# 最小二乘法的极大似然解释

最开始学习机器学习的时候，首先遇到的就是回归算法，回归算法里最最重要的就是最小二乘法，为什么损失函数要用平方和，而且还得是最小？仔细想想最小二乘法视乎很合理，但是合理在哪，怎么用数学方法来证明它合理。

# Stanford 机器学习练习 Part 3 Neural Networks: Representation

从神经网络开始，感觉自己慢慢跟不上课程的节奏了，一些代码好多参考了别人的代码，而且，让我现在单独写也不一定写的出来了。学习就是一件慢慢积累的过程，两年前我学算法的时候，好多算法都完全看不懂，但后来，看的多了，做的多了，有一天就茅塞顿开。所有的困难都是一时的，只要坚持下去，一切问题都会解决的。没忍住发了点鸡汤文。

关于神经网络，我表示现在的理解就是每一层都每个神经元都是依靠logistics regression得出的，所以必须先要掌握logistics regression。关于如何训练这个模型的算法（比如 反向传播算法），我也在这周的课程中慢慢学。其中涉及到好多数据处理的技巧，这些都是得通过练习才能学会的。

下面是我参考别人代码写的第四周神经网络的编程练习，仅作为参考。

lrCostFunction.m

function [J, grad] = lrCostFunction(theta, X, y, lambda)
%LRCOSTFUNCTION Compute cost and gradient for logistic regression with
%regularization
%   J = LRCOSTFUNCTION(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;

% ====================== 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
%
% Hint: The computation of the cost function and gradients can be
%       efficiently vectorized. For example, consider the computation
%
%           sigmoid(X * theta)
%
%       Each row of the resulting matrix will contain the value of the
%       prediction for that example. You can make use of this to vectorize
%       the cost function and gradient computations.
%
% Hint: When computing the gradient of the regularized cost function,
%       there're many possible vectorized solutions, but one solution
%       looks like:
%           temp = theta;
%           temp(1) = 0;   % because we don't add anything for j = 0
%

h = sigmoid(X*theta);
J = m^-1 * sum(((-1) * y.*log(h)).-((1- y).*log(1 - h)));
theta(1) = 0;
tmp = lambda/(2*m)*sum(theta.^2);
J = J + tmp;
grad = m^-1 * ((h.-y)'*X)' + lambda/m * theta;

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

end

oneVsAll.m

function [all_theta] = oneVsAll(X, y, num_labels, lambda)
%ONEVSALL trains multiple logistic regression classifiers and returns all
%the classifiers in a matrix all_theta, where the i-th row of all_theta
%corresponds to the classifier for label i
%   [all_theta] = ONEVSALL(X, y, num_labels, lambda) trains num_labels
%   logisitc regression classifiers and returns each of these classifiers
%   in a matrix all_theta, where the i-th row of all_theta corresponds
%   to the classifier for label i

% Some useful variables
m = size(X, 1);
n = size(X, 2);

% You need to return the following variables correctly
all_theta = zeros(num_labels, n + 1);

% Add ones to the X data matrix
X = [ones(m, 1) X];

% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the following code to train num_labels
%               logistic regression classifiers with regularization
%               parameter lambda.
%
% Hint: theta(:) will return a column vector.
%
% Hint: You can use y == c to obtain a vector of 1's and 0's that tell use
%       whether the ground truth is true/false for this class.
%
% Note: For this assignment, we recommend using fmincg to optimize the cost
%       function. It is okay to use a for-loop (for c = 1:num_labels) to
%       loop over the different classes.
%
%       fmincg works similarly to fminunc, but is more efficient when we
%       are dealing with large number of parameters.
%
% Example Code for fmincg:
%
%     % Set Initial theta
%     initial_theta = zeros(n + 1, 1);
%
%     % Set options for fminunc
%     options = optimset('GradObj', 'on', 'MaxIter', 50);
%
%     % Run fmincg to obtain the optimal theta
%     % This function will return theta and the cost
%     [theta] = ...
%         fmincg (@(t)(lrCostFunction(t, X, (y == c), lambda)), ...
%                 initial_theta, options);
%
for k=1:num_labels
initial_theta = zeros(n + 1, 1);
options = optimset('GradObj', 'on', 'MaxIter', 50);
[theta] = fmincg (@(t)(lrCostFunction(t, X, (y == k), lambda)),initial_theta, options);
all_theta(k,:) = theta';
end
% =========================================================================

end


predictOneVsAll.m

function p = predictOneVsAll(all_theta, X)
%PREDICT Predict the label for a trained one-vs-all classifier. The labels
%are in the range 1..K, where K = size(all_theta, 1).
%  p = PREDICTONEVSALL(all_theta, X) will return a vector of predictions
%  for each example in the matrix X. Note that X contains the examples in
%  rows. all_theta is a matrix where the i-th row is a trained logistic
%  regression theta vector for the i-th class. You should set p to a vector
%  of values from 1..K (e.g., p = [1; 3; 1; 2] predicts classes 1, 3, 1, 2
%  for 4 examples)

m = size(X, 1);
num_labels = size(all_theta, 1);

% You need to return the following variables correctly
p = zeros(size(X, 1), 1);

% Add ones to the X data matrix
X = [ones(m, 1) X];

% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
%               your learned logistic regression parameters (one-vs-all).
%               You should set p to a vector of predictions (from 1 to
%               num_labels).
%
% Hint: This code can be done all vectorized using the max function.
%       In particular, the max function can also return the index of the
%       are in rows, then, you can use max(A, [], 2) to obtain the max
%       for each row.
%
[c,i] = max(sigmoid(X * all_theta'), [], 2);
p = i;

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

end


predict.m

function p = predict(Theta1, Theta2, X)
%PREDICT Predict the label of an input given a trained neural network
%   p = PREDICT(Theta1, Theta2, X) outputs the predicted label of X given the
%   trained weights of a neural network (Theta1, Theta2)

% Useful values
m = size(X, 1);
num_labels = size(Theta2, 1);

% You need to return the following variables correctly
p = zeros(size(X, 1), 1);

% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
%               your learned neural network. You should set p to a
%               vector containing labels between 1 to num_labels.
%
% Hint: The max function might come in useful. In particular, the max
%       function can also return the index of the max element, for more
%       information see 'help max'. If your examples are in rows, then, you
%       can use max(A, [], 2) to obtain the max for each row.
%

X = [ones(m, 1) X];
z2 = Theta1 * X';
a2 = sigmoid(z2);
a2 = [ones(1, m);a2];
z3 = Theta2 * a2;
a3 = sigmoid(z3);
output =a3';
[c,i] = max(output, [], 2);
p = i;

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

end


# Stanford 机器学习练习 Part 1 Linear Regression

warmUpExercise.m

function A = warmUpExercise()
%WARMUPEXERCISE Example function in octave
%   A = WARMUPEXERCISE() is an example function that returns the 5x5 identity matrix
A = [];

% ============= YOUR CODE HERE ==============
% Instructions: Return the 5x5 identity matrix
%               In octave, we return values by defining which variables
%               represent the return values (at the top of the file)
%               and then set them accordingly.
A = eye(5);
% ===========================================
end

computeCost.m

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);
% =========================================================================
end

plotData.m

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', 5);
xlabel("x");
ylabel("y");
% ============================================================
end

function [theta, J_history] = gradientDescent(X, y, theta, alpha, num_iters)
%   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.
%
temp = 0;
temp = temp + alpha/m * X' * (y - X * theta);
theta = theta + temp;
% ============================================================
% Save the cost J in every iteration
J_history(iter) = computeCost(X, y, theta);
end
end

ex1.m

%% 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
%     computeCost.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 % %% Initialization clear ; close all; clc %% ==================== 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 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); featureNormalize.m 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; m = size(X, 2); mu = zeros(1, size(X, 2)); mu = mean(X); sigma = std(X); for i = 1:m X_norm(:,i) = (X(:,i).-mu(i))./sigma(i); end % ====================== 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. % % ============================================================ end computeCostMulti.m 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 = 1/(2*m) * ( X * theta - y)' * (X*theta - y); % ========================================================================= end gradientDescentMulti.m 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); temp = zeros(feature_number,1); for iter = 1:num_iters temp = alpha/m * X' * (y - X*theta); theta = theta + temp; J_history(iter) = computeCostMulti(X, y, theta); % ====================== 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. % % ============================================================ % Save the cost J in every iteration J_history(iter) = computeCostMulti(X, y, theta); end end normalEqn.m 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 ex1_multi.m %% 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); % 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 = 400; % 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 = 0; % You should change this % ============================================================ 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.
%

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 = 0; % You should change this

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

fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...
'(using normal equations):\n \$%f\n'], price);