保姆级教程吴恩达机器学习ex5Matlab代码解析
%%Machine learning From Andrew
%%By youknowwho3_3 in CSDN #GirlsHelpGirls#DOUBANEZU
%%Regularized Linear Regression and Bias vs. Variance
%implement the backpropagation algorithm for neural networks and
%apply it to the task of hand-written digit recognition
%1. Regularized Linear Regression
% 1.1 Visualizing the dataset
% 1.2 Regularized linear regression cost function
% 1.3 Regularized linear regression gradient
% 1.4 Fitting linear regression
%2. Bias-variance
% 2.1 Learning curves
%3. Polynomial regression
% 3.1 Learning Polynomial Regression
% 3.2 Optional (ungraded) exercise: Adjusting the regularization parameter
% 3.3 Selecting lambda using a cross validation set
% 3.4 Optional (ungraded) exercise: Computing test set error
% 3.5 Optional (ungraded) exercise: Plotting learning curves with randomly selected examples
%%
%%1. Regularized Linear Regression
%%1.1 Visualizing the dataset
% Load from ex5data1:
% You will have X, y, Xval, yval, Xtest, ytest in your environment
load ('ex5data1.mat');
%size(X)=12*1
%size(Xtest)=21*1
%size(Xval)=21*1
%size(y)=12*1
%size(ytest)=21*1
%size(yval)=21*1
% m = Number of examples
m = size(X, 1);
% Plot training data
figure;
plot(X, y, 'rx', 'MarkerSize', 10, 'LineWidth', 1.5);
xlabel('Change in water level (x)');
ylabel('Water flowing out of the dam (y)');
%%1.2 Regularized linear regression cost function
theta = [1 ; 1];%size(theta)=2*1
J = linearRegCostFunction([ones(m, 1) X], y, theta, 1);
fprintf('The Expected Cost at theta = [1 ; 1]: 303.993192');
fprintf('Cost at theta = [1 ; 1]: %f', J);
%%1.3 Regularized linear regression gradient
[J, Grad] = linearRegCostFunction([ones(m, 1) X], y, theta, 1);
fprintf(['The Expected Gradient at theta = [1 ; 1]: [-15.30; 598.250]\n']);
fprintf('Gradient at theta = [1 ; 1]: [%f; %f]\n',Grad(1), Grad(2));
%%1.4 Fitting linear regression
% Train linear regression with lambda = 0
lambda = 0;
[theta] = trainLinearReg([ones(m, 1) X], y, lambda);
% Plot fit over the data
figure;
plot(X, y, 'rx', 'MarkerSize', 10, 'LineWidth', 1.5);
xlabel('Change in water level (x)');
ylabel('Water flowing out of the dam (y)');
hold on;
plot(X, [ones(m, 1) X]*theta,'b--', 'LineWidth', 2)
%blue的线这么写
hold off;
%%2. Bias-variance
%%2.1 Learning curves
lambda = 0;
[error_train, error_val] = learningCurve([ones(m, 1) X], y, [ones(size(Xval, 1), 1) Xval], yval, lambda);
plot(1:m, error_train, 1:m, error_val);
title('Learning curve for linear regression')
legend('Train', 'Cross Validation')
xlabel('Number of training examples')
ylabel('Error')
axis([0 13 0 150])
fprintf('# Training Examples\tTrain Error\tCross Validation Error\n');
for i = 1:m
fprintf(' \t%d\t\t%f\t%f\n', i, error_train(i), error_val(i));
end
%%3. Polynomial regression多项式回归
p = 8;
% Map X onto Polynomial Features and Normalize
X_poly = polyFeatures(X, p);
[X_poly, mu, sigma] = featureNormalize(X_poly); % Normalize
X_poly = [ones(m, 1), X_poly]; % Add Ones
% Map X_poly_test and normalize (using mu and sigma)
X_poly_test = polyFeatures(Xtest, p);
X_poly_test = X_poly_test-mu; % uses implicit expansion instead of bsxfun
X_poly_test = X_poly_test./sigma; % uses implicit expansion instead of bsxfun
X_poly_test = [ones(size(X_poly_test, 1), 1), X_poly_test]; % Add Ones
% Map X_poly_val and normalize (using mu and sigma)
X_poly_val = polyFeatures(Xval, p);
X_poly_val = X_poly_val-mu; % uses implicit expansion instead of bsxfun
X_poly_val = X_poly_val./sigma; % uses implicit expansion instead of bsxfun
X_poly_val = [ones(size(X_poly_val, 1), 1), X_poly_val]; % Add Ones
fprintf('Normalized Training Example 1:\n');
fprintf(' %f \n', X_poly(1, :));
% Train the model
lambda = 0;
[theta] = trainLinearReg(X_poly, y, lambda);
% Plot training data and fit
plot(X, y, 'rx', 'MarkerSize', 10, 'LineWidth', 1.5);
plotFit(min(X), max(X), mu, sigma, theta, p);
xlabel('Change in water level (x)');
ylabel('Water flowing out of the dam (y)');
title (sprintf('Polynomial Regression Fit (lambda = %f)', lambda));
[error_train, error_val] = learningCurve(X_poly, y, X_poly_val, yval, lambda);
plot(1:m, error_train, 1:m, error_val);
title(sprintf('Polynomial Regression Learning Curve (lambda = %f)', lambda));
xlabel('Number of training examples')
ylabel('Error')
axis([0 13 0 100])
legend('Train', 'Cross Validation')
[lambda_vec, error_train, error_val] = validationCurve(X_poly, y, X_poly_val, yval);
plot(lambda_vec, error_train, lambda_vec, error_val);
legend('Train', 'Cross Validation');
xlabel('lambda');
ylabel('Error');
for i = 1:length(lambda_vec)
if i == 1
fprintf('lambda\t\tTrain Error\tValidation Error\n');
end
fprintf('%f\t%f\t%f\n',lambda_vec(i), error_train(i), error_val(i));
end
%function validationCurve.m
function [lambda_vec, error_train, error_val]= validationCurve(X, y, Xval, yval)
% Selected values of lambda (you should not change this)
lambda_vec = [0 0.001 0.003 0.01 0.03 0.1 0.3 1 3 10]';
% You need to return these variables correctly.
error_train = zeros(length(lambda_vec), 1);
error_val = zeros(length(lambda_vec), 1);
for i = 1:length(lambda_vec)
theta = trainLinearReg(X, y, lambda_vec(i));
error_train(i) = linearRegCostFunction(X, y, theta, 0);
error_val(i) = linearRegCostFunction(Xval, yval, theta, 0);
end
end%Hypothesis of Polynomial regression:
%function polyFeatures.m
function X_poly=polyFeatures(X, p)
%Specifically, when a training set of size is passed into the function,
% the function should return a matrix X_poly,
% where column 1 holds the original values of X,
% column 2 holds the values of X.^2, column 3 holds the values of X.^3, and so on.
% Note that you don't have to account for the zero-th power in this function.
% Now that you have a function that will map features to a higher dimension, the code in the next section will apply it to the training set, the test set, and the cross validation set (which you haven't used yet).
X_poly=X;
for i=2:p
X_poly=[X_poly,X.^i];
end
%{
% You need to return the following variables correctly.
X_poly = zeros(numel(X), p);
% ====================== YOUR CODE HERE ======================
% Instructions: Given a vector X, return a matrix X_poly where the p-th
% column of X contains the values of X to the p-th power.
for i = 1:p
X_poly(:,i) = X.^i;
end
%}
endRegularized Linear Regression Cost Function:
The partial derivative(偏导数) of Regularized Linear Regression's cost for
%function linearRegCostFunction
function [J,Grad] = linearRegCostFunction(X, y, theta, lambda)
m=length(y);%m=12 size(X)=12*2
h=X*theta;%size(h)=12*1
J=0;
Grad = zeros(size(theta));
J=1/(2*m)*sum((h-y).^2)+lambda/(2*m)*sum(theta(2:end).^2);
%size((h-y).^2)=12*1 所以要sum
%theta(2:end) theta第二行到最后
Grad(1) = (1/m)*(X(:,1)'*(h-y)); % scalar == 1x1
Grad(2:end) = (1/m)*(X(:,2:end)'*(h-y))+(lambda/m)*theta(2:end); % n x 1
%Grad=1/m*(h-y)'*X+lambda/m*theta(2:end);
%if j=0 即theta=1*n,theta(2:end)=0,即*0,但是X却是所有的j
Grad=Grad(:);
end
%function learningCurve
function [error_train,error_val] = learningCurve(X, y, Xval, yval, lambda)
m=size(X,1);
error_train=zeros(m,1);
error_val=zeros(m,1);
for i = 1:m%用多少个数据来train
theta = trainLinearReg(X(1:i,:), y(1:i), lambda);
error_train(i) = linearRegCostFunction(X(1:i,:), y(1:i), theta, 0);
error_val(i) = linearRegCostFunction(Xval, yval, theta, 0);
end
end
function [theta] = trainLinearReg(X, y, lambda)
%TRAINLINEARREG Trains linear regression given a dataset (X, y) and a
%regularization parameter lambda
% [theta] = TRAINLINEARREG (X, y, lambda) trains linear regression using
% the dataset (X, y) and regularization parameter lambda. Returns the
% trained parameters theta.
%
% Initialize Theta
%size(X)=12*2
initial_theta = zeros(size(X, 2), 1);
% Create "short hand" for the cost function to be minimized
costFunction = @(t)linearRegCostFunction(X, y, t, lambda);
% Now, costFunction is a function that takes in only one argument
options = optimset('MaxIter', 200, 'GradObj', 'on');
% Minimize using fmincg
theta = fmincg(costFunction, initial_theta, options);
end
function [X, fX, i] = fmincg(f, X, options, P1, P2, P3, P4, P5)
% Minimize a continuous differentialble multivariate function. Starting point
% is given by "X" (D by 1), and the function named in the string "f", must
% return a function value and a vector of partial derivatives. The Polack-
% Ribiere flavour of conjugate gradients is used to compute search directions,
% and a line search using quadratic and cubic polynomial approximations and the
% Wolfe-Powell stopping criteria is used together with the slope ratio method
% for guessing initial step sizes. Additionally a bunch of checks are made to
% make sure that exploration is taking place and that extrapolation will not
% be unboundedly large. The "length" gives the length of the run: if it is
% positive, it gives the maximum number of line searches, if negative its
% absolute gives the maximum allowed number of function evaluations. You can
% (optionally) give "length" a second component, which will indicate the
% reduction in function value to be expected in the first line-search (defaults
% to 1.0). The function returns when either its length is up, or if no further
% progress can be made (ie, we are at a minimum, or so close that due to
% numerical problems, we cannot get any closer). If the function terminates
% within a few iterations, it could be an indication that the function value
% and derivatives are not consistent (ie, there may be a bug in the
% implementation of your "f" function). The function returns the found
% solution "X", a vector of function values "fX" indicating the progress made
% and "i" the number of iterations (line searches or function evaluations,
% depending on the sign of "length") used.
%
% Usage: [X, fX, i] = fmincg(f, X, options, P1, P2, P3, P4, P5)
%
% See also: checkgrad
%
% Copyright (C) 2001 and 2002 by Carl Edward Rasmussen. Date 2002-02-13
%
%
% (C) Copyright 1999, 2000 & 2001, Carl Edward Rasmussen
%
% Permission is granted for anyone to copy, use, or modify these
% programs and accompanying documents for purposes of research or
% education, provided this copyright notice is retained, and note is
% made of any changes that have been made.
%
% These programs and documents are distributed without any warranty,
% express or implied. As the programs were written for research
% purposes only, they have not been tested to the degree that would be
% advisable in any important application. All use of these programs is
% entirely at the user's own risk.
%
% [ml-class] Changes Made:
% 1) Function name and argument specifications
% 2) Output display
%
% Read options
if exist('options', 'var') && ~isempty(options) && isfield(options, 'MaxIter')
length = options.MaxIter;
else
length = 100;
end
RHO = 0.01; % a bunch of constants for line searches
SIG = 0.5; % RHO and SIG are the constants in the Wolfe-Powell conditions
INT = 0.1; % don't reevaluate within 0.1 of the limit of the current bracket
EXT = 3.0; % extrapolate maximum 3 times the current bracket
MAX = 20; % max 20 function evaluations per line search
RATIO = 100; % maximum allowed slope ratio
argstr = ['feval(f, X']; % compose string used to call function
for i = 1:(nargin - 3)
argstr = [argstr, ',P', int2str(i)];
end
argstr = [argstr, ')'];
if max(size(length)) == 2
red=length(2);
length=length(1);
else red=1;
end
S=['Iteration '];
i = 0; % zero the run length counter
ls_failed = 0; % no previous line search has failed
fX = [];
[f1 df1] = eval(argstr); % get function value and gradient
i = i + (length<0); % count epochs?!
s = -df1; % search direction is steepest
d1 = -s'*s; % this is the slope
z1 = red/(1-d1); % initial step is red/(|s|+1)
while i < abs(length) % while not finished
i = i + (length>0); % count iterations?!
X0 = X; f0 = f1; df0 = df1; % make a copy of current values
X = X + z1*s; % begin line search
[f2 df2] = eval(argstr);
i = i + (length<0); % count epochs?!
d2 = df2'*s;
f3 = f1; d3 = d1; z3 = -z1; % initialize point 3 equal to point 1
if length>0, M = MAX; else M = min(MAX, -length-i); end
success = 0; limit = -1; % initialize quanteties
while 1
while ((f2 > f1+z1*RHO*d1) || (d2 > -SIG*d1)) && (M > 0)
limit = z1; % tighten the bracket
if f2 > f1
z2 = z3 - (0.5*d3*z3*z3)/(d3*z3+f2-f3); % quadratic fit
else
A = 6*(f2-f3)/z3+3*(d2+d3); % cubic fit
B = 3*(f3-f2)-z3*(d3+2*d2);
z2 = (sqrt(B*B-A*d2*z3*z3)-B)/A; % numerical error possible - ok!
end
if isnan(z2) || isinf(z2)
z2 = z3/2; % if we had a numerical problem then bisect
end
z2 = max(min(z2, INT*z3),(1-INT)*z3); % don't accept too close to limits
z1 = z1 + z2; % update the step
X = X + z2*s;
[f2 df2] = eval(argstr);
M = M - 1; i = i + (length<0); % count epochs?!
d2 = df2'*s;
z3 = z3-z2; % z3 is now relative to the location of z2
end
if f2 > f1+z1*RHO*d1 || d2 > -SIG*d1
break; % this is a failure
elseif d2 > SIG*d1
success = 1; break; % success
elseif M == 0
break; % failure
end
A = 6*(f2-f3)/z3+3*(d2+d3); % make cubic extrapolation
B = 3*(f3-f2)-z3*(d3+2*d2);
z2 = -d2*z3*z3/(B+sqrt(B*B-A*d2*z3*z3)); % num. error possible - ok!
if ~isreal(z2) || isnan(z2) || isinf(z2) || z2 < 0 % num prob or wrong sign?
if limit < -0.5 % if we have no upper limit
z2 = z1 * (EXT-1); % the extrapolate the maximum amount
else
z2 = (limit-z1)/2; % otherwise bisect
end
elseif (limit > -0.5) && (z2+z1 > limit) % extraplation beyond max?
z2 = (limit-z1)/2; % bisect
elseif (limit < -0.5) && (z2+z1 > z1*EXT) % extrapolation beyond limit
z2 = z1*(EXT-1.0); % set to extrapolation limit
elseif z2 < -z3*INT
z2 = -z3*INT;
elseif (limit > -0.5) && (z2 < (limit-z1)*(1.0-INT)) % too close to limit?
z2 = (limit-z1)*(1.0-INT);
end
f3 = f2; d3 = d2; z3 = -z2; % set point 3 equal to point 2
z1 = z1 + z2; X = X + z2*s; % update current estimates
[f2 df2] = eval(argstr);
M = M - 1; i = i + (length<0); % count epochs?!
d2 = df2'*s;
end % end of line search
if success % if line search succeeded
f1 = f2; fX = [fX' f1]';
fprintf('%s %4i | Cost: %4.6e\r', S, i, f1);
s = (df2'*df2-df1'*df2)/(df1'*df1)*s - df2; % Polack-Ribiere direction
tmp = df1; df1 = df2; df2 = tmp; % swap derivatives
d2 = df1'*s;
if d2 > 0 % new slope must be negative
s = -df1; % otherwise use steepest direction
d2 = -s'*s;
end
z1 = z1 * min(RATIO, d1/(d2-realmin)); % slope ratio but max RATIO
d1 = d2;
ls_failed = 0; % this line search did not fail
else
X = X0; f1 = f0; df1 = df0; % restore point from before failed line search
if ls_failed || i > abs(length) % line search failed twice in a row
break; % or we ran out of time, so we give up
end
tmp = df1; df1 = df2; df2 = tmp; % swap derivatives
s = -df1; % try steepest
d1 = -s'*s;
z1 = 1/(1-d1);
ls_failed = 1; % this line search failed
end
if exist('OCTAVE_VERSION')
fflush(stdout);
end
end
fprintf('\n');
end
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.
mu = mean(X);
X_norm = bsxfun(@minus, X, mu);
sigma = std(X_norm);
X_norm = bsxfun(@rdivide, X_norm, sigma);
% ============================================================
end
function plotFit(min_x, max_x, mu, sigma, theta, p)
%PLOTFIT Plots a learned polynomial regression fit over an existing figure.
%Also works with linear regression.
% PLOTFIT(min_x, max_x, mu, sigma, theta, p) plots the learned polynomial
% fit with power p and feature normalization (mu, sigma).
% Hold on to the current figure
hold on;
% We plot a range slightly bigger than the min and max values to get
% an idea of how the fit will vary outside the range of the data points
x = (min_x - 15: 0.05 : max_x + 25)';
% Map the X values
X_poly = polyFeatures(x, p);
X_poly = bsxfun(@minus, X_poly, mu);
X_poly = bsxfun(@rdivide, X_poly, sigma);
% Add ones
X_poly = [ones(size(x, 1), 1) X_poly];
% Plot
plot(x, X_poly * theta, '--', 'LineWidth', 2)
% Hold off to the current figure
hold off
end