CheeseZH: Stanford University: Machine Learning Ex2:Logistic Regression

1. Sigmoid Function

In Logisttic Regression, the hypothesis is defined as:

where function g is the sigmoid function. The sigmoid function is defined as:


2.Cost function and gradient

The cost function in logistic regression is:


the gradient of the cost is a vector of the same length as θ  where jth element(for j=0,1,...,n) is defined as follows:


3. Regularized Cost function and gradient

Recall that the regularized cost function in logistic regression is:


The gradient of the cost function is a vector where the jth element is defined as follows:

for j=0:


for j>=1:


 

Here are the code files:

ex2_data1.txt

 

34.62365962451697,78.0246928153624,0
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35.84740876993872,72.90219802708364,0
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79.0327360507101,75.3443764369103,1
45.08327747668339,56.3163717815305,0
61.10666453684766,96.51142588489624,1
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84.43281996120035,43.53339331072109,1
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82.30705337399482,76.48196330235604,1
69.36458875970939,97.71869196188608,1
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53.9710521485623,89.20735013750205,1
69.07014406283025,52.74046973016765,1
67.94685547711617,46.67857410673128,0
70.66150955499435,92.92713789364831,1
76.97878372747498,47.57596364975532,1
67.37202754570876,42.83843832029179,0
89.67677575072079,65.79936592745237,1
50.534788289883,48.85581152764205,0
34.21206097786789,44.20952859866288,0
77.9240914545704,68.9723599933059,1
62.27101367004632,69.95445795447587,1
80.1901807509566,44.82162893218353,1
93.114388797442,38.80067033713209,0
61.83020602312595,50.25610789244621,0
38.78580379679423,64.99568095539578,0
61.379289447425,72.80788731317097,1
85.40451939411645,57.05198397627122,1
52.10797973193984,63.12762376881715,0
52.04540476831827,69.43286012045222,1
40.23689373545111,71.16774802184875,0
54.63510555424817,52.21388588061123,0
33.91550010906887,98.86943574220611,0
64.17698887494485,80.90806058670817,1
74.78925295941542,41.57341522824434,0
34.1836400264419,75.2377203360134,0
83.90239366249155,56.30804621605327,1
51.54772026906181,46.85629026349976,0
94.44336776917852,65.56892160559052,1
82.36875375713919,40.61825515970618,0
51.04775177128865,45.82270145776001,0
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77.19303492601364,70.45820000180959,1
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91.56497449807442,88.69629254546599,1
79.94481794066932,74.16311935043758,1
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90.54671411399852,43.39060180650027,1
34.52451385320009,60.39634245837173,0
50.2864961189907,49.80453881323059,0
49.58667721632031,59.80895099453265,0
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32.57720016809309,95.59854761387875,0
74.24869136721598,69.82457122657193,1
71.79646205863379,78.45356224515052,1
75.3956114656803,85.75993667331619,1
35.28611281526193,47.02051394723416,0
56.25381749711624,39.26147251058019,0
30.05882244669796,49.59297386723685,0
44.66826172480893,66.45008614558913,0
66.56089447242954,41.09209807936973,0
40.45755098375164,97.53518548909936,1
49.07256321908844,51.88321182073966,0
80.27957401466998,92.11606081344084,1
66.74671856944039,60.99139402740988,1
32.72283304060323,43.30717306430063,0
64.0393204150601,78.03168802018232,1
72.34649422579923,96.22759296761404,1
60.45788573918959,73.09499809758037,1
58.84095621726802,75.85844831279042,1
99.82785779692128,72.36925193383885,1
47.26426910848174,88.47586499559782,1
50.45815980285988,75.80985952982456,1
60.45555629271532,42.50840943572217,0
82.22666157785568,42.71987853716458,0
88.9138964166533,69.80378889835472,1
94.83450672430196,45.69430680250754,1
67.31925746917527,66.58935317747915,1
57.23870631569862,59.51428198012956,1
80.36675600171273,90.96014789746954,1
68.46852178591112,85.59430710452014,1
42.0754545384731,78.84478600148043,0
75.47770200533905,90.42453899753964,1
78.63542434898018,96.64742716885644,1
52.34800398794107,60.76950525602592,0
94.09433112516793,77.15910509073893,1
90.44855097096364,87.50879176484702,1
55.48216114069585,35.57070347228866,0
74.49269241843041,84.84513684930135,1
89.84580670720979,45.35828361091658,1
83.48916274498238,48.38028579728175,1
42.2617008099817,87.10385094025457,1
99.31500880510394,68.77540947206617,1
55.34001756003703,64.9319380069486,1
74.77589300092767,89.52981289513276,1
View Code

 

ex2.m

  1 %% Machine Learning Online Class - Exercise 2: Logistic Regression
  2 %
  3 %  Instructions
  4 %  ------------
  5 % 
  6 %  This file contains code that helps you get started on the logistic
  7 %  regression exercise. You will need to complete the following functions 
  8 %  in this exericse:
  9 %
 10 %     sigmoid.m
 11 %     costFunction.m
 12 %     predict.m
 13 %     costFunctionReg.m
 14 %
 15 %  For this exercise, you will not need to change any code in this file,
 16 %  or any other files other than those mentioned above.
 17 %
 18 
 19 %% Initialization
 20 clear ; close all; clc
 21 
 22 %% Load Data
 23 %  The first two columns contains the exam scores and the third column
 24 %  contains the label.
 25 
 26 data = load('ex2data1.txt');
 27 X = data(:, [1, 2]); y = data(:, 3);
 28 
 29 %% ==================== Part 1: Plotting ====================
 30 %  We start the exercise by first plotting the data to understand the 
 31 %  the problem we are working with.
 32 
 33 fprintf(['Plotting data with + indicating (y = 1) examples and o ' ...
 34          'indicating (y = 0) examples.\n']);
 35 
 36 plotData(X, y);
 37 
 38 % Put some labels 
 39 hold on;
 40 % Labels and Legend
 41 xlabel('Exam 1 score')
 42 ylabel('Exam 2 score')
 43 
 44 % Specified in plot order
 45 legend('Admitted', 'Not admitted')
 46 hold off;
 47 
 48 fprintf('\nProgram paused. Press enter to continue.\n');
 49 pause;
 50 
 51 
 52 %% ============ Part 2: Compute Cost and Gradient ============
 53 %  In this part of the exercise, you will implement the cost and gradient
 54 %  for logistic regression. You neeed to complete the code in 
 55 %  costFunction.m
 56 
 57 %  Setup the data matrix appropriately, and add ones for the intercept term
 58 [m, n] = size(X);
 59 
 60 % Add intercept term to x and X_test
 61 X = [ones(m, 1) X];
 62 
 63 % Initialize fitting parameters
 64 initial_theta = zeros(n + 1, 1);
 65 
 66 % Compute and display initial cost and gradient
 67 [cost, grad] = costFunction(initial_theta, X, y);
 68 
 69 fprintf('Cost at initial theta (zeros): %f\n', cost);
 70 fprintf('Gradient at initial theta (zeros): \n');
 71 fprintf(' %f \n', grad);
 72 
 73 fprintf('\nProgram paused. Press enter to continue.\n');
 74 pause;
 75 
 76 
 77 %% ============= Part 3: Optimizing using fminunc  =============
 78 %  In this exercise, you will use a built-in function (fminunc) to find the
 79 %  optimal parameters theta.
 80 
 81 %  Set options for fminunc
 82 options = optimset('GradObj', 'on', 'MaxIter', 400);
 83 
 84 %  Run fminunc to obtain the optimal theta
 85 %  This function will return theta and the cost 
 86 [theta, cost] = ...
 87     fminunc(@(t)(costFunction(t, X, y)), initial_theta, options);
 88 
 89 % Print theta to screen
 90 fprintf('Cost at theta found by fminunc: %f\n', cost);
 91 fprintf('theta: \n');
 92 fprintf(' %f \n', theta);
 93 
 94 % Plot Boundary
 95 plotDecisionBoundary(theta, X, y);
 96 
 97 % Put some labels 
 98 hold on;
 99 % Labels and Legend
100 xlabel('Exam 1 score')
101 ylabel('Exam 2 score')
102 
103 % Specified in plot order
104 legend('Admitted', 'Not admitted')
105 hold off;
106 
107 fprintf('\nProgram paused. Press enter to continue.\n');
108 pause;
109 
110 %% ============== Part 4: Predict and Accuracies ==============
111 %  After learning the parameters, you'll like to use it to predict the outcomes
112 %  on unseen data. In this part, you will use the logistic regression model
113 %  to predict the probability that a student with score 45 on exam 1 and 
114 %  score 85 on exam 2 will be admitted.
115 %
116 %  Furthermore, you will compute the training and test set accuracies of 
117 %  our model.
118 %
119 %  Your task is to complete the code in predict.m
120 
121 %  Predict probability for a student with score 45 on exam 1 
122 %  and score 85 on exam 2 
123 
124 prob = sigmoid([1 45 85] * theta);
125 fprintf(['For a student with scores 45 and 85, we predict an admission ' ...
126          'probability of %f\n\n'], prob);
127 
128 % Compute accuracy on our training set
129 p = predict(theta, X);
130 
131 fprintf('Train Accuracy: %f\n', mean(double(p == y)) * 100);
132 
133 fprintf('\nProgram paused. Press enter to continue.\n');
134 pause;
View Code

sigmoid.m

 1 function g = sigmoid(z)
 2 %SIGMOID Compute sigmoid functoon
 3 %   J = SIGMOID(z) computes the sigmoid of z.
 4 
 5 % You need to return the following variables correctly 
 6 g = zeros(size(z));
 7 
 8 % ====================== YOUR CODE HERE ======================
 9 % Instructions: Compute the sigmoid of each value of z (z can be a matrix,
10 %               vector or scalar).
11 
12 
13 g = 1./(1+exp(-z));
14 
15 
16 % =============================================================
17 
18 end
View Code

costFunction.m

 1 function [J, grad] = costFunction(theta, X, y)
 2 %COSTFUNCTION Compute cost and gradient for logistic regression
 3 %   J = COSTFUNCTION(theta, X, y) computes the cost of using theta as the
 4 %   parameter for logistic regression and the gradient of the cost
 5 %   w.r.t. to the parameters.
 6 
 7 % Initialize some useful values
 8 m = length(y); % number of training examples
 9 
10 % You need to return the following variables correctly 
11 J = 0;
12 grad = zeros(size(theta));
13 
14 % ====================== YOUR CODE HERE ======================
15 % Instructions: Compute the cost of a particular choice of theta.
16 %               You should set J to the cost.
17 %               Compute the partial derivatives and set grad to the partial
18 %               derivatives of the cost w.r.t. each parameter in theta
19 %
20 % Note: grad should have the same dimensions as theta
21 %
22 hx = sigmoid(X*theta);  % m x 1
23 J = -1/m*(y'*log(hx)+((1-y)'*log(1-hx)));
24 grad = 1/m*X'*(hx-y);
25 
26 
27 
28 
29 
30 
31 % =============================================================
32 
33 end
View Code

predict.m

 1 function p = predict(theta, X)
 2 %PREDICT Predict whether the label is 0 or 1 using learned logistic 
 3 %regression parameters theta
 4 %   p = PREDICT(theta, X) computes the predictions for X using a 
 5 %   threshold at 0.5 (i.e., if sigmoid(theta'*x) >= 0.5, predict 1)
 6 
 7 m = size(X, 1); % Number of training examples
 8 
 9 % You need to return the following variables correctly
10 p = zeros(m, 1);
11 
12 % ====================== YOUR CODE HERE ======================
13 % Instructions: Complete the following code to make predictions using
14 %               your learned logistic regression parameters. 
15 %               You should set p to a vector of 0's and 1's
16 %
17 
18 p = sigmoid(X*theta)>=0.5;
19 
20 
21 
22 
23 % =========================================================================
24 
25 
26 end
View Code

costFunctionReg.m

 1 function [J, grad] = costFunctionReg(theta, X, y, lambda)
 2 %COSTFUNCTIONREG Compute cost and gradient for logistic regression with regularization
 3 %   J = COSTFUNCTIONREG(theta, X, y, lambda) computes the cost of using
 4 %   theta as the parameter for regularized logistic regression and the
 5 %   gradient of the cost w.r.t. to the parameters. 
 6 
 7 % Initialize some useful values
 8 m = length(y); % number of training examples
 9 
10 % You need to return the following variables correctly 
11 J = 0;
12 grad = zeros(size(theta));
13 
14 % ====================== YOUR CODE HERE ======================
15 % Instructions: Compute the cost of a particular choice of theta.
16 %               You should set J to the cost.
17 %               Compute the partial derivatives and set grad to the partial
18 %               derivatives of the cost w.r.t. each parameter in theta
19 hx = sigmoid(X*theta);
20 reg = lambda/(2*m)*sum(theta(2:size(theta),:).^2);
21 J = -1/m*(y'*log(hx)+(1-y)'*log(1-hx)) + reg;
22 theta(1) = 0;
23 grad = 1/m*X'*(hx-y)+lambda/m*theta;
24 
25 
26 % =============================================================
27 
28 end
View Code

 

 

 

 

 

 

posted @ 2015-06-25 20:06  ZH奶酪  阅读(2058)  评论(0)  编辑  收藏  举报