# Chapter 20 – Coding Example#

Data Science and Machine Learning for Geoscientists

import numpy as np
import random
class Network(object):

def __init__(self, sizes):
self.num_layers = len(sizes)    # sizes = [2,3,1] --> it has 2 inputs, 3 neurons in the second layer, and 1 neuron in the last layer
self.sizes = sizes    # len of sizes is how many layers it has, it this case, 3 layers
self.biases = [np.random.randn(y, 1) for y in sizes[1:]]    # y = 3 and 1
self.weights = [np.random.randn(y, x)     # np.random.randn(y, 1)  creates an array containing y lists, each has 1 random value
for x, y in zip(sizes[:-1], sizes[1:])]    # np.random.randn mean 0 and variance 1
# 2,3          3,1
def feedforward(self, a):
"""Return the output of the network if "a" is input."""
for b, w in zip(self.biases, self.weights):
a = sigmoid(np.dot(w, a)+b)
return a    # a is an array that contains

def SGD(self, training_data, epochs, mini_batch_size, eta,
test_data=None):
"""Train the neural network using mini-batch stochastic
gradient descent.  The "training_data" is a list of tuples
"(x, y)" representing the training inputs and the desired
outputs.  The other non-optional parameters are
self-explanatory.  If "test_data" is provided then the
network will be evaluated against the test data after each
epoch, and partial progress printed out.  This is useful for
tracking progress, but slows things down substantially."""
if test_data: n_test = len(test_data)
n = len(training_data)
for j in range(epochs):
random.shuffle(training_data)
mini_batches = [
training_data[k:k+mini_batch_size]
for k in range(0, n, mini_batch_size)]
for mini_batch in mini_batches:
self.update_mini_batch(mini_batch, eta)
if test_data:
print("Epoch {0}: {1} / {2}".format(
j, self.evaluate(test_data), n_test))
else:
print("Epoch {0} complete".format(j))

def update_mini_batch(self, mini_batch, eta):
"""Update the network's weights and biases by applying
gradient descent using backpropagation to a single mini batch.
The "mini_batch" is a list of tuples "(x, y)", and "eta"
is the learning rate."""
nabla_b = [np.zeros(b.shape) for b in self.biases]
nabla_w = [np.zeros(w.shape) for w in self.weights]
for x, y in mini_batch:
delta_nabla_b, delta_nabla_w = self.backprop(x, y)
nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]
nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]
self.weights = [w-(eta/len(mini_batch))*nw
for w, nw in zip(self.weights, nabla_w)]
self.biases = [b-(eta/len(mini_batch))*nb
for b, nb in zip(self.biases, nabla_b)]

def backprop(self, x, y):
"""Return a tuple (nabla_b, nabla_w) representing the
gradient for the cost function C_x.  nabla_b and
nabla_w are layer-by-layer lists of numpy arrays, similar
to self.biases and self.weights."""
nabla_b = [np.zeros(b.shape) for b in self.biases]
nabla_w = [np.zeros(w.shape) for w in self.weights]
# feedforward
activation = x
activations = [x] # list to store all the activations, layer by layer
zs = [] # list to store all the z vectors, layer by layer
for b, w in zip(self.biases, self.weights):
z = np.dot(w, activation)+b
zs.append(z)
activation = sigmoid(z)
activations.append(activation)
# backward pass
delta = self.cost_derivative(activations[-1], y) * sigmoid_prime(zs[-1])
nabla_b[-1] = delta
nabla_w[-1] = np.dot(delta, activations[-2].transpose())
# Note that the variable l in the loop below is used a little
# differently to the notation in Chapter 2 of the book.  Here,
# l = 1 means the last layer of neurons, l = 2 is the
# second-last layer, and so on.  It's a renumbering of the
# scheme in the book, used here to take advantage of the fact
# that Python can use negative indices in lists.
for l in range(2, self.num_layers):
z = zs[-l]
sp = sigmoid_prime(z)
delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
nabla_b[-l] = delta
nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())
return (nabla_b, nabla_w)

def evaluate(self, test_data):
"""Return the number of test inputs for which the neural
network outputs the correct result. Note that the neural
network's output is assumed to be the index of whichever
neuron in the final layer has the highest activation."""
test_results = [(np.argmax(self.feedforward(x)), y)
for (x, y) in test_data]
return sum(int(x == y) for (x, y) in test_results)

def cost_derivative(self, output_activations, y):
"""Return the vector of partial derivatives \partial C_x /
\partial a for the output activations."""
return (output_activations-y)

def sigmoid(z):
return 1.0/(1.0+np.exp(-z))
def sigmoid_prime(z):
"""Derivative of the sigmoid function."""
return sigmoid(z)*(1-sigmoid(z))

sizes = [2, 3, 1]
for y in sizes[1:]:
#print(y)
pass
np.random.randn(3, 1)

array([[ 2.06118074],
[-0.42243013],
[-0.89537547]])

net = Network([2, 3, 1])

"""
~~~~~~~~~~~~

A library to load the MNIST image data.  For details of the data
structures that are returned, see the doc strings for load_data
and load_data_wrapper.  In practice, load_data_wrapper is the
function usually called by our neural network code.
"""

#### Libraries
# Standard library
import pickle
import gzip

# Third-party libraries
import numpy as np

"""Return the MNIST data as a tuple containing the training data,
the validation data, and the test data.

The training_data is returned as a tuple with two entries.
The first entry contains the actual training images.  This is a
numpy ndarray with 50,000 entries.  Each entry is, in turn, a
numpy ndarray with 784 values, representing the 28 * 28 = 784
pixels in a single MNIST image.

The second entry in the training_data tuple is a numpy ndarray
containing 50,000 entries.  Those entries are just the digit
values (0...9) for the corresponding images contained in the first
entry of the tuple.

The validation_data and test_data are similar, except
each contains only 10,000 images.

This is a nice data format, but for use in neural networks it's
helpful to modify the format of the training_data a little.
That's done in the wrapper function load_data_wrapper(), see
below.
"""
f = gzip.open('mnist.pkl.gz', 'rb')
training_data, validation_data, test_data = pickle.load(f, encoding='latin1')
f.close()
return (training_data, validation_data, test_data)

"""Return a tuple containing (training_data, validation_data,
test_data). Based on load_data, but the format is more
convenient for use in our implementation of neural networks.

In particular, training_data is a list containing 50,000
2-tuples (x, y).  x is a 784-dimensional numpy.ndarray
containing the input image.  y is a 10-dimensional
numpy.ndarray representing the unit vector corresponding to the
correct digit for x.

validation_data and test_data are lists containing 10,000
2-tuples (x, y).  In each case, x is a 784-dimensional
numpy.ndarry containing the input image, and y is the
corresponding classification, i.e., the digit values (integers)
corresponding to x.

Obviously, this means we're using slightly different formats for
the training data and the validation / test data.  These formats
turn out to be the most convenient for use in our neural network
code."""
training_inputs = [np.reshape(x, (784, 1)) for x in tr_d]
training_results = [vectorized_result(y) for y in tr_d]
training_data = list(zip(training_inputs, training_results))
validation_inputs = [np.reshape(x, (784, 1)) for x in va_d]
validation_data = list(zip(validation_inputs, va_d))
test_inputs = [np.reshape(x, (784, 1)) for x in te_d]
test_data = list(zip(test_inputs, te_d))
return (training_data, validation_data, test_data)

def vectorized_result(j):
"""Return a 10-dimensional unit vector with a 1.0 in the jth
position and zeroes elsewhere.  This is used to convert a digit
(0...9) into a corresponding desired output from the neural
network."""
e = np.zeros((10, 1))
e[j] = 1.0
return e

training_data, validation_data, test_data = load_data_wrapper()

net = Network([784, 30, 10])
net.SGD(training_data, 30, 10, 3.0, test_data=test_data)

Epoch 0: 8281 / 10000
Epoch 1: 8372 / 10000
Epoch 2: 9320 / 10000
Epoch 3: 9359 / 10000
Epoch 4: 9395 / 10000
Epoch 5: 9395 / 10000
Epoch 6: 9416 / 10000
Epoch 7: 9437 / 10000
Epoch 8: 9469 / 10000
Epoch 9: 9444 / 10000
Epoch 10: 9464 / 10000
Epoch 11: 9471 / 10000
Epoch 12: 9450 / 10000
Epoch 13: 9488 / 10000
Epoch 14: 9478 / 10000
Epoch 15: 9465 / 10000
Epoch 16: 9476 / 10000
Epoch 17: 9482 / 10000
Epoch 18: 9490 / 10000
Epoch 19: 9486 / 10000
Epoch 20: 9505 / 10000
Epoch 21: 9497 / 10000
Epoch 22: 9496 / 10000
Epoch 23: 9491 / 10000
Epoch 24: 9494 / 10000
Epoch 25: 9508 / 10000
Epoch 26: 9506 / 10000
Epoch 27: 9496 / 10000
Epoch 28: 9493 / 10000
Epoch 29: 9518 / 10000