【转载】用OCTAVE实现一元线性回归的梯度下降算法

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原文地址：http://www.cnblogs.com/KID-XiaoYuan/p/7247481.html

STEP1 PLOTTING THE DATA

在处理数据之前，我们通常要了解数据，对于这次的数据集合，我们可以通过离散的点来描绘它，在一个2D的平面里把它画出来。

ex1data1.txt

我们把ex1data1中的内容读取到X变量和y变量中，用m表示数据长度。

data = load('ex1data1.txt');

X = data(:,1);

y = data(:,2);

m = length(y);

接下来通过图像描绘出来。

plot(x,y,'rx','MakerSize',10);

ylabel('Profit in $10,000s');

xlabel('Population of City in 10,000s');

现在我们得到图像如图所示，就是原始的数据的直观表示。

STEP2 GRADIENT DESCENT

现在，我们通过梯度下降法对参数θ进行线性回归。

依照我们之前所得出步骤方法

迭代更新

计算θ值函数：

function J = computeCost(X, y, theta)

%COMPUTECOST Compute cost for linear regression

% J = COMPUTECOST(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); % X(79,2) theta(2,1)

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

end

接下来是梯度下降函数

function [theta, J_history] = gradientDescent(X, y, theta, alpha, num_iters)

%GRADIENTDESCENT Performs gradient descent to learn theta

% theta = GRADIENTDESENT(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);

theta_s=theta;

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.

%

theta(1) = theta(1) - alpha / m * sum(X * theta_s - y);

theta(2) = theta(2) - alpha / m * sum((X * theta_s - y) .* X(:,2));

% 必须同时更新theta(1)和theta(2)，所以不能用X * theta,而要用theta_s存储上次结果。

theta_s=theta;

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

% Save the cost J in every iteration

J_history(iter) = computeCost(X, y, theta);

end

J_history

end

绘图函数：

function plotData(x, y)

%PLOTDATA Plots the data points x and y into a new figure

% PLOTDATA(x,y) plots the data points and gives the figure axes labels of

% population and profit.

% ====================== YOUR CODE HERE ======================

% Instructions: Plot the training data into a figure using the

% "figure" and "plot" commands. Set the axes labels using

% the "xlabel" and "ylabel" commands. Assume the

% population and revenue data have been passed in

% as the x and y arguments of this function.

%

% Hint: You can use the 'rx' option with plot to have the markers

% appear as red crosses. Furthermore, you can make the

% markers larger by using plot(..., 'rx', 'MarkerSize', 10);

figure; % open a new figure window

plot(x, y, 'rx', 'MarkerSize', 10); % Plot the data

ylabel('Profit in $10,000s'); % Set the y axis label

xlabel('Population of City in 10,000s'); % Set the x axis label

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

end

根据以上函数，我们进行线性回归：

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<br>%% Machine Learning Online Class - Exercise 1: Linear Regression

% Instructions

% ------------

%

% This file contains code that helps you get started on the

% linear exercise. You will need to complete the following functions

% in this exericse:

%

% warmUpExercise.m

% plotData.m

% gradientDescent.m

% computeCost.m

% gradientDescentMulti.m

% computeCostMulti.m

% featureNormalize.m

% normalEqn.m

%

% For this exercise, you will not need to change any code in this file,

% or any other files other than those mentioned above.

%

% x refers to the population size in 10,000s

% y refers to the profit in $10,000s

%

%% ==================== Part 1: Basic Function ====================

% Complete warmUpExercise.m

fprintf('Running warmUpExercise ... \n');

fprintf('5x5 Identity Matrix: \n');

warmUpExercise()

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

pause;

%% ======================= Part 2: Plotting =======================

fprintf('Plotting Data ...\n')

data = load('ex1data1.txt');

X = data(:, 1); y = data(:, 2);

m = length(y); % number of training examples

% Plot Data

% Note: You have to complete the code in plotData.m

plotData(X, y);

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

pause;

%% =================== Part 3: Gradient descent ===================

fprintf('Running Gradient Descent ...\n')

X = [ones(m, 1), data(:,1)]; % Add a column of ones to x

theta = zeros(2, 1); % initialize fitting parameters

% Some gradient descent settings

iterations = 1500;

alpha = 0.01;

% compute and display initial cost

computeCost(X, y, theta)

% run gradient descent

theta = gradientDescent(X, y, theta, alpha, iterations);

% print theta to screen

fprintf('Theta found by gradient descent: ');

fprintf('%f %f \n', theta(1), theta(2));

% Plot the linear fit

hold on; % keep previous plot visible

plot(X(:,2), X*theta, '-')

legend('Training data', 'Linear regression')

hold off % don't overlay any more plots on this figure

% Predict values for population sizes of 35,000 and 70,000

predict1 = [1, 3.5] *theta;

fprintf('For population = 35,000, we predict a profit of %f\n',...

predict1*10000);

predict2 = [1, 7] * theta;

fprintf('For population = 70,000, we predict a profit of %f\n',...

predict2*10000);

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

pause;

%% ============= Part 4: Visualizing J(theta_0, theta_1) =============

fprintf('Visualizing J(theta_0, theta_1) ...\n')

% Grid over which we will calculate J

theta0_vals = linspace(-10, 10, 100);

theta1_vals = linspace(-1, 4, 100);

% initialize J_vals to a matrix of 0's

J_vals = zeros(length(theta0_vals), length(theta1_vals));

% Fill out J_vals

for i = 1:length(theta0_vals)

for j = 1:length(theta1_vals)

t = [theta0_vals(i); theta1_vals(j)];

J_vals(i,j) = computeCost(X, y, t);

end

% Because of the way meshgrids work in the surf command, we need to

% transpose J_vals before calling surf, or else the axes will be flipped

J_vals = J_vals';

% Surface plot

figure;

surf(theta0_vals, theta1_vals, J_vals)

xlabel('\theta_0'); ylabel('\theta_1');

% Contour plot

figure;

% Plot J_vals as 15 contours spaced logarithmically between 0.01 and 100

contour(theta0_vals, theta1_vals, J_vals, logspace(-2, 3, 20))

xlabel('\theta_0'); ylabel('\theta_1');

hold on;

plot(theta(1), theta(2), 'rx', 'MarkerSize', 10, 'LineWidth', 2);

如图所示，绘制出线性回归函数。

这时所绘制2D等高线图梯度下降表面图：

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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)); % mean value 均值 size(X,2) 列数

sigma = zeros(1, size(X, 2)); % standard deviation 标准差

% ====================== 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); % mean value

sigma = std(X); % standard deviation

X_norm = (X - repmat(mu,size(X,1),1)) ./ repmat(sigma,size(X,1),1);

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

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

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] = 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 = pinv( X' * X ) * X' * y;

% -------------------------------------------------------------

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

end

%% Machine Learning Online Class

% Exercise 1: Linear regression with multiple variables

%

% Instructions

% ------------

%

% This file contains code that helps you get started on the

% linear regression exercise.

%

% You will need to complete the following functions in this

% exericse:

%

% warmUpExercise.m

% plotData.m

% gradientDescent.m

% computeCost.m

% gradientDescentMulti.m

% computeCostMulti.m

% featureNormalize.m

% normalEqn.m

%

% For this part of the exercise, you will need to change some

% parts of the code below for various experiments (e.g., changing

% learning rates).

%

%% Initialization

%% ================ Part 1: Feature Normalization ================

%% Clear and Close Figures

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); % 均值0，标准差1

% 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 = 0.01;

num_iters = 8500;

% 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), J_history, '-b', '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.

price = [1 (([1650 3]-mu) ./ sigma)] * theta ;

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

fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...

'(using gradient descent):\n $%f\n'], price);

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

pause;

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

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

% ====================== YOUR CODE HERE ======================

% Instructions: The following code computes the closed form

% solution for linear regression using the normal

% equations. You should complete the code in

% normalEqn.m

%

% After doing so, you should complete this code

% to predict the price of a 1650 sq-ft, 3 br house.

%

%% Load Data

data = csvread('ex1data2.txt');

X = data(:, 1:2);

y = data(:, 3);

m = length(y);

% Add intercept term to X

X = [ones(m, 1) X];

% Calculate the parameters from the normal equation

theta = normalEqn(X, y);

% Display normal equation's result

fprintf('Theta computed from the normal equations: \n');

fprintf(' %f \n', theta);

fprintf('\n');

% Estimate the price of a 1650 sq-ft, 3 br house

% ====================== YOUR CODE HERE ======================

price = [1 1650 3] * theta ;

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

fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...

'(using normal equations):\n $%f\n'], price);

处理前：

处理后：

回归过程如图所示：

至此，我们通过梯度下降法解决了此问题，我们还可以通过之前所说的数学方法来解决，但是对于数据太大的情况（通常大于10000），我们就会通过梯度下降法来解决了

根据以上函数，我们进行线性回归：

正文

STEP1 PLOTTING THE DATA

STEP2 GRADIENT DESCENT

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