Commit d4370156 authored by RACHIT BANSAL's avatar RACHIT BANSAL

add Anomaly_detection_and_recommender_systems

parents
function checkCostFunction(lambda)
%CHECKCOSTFUNCTION Creates a collaborative filering problem
%to check your cost function and gradients
% CHECKCOSTFUNCTION(lambda) Creates a collaborative filering problem
% to check your cost function and gradients, it will output the
% analytical gradients produced by your code and the numerical gradients
% (computed using computeNumericalGradient). These two gradient
% computations should result in very similar values.
% Set lambda
if ~exist('lambda', 'var') || isempty(lambda)
lambda = 0;
end
%% Create small problem
X_t = rand(4, 3);
Theta_t = rand(5, 3);
% Zap out most entries
Y = X_t * Theta_t';
Y(rand(size(Y)) > 0.5) = 0;
R = zeros(size(Y));
R(Y ~= 0) = 1;
%% Run Gradient Checking
X = randn(size(X_t));
Theta = randn(size(Theta_t));
num_users = size(Y, 2);
num_movies = size(Y, 1);
num_features = size(Theta_t, 2);
numgrad = computeNumericalGradient( ...
@(t) cofiCostFunc(t, Y, R, num_users, num_movies, ...
num_features, lambda), [X(:); Theta(:)]);
[cost, grad] = cofiCostFunc([X(:); Theta(:)], Y, R, num_users, ...
num_movies, num_features, lambda);
disp([numgrad grad]);
fprintf(['The above two columns you get should be very similar.\n' ...
'(Left-Your Numerical Gradient, Right-Analytical Gradient)\n\n']);
diff = norm(numgrad-grad)/norm(numgrad+grad);
fprintf(['If your cost function implementation is correct, then \n' ...
'the relative difference will be small (less than 1e-9). \n' ...
'\nRelative Difference: %g\n'], diff);
end
\ No newline at end of file
function [J, grad] = cofiCostFunc(params, Y, R, num_users, num_movies, ...
num_features, lambda)
%COFICOSTFUNC Collaborative filtering cost function
% [J, grad] = COFICOSTFUNC(params, Y, R, num_users, num_movies, ...
% num_features, lambda) returns the cost and gradient for the
% collaborative filtering problem.
%
% Unfold the U and W matrices from params
X = reshape(params(1:num_movies*num_features), num_movies, num_features);
Theta = reshape(params(num_movies*num_features+1:end), ...
num_users, num_features);
% You need to return the following values correctly
J = 0;
X_grad = zeros(size(X));
Theta_grad = zeros(size(Theta));
% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost function and gradient for collaborative
% filtering. Concretely, you should first implement the cost
% function (without regularization) and make sure it is
% matches our costs. After that, you should implement the
% gradient and use the checkCostFunction routine to check
% that the gradient is correct. Finally, you should implement
% regularization.
%
% Notes: X - num_movies x num_features matrix of movie features
% Theta - num_users x num_features matrix of user features
% Y - num_movies x num_users matrix of user ratings of movies
% R - num_movies x num_users matrix, where R(i, j) = 1 if the
% i-th movie was rated by the j-th user
%
% You should set the following variables correctly:
%
% X_grad - num_movies x num_features matrix, containing the
% partial derivatives w.r.t. to each element of X
% Theta_grad - num_users x num_features matrix, containing the
% partial derivatives w.r.t. to each element of Theta
%
J=sum(sum(((X*(Theta')-Y).^2).*R))/2+lambda*(sum(sum(Theta.^2)))/2 +lambda*(sum(sum(X.^2)))/2;
for i=1:num_movies
idx = find(R(i, :)==1);
Thetatemp = Theta(idx, :);
Ytemp = Y(i, idx);
X_grad(i, :) = (X(i, :) * (Thetatemp') - Ytemp) * Thetatemp;
end
for i=1:num_users
idx = find(R(:,i)==1);
Xtemp = X(idx, :);
Ytemp = Y(idx, i);
Theta_grad(i, :) = (Theta(i, :) * (Xtemp') - Ytemp') * Xtemp;
end
X_grad=X_grad+lambda.*X;
Theta_grad=Theta_grad+lambda.*Theta;
% =============================================================
grad = [X_grad(:); Theta_grad(:)];
end
function numgrad = computeNumericalGradient(J, theta)
%COMPUTENUMERICALGRADIENT Computes the gradient using "finite differences"
%and gives us a numerical estimate of the gradient.
% numgrad = COMPUTENUMERICALGRADIENT(J, theta) computes the numerical
% gradient of the function J around theta. Calling y = J(theta) should
% return the function value at theta.
% Notes: The following code implements numerical gradient checking, and
% returns the numerical gradient.It sets numgrad(i) to (a numerical
% approximation of) the partial derivative of J with respect to the
% i-th input argument, evaluated at theta. (i.e., numgrad(i) should
% be the (approximately) the partial derivative of J with respect
% to theta(i).)
%
numgrad = zeros(size(theta));
perturb = zeros(size(theta));
e = 1e-4;
for p = 1:numel(theta)
% Set perturbation vector
perturb(p) = e;
loss1 = J(theta - perturb);
loss2 = J(theta + perturb);
% Compute Numerical Gradient
numgrad(p) = (loss2 - loss1) / (2*e);
perturb(p) = 0;
end
end
function [mu sigma2] = estimateGaussian(X)
%ESTIMATEGAUSSIAN This function estimates the parameters of a
%Gaussian distribution using the data in X
% [mu sigma2] = estimateGaussian(X),
% The input X is the dataset with each n-dimensional data point in one row
% The output is an n-dimensional vector mu, the mean of the data set
% and the variances sigma^2, an n x 1 vector
%
% Useful variables
[m, n] = size(X);
% You should return these values correctly
mu = zeros(n, 1);
sigma2 = zeros(n, 1);
% ====================== YOUR CODE HERE ======================
% Instructions: Compute the mean of the data and the variances
% In particular, mu(i) should contain the mean of
% the data for the i-th feature and sigma2(i)
% should contain variance of the i-th feature.
%
mu=(sum(X))'/m;
for i=1:n
for j=1:m
sigma2(i)=sigma2(i)+ (X(j,i)-mu(i)).^2;
end
end
sigma2=sigma2/m;
% =============================================================
end
%% Machine Learning Online Class
% Exercise 8 | Anomaly Detection and Collaborative Filtering
%
% Instructions
% ------------
%
% This file contains code that helps you get started on the
% exercise. You will need to complete the following functions:
%
% estimateGaussian.m
% selectThreshold.m
% cofiCostFunc.m
%
% For this exercise, you will not need to change any code in this file,
% or any other files other than those mentioned above.
%
%% Initialization
clear ; close all; clc
%% ================== Part 1: Load Example Dataset ===================
% We start this exercise by using a small dataset that is easy to
% visualize.
%
% Our example case consists of 2 network server statistics across
% several machines: the latency and throughput of each machine.
% This exercise will help us find possibly faulty (or very fast) machines.
%
fprintf('Visualizing example dataset for outlier detection.\n\n');
% The following command loads the dataset. You should now have the
% variables X, Xval, yval in your environment
load('ex8data1.mat');
% Visualize the example dataset
plot(X(:, 1), X(:, 2), 'bx');
axis([0 30 0 30]);
xlabel('Latency (ms)');
ylabel('Throughput (mb/s)');
fprintf('Program paused. Press enter to continue.\n');
pause
%% ================== Part 2: Estimate the dataset statistics ===================
% For this exercise, we assume a Gaussian distribution for the dataset.
%
% We first estimate the parameters of our assumed Gaussian distribution,
% then compute the probabilities for each of the points and then visualize
% both the overall distribution and where each of the points falls in
% terms of that distribution.
%
fprintf('Visualizing Gaussian fit.\n\n');
% Estimate my and sigma2
[mu sigma2] = estimateGaussian(X);
% Returns the density of the multivariate normal at each data point (row)
% of X
p = multivariateGaussian(X, mu, sigma2);
% Visualize the fit
visualizeFit(X, mu, sigma2);
xlabel('Latency (ms)');
ylabel('Throughput (mb/s)');
fprintf('Program paused. Press enter to continue.\n');
pause;
%% ================== Part 3: Find Outliers ===================
% Now you will find a good epsilon threshold using a cross-validation set
% probabilities given the estimated Gaussian distribution
%
pval = multivariateGaussian(Xval, mu, sigma2);
[epsilon F1] = selectThreshold(yval, pval);
fprintf('Best epsilon found using cross-validation: %e\n', epsilon);
fprintf('Best F1 on Cross Validation Set: %f\n', F1);
fprintf(' (you should see a value epsilon of about 8.99e-05)\n');
fprintf(' (you should see a Best F1 value of 0.875000)\n\n');
% Find the outliers in the training set and plot the
outliers = find(p < epsilon);
% Draw a red circle around those outliers
hold on
plot(X(outliers, 1), X(outliers, 2), 'ro', 'LineWidth', 2, 'MarkerSize', 10);
hold off
fprintf('Program paused. Press enter to continue.\n');
pause;
%% ================== Part 4: Multidimensional Outliers ===================
% We will now use the code from the previous part and apply it to a
% harder problem in which more features describe each datapoint and only
% some features indicate whether a point is an outlier.
%
% Loads the second dataset. You should now have the
% variables X, Xval, yval in your environment
load('ex8data2.mat');
% Apply the same steps to the larger dataset
[mu sigma2] = estimateGaussian(X);
% Training set
p = multivariateGaussian(X, mu, sigma2);
% Cross-validation set
pval = multivariateGaussian(Xval, mu, sigma2);
% Find the best threshold
[epsilon F1] = selectThreshold(yval, pval);
fprintf('Best epsilon found using cross-validation: %e\n', epsilon);
fprintf('Best F1 on Cross Validation Set: %f\n', F1);
fprintf(' (you should see a value epsilon of about 1.38e-18)\n');
fprintf(' (you should see a Best F1 value of 0.615385)\n');
fprintf('# Outliers found: %d\n\n', sum(p < epsilon));
%% Machine Learning Online Class
% Exercise 8 | Anomaly Detection and Collaborative Filtering
%
% Instructions
% ------------
%
% This file contains code that helps you get started on the
% exercise. You will need to complete the following functions:
%
% estimateGaussian.m
% selectThreshold.m
% cofiCostFunc.m
%
% For this exercise, you will not need to change any code in this file,
% or any other files other than those mentioned above.
%
%% =============== Part 1: Loading movie ratings dataset ================
% You will start by loading the movie ratings dataset to understand the
% structure of the data.
%
fprintf('Loading movie ratings dataset.\n\n');
% Load data
load ('ex8_movies.mat');
% Y is a 1682x943 matrix, containing ratings (1-5) of 1682 movies on
% 943 users
%
% R is a 1682x943 matrix, where R(i,j) = 1 if and only if user j gave a
% rating to movie i
% From the matrix, we can compute statistics like average rating.
fprintf('Average rating for movie 1 (Toy Story): %f / 5\n\n', ...
mean(Y(1, R(1, :))));
% We can "visualize" the ratings matrix by plotting it with imagesc
imagesc(Y);
ylabel('Movies');
xlabel('Users');
fprintf('\nProgram paused. Press enter to continue.\n');
pause;
%% ============ Part 2: Collaborative Filtering Cost Function ===========
% You will now implement the cost function for collaborative filtering.
% To help you debug your cost function, we have included set of weights
% that we trained on that. Specifically, you should complete the code in
% cofiCostFunc.m to return J.
% Load pre-trained weights (X, Theta, num_users, num_movies, num_features)
load ('ex8_movieParams.mat');
% Reduce the data set size so that this runs faster
num_users = 4; num_movies = 5; num_features = 3;
X = X(1:num_movies, 1:num_features);
Theta = Theta(1:num_users, 1:num_features);
Y = Y(1:num_movies, 1:num_users);
R = R(1:num_movies, 1:num_users);
% Evaluate cost function
J = cofiCostFunc([X(:) ; Theta(:)], Y, R, num_users, num_movies, ...
num_features, 0);
fprintf(['Cost at loaded parameters: %f '...
'\n(this value should be about 22.22)\n'], J);
fprintf('\nProgram paused. Press enter to continue.\n');
pause;
%% ============== Part 3: Collaborative Filtering Gradient ==============
% Once your cost function matches up with ours, you should now implement
% the collaborative filtering gradient function. Specifically, you should
% complete the code in cofiCostFunc.m to return the grad argument.
%
fprintf('\nChecking Gradients (without regularization) ... \n');
% Check gradients by running checkNNGradients
checkCostFunction;
fprintf('\nProgram paused. Press enter to continue.\n');
pause;
%% ========= Part 4: Collaborative Filtering Cost Regularization ========
% Now, you should implement regularization for the cost function for
% collaborative filtering. You can implement it by adding the cost of
% regularization to the original cost computation.
%
% Evaluate cost function
J = cofiCostFunc([X(:) ; Theta(:)], Y, R, num_users, num_movies, ...
num_features, 1.5);
fprintf(['Cost at loaded parameters (lambda = 1.5): %f '...
'\n(this value should be about 31.34)\n'], J);
fprintf('\nProgram paused. Press enter to continue.\n');
pause;
%% ======= Part 5: Collaborative Filtering Gradient Regularization ======
% Once your cost matches up with ours, you should proceed to implement
% regularization for the gradient.
%
%
fprintf('\nChecking Gradients (with regularization) ... \n');
% Check gradients by running checkNNGradients
checkCostFunction(1.5);
fprintf('\nProgram paused. Press enter to continue.\n');
pause;
%% ============== Part 6: Entering ratings for a new user ===============
% Before we will train the collaborative filtering model, we will first
% add ratings that correspond to a new user that we just observed. This
% part of the code will also allow you to put in your own ratings for the
% movies in our dataset!
%
movieList = loadMovieList();
% Initialize my ratings
my_ratings = zeros(1682, 1);
% Check the file movie_idx.txt for id of each movie in our dataset
% For example, Toy Story (1995) has ID 1, so to rate it "4", you can set
my_ratings(1) = 4;
% Or suppose did not enjoy Silence of the Lambs (1991), you can set
my_ratings(98) = 2;
% We have selected a few movies we liked / did not like and the ratings we
% gave are as follows:
my_ratings(7) = 3;
my_ratings(12)= 5;
my_ratings(54) = 4;
my_ratings(64)= 5;
my_ratings(66)= 3;
my_ratings(69) = 5;
my_ratings(183) = 4;
my_ratings(226) = 5;
my_ratings(355)= 5;
fprintf('\n\nNew user ratings:\n');
for i = 1:length(my_ratings)
if my_ratings(i) > 0
fprintf('Rated %d for %s\n', my_ratings(i), ...
movieList{i});
end
end
fprintf('\nProgram paused. Press enter to continue.\n');
pause;
%% ================== Part 7: Learning Movie Ratings ====================
% Now, you will train the collaborative filtering model on a movie rating
% dataset of 1682 movies and 943 users
%
fprintf('\nTraining collaborative filtering...\n');
% Load data
load('ex8_movies.mat');
% Y is a 1682x943 matrix, containing ratings (1-5) of 1682 movies by
% 943 users
%
% R is a 1682x943 matrix, where R(i,j) = 1 if and only if user j gave a
% rating to movie i
% Add our own ratings to the data matrix
Y = [my_ratings Y];
R = [(my_ratings ~= 0) R];
% Normalize Ratings
[Ynorm, Ymean] = normalizeRatings(Y, R);
% Useful Values
num_users = size(Y, 2);
num_movies = size(Y, 1);
num_features = 10;
% Set Initial Parameters (Theta, X)
X = randn(num_movies, num_features);
Theta = randn(num_users, num_features);
initial_parameters = [X(:); Theta(:)];
% Set options for fmincg
options = optimset('GradObj', 'on', 'MaxIter', 100);
% Set Regularization
lambda = 10;
theta = fmincg (@(t)(cofiCostFunc(t, Ynorm, R, num_users, num_movies, ...
num_features, lambda)), ...
initial_parameters, options);
% Unfold the returned theta back into U and W
X = reshape(theta(1:num_movies*num_features), num_movies, num_features);
Theta = reshape(theta(num_movies*num_features+1:end), ...
num_users, num_features);
fprintf('Recommender system learning completed.\n');
fprintf('\nProgram paused. Press enter to continue.\n');
pause;
%% ================== Part 8: Recommendation for you ====================
% After training the model, you can now make recommendations by computing
% the predictions matrix.
%
p = X * Theta';
my_predictions = p(:,1) + Ymean;
movieList = loadMovieList();
[r, ix] = sort(my_predictions, 'descend');
fprintf('\nTop recommendations for you:\n');
for i=1:10
j = ix(i);
fprintf('Predicting rating %.1f for movie %s\n', my_predictions(j), ...
movieList{j});
end
fprintf('\n\nOriginal ratings provided:\n');
for i = 1:length(my_ratings)
if my_ratings(i) > 0
fprintf('Rated %d for %s\n', my_ratings(i), ...
movieList{i});
end
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