Added solution to assignments

machine-learning-ex5/ex5/ex5.m

+%% Machine Learning Online Class
+%  Exercise 5 | Regularized Linear Regression and Bias-Variance
+%
+%  Instructions
+%  ------------
+% 
+%  This file contains code that helps you get started on the
+%  exercise. You will need to complete the following functions:
+%
+%     linearRegCostFunction.m
+%     learningCurve.m
+%     validationCurve.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: Loading and Visualizing Data =============
+%  We start the exercise by first loading and visualizing the dataset. 
+%  The following code will load the dataset into your environment and plot
+%  the data.
+%
+
+% Load Training Data
+fprintf('Loading and Visualizing Data ...\n')
+
+% Load from ex5data1: 
+% You will have X, y, Xval, yval, Xtest, ytest in your environment
+load ('ex5data1.mat');
+
+% m = Number of examples
+m = size(X, 1);
+
+% Plot training data
+plot(X, y, 'rx', 'MarkerSize', 10, 'LineWidth', 1.5);
+xlabel('Change in water level (x)');
+ylabel('Water flowing out of the dam (y)');
+
+fprintf('Program paused. Press enter to continue.\n');
+pause;
+
+%% =========== Part 2: Regularized Linear Regression Cost =============
+%  You should now implement the cost function for regularized linear 
+%  regression. 
+%
+
+theta = [1 ; 1];
+J = linearRegCostFunction([ones(m, 1) X], y, theta, 1);
+
+fprintf(['Cost at theta = [1 ; 1]: %f '...
+         '\n(this value should be about 303.993192)\n'], J);
+
+fprintf('Program paused. Press enter to continue.\n');
+pause;
+
+%% =========== Part 3: Regularized Linear Regression Gradient =============
+%  You should now implement the gradient for regularized linear 
+%  regression.
+%
+
+theta = [1 ; 1];
+[J, grad] = linearRegCostFunction([ones(m, 1) X], y, theta, 1);
+
+fprintf(['Gradient at theta = [1 ; 1]:  [%f; %f] '...
+         '\n(this value should be about [-15.303016; 598.250744])\n'], ...
+         grad(1), grad(2));
+
+fprintf('Program paused. Press enter to continue.\n');
+pause;
+
+
+%% =========== Part 4: Train Linear Regression =============
+%  Once you have implemented the cost and gradient correctly, the
+%  trainLinearReg function will use your cost function to train 
+%  regularized linear regression.
+% 
+%  Write Up Note: The data is non-linear, so this will not give a great 
+%                 fit.
+%
+
+%  Train linear regression with lambda = 0
+lambda = 0;
+[theta] = trainLinearReg([ones(m, 1) X], y, lambda);
+
+%  Plot fit over the data
+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, '--', 'LineWidth', 2)
+hold off;
+
+fprintf('Program paused. Press enter to continue.\n');
+pause;
+
+
+%% =========== Part 5: Learning Curve for Linear Regression =============
+%  Next, you should implement the learningCurve function. 
+%
+%  Write Up Note: Since the model is underfitting the data, we expect to
+%                 see a graph with "high bias" -- Figure 3 in ex5.pdf 
+%
+
+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
+
+fprintf('Program paused. Press enter to continue.\n');
+pause;
+
+%% =========== Part 6: Feature Mapping for Polynomial Regression =============
+%  One solution to this is to use polynomial regression. You should now
+%  complete polyFeatures to map each example into its powers
+%
+
+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 = bsxfun(@minus, X_poly_test, mu);
+X_poly_test = bsxfun(@rdivide, X_poly_test, sigma);
+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 = bsxfun(@minus, X_poly_val, mu);
+X_poly_val = bsxfun(@rdivide, X_poly_val, sigma);
+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, :));
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+
+
+%% =========== Part 7: Learning Curve for Polynomial Regression =============
+%  Now, you will get to experiment with polynomial regression with multiple
+%  values of lambda. The code below runs polynomial regression with 
+%  lambda = 0. You should try running the code with different values of
+%  lambda to see how the fit and learning curve change.
+%
+
+lambda = 0;
+[theta] = trainLinearReg(X_poly, y, lambda);
+
+% Plot training data and fit
+figure(1);
+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));
+
+figure(2);
+[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')
+
+fprintf('Polynomial Regression (lambda = %f)\n\n', lambda);
+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
+
+fprintf('Program paused. Press enter to continue.\n');
+pause;
+
+%% =========== Part 8: Validation for Selecting Lambda =============
+%  You will now implement validationCurve to test various values of 
+%  lambda on a validation set. You will then use this to select the
+%  "best" lambda value.
+%
+
+[lambda_vec, error_train, error_val] = ...
+    validationCurve(X_poly, y, X_poly_val, yval);
+
+close all;
+plot(lambda_vec, error_train, lambda_vec, error_val);
+legend('Train', 'Cross Validation');
+xlabel('lambda');
+ylabel('Error');
+
+fprintf('lambda\t\tTrain Error\tValidation Error\n');
+for i = 1:length(lambda_vec)
+	fprintf(' %f\t%f\t%f\n', ...
+            lambda_vec(i), error_train(i), error_val(i));
+end
+
+fprintf('Program paused. Press enter to continue.\n');
+pause;

machine-learning-ex5/ex5/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.
+
+mu = mean(X);
+X_norm = bsxfun(@minus, X, mu);
+
+sigma = std(X_norm);
+X_norm = bsxfun(@rdivide, X_norm, sigma);
+
+
+% ============================================================
+
+end

machine-learning-ex5/ex5/fmincg.m

+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');

machine-learning-ex5/ex5/learningCurve.m

+function [error_train, error_val] = ...
+    learningCurve(X, y, Xval, yval, lambda)
+%LEARNINGCURVE Generates the train and cross validation set errors needed 
+%to plot a learning curve
+%   [error_train, error_val] = ...
+%       LEARNINGCURVE(X, y, Xval, yval, lambda) returns the train and
+%       cross validation set errors for a learning curve. In particular, 
+%       it returns two vectors of the same length - error_train and 
+%       error_val. Then, error_train(i) contains the training error for
+%       i examples (and similarly for error_val(i)).
+%
+%   In this function, you will compute the train and test errors for
+%   dataset sizes from 1 up to m. In practice, when working with larger
+%   datasets, you might want to do this in larger intervals.
+%
+
+% Number of training examples
+m = size(X, 1);
+
+% You need to return these values correctly
+error_train = zeros(m, 1);
+error_val   = zeros(m, 1);
+
+% ====================== YOUR CODE HERE ======================
+% Instructions: Fill in this function to return training errors in 
+%               error_train and the cross validation errors in error_val. 
+%               i.e., error_train(i) and 
+%               error_val(i) should give you the errors
+%               obtained after training on i examples.
+%
+% Note: You should evaluate the training error on the first i training
+%       examples (i.e., X(1:i, :) and y(1:i)).
+%
+%       For the cross-validation error, you should instead evaluate on
+%       the _entire_ cross validation set (Xval and yval).
+%
+% Note: If you are using your cost function (linearRegCostFunction)
+%       to compute the training and cross validation error, you should 
+%       call the function with the lambda argument set to 0. 
+%       Do note that you will still need to use lambda when running
+%       the training to obtain the theta parameters.
+%
+% Hint: You can loop over the examples with the following:
+%
+%       for i = 1:m
+%           % Compute train/cross validation errors using training examples 
+%           % X(1:i, :) and y(1:i), storing the result in 
+%           % error_train(i) and error_val(i)
+%           ....
+%           
+%       end
+%
+
+% ---------------------- Sample Solution ----------------------
+for i = 1:m
+    X_train = X(1:i,:);
+    y_train = y(1:i);
+    
+    theta = trainLinearReg(X_train, y_train, lambda);
+    
+    error_train(i) = linearRegCostFunction(X_train, y_train, theta, 0);
+    error_val(i) = linearRegCostFunction(Xval, yval, theta, 0);
+end
+% -------------------------------------------------------------
+
+% =========================================================================
+
+end

machine-learning-ex5/ex5/lib/jsonlab/AUTHORS.txt

+The author of "jsonlab" toolbox is Qianqian Fang. Qianqian
+is currently an Assistant Professor at Massachusetts General Hospital, 
+Harvard Medical School.
+
+Address: Martinos Center for Biomedical Imaging, 
+         Massachusetts General Hospital, 
+         Harvard Medical School
+         Bldg 149, 13th St, Charlestown, MA 02129, USA
+URL: http://nmr.mgh.harvard.edu/~fangq/
+Email: <fangq at nmr.mgh.harvard.edu> or <fangqq at gmail.com>
+
+
+The script loadjson.m was built upon previous works by
+
+- Nedialko Krouchev: http://www.mathworks.com/matlabcentral/fileexchange/25713
+       date: 2009/11/02
+- François Glineur: http://www.mathworks.com/matlabcentral/fileexchange/23393
+       date: 2009/03/22
+- Joel Feenstra: http://www.mathworks.com/matlabcentral/fileexchange/20565
+       date: 2008/07/03
+
+
+This toolbox contains patches submitted by the following contributors:
+
+- Blake Johnson <bjohnso at bbn.com>
+  part of revision 341
+
+- Niclas Borlin <Niclas.Borlin at cs.umu.se>
+  various fixes in revision 394, including
+  - loadjson crashes for all-zero sparse matrix.
+  - loadjson crashes for empty sparse matrix.
+  - Non-zero size of 0-by-N and N-by-0 empty matrices is lost after savejson/loadjson.
+  - loadjson crashes for sparse real column vector.
+  - loadjson crashes for sparse complex column vector.
+  - Data is corrupted by savejson for sparse real row vector.
+  - savejson crashes for sparse complex row vector. 
+
+- Yul Kang <yul.kang.on at gmail.com>
+  patches for svn revision 415.
+  - savejson saves an empty cell array as [] instead of null
+  - loadjson differentiates an empty struct from an empty array

machine-learning-ex5/ex5/lib/jsonlab/ChangeLog.txt

+============================================================================
+
+   JSONlab - a toolbox to encode/decode JSON/UBJSON files in MATLAB/Octave
+
+----------------------------------------------------------------------------
+
+JSONlab ChangeLog (key features marked by *):
+
+== JSONlab 1.0 (codename: Optimus - Final), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2015/01/02 polish help info for all major functions, update examples, finalize 1.0
+ 2014/12/19 fix a bug to strictly respect NoRowBracket in savejson
+
+== JSONlab 1.0.0-RC2 (codename: Optimus - RC2), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2014/11/22 show progress bar in loadjson ('ShowProgress') 
+ 2014/11/17 add Compact option in savejson to output compact JSON format ('Compact')
+ 2014/11/17 add FastArrayParser in loadjson to specify fast parser applicable levels
+ 2014/09/18 start official github mirror: https://github.com/fangq/jsonlab
+
+== JSONlab 1.0.0-RC1 (codename: Optimus - RC1), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2014/09/17  fix several compatibility issues when running on octave versions 3.2-3.8
+ 2014/09/17  support 2D cell and struct arrays in both savejson and saveubjson
+ 2014/08/04  escape special characters in a JSON string
+ 2014/02/16  fix a bug when saving ubjson files
+
+== JSONlab 0.9.9 (codename: Optimus - beta), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2014/01/22  use binary read and write in saveubjson and loadubjson
+
+== JSONlab 0.9.8-1 (codename: Optimus - alpha update 1), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2013/10/07 better round-trip conservation for empty arrays and structs (patch submitted by Yul Kang)
+
+== JSONlab 0.9.8 (codename: Optimus - alpha), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+ 2013/08/23 *universal Binary JSON (UBJSON) support, including both saveubjson and loadubjson
+
+== JSONlab 0.9.1 (codename: Rodimus, update 1), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+ 2012/12/18 *handling of various empty and sparse matrices (fixes submitted by Niclas Borlin)
+
+== JSONlab 0.9.0 (codename: Rodimus), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2012/06/17 *new format for an invalid leading char, unpacking hex code in savejson
+ 2012/06/01  support JSONP in savejson
+ 2012/05/25  fix the empty cell bug (reported by Cyril Davin)
+ 2012/04/05  savejson can save to a file (suggested by Patrick Rapin)
+
+== JSONlab 0.8.1 (codename: Sentiel, Update 1), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2012/02/28  loadjson quotation mark escape bug, see http://bit.ly/yyk1nS
+ 2012/01/25  patch to handle root-less objects, contributed by Blake Johnson
+
+== JSONlab 0.8.0 (codename: Sentiel), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2012/01/13 *speed up loadjson by 20 fold when parsing large data arrays in matlab
+ 2012/01/11  remove row bracket if an array has 1 element, suggested by Mykel Kochenderfer
+ 2011/12/22 *accept sequence of 'param',value input in savejson and loadjson
+ 2011/11/18  fix struct array bug reported by Mykel Kochenderfer
+
+== JSONlab 0.5.1 (codename: Nexus Update 1), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2011/10/21  fix a bug in loadjson, previous code does not use any of the acceleration
+ 2011/10/20  loadjson supports JSON collections - concatenated JSON objects
+
+== JSONlab 0.5.0 (codename: Nexus), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2011/10/16  package and release jsonlab 0.5.0
+ 2011/10/15 *add json demo and regression test, support cpx numbers, fix double quote bug
+ 2011/10/11 *speed up readjson dramatically, interpret _Array* tags, show data in root level
+ 2011/10/10  create jsonlab project, start jsonlab website, add online documentation
+ 2011/10/07 *speed up savejson by 25x using sprintf instead of mat2str, add options support
+ 2011/10/06 *savejson works for structs, cells and arrays
+ 2011/09/09  derive loadjson from JSON parser from MATLAB Central, draft savejson.m

machine-learning-ex5/ex5/lib/jsonlab/LICENSE_BSD.txt

+Copyright 2011-2015 Qianqian Fang <fangq at nmr.mgh.harvard.edu>. All rights reserved.
+
+Redistribution and use in source and binary forms, with or without modification, are
+permitted provided that the following conditions are met:
+
+   1. Redistributions of source code must retain the above copyright notice, this list of
+      conditions and the following disclaimer.
+
+   2. Redistributions in binary form must reproduce the above copyright notice, this list
+      of conditions and the following disclaimer in the documentation and/or other materials
+      provided with the distribution.
+
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS ''AS IS'' AND ANY EXPRESS OR IMPLIED
+WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND
+FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDERS 
+OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
+SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON
+ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF
+ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+The views and conclusions contained in the software and documentation are those of the
+authors and should not be interpreted as representing official policies, either expressed
+or implied, of the copyright holders.

machine-learning-ex5/ex5/lib/jsonlab/jsonopt.m

+function val=jsonopt(key,default,varargin)
+%
+% val=jsonopt(key,default,optstruct)
+%
+% setting options based on a struct. The struct can be produced
+% by varargin2struct from a list of 'param','value' pairs
+%
+% authors:Qianqian Fang (fangq<at> nmr.mgh.harvard.edu)
+%
+% $Id: loadjson.m 371 2012-06-20 12:43:06Z fangq $
+%
+% input:
+%      key: a string with which one look up a value from a struct
+%      default: if the key does not exist, return default
+%      optstruct: a struct where each sub-field is a key 
+%
+% output:
+%      val: if key exists, val=optstruct.key; otherwise val=default
+%
+% license:
+%     BSD, see LICENSE_BSD.txt files for details
+%
+% -- this function is part of jsonlab toolbox (http://iso2mesh.sf.net/cgi-bin/index.cgi?jsonlab)
+% 
+
+val=default;
+if(nargin<=2) return; end
+opt=varargin{1};
+if(isstruct(opt) && isfield(opt,key))
+    val=getfield(opt,key);
+end
+

machine-learning-ex5/ex5/lib/jsonlab/mergestruct.m

+function s=mergestruct(s1,s2)
+%
+% s=mergestruct(s1,s2)
+%
+% merge two struct objects into one
+%
+% authors:Qianqian Fang (fangq<at> nmr.mgh.harvard.edu)
+% date: 2012/12/22
+%
+% input:
+%      s1,s2: a struct object, s1 and s2 can not be arrays
+%
+% output:
+%      s: the merged struct object. fields in s1 and s2 will be combined in s.
+%
+% license:
+%     BSD, see LICENSE_BSD.txt files for details 
+%
+% -- this function is part of jsonlab toolbox (http://iso2mesh.sf.net/cgi-bin/index.cgi?jsonlab)
+%
+
+if(~isstruct(s1) || ~isstruct(s2))
+    error('input parameters contain non-struct');
+end
+if(length(s1)>1 || length(s2)>1)
+    error('can not merge struct arrays');
+end
+fn=fieldnames(s2);
+s=s1;
+for i=1:length(fn)              
+    s=setfield(s,fn{i},getfield(s2,fn{i}));
+end
+

machine-learning-ex5/ex5/lib/jsonlab/varargin2struct.m

+function opt=varargin2struct(varargin)
+%
+% opt=varargin2struct('param1',value1,'param2',value2,...)
+%   or
+% opt=varargin2struct(...,optstruct,...)
+%
+% convert a series of input parameters into a structure
+%
+% authors:Qianqian Fang (fangq<at> nmr.mgh.harvard.edu)
+% date: 2012/12/22
+%
+% input:
+%      'param', value: the input parameters should be pairs of a string and a value
+%       optstruct: if a parameter is a struct, the fields will be merged to the output struct
+%
+% output:
+%      opt: a struct where opt.param1=value1, opt.param2=value2 ...
+%
+% license:
+%     BSD, see LICENSE_BSD.txt files for details 
+%
+% -- this function is part of jsonlab toolbox (http://iso2mesh.sf.net/cgi-bin/index.cgi?jsonlab)
+%
+
+len=length(varargin);
+opt=struct;
+if(len==0) return; end
+i=1;
+while(i<=len)
+    if(isstruct(varargin{i}))
+        opt=mergestruct(opt,varargin{i});
+    elseif(ischar(varargin{i}) && i<len)
+        opt=setfield(opt,varargin{i},varargin{i+1});
+        i=i+1;
+    else
+        error('input must be in the form of ...,''name'',value,... pairs or structs');
+    end
+    i=i+1;
+end
+

machine-learning-ex5/ex5/lib/makeValidFieldName.m

+function str = makeValidFieldName(str)
+% From MATLAB doc: field names must begin with a letter, which may be
+% followed by any combination of letters, digits, and underscores.
+% Invalid characters will be converted to underscores, and the prefix
+% "x0x[Hex code]_" will be added if the first character is not a letter.
+    isoct=exist('OCTAVE_VERSION','builtin');
+    pos=regexp(str,'^[^A-Za-z]','once');
+    if(~isempty(pos))
+        if(~isoct)
+            str=regexprep(str,'^([^A-Za-z])','x0x${sprintf(''%X'',unicode2native($1))}_','once');
+        else
+            str=sprintf('x0x%X_%s',char(str(1)),str(2:end));
+        end
+    end
+    if(isempty(regexp(str,'[^0-9A-Za-z_]', 'once' ))) return;  end
+    if(~isoct)
+        str=regexprep(str,'([^0-9A-Za-z_])','_0x${sprintf(''%X'',unicode2native($1))}_');
+    else
+        pos=regexp(str,'[^0-9A-Za-z_]');
+        if(isempty(pos)) return; end
+        str0=str;
+        pos0=[0 pos(:)' length(str)];
+        str='';
+        for i=1:length(pos)
+            str=[str str0(pos0(i)+1:pos(i)-1) sprintf('_0x%X_',str0(pos(i)))];
+        end
+        if(pos(end)~=length(str))
+            str=[str str0(pos0(end-1)+1:pos0(end))];
+        end
+    end

machine-learning-ex5/ex5/lib/submitWithConfiguration.m

+function submitWithConfiguration(conf)
+  addpath('./lib/jsonlab');
+
+  parts = parts(conf);
+
+  fprintf('== Submitting solutions | %s...\n', conf.itemName);
+
+  tokenFile = 'token.mat';
+  if exist(tokenFile, 'file')
+    load(tokenFile);
+    [email token] = promptToken(email, token, tokenFile);
+  else
+    [email token] = promptToken('', '', tokenFile);
+  end
+
+  if isempty(token)
+    fprintf('!! Submission Cancelled\n');
+    return
+  end
+
+  try
+    response = submitParts(conf, email, token, parts);
+  catch
+    e = lasterror();
+    fprintf('\n!! Submission failed: %s\n', e.message);
+    fprintf('\n\nFunction: %s\nFileName: %s\nLineNumber: %d\n', ...
+      e.stack(1,1).name, e.stack(1,1).file, e.stack(1,1).line);
+    fprintf('\nPlease correct your code and resubmit.\n');
+    return
+  end
+
+  if isfield(response, 'errorMessage')
+    fprintf('!! Submission failed: %s\n', response.errorMessage);
+  elseif isfield(response, 'errorCode')
+    fprintf('!! Submission failed: %s\n', response.message);
+  else
+    showFeedback(parts, response);
+    save(tokenFile, 'email', 'token');
+  end
+end
+
+function [email token] = promptToken(email, existingToken, tokenFile)
+  if (~isempty(email) && ~isempty(existingToken))
+    prompt = sprintf( ...
+      'Use token from last successful submission (%s)? (Y/n): ', ...
+      email);
+    reenter = input(prompt, 's');
+
+    if (isempty(reenter) || reenter(1) == 'Y' || reenter(1) == 'y')
+      token = existingToken;
+      return;
+    else
+      delete(tokenFile);
+    end
+  end
+  email = input('Login (email address): ', 's');
+  token = input('Token: ', 's');
+end
+
+function isValid = isValidPartOptionIndex(partOptions, i)
+  isValid = (~isempty(i)) && (1 <= i) && (i <= numel(partOptions));
+end
+
+function response = submitParts(conf, email, token, parts)
+  body = makePostBody(conf, email, token, parts);
+  submissionUrl = submissionUrl();
+
+  responseBody = getResponse(submissionUrl, body);
+  jsonResponse = validateResponse(responseBody);
+  response = loadjson(jsonResponse);
+end
+
+function body = makePostBody(conf, email, token, parts)
+  bodyStruct.assignmentSlug = conf.assignmentSlug;
+  bodyStruct.submitterEmail = email;
+  bodyStruct.secret = token;
+  bodyStruct.parts = makePartsStruct(conf, parts);
+
+  opt.Compact = 1;
+  body = savejson('', bodyStruct, opt);
+end
+
+function partsStruct = makePartsStruct(conf, parts)
+  for part = parts
+    partId = part{:}.id;
+    fieldName = makeValidFieldName(partId);
+    outputStruct.output = conf.output(partId);
+    partsStruct.(fieldName) = outputStruct;
+  end
+end
+
+function [parts] = parts(conf)
+  parts = {};
+  for partArray = conf.partArrays
+    part.id = partArray{:}{1};
+    part.sourceFiles = partArray{:}{2};
+    part.name = partArray{:}{3};
+    parts{end + 1} = part;
+  end
+end
+
+function showFeedback(parts, response)
+  fprintf('== \n');
+  fprintf('== %43s | %9s | %-s\n', 'Part Name', 'Score', 'Feedback');
+  fprintf('== %43s | %9s | %-s\n', '---------', '-----', '--------');
+  for part = parts
+    score = '';
+    partFeedback = '';
+    partFeedback = response.partFeedbacks.(makeValidFieldName(part{:}.id));
+    partEvaluation = response.partEvaluations.(makeValidFieldName(part{:}.id));
+    score = sprintf('%d / %3d', partEvaluation.score, partEvaluation.maxScore);
+    fprintf('== %43s | %9s | %-s\n', part{:}.name, score, partFeedback);
+  end
+  evaluation = response.evaluation;
+  totalScore = sprintf('%d / %d', evaluation.score, evaluation.maxScore);
+  fprintf('==                                   --------------------------------\n');
+  fprintf('== %43s | %9s | %-s\n', '', totalScore, '');
+  fprintf('== \n');
+end
+
+% use urlread or curl to send submit results to the grader and get a response
+function response = getResponse(url, body)
+% try using urlread() and a secure connection
+  params = {'jsonBody', body};
+  [response, success] = urlread(url, 'post', params);
+
+  if (success == 0)
+    % urlread didn't work, try curl & the peer certificate patch
+    if ispc
+      % testing note: use 'jsonBody =' for a test case
+      json_command = sprintf('echo jsonBody=%s | curl -k -X POST -d @- %s', body, url);
+    else
+      % it's linux/OS X, so use the other form
+      json_command = sprintf('echo ''jsonBody=%s'' | curl -k -X POST -d @- %s', body, url);
+    end
+    % get the response body for the peer certificate patch method
+    [code, response] = system(json_command);
+    % test the success code
+    if (code ~= 0)
+      fprintf('[error] submission with curl() was not successful\n');
+    end
+  end
+end
+
+% validate the grader's response
+function response = validateResponse(resp)
+  % test if the response is json or an HTML page
+  isJson = length(resp) > 0 && resp(1) == '{';
+  isHtml = findstr(lower(resp), '<html');
+
+  if (isJson)
+    response = resp;
+  elseif (isHtml)
+    % the response is html, so it's probably an error message
+    printHTMLContents(resp);
+    error('Grader response is an HTML message');
+  else
+    error('Grader sent no response');
+  end
+end
+
+% parse a HTML response and print it's contents
+function printHTMLContents(response)
+  strippedResponse = regexprep(response, '<[^>]+>', ' ');
+  strippedResponse = regexprep(strippedResponse, '[\t ]+', ' ');
+  fprintf(strippedResponse);
+end
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+% Service configuration
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+function submissionUrl = submissionUrl()
+  submissionUrl = 'https://www-origin.coursera.org/api/onDemandProgrammingImmediateFormSubmissions.v1';
+end

machine-learning-ex5/ex5/linearRegCostFunction.m

+function [J, grad] = linearRegCostFunction(X, y, theta, lambda)
+%LINEARREGCOSTFUNCTION Compute cost and gradient for regularized linear 
+%regression with multiple variables
+%   [J, grad] = LINEARREGCOSTFUNCTION(X, y, theta, lambda) computes the 
+%   cost of using theta as the parameter for linear regression to fit the 
+%   data points in X and y. Returns the cost in J and the gradient in grad
+
+% Initialize some useful values
+m = length(y); % number of training examples
+
+% You need to return the following variables correctly 
+J = 0;
+grad = zeros(size(theta));
+
+% ====================== YOUR CODE HERE ======================
+% Instructions: Compute the cost and gradient of regularized linear 
+%               regression for a particular choice of theta.
+%
+%               You should set J to the cost and grad to the gradient.
+%
+diff_h_theta = X*theta - y; % 12x1
+J = (sum(diff_h_theta.^2) + lambda*sum((theta(2:end)).^2))/(2*m);
+
+grad = (X')*diff_h_theta/m;
+grad(2:end) = grad(2:end) + (lambda/m)*(theta(2:end));
+% =========================================================================
+
+grad = grad(:);
+
+end

machine-learning-ex5/ex5/plotFit.m

+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

machine-learning-ex5/ex5/polyFeatures.m

+function [X_poly] = polyFeatures(X, p)
+%POLYFEATURES Maps X (1D vector) into the p-th power
+%   [X_poly] = POLYFEATURES(X, p) takes a data matrix X (size m x 1) and
+%   maps each example into its polynomial features where
+%   X_poly(i, :) = [X(i) X(i).^2 X(i).^3 ...  X(i).^p];
+%
+
+
+% You need to return the following variables correctly.
+X_poly = zeros(numel(X), p);
+m = length(X);
+
+% ====================== 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.
+%
+% 
+X_poly(:,1) = X(:,1);
+for i = 2:p
+    X_poly(:, i) = X_poly(:, i-1).*(X_poly(:, 1));
+end
+% =========================================================================
+
+end

machine-learning-ex5/ex5/submit.m

+function submit()
+  addpath('./lib');
+
+  conf.assignmentSlug = 'regularized-linear-regression-and-bias-variance';
+  conf.itemName = 'Regularized Linear Regression and Bias/Variance';
+  conf.partArrays = { ...
+    { ...
+      '1', ...
+      { 'linearRegCostFunction.m' }, ...
+      'Regularized Linear Regression Cost Function', ...
+    }, ...
+    { ...
+      '2', ...
+      { 'linearRegCostFunction.m' }, ...
+      'Regularized Linear Regression Gradient', ...
+    }, ...
+    { ...
+      '3', ...
+      { 'learningCurve.m' }, ...
+      'Learning Curve', ...
+    }, ...
+    { ...
+      '4', ...
+      { 'polyFeatures.m' }, ...
+      'Polynomial Feature Mapping', ...
+    }, ...
+    { ...
+      '5', ...
+      { 'validationCurve.m' }, ...
+      'Validation Curve', ...
+    }, ...
+  };
+  conf.output = @output;
+
+  submitWithConfiguration(conf);
+end
+
+function out = output(partId, auxstring)
+  % Random Test Cases
+  X = [ones(10,1) sin(1:1.5:15)' cos(1:1.5:15)'];
+  y = sin(1:3:30)';
+  Xval = [ones(10,1) sin(0:1.5:14)' cos(0:1.5:14)'];
+  yval = sin(1:10)';
+  if partId == '1'
+    [J] = linearRegCostFunction(X, y, [0.1 0.2 0.3]', 0.5);
+    out = sprintf('%0.5f ', J);
+  elseif partId == '2'
+    [J, grad] = linearRegCostFunction(X, y, [0.1 0.2 0.3]', 0.5);
+    out = sprintf('%0.5f ', grad);
+  elseif partId == '3'
+    [error_train, error_val] = ...
+        learningCurve(X, y, Xval, yval, 1);
+    out = sprintf('%0.5f ', [error_train(:); error_val(:)]);
+  elseif partId == '4'
+    [X_poly] = polyFeatures(X(2,:)', 8);
+    out = sprintf('%0.5f ', X_poly);
+  elseif partId == '5'
+    [lambda_vec, error_train, error_val] = ...
+        validationCurve(X, y, Xval, yval);
+    out = sprintf('%0.5f ', ...
+        [lambda_vec(:); error_train(:); error_val(:)]);
+  end 
+end

machine-learning-ex5/ex5/token.txt

+OVWg4dNfO9jCT1Mj

machine-learning-ex5/ex5/trainLinearReg.m

+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
+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

machine-learning-ex5/ex5/validationCurve.m

+function [lambda_vec, error_train, error_val] = ...
+    validationCurve(X, y, Xval, yval)
+%VALIDATIONCURVE Generate the train and validation errors needed to
+%plot a validation curve that we can use to select lambda
+%   [lambda_vec, error_train, error_val] = ...
+%       VALIDATIONCURVE(X, y, Xval, yval) returns the train
+%       and validation errors (in error_train, error_val)
+%       for different values of lambda. You are given the training set (X,
+%       y) and validation set (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);
+
+% ====================== YOUR CODE HERE ======================
+% Instructions: Fill in this function to return training errors in 
+%               error_train and the validation errors in error_val. The 
+%               vector lambda_vec contains the different lambda parameters 
+%               to use for each calculation of the errors, i.e, 
+%               error_train(i), and error_val(i) should give 
+%               you the errors obtained after training with 
+%               lambda = lambda_vec(i)
+%
+% Note: You can loop over lambda_vec with the following:
+%
+%       for i = 1:length(lambda_vec)
+%           lambda = lambda_vec(i);
+%           % Compute train / val errors when training linear 
+%           % regression with regularization parameter lambda
+%           % You should store the result in error_train(i)
+%           % and error_val(i)
+%           ....
+%           
+%       end
+%
+%
+for i = 1:length(lambda_vec)
+    lambda = lambda_vec(i);
+    
+    theta = trainLinearReg(X, y, lambda);
+    
+    error_train(i) = linearRegCostFunction(X, y, theta, 0);
+    error_val(i) = linearRegCostFunction(Xval, yval, theta, 0);
+end
+% =========================================================================
+
+end