do when
% working with learning algorithms.
% You need to set these values correctly
X_norm = X;
mu = zeros(1, size(X, 2));
sigma = zeros(1, size(X, 2));
% ====================== YOUR CODE HERE ======================
% Instructions: First, for each feature dimension, compute the mean
% of the feature and subtract it from the dataset,
% storing the mean value in mu. Next, compute the
% standard deviation of each feature and divide
% each feature by it's standard deviation, storing
% the standard deviation in sigma.
%
% Note that X is a matrix where each column is a
% feature and each row is an example. You need
% to perform the normalization separately for
% each feature.
%
% Hint: You might find the 'mean' and 'std' functions useful.
%
mu = mean(X_norm); % 求X_norm每列的均值，即每个特征的均值
sigma = std(X_norm); % 标准差，std (X) = sqrt ( 1/(N-1) SUM_i (X(i) - mean(X))^2 )
% 求每列的标准差
X_norm = (X_norm - mu) ./ sigma; % 注意"./"
% ============================================================
end
function J = computeCostMulti(X, y, theta)
%COMPUTECOSTMULTI Compute cost for linear regression with multiple variables
% J = COMPUTECOSTMULTI(X, y, theta) computes the cost of using theta as the
% parameter for linear regression to fit the data points in X and y
% Initialize some useful values
m = length(y); % number of training examples
% You need to return the following variables correctly
J = 0;
% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost of a particular choice of theta
% You should set J to the cost.
J = sum((X * theta - y) .^ 2) / 2 / m;
% =========================================================================
end
function [theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters)
%GRADIENTDESCENTMULTI Performs gradient descent to learn theta
% theta = GRADIENTDESCENTMULTI(x, y, theta, alpha, num_iters) updates theta by
% taking num_iters gradient steps with learning rate alpha
% Initialize some useful values
m = length(y); % number of training examples
J_history = zeros(num_iters, 1);
for iter = 1:num_iters
% ====================== YOUR CODE HERE ======================
% Instructions: Perform a single gradient step on the parameter vector
% theta.
%
% Hint: While debugging, it can be useful to print out the values
% of the cost function (computeCostMulti) and gradient here.
%
theta = theta - alpha / m * (X' * (X * theta - y));
% ============================================================
% Save the cost J in every iteration
J_history(iter) = computeCostMulti(X, y, theta);
end
end
% 正规方程
function [theta] = normalEqn(X, y)
%NORMALEQN Computes the closed-form solution to linear regression
% NORMALEQN(X,y) computes the closed-form solution to linear
% regression using the normal equations.
theta = zeros(size(X, 2), 1);
% ====================== YOUR CODE HERE ======================
% Instructions: Complete the code to compute the closed form solution
% to linear regression and put the result in theta.
%
% ---------------------- Sample Solution ----------------------
theta = inv(X' * X) * X' * y; % 用正规方程求theta
% -------------------------------------------------------------
% ============================================================
end
% 在ex1_multi.m 最底部
% Estimate the price of a 1650 sq-ft, 3 br house
% ====================== YOUR CODE HERE ======================
price = [1, 1650, 3] * theta; % You should change this
% ============================================================