应用回归分析多元线性回归_应用多元线性回归时?

clear ; close all; clc

fprintf('Loading data ...n');

%% Load Data
data = load('ex1data2.txt');
X = data(:, 1:2);
y = data(:, 3);
m = length(y);

% Print out some data points
fprintf('First 10 examples from the dataset: n');
fprintf(' x = [%.0f %.0f], y = %.0f n', [X(1:10,:) y(1:10,:)]');

fprintf('Program paused. Press enter to continue.n');
pause;

% Scale features and set them to zero mean
fprintf('Normalizing Features ...n');

[X ,mu ,sigma] = featureNormalize(X);

% Add intercept term to X
X = [ones(m, 1) X];


%% ================ Part 2: Gradient Descent ================

% ====================== YOUR CODE HERE ======================
% Instructions: We have provided you with the following starter
%               code that runs gradient descent with a particular
%               learning rate (alpha). 
%
%               Your task is to first make sure that your functions - 
%               computeCost and gradientDescent already work with 
%               this starter code and support multiple variables.
%
%               After that, try running gradient descent with 
%               different values of alpha and see which one gives
%               you the best result.
%
%               Finally, you should complete the code at the end
%               to predict the price of a 1650 sq-ft, 3 br house.
%
% Hint: By using the 'hold on' command, you can plot multiple
%       graphs on the same figure.
%
% Hint: At prediction, make sure you do the same feature normalization.
%

fprintf('Running gradient descent ...n');

% Choose some alpha value
alpha = [0.03,0.1,0.01];
num_iters = 30;

% Init Theta and Run Gradient Descent 
theta = zeros(3, 1);
[theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters);

% Plot the convergence graph
figure;
plot(1:numel(J_history(:,1)), J_history(:,1), '-b', 'LineWidth', 2);
hold on 
plot(1:numel(J_history(:,2)), J_history(:,2), '-r', 'LineWidth', 2);
hold on 
plot(1:numel(J_history(:,3)), J_history(:,3), '-g', 'LineWidth', 2);
xlabel('Number of iterations');
ylabel('Cost J');

% Display gradient descent's result
fprintf('Theta computed from gradient descent: n');
fprintf(' %f n', theta);
fprintf('n');

% Estimate the price of a 1650 sq-ft, 3 br house
% ====================== YOUR CODE HERE ======================
% Recall that the first column of X is all-ones. Thus, it does
% not need to be normalized.
U = [1650,3];  % You should change this
for i = 1:2
    U(i)=(U(i)-mu(i))/sigma(i);
end
U = [1 U];
price = U*theta;


% ============================================================

fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...
         '(using gradient descent):n $%fn'], price);

fprintf('Program paused. Press enter to continue.n');
pause;

%% ================ Part 3: Normal Equations ================

fprintf('Solving with normal equations...n');

function [X_norm, mu, sigma] = featureNormalize(X)
    X_norm = X;
    mu = mean(X);
    sigma = std(X);
    for i = 1:2
        X_norm(:,i) = (X(:,i)-mu(i))./sigma(i);
    end
end

    均值

    标准差

function J = computeCostMulti(X, y, theta)
    m = length(y);
    J = (X*theta-y)'*(X*theta-y);
end

function [theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters)
    m = length(y);
    J_history = zeros(num_iters, 3);
    for iter = 1:num_iters
        for i = 1:3
            theta = theta-X'*(X*theta-y)*alpha(i)/m;
            J_history(iter,i) = computeCostMulti(X, y, theta);
        end
    end
end

此处我设置了三个学习率，直接在函数中一次计算。

可以看到梯度下降效果明显