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annotate tvii/logistic_regression.py @ 50:4b20694b8a16

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add module + test for uniqueness
author Jeff Hammel <k0scist@gmail.com>
date Sun, 17 Sep 2017 14:28:36 -0700
parents 4d173452377e
children 0807ac8992ba
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1 """
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2 z = w'x + b
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3 a = sigmoid(z)
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4 L(a,y) = -(y*log(a) + (1-y)*log(1-a))
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6 [| | | ]
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7 X = [x1 x2 x3]
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8 [| | | ]
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9
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10 [z1 z2 z3 .. zm] = w'*X + [b b b b ] = [w'*x1+b + w'*x2+b ...]
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11 """
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14 import numpy as np
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15 from .sigmoid import sigmoid
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18 def loss(a, y):
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19 # UNTESTED!
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20 # derivative = -(y/a) + (1-y)/(1-a)
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21 return -y*np.log(a) - (1-y)*np.log(1-a)
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23 def propagate(w, b, X, Y):
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24 """
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25 Implement the cost function and its gradient for the propagation:
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26 Forward Propagation:
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27 - You get X
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28 - You compute $A = \sigma(w^T X + b) = (a^{(0)}, a^{(1)}, ..., a^{(m-1)}, a^{(m)})$
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29 - You calculate the cost function: $J = -\frac{1}{m}\sum_{i=1}^{m}y^{(i)}\log(a^{(i)})+(1-y^{(i)})\log(1-a^{(i)})$
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30
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31 Here are the two formulas you will be using:
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32
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33 $$ \frac{\partial J}{\partial w} = \frac{1}{m}X(A-Y)^T\tag{7}$$
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34 $$ \frac{\partial J}{\partial b} = \frac{1}{m} \sum_{i=1}^m (a^{(i)}-y^{(i)})\tag{8}$$
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35
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36 Arguments:
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37 w -- weights, a numpy array of size (num_px * num_px * 3, 1)
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38 b -- bias, a scalar
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39 X -- data of size (num_px * num_px * 3, number of examples)
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40 Y -- true "label" vector (containing 0 if non-cat, 1 if cat) of size (1, number of examples)
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41
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42 Return:
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43 cost -- negative log-likelihood cost for logistic regression
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44 dw -- gradient of the loss with respect to w, thus same shape as w
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45 db -- gradient of the loss with respect to b, thus same shape as b
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46
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47 Tips:
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48 - Write your code step by step for the propagation. np.log(), np.dot()
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49 """
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50
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53 # FORWARD PROPAGATION (FROM X TO COST)
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54 cost = cost_function(w, b, X, Y) # compute cost
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55
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56 # BACKWARD PROPAGATION (TO FIND GRADIENT)
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57 m = X.shape[1]
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58 A = sigmoid(np.dot(w.T, X) + b) # compute activation
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59 dw = (1./m)*np.dot(X, (A - Y).T)
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60 db = (1./m)*np.sum(A - Y)
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61
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62 # sanity check
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63 assert(A.shape[1] == m)
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64 assert(dw.shape == w.shape), "dw.shape is {}; w.shape is {}".format(dw.shape, w.shape)
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65 assert(db.dtype == float)
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66 cost = np.squeeze(cost)
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67 assert(cost.shape == ())
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68
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69 # return gradients
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70 grads = {"dw": dw,
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71 "db": db}
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72 return grads, cost
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74
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75 def cost_function(w, b, X, Y):
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76 """
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77 Cost function for binary classification
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78 yhat = sigmoid(W.T*x + b)
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79 interpret yhat thhe probably that y=1
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80
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81 Loss function:
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82 y log(yhat) + (1 - y) log(1 - yhat)
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83 """
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84
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85 m = X.shape[1]
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86 A = sigmoid(np.dot(w.T, X) + b)
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87 cost = np.sum(Y*np.log(A) + (1 - Y)*np.log(1 - A))
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88 return (-1./m)*cost
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90
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91 def optimize(w, b, X, Y, num_iterations, learning_rate, print_cost = False):
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92 """
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93 This function optimizes w and b by running a gradient descent algorithm
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94
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95 Arguments:
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96 w -- weights, a numpy array of size (num_px * num_px * 3, 1)
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97 b -- bias, a scalar
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98 X -- data of shape (num_px * num_px * 3, number of examples)
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99 Y -- true "label" vector (containing 0 if non-cat, 1 if cat), of shape (1, number of examples)
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100 num_iterations -- number of iterations of the optimization loop
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101 learning_rate -- learning rate of the gradient descent update rule
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102 print_cost -- True to print the loss every 100 steps
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103
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104 Returns:
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105 params -- dictionary containing the weights w and bias b
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106 grads -- dictionary containing the gradients of the weights and bias with respect to the cost function
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107 costs -- list of all the costs computed during the optimization, this will be used to plot the learning curve.
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108
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109 Tips:
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110 You basically need to write down two steps and iterate through them:
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111 1) Calculate the cost and the gradient for the current parameters. Use propagate().
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112 2) Update the parameters using gradient descent rule for w and b.
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113 """
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114
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115 costs = []
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116
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117 for i in range(num_iterations):
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118
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119 # Cost and gradient calculation
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120 grads, cost = propagate(w, b, X, Y)
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121
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122 # Retrieve derivatives from grads
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123 dw = grads["dw"]
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124 db = grads["db"]
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125
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126 # gradient descent
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127 w = w - learning_rate*dw
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128 b = b - learning_rate*db
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129
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130 # Record the costs
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131 if i % 100 == 0:
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132 costs.append(cost)
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133
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134 # Print the cost every 100 training examples
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135 if print_cost and not (i % 100):
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136 print ("Cost after iteration %i: %f" %(i, cost))
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137
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138 # package data for return
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139 params = {"w": w,
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140 "b": b}
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141 grads = {"dw": dw,
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142 "db": db}
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143 return params, grads, costs
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144
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145
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146 def predict(w, b, X):
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147 '''
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148 Predict whether the label is 0 or 1 using learned logistic regression parameters (w, b)
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149
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150 Arguments:
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151 w -- weights, a numpy array of size (num_px * num_px * 3, 1)
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152 b -- bias, a scalar
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153 X -- data of size (num_px * num_px * 3, number of examples)
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154
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155 Returns:
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156 Y_prediction -- a numpy array (vector) containing all predictions (0/1) for the examples in X
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157 '''
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158
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159 m = X.shape[1]
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160 Y_prediction = np.zeros((1,m))
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161 w = w.reshape(X.shape[0], 1)
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162
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163 # Compute vector "A" predicting the probabilities
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164 A = sigmoid(np.dot(w.T, X) + b)
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165
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166 for i in range(A.shape[1]):
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167 # Convert probabilities A[0,i] to actual predictions p[0,i]
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168 Y_prediction[0][i] = 0 if A[0][i] <= 0.5 else 1
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169
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170 assert(Y_prediction.shape == (1, m))
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171
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172 return Y_prediction
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173
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174
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175 def model(X_train, Y_train, X_test, Y_test, num_iterations = 2000, learning_rate = 0.5, print_cost = False):
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176 """
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177 Builds the logistic regression model by calling the function you've implemented previously
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178
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179 Arguments:
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180 X_train -- training set represented by a numpy array of shape (num_px * num_px * 3, m_train)
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181 Y_train -- training labels represented by a numpy array (vector) of shape (1, m_train)
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182 X_test -- test set represented by a numpy array of shape (num_px * num_px * 3, m_test)
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183 Y_test -- test labels represented by a numpy array (vector) of shape (1, m_test)
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184 num_iterations -- hyperparameter representing the number of iterations to optimize the parameters
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185 learning_rate -- hyperparameter representing the learning rate used in the update rule of optimize()
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186 print_cost -- Set to true to print the cost every 100 iterations
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187
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188 Returns:
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189 d -- dictionary containing information about the model.
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190 """
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191
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192 # initialize parameters with zeros
33
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193 w = np.zeros((X_train.shape[0], 1))
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194 b = 0
32
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195
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196 # Gradient descent
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197 parameters, grads, costs = optimize(w, b, X_train, Y_train, num_iterations, learning_rate, print_cost=print_cost)
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198
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199 # Retrieve parameters w and b from dictionary "parameters"
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200 w = parameters["w"]
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201 b = parameters["b"]
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202
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203 # Predict test/train set examples
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204 Y_prediction_test = predict(w, b, X_test)
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205 Y_prediction_train = predict(w, b, X_train)
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206
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207
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208 # Print train/test Errors
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209 print("train accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_train - Y_train)) * 100))
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210 print("test accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_test - Y_test)) * 100))
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211
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212 d = {"costs": costs,
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213 "Y_prediction_test": Y_prediction_test,
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214 "Y_prediction_train" : Y_prediction_train,
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215 "w" : w,
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216 "b" : b,
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217 "learning_rate" : learning_rate,
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218 "num_iterations": num_iterations}
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219 return d