Planar data classification with one hidden layer

You will learn how to:

  • Implement a 2-class classification neural network with a single hidden layer
  • Use units with a non-linear activation function, such as tanh
  • Compute the cross entropy loss
  • Implement forward and backward propagation

Packages

Let’s first import all the packages that you will need during this assignment.

  • numpy is the fundamental package for scientific computing with Python.
  • sklearn provides simple and efficient tools for data mining and data analysis.
  • matplotlib is a library for plotting graphs in Python.
  • testCases provides some test examples to assess the correctness of your functions.
  • planar_utils provide various useful functions used in this assignment.
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# Package imports
import numpy as np
import matplotlib.pyplot as plt
from testCases import *
import sklearn
import sklearn.datasets
import sklearn.linear_model
from planar_utils import plot_decision_boundary, sigmoid, load_planar_dataset, load_extra_datasets

%matplotlib inline

np.random.seed(1) # set a seed so that the results are consistent

Dataset

First, let’s get the dataset you will work on. The following code will load a “flower” 2-class dataset into variables X and Y.

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X, Y = load_planar_dataset()

Visualize the dataset using matplotlib. The data looks like a “flower” with some red (label y=0) and some blue (y=1) points. Your goal is to build a model to fit this data.

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# Visualize the data:
plt.scatter(X[0, :], X[1, :], c=Y, s=40, cmap=plt.cm.Spectral)

img

How many training examples do you have? In addition, what is the shape of the variables X and Y?

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### START CODE HERE ### (≈ 3 lines of code)
shape_X = X.shape
shape_Y = Y.shape
m = X.shape[1] # training set size
### END CODE HERE ###

print ('The shape of X is: ' + str(shape_X))
print ('The shape of Y is: ' + str(shape_Y))
print ('I have m = %d training examples!' % (m))

"""
out:
The shape of X is: (2, 400)
The shape of Y is: (1, 400)
I have m = 400 training examples!
"""

Simple Logistic Regression

Before building a full neural network, lets first see how logistic regression performs on this problem. You can use sklearn’s built-in functions to do that.

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# Train the logistic regression classifier
clf = sklearn.linear_model.LogisticRegressionCV()
clf.fit(X.T, Y.T)

"""
out:
LogisticRegressionCV(Cs=10, class_weight=None, cv=None, dual=False,
fit_intercept=True, intercept_scaling=1.0, max_iter=100,
multi_class='ovr', n_jobs=1, penalty='l2', random_state=None,
refit=True, scoring=None, solver='lbfgs', tol=0.0001, verbose=0)
"""

You can now plot the decision boundary of these models.

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# Plot the decision boundary for logistic regression
plot_decision_boundary(lambda x: clf.predict(x), X, Y)
plt.title("Logistic Regression")

# Print accuracy
LR_predictions = clf.predict(X.T)
print ('Accuracy of logistic regression: %d ' % float((np.dot(Y,LR_predictions) + np.dot(1-Y,1-LR_predictions))/float(Y.size)*100) +
'% ' + "(percentage of correctly labelled datapoints)")

out:
Accuracy of logistic regression: 47 % (percentage of correctly labelled datapoints)

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The dataset is not linearly separable, so logistic regression doesn’t perform well. Hopefully a neural network will do better. Let’s try this now!

Neural Network model

Logistic regression did not work well on the “flower dataset”. You are going to train a Neural Network with a single hidden layer.

img

The general methodology to build a Neural Network is to:

  • Define the neural network structure ( # of input units, # of hidden units, etc).
  • Initialize the model’s parameters
    Loop:
    • Implement forward propagation
    • Compute loss
    • Implement backward propagation to get the gradients
    • Update parameters (gradient descent)

Defining the neural network structure

Define three cariables:

  • n_x: the size of the input layer
  • n_h: the size of the hidden layer (set this to 4)
  • n_y: the size of the output layer
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# GRADED FUNCTION: layer_sizes

def layer_sizes(X, Y):
"""
Arguments:
X -- input dataset of shape (input size, number of examples)
Y -- labels of shape (output size, number of examples)

Returns:
n_x -- the size of the input layer
n_h -- the size of the hidden layer
n_y -- the size of the output layer
"""
### START CODE HERE ### (≈ 3 lines of code)
n_x = X.shape[0] # size of input layer
n_h = 4
n_y = Y.shape[0] # size of output layer
### END CODE HERE ###
return (n_x, n_h, n_y)

"""
X_assess, Y_assess = layer_sizes_test_case()
(n_x, n_h, n_y) = layer_sizes(X_assess, Y_assess)
print("The size of the input layer is: n_x = " + str(n_x))
print("The size of the hidden layer is: n_h = " + str(n_h))
print("The size of the output layer is: n_y = " + str(n_y))

out:
The size of the input layer is: n_x = 5
The size of the hidden layer is: n_h = 4
The size of the output layer is: n_y = 2
"""

Initialize the model’s parameters

  • Make sure your parameters’ sizes are right. Refer to the neural network figure above if needed.
  • You will initialize the weights matrices with random values.
    • Use: np.random.randn(a,b) * 0.01 to randomly initialize a matrix of shape (a,b).
  • You will initialize the bias vectors as zeros.
    • Use: np.zeros((a,b)) to initialize a matrix of shape (a,b) with zeros.
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# GRADED FUNCTION: initialize_parameters

def initialize_parameters(n_x, n_h, n_y):
"""
Argument:
n_x -- size of the input layer
n_h -- size of the hidden layer
n_y -- size of the output layer

Returns:
params -- python dictionary containing your parameters:
W1 -- weight matrix of shape (n_h, n_x)
b1 -- bias vector of shape (n_h, 1)
W2 -- weight matrix of shape (n_y, n_h)
b2 -- bias vector of shape (n_y, 1)
"""

np.random.seed(2) # we set up a seed so that your output matches ours although the initialization is random.

### START CODE HERE ### (≈ 4 lines of code)
W1 = np.random.randn(n_h, n_x) * 0.01
b1 = np.zeros((n_h, 1))
W2 = np.random.randn(n_y, n_h) * 0.01
b2 = np.zeros((n_y, 1))
### END CODE HERE ###

assert (W1.shape == (n_h, n_x))
assert (b1.shape == (n_h, 1))
assert (W2.shape == (n_y, n_h))
assert (b2.shape == (n_y, 1))

parameters = {"W1": W1,
"b1": b1,
"W2": W2,
"b2": b2}

return parameters

"""
n_x, n_h, n_y = initialize_parameters_test_case()

parameters = initialize_parameters(n_x, n_h, n_y)
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))

out:
W1 = [[-0.00416758 -0.00056267]
[-0.02136196 0.01640271]
[-0.01793436 -0.00841747]
[ 0.00502881 -0.01245288]]
b1 = [[ 0.]
[ 0.]
[ 0.]
[ 0.]]
W2 = [[-0.01057952 -0.00909008 0.00551454 0.02292208]]
b2 = [[ 0.]]
"""

The Loop

  • Look above at the mathematical representation of your classifier.
  • You can use the function sigmoid(). It is built-in (imported) in the notebook.
  • You can use the function np.tanh(). It is part of the numpy library.
  • The steps you have to implement are:
    • Retrieve each parameter from the dictionary “parameters” (which is the output of initialize_parameters()) by using parameters[“..”].
    • Implement Forward Propagation. Compute Z[1],A[1],Z[2] and A[2] (the vector of all your predictions on all the examples in the training set).
  • Values needed in the backpropagation are stored in “cache”. The cache will be given as an input to the backpropagation function.
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# GRADED FUNCTION: forward_propagation

def forward_propagation(X, parameters):
"""
Argument:
X -- input data of size (n_x, m)
parameters -- python dictionary containing your parameters (output of initialization function)

Returns:
A2 -- The sigmoid output of the second activation
cache -- a dictionary containing "Z1", "A1", "Z2" and "A2"
"""
# Retrieve each parameter from the dictionary "parameters"
### START CODE HERE ### (≈ 4 lines of code)
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]
### END CODE HERE ###

# Implement Forward Propagation to calculate A2 (probabilities)
### START CODE HERE ### (≈ 4 lines of code)
Z1 = np.dot(W1, X) + b1
A1 = np.tanh(Z1)
Z2 = np.dot(W2, A1) + b2
A2 = sigmoid(Z2)
### END CODE HERE ###

assert(A2.shape == (1, X.shape[1]))

cache = {"Z1": Z1,
"A1": A1,
"Z2": Z2,
"A2": A2}

return A2, cache

"""
X_assess, parameters = forward_propagation_test_case()

A2, cache = forward_propagation(X_assess, parameters)

# Note: we use the mean here just to make sure that your output matches ours.
print(np.mean(cache['Z1']) ,np.mean(cache['A1']),np.mean(cache['Z2']),np.mean(cache['A2']))

out:
-0.000499755777742 -0.000496963353232 0.000438187450959 0.500109546852
"""

img

Implement compute_cost() to compute the value of the cost J.

img

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# GRADED FUNCTION: compute_cost

def compute_cost(A2, Y, parameters):
"""
Computes the cross-entropy cost given in equation (13)

Arguments:
A2 -- The sigmoid output of the second activation, of shape (1, number of examples)
Y -- "true" labels vector of shape (1, number of examples)
parameters -- python dictionary containing your parameters W1, b1, W2 and b2

Returns:
cost -- cross-entropy cost given equation (13)
"""

m = Y.shape[1] # number of example

# Compute the cross-entropy cost
### START CODE HERE ### (≈ 2 lines of code)
logprobs = np.multiply(np.log(A2), Y) + np.multiply(np.log(1-A2), 1-Y)
cost = -1/m * np.sum(logprobs)
### END CODE HERE ###

cost = np.squeeze(cost) # makes sure cost is the dimension we expect.
# E.g., turns [[17]] into 17
assert(isinstance(cost, float))

return cost

"""
A2, Y_assess, parameters = compute_cost_test_case()

print("cost = " + str(compute_cost(A2, Y_assess, parameters)))

out:
cost = 0.692919893776
"""

Using the cache computed during forward propagation, you can now implement backward propagation.

Implement the function backward_propagation().

Backpropagation is usually the hardest (most mathematical) part in deep learning. To help you, here again is the slide from the lecture on backpropagation. You’ll want to use the six equations on the right of this slide, since you are building a vectorized implementation.

img

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# GRADED FUNCTION: backward_propagation

def backward_propagation(parameters, cache, X, Y):
"""
Implement the backward propagation using the instructions above.

Arguments:
parameters -- python dictionary containing our parameters
cache -- a dictionary containing "Z1", "A1", "Z2" and "A2".
X -- input data of shape (2, number of examples)
Y -- "true" labels vector of shape (1, number of examples)

Returns:
grads -- python dictionary containing your gradients with respect to different parameters
"""
m = X.shape[1]

# First, retrieve W1 and W2 from the dictionary "parameters".
### START CODE HERE ### (≈ 2 lines of code)
W1 = parameters["W1"]
W2 = parameters["W2"]
### END CODE HERE ###

# Retrieve also A1 and A2 from dictionary "cache".
### START CODE HERE ### (≈ 2 lines of code)
A1 = cache["A1"]
A2 = cache["A2"]
### END CODE HERE ###

# Backward propagation: calculate dW1, db1, dW2, db2.
### START CODE HERE ### (≈ 6 lines of code, corresponding to 6 equations on slide above)
dZ2= A2 - Y
dW2 = 1/m * np.dot(dZ2, A1.T)
db2 = 1/m * np.sum(dZ2, axis=1, keepdims=True)
dZ1 = np.dot(W2.T, dZ2) * (1 - np.power(A1, 2))
dW1 = 1/m * np.dot(dZ1, X.T)
db1 = 1/m * np.sum(dZ1, axis=1, keepdims=True)
### END CODE HERE ###

grads = {"dW1": dW1,
"db1": db1,
"dW2": dW2,
"db2": db2}

return grads

"""
parameters, cache, X_assess, Y_assess = backward_propagation_test_case()

grads = backward_propagation(parameters, cache, X_assess, Y_assess)
print ("dW1 = "+ str(grads["dW1"]))
print ("db1 = "+ str(grads["db1"]))
print ("dW2 = "+ str(grads["dW2"]))
print ("db2 = "+ str(grads["db2"]))

out:
dW1 = [[ 0.01018708 -0.00708701]
[ 0.00873447 -0.0060768 ]
[-0.00530847 0.00369379]
[-0.02206365 0.01535126]]
db1 = [[-0.00069728]
[-0.00060606]
[ 0.000364 ]
[ 0.00151207]]
dW2 = [[ 0.00363613 0.03153604 0.01162914 -0.01318316]]
db2 = [[ 0.06589489]]
"""

Implement the update rule. Use gradient descent. You have to use (dW1, db1, dW2, db2) in order to update (W1, b1, W2, b2).

General gradient descent rule: θ=θ−α*(∂J/∂θ) where α is the learning rate and θ represents a parameter.

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# GRADED FUNCTION: update_parameters

def update_parameters(parameters, grads, learning_rate = 1.2):
"""
Updates parameters using the gradient descent update rule given above

Arguments:
parameters -- python dictionary containing your parameters
grads -- python dictionary containing your gradients

Returns:
parameters -- python dictionary containing your updated parameters
"""
# Retrieve each parameter from the dictionary "parameters"
### START CODE HERE ### (≈ 4 lines of code)
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]
### END CODE HERE ###

# Retrieve each gradient from the dictionary "grads"
### START CODE HERE ### (≈ 4 lines of code)
dW1 = grads["dW1"]
db1 = grads["db1"]
dW2 = grads["dW2"]
db2 = grads["db2"]
## END CODE HERE ###

# Update rule for each parameter
### START CODE HERE ### (≈ 4 lines of code)
W1 = W1 - learning_rate * dW1
b1 = b1 - learning_rate * db1
W2 = W2 - learning_rate * dW2
b2 = b2 - learning_rate * db2
### END CODE HERE ###

parameters = {"W1": W1,
"b1": b1,
"W2": W2,
"b2": b2}

return parameters

"""
parameters, grads = update_parameters_test_case()
parameters = update_parameters(parameters, grads)

print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))

out:
W1 = [[-0.00643025 0.01936718]
[-0.02410458 0.03978052]
[-0.01653973 -0.02096177]
[ 0.01046864 -0.05990141]]
b1 = [[ -1.02420756e-06]
[ 1.27373948e-05]
[ 8.32996807e-07]
[ -3.20136836e-06]]
W2 = [[-0.01041081 -0.04463285 0.01758031 0.04747113]]
b2 = [[ 0.00010457]]
"""

Integrate parts 4.1, 4.2 and 4.3 in nn_model()

Build your neural network model in nn_model(). The neural network model has to use the previous functions in the right order.

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# GRADED FUNCTION: nn_model

def nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False):
"""
Arguments:
X -- dataset of shape (2, number of examples)
Y -- labels of shape (1, number of examples)
n_h -- size of the hidden layer
num_iterations -- Number of iterations in gradient descent loop
print_cost -- if True, print the cost every 1000 iterations

Returns:
parameters -- parameters learnt by the model. They can then be used to predict.
"""

np.random.seed(3)
n_x = layer_sizes(X, Y)[0]
n_y = layer_sizes(X, Y)[2]

# Initialize parameters, then retrieve W1, b1, W2, b2. Inputs: "n_x, n_h, n_y". Outputs = "W1, b1, W2, b2, parameters".
### START CODE HERE ### (≈ 5 lines of code)
parameters = initialize_parameters(n_x, n_h, n_y)
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]
### END CODE HERE ###

# Loop (gradient descent)

for i in range(0, num_iterations):

### START CODE HERE ### (≈ 4 lines of code)
# Forward propagation. Inputs: "X, parameters". Outputs: "A2, cache".
A2, cache = forward_propagation(X, parameters)

# Cost function. Inputs: "A2, Y, parameters". Outputs: "cost".
cost = compute_cost(A2, Y, parameters)

# Backpropagation. Inputs: "parameters, cache, X, Y". Outputs: "grads".
grads = backward_propagation(parameters, cache, X, Y)

# Gradient descent parameter update. Inputs: "parameters, grads". Outputs: "parameters".
parameters = update_parameters(parameters, grads)

### END CODE HERE ###

# Print the cost every 1000 iterations
if print_cost and i % 1000 == 0:
print ("Cost after iteration %i: %f" %(i, cost))

return parameters


"""
X_assess, Y_assess = nn_model_test_case()

parameters = nn_model(X_assess, Y_assess, 4, num_iterations=10000, print_cost=False)
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))

out:
W1 = [[-4.18494482 5.33220319]
[-7.52989354 1.24306197]
[-4.19295428 5.32631786]
[ 7.52983748 -1.24309404]]
b1 = [[ 2.32926815]
[ 3.7945905 ]
[ 2.33002544]
[-3.79468791]]
W2 = [[-6033.83672179 -6008.12981272 -6033.10095329 6008.06636901]]
b2 = [[-52.66607704]]
"""

Predictions

img

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# GRADED FUNCTION: predict

def predict(parameters, X):
"""
Using the learned parameters, predicts a class for each example in X

Arguments:
parameters -- python dictionary containing your parameters
X -- input data of size (n_x, m)

Returns
predictions -- vector of predictions of our model (red: 0 / blue: 1)
"""

# Computes probabilities using forward propagation, and classifies to 0/1 using 0.5 as the threshold.
### START CODE HERE ### (≈ 2 lines of code)
A2, cache = forward_propagation(X, parameters)
predictions = (A2 > 0.5)
### END CODE HERE ###

return predictions

"""
parameters, X_assess = predict_test_case()

predictions = predict(parameters, X_assess)
print("predictions mean = " + str(np.mean(predictions)))

out:
predictions mean = 0.666666666667
"""

It is time to run the model and see how it performs on a planar dataset. Run the following code to test your model with a single hidden layer of $n_h$ hidden units.

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# Build a model with a n_h-dimensional hidden layer
parameters = nn_model(X, Y, n_h = 4, num_iterations = 10000, print_cost=True)

# Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), X, Y)
plt.title("Decision Boundary for hidden layer size " + str(4))

"""
out:
Cost after iteration 0: 0.693048
Cost after iteration 1000: 0.288083
Cost after iteration 2000: 0.254385
Cost after iteration 3000: 0.233864
Cost after iteration 4000: 0.226792
Cost after iteration 5000: 0.222644
Cost after iteration 6000: 0.219731
Cost after iteration 7000: 0.217504
Cost after iteration 8000: 0.219504
Cost after iteration 9000: 0.218571
"""

img

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# Print accuracy
predictions = predict(parameters, X)
print ('Accuracy: %d' % float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100) + '%')

"""
out:
Accuracy: 90%
"""

Accuracy is really high compared to Logistic Regression. The model has learnt the leaf patterns of the flower! Neural networks are able to learn even highly non-linear decision boundaries, unlike logistic regression.

Tuning hidden layer size

Run the following code. It may take 1-2 minutes. You will observe different behaviors of the model for various hidden layer sizes.

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# This may take about 2 minutes to run

plt.figure(figsize=(16, 32))
hidden_layer_sizes = [1, 2, 3, 4, 5, 20, 50]
for i, n_h in enumerate(hidden_layer_sizes):
plt.subplot(5, 2, i+1)
plt.title('Hidden Layer of size %d' % n_h)
parameters = nn_model(X, Y, n_h, num_iterations = 5000)
plot_decision_boundary(lambda x: predict(parameters, x.T), X, Y)
predictions = predict(parameters, X)
accuracy = float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100)
print ("Accuracy for {} hidden units: {} %".format(n_h, accuracy))

out:
Accuracy for 1 hidden units: 67.5 %
Accuracy for 2 hidden units: 67.25 %
Accuracy for 3 hidden units: 90.75 %
Accuracy for 4 hidden units: 90.5 %
Accuracy for 5 hidden units: 91.25 %
Accuracy for 20 hidden units: 90.0 %
Accuracy for 50 hidden units: 90.75 %

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Performance on other datasets

If you want, you can rerun the whole notebook (minus the dataset part) for each of the following datasets.

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# Datasets
noisy_circles, noisy_moons, blobs, gaussian_quantiles, no_structure = load_extra_datasets()

datasets = {"noisy_circles": noisy_circles,
"noisy_moons": noisy_moons,
"blobs": blobs,
"gaussian_quantiles": gaussian_quantiles}

### START CODE HERE ### (choose your dataset)
dataset = "noisy_moons"
### END CODE HERE ###

X, Y = datasets[dataset]
X, Y = X.T, Y.reshape(1, Y.shape[0])

# make blobs binary
if dataset == "blobs":
Y = Y%2

# Visualize the data
plt.scatter(X[0, :], X[1, :], c=Y, s=40, cmap=plt.cm.Spectral);

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Conclusion

You’ve learnt to:

  • Build a complete neural network with a hidden layer
  • Make a good use of a non-linear unit
  • Implemented forward propagation and backpropagation, and trained a neural network
  • See the impact of varying the hidden layer size, including overfitting.
-------------本文结束感谢您的阅读-------------

本文标题:Planar data classification with one hidden layer

文章作者:丁鹏

发布时间:2018年01月02日 - 22:01

最后更新:2018年03月20日 - 11:03

原始链接:http://deepon.me/2018/01/02/Planar-data-classification-with-one-hidden-layer/

许可协议: 署名-非商业性使用-禁止演绎 4.0 国际 转载请保留原文链接及作者。

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