Neural Network



Using MNIST

MNIST Characters


The MNIST database (Modified National Institute of Standards and Technology database) of handwritten digits consists of a training set of 60,000 examples, and a test set of 10,000 examples. It is a subset of a larger set available from NIST. Additionally, the black and white images from NIST were size-normalized and centered to fit into a 28x28 pixel bounding box and anti-aliased, which introduced grayscale levels.

This database is well liked for training and testing in the field of machine learning and image processing. It is a remixed subset of the original NIST datasets. One half of the 60,000 training images consist of images from NIST's testing dataset and the other half from Nist's training set. The 10,000 images from the testing set are similarly assembled.

The MNIST dataset is used by researchers to test and compare their research results with others. The lowest error rates in literature are as low as 0.21 percent.1



Reading the MNIST data set

The images from the data set have the size 28 x 28. They are saved in the csv data files mnist_train.csv and mnist_test.csv.

Every line of these files consists of an image, i.e. 785 numbers between 0 and 255.

The first number of each line is the label, i.e. the digit which is depicted in the image. The following 784 numbers are the pixels of the 28 x 28 image.

import numpy as np
import matplotlib.pyplot as plt

image_size = 28 # width and length
no_of_different_labels = 10 #  i.e. 0, 1, 2, 3, ..., 9
image_pixels = image_size * image_size
data_path = "data/mnist/"
train_data = np.loadtxt(data_path + "mnist_train.csv", 
                        delimiter=",")
test_data = np.loadtxt(data_path + "mnist_test.csv", 
                       delimiter=",") 
test_data[:10]
Output:
array([[7., 0., 0., ..., 0., 0., 0.],
       [2., 0., 0., ..., 0., 0., 0.],
       [1., 0., 0., ..., 0., 0., 0.],
       ...,
       [9., 0., 0., ..., 0., 0., 0.],
       [5., 0., 0., ..., 0., 0., 0.],
       [9., 0., 0., ..., 0., 0., 0.]])
test_data[test_data==255]
test_data.shape
Output:
(10000, 785)

The images of the MNIST dataset are greyscale and the pixels range between 0 and 255 including both bounding values. We will map these values into an interval from [0.01, 1] by multiplying each pixel by 0.99 / 255 and adding 0.01 to the result. This way, we avoid 0 values as inputs, which are capable of preventing weight updates, as we we seen in the introductory chapter.

fac = 0.99 / 255
train_imgs = np.asfarray(train_data[:, 1:]) * fac + 0.01
test_imgs = np.asfarray(test_data[:, 1:]) * fac + 0.01

train_labels = np.asfarray(train_data[:, :1])
test_labels = np.asfarray(test_data[:, :1])

We need the labels in our calculations in a one-hot representation. We have 10 digits from 0 to 9, i.e. lr = np.arange(10).

Turning a label into one-hot representation can be achieved with the command: (lr==label).astype(np.int)

We demonstrate this in the following:

import numpy as np

lr = np.arange(10)

for label in range(10):
    one_hot = (lr==label).astype(np.int)
    print("label: ", label, " in one-hot representation: ", one_hot)
label:  0  in one-hot representation:  [1 0 0 0 0 0 0 0 0 0]
label:  1  in one-hot representation:  [0 1 0 0 0 0 0 0 0 0]
label:  2  in one-hot representation:  [0 0 1 0 0 0 0 0 0 0]
label:  3  in one-hot representation:  [0 0 0 1 0 0 0 0 0 0]
label:  4  in one-hot representation:  [0 0 0 0 1 0 0 0 0 0]
label:  5  in one-hot representation:  [0 0 0 0 0 1 0 0 0 0]
label:  6  in one-hot representation:  [0 0 0 0 0 0 1 0 0 0]
label:  7  in one-hot representation:  [0 0 0 0 0 0 0 1 0 0]
label:  8  in one-hot representation:  [0 0 0 0 0 0 0 0 1 0]
label:  9  in one-hot representation:  [0 0 0 0 0 0 0 0 0 1]

We are ready now to turn our labelled images into one-hot representations. Instead of zeroes and one, we create 0.01 and 0.99, which will be better for our calculations:

lr = np.arange(no_of_different_labels)

# transform labels into one hot representation
train_labels_one_hot = (lr==train_labels).astype(np.float)
test_labels_one_hot = (lr==test_labels).astype(np.float)

# we don't want zeroes and ones in the labels neither:
train_labels_one_hot[train_labels_one_hot==0] = 0.01
train_labels_one_hot[train_labels_one_hot==1] = 0.99
test_labels_one_hot[test_labels_one_hot==0] = 0.01
test_labels_one_hot[test_labels_one_hot==1] = 0.99

Before we start using the MNIST data sets with our neural network, we will have a look at some images:

for i in range(10):
    img = train_imgs[i].reshape((28,28))
    plt.imshow(img, cmap="Greys")
    plt.show()



Dumping the Data for Faster Reload

You may have noticed that it is quite slow to read in the data from the csv files.

We will save the data in binary format with the dump function from the pickle module:

import pickle

with open("data/mnist/pickled_mnist.pkl", "bw") as fh:
    data = (train_imgs, 
            test_imgs, 
            train_labels,
            test_labels)
    pickle.dump(data, fh)
    

We are able now to read in the data by using pickle.load. This is a lot faster than using loadtxt on the csv files:

import pickle

with open("data/mnist/pickled_mnist.pkl", "br") as fh:
    data = pickle.load(fh)

train_imgs = data[0]
test_imgs = data[1]
train_labels = data[2]
test_labels = data[3]

train_labels_one_hot = (lr==train_labels).astype(np.float)
test_labels_one_hot = (lr==test_labels).astype(np.float)


image_size = 28 # width and length
no_of_different_labels = 10 #  i.e. 0, 1, 2, 3, ..., 9
image_pixels = image_size * image_size



Classifying the Data

We will use the following neuronal network class for our first classification:

import numpy as np

@np.vectorize
def sigmoid(x):
    return 1 / (1 + np.e ** -x)
activation_function = sigmoid

from scipy.stats import truncnorm

def truncated_normal(mean=0, sd=1, low=0, upp=10):
    return truncnorm((low - mean) / sd, 
                     (upp - mean) / sd, 
                     loc=mean, 
                     scale=sd)


class NeuralNetwork:
    
    def __init__(self, 
                 no_of_in_nodes, 
                 no_of_out_nodes, 
                 no_of_hidden_nodes,
                 learning_rate):
        self.no_of_in_nodes = no_of_in_nodes
        self.no_of_out_nodes = no_of_out_nodes
        self.no_of_hidden_nodes = no_of_hidden_nodes
        self.learning_rate = learning_rate 
        self.create_weight_matrices()
        
    def create_weight_matrices(self):
        """ 
        A method to initialize the weight 
        matrices of the neural network
        """
        rad = 1 / np.sqrt(self.no_of_in_nodes)
        X = truncated_normal(mean=0, 
                             sd=1, 
                             low=-rad, 
                             upp=rad)
        self.wih = X.rvs((self.no_of_hidden_nodes, 
                                       self.no_of_in_nodes))
        rad = 1 / np.sqrt(self.no_of_hidden_nodes)
        X = truncated_normal(mean=0, sd=1, low=-rad, upp=rad)
        self.who = X.rvs((self.no_of_out_nodes, 
                                         self.no_of_hidden_nodes))
        
    
    def train(self, input_vector, target_vector):
        """
        input_vector and target_vector can 
        be tuple, list or ndarray
        """
        
        input_vector = np.array(input_vector, ndmin=2).T
        target_vector = np.array(target_vector, ndmin=2).T
        
        output_vector1 = np.dot(self.wih, 
                                input_vector)
        output_hidden = activation_function(output_vector1)
        
        output_vector2 = np.dot(self.who, 
                                output_hidden)
        output_network = activation_function(output_vector2)
        
        output_errors = target_vector - output_network
        # update the weights:
        tmp = output_errors * output_network \
              * (1.0 - output_network)     
        tmp = self.learning_rate  * np.dot(tmp, 
                                           output_hidden.T)
        self.who += tmp


        # calculate hidden errors:
        hidden_errors = np.dot(self.who.T, 
                               output_errors)
        # update the weights:
        tmp = hidden_errors * output_hidden * \
              (1.0 - output_hidden)
        self.wih += self.learning_rate \
                          * np.dot(tmp, input_vector.T)
        

        
    
    def run(self, input_vector):
        # input_vector can be tuple, list or ndarray
        input_vector = np.array(input_vector, ndmin=2).T

        output_vector = np.dot(self.wih, 
                               input_vector)
        output_vector = activation_function(output_vector)
        
        output_vector = np.dot(self.who, 
                               output_vector)
        output_vector = activation_function(output_vector)
    
        return output_vector
            
    def confusion_matrix(self, data_array, labels):
        cm = np.zeros((10, 10), int)
        for i in range(len(data_array)):
            res = self.run(data_array[i])
            res_max = res.argmax()
            target = labels[i][0]
            cm[res_max, int(target)] += 1
        return cm    

    def precision(self, label, confusion_matrix):
        col = confusion_matrix[:, label]
        return confusion_matrix[label, label] / col.sum()
    
    def recall(self, label, confusion_matrix):
        row = confusion_matrix[label, :]
        return confusion_matrix[label, label] / row.sum()
        
    
    def evaluate(self, data, labels):
        corrects, wrongs = 0, 0
        for i in range(len(data)):
            res = self.run(data[i])
            res_max = res.argmax()
            if res_max == labels[i]:
                corrects += 1
            else:
                wrongs += 1
        return corrects, wrongs
            
ANN = NeuralNetwork(no_of_in_nodes = image_pixels, 
                    no_of_out_nodes = 10, 
                    no_of_hidden_nodes = 100,
                    learning_rate = 0.1)
    
    
for i in range(len(train_imgs)):
    ANN.train(train_imgs[i], train_labels_one_hot[i])
for i in range(20):
    res = ANN.run(test_imgs[i])
    print(test_labels[i], np.argmax(res), np.max(res))
    
[7.] 7 0.9980952178965483
[2.] 2 0.9822212212838083
[1.] 1 0.9971043314493291
[0.] 0 0.9968193574802353
[4.] 4 0.9863130710838649
[1.] 1 0.9967789929522022
[4.] 4 0.9837692151630008
[9.] 9 0.9828059684397105
[5.] 6 0.7183407051482742
[9.] 9 0.9862589613724044
[0.] 0 0.9943681077245705
[6.] 6 0.9111256740230744
[9.] 9 0.9952070486287015
[0.] 0 0.9963298568809852
[1.] 1 0.9985038293964573
[5.] 5 0.9054505654450581
[9.] 9 0.9978209006348177
[7.] 7 0.9978257082434695
[3.] 3 0.9109504057097583
[4.] 4 0.996328871128103
corrects, wrongs = ANN.evaluate(train_imgs, train_labels)
print("accuracy train: ", corrects / ( corrects + wrongs))
corrects, wrongs = ANN.evaluate(test_imgs, test_labels)
print("accuracy: test", corrects / ( corrects + wrongs))

cm = ANN.confusion_matrix(train_imgs, train_labels)
print(cm)

for i in range(10):
    print("digit: ", i, "precision: ", ANN.precision(i, cm), "recall: ", ANN.recall(i, cm))
accuracy train:  0.9479333333333333
accuracy: test 0.946
[[5821    2   44   28   12   49   43   19   22   19]
 [   0 6612   39   19   10   18   12   48  106   12]
 [   3   30 5569   69   15   23    3   66   20   10]
 [   6   37   87 5818    2  156    3   22  137   76]
 [   4   10   41    4 5378   18    6   29   15   61]
 [   5    2    3   50    2 4941   28    1   18    6]
 [  35    5   62   18   71   84 5803    6   57    4]
 [   2    7   37   39    9    6    0 5861    2   28]
 [  37   19   62   46    8   51   19   10 5363   23]
 [  10   18   14   40  335   75    1  203  111 5710]]
digit:  0 precision:  0.9827789971298329 recall:  0.9607195906915332
digit:  1 precision:  0.9807178878671018 recall:  0.9616055846422339
digit:  2 precision:  0.9347096341054045 recall:  0.9588498622589532
digit:  3 precision:  0.9489479693361605 recall:  0.9170870113493065
digit:  4 precision:  0.920575145498117 recall:  0.9662234998203377
digit:  5 precision:  0.9114554510237963 recall:  0.977254746835443
digit:  6 precision:  0.9805677593781683 recall:  0.9443449959316518
digit:  7 precision:  0.9355147645650439 recall:  0.9783007845100985
digit:  8 precision:  0.9165954537685865 recall:  0.9512238382405108
digit:  9 precision:  0.9598251807026391 recall:  0.876170016878932

Multiple Runs

We can repeat the training multiple times. Each run is called an "epoch".

epochs = 3

NN = NeuralNetwork(no_of_in_nodes = image_pixels, 
                   no_of_out_nodes = 10, 
                   no_of_hidden_nodes = 100,
                   learning_rate = 0.1)


for epoch in range(epochs):  
    print("epoch: ", epoch)
    for i in range(len(train_imgs)):
        NN.train(train_imgs[i], 
                 train_labels_one_hot[i])
  
    corrects, wrongs = NN.evaluate(train_imgs, train_labels)
    print("accuracy train: ", corrects / ( corrects + wrongs))
    corrects, wrongs = NN.evaluate(test_imgs, test_labels)
    print("accuracy: test", corrects / ( corrects + wrongs))
epoch:  0
accuracy train:  0.9460166666666666
accuracy: test 0.9449
epoch:  1
accuracy train:  0.9633833333333334
accuracy: test 0.9581
epoch:  2
accuracy train:  0.9707
accuracy: test 0.9622

We want to do the multiple training of the training set inside of our network. To this purpose we rewrite the method train and add a method train_single. train_single is more or less what we called 'train' before. Whereas the new 'train' method is doing the epoch counting. For testing purposes, we save the weight matrices after each epoch in
the list intermediate_weights. This list is returned as the output of train:

import numpy as np

@np.vectorize
def sigmoid(x):
    return 1 / (1 + np.e ** -x)
activation_function = sigmoid

from scipy.stats import truncnorm

def truncated_normal(mean=0, sd=1, low=0, upp=10):
    return truncnorm((low - mean) / sd, 
                     (upp - mean) / sd, 
                     loc=mean, 
                     scale=sd)


class NeuralNetwork:
    
    def __init__(self, 
                 no_of_in_nodes, 
                 no_of_out_nodes, 
                 no_of_hidden_nodes,
                 learning_rate):
        self.no_of_in_nodes = no_of_in_nodes
        self.no_of_out_nodes = no_of_out_nodes
        self.no_of_hidden_nodes = no_of_hidden_nodes
        self.learning_rate = learning_rate 
        self.create_weight_matrices()
        
    def create_weight_matrices(self):
        """ A method to initialize the weight matrices of the neural network"""
        rad = 1 / np.sqrt(self.no_of_in_nodes)
        X = truncated_normal(mean=0, 
                             sd=1, 
                             low=-rad, 
                             upp=rad)
        self.wih = X.rvs((self.no_of_hidden_nodes, 
                                       self.no_of_in_nodes))
        rad = 1 / np.sqrt(self.no_of_hidden_nodes)
        X = truncated_normal(mean=0, 
                             sd=1, 
                             low=-rad, 
                             upp=rad)
        self.who = X.rvs((self.no_of_out_nodes, 
                                        self.no_of_hidden_nodes))
        
    
    def train_single(self, input_vector, target_vector):
        """
        input_vector and target_vector can be tuple, 
        list or ndarray
        """
        
        output_vectors = []
        input_vector = np.array(input_vector, ndmin=2).T
        target_vector = np.array(target_vector, ndmin=2).T

        
        output_vector1 = np.dot(self.wih, 
                                input_vector)
        output_hidden = activation_function(output_vector1)
        
        output_vector2 = np.dot(self.who, 
                                output_hidden)
        output_network = activation_function(output_vector2)
        
        output_errors = target_vector - output_network
        # update the weights:
        tmp = output_errors * output_network * \
              (1.0 - output_network)     
        tmp = self.learning_rate  * np.dot(tmp, 
                                           output_hidden.T)
        self.who += tmp


        # calculate hidden errors:
        hidden_errors = np.dot(self.who.T, 
                               output_errors)
        # update the weights:
        tmp = hidden_errors * output_hidden * (1.0 - output_hidden)
        self.wih += self.learning_rate * np.dot(tmp, input_vector.T)
        

    def train(self, data_array, 
              labels_one_hot_array,
              epochs=1,
              intermediate_results=False):
        intermediate_weights = []
        for epoch in range(epochs):  
            print("*", end="")
            for i in range(len(data_array)):
                self.train_single(data_array[i], 
                                  labels_one_hot_array[i])
            if intermediate_results:
                intermediate_weights.append((self.wih.copy(), 
                                             self.who.copy()))
        return intermediate_weights        
            
    def confusion_matrix(self, data_array, labels):
        cm = {}
        for i in range(len(data_array)):
            res = self.run(data_array[i])
            res_max = res.argmax()
            target = labels[i][0]
            if (target, res_max) in cm:
                cm[(target, res_max)] += 1
            else:
                cm[(target, res_max)] = 1
        return cm
        
    
    def run(self, input_vector):
        """ input_vector can be tuple, list or ndarray """
        
        input_vector = np.array(input_vector, ndmin=2).T

        output_vector = np.dot(self.wih, 
                               input_vector)
        output_vector = activation_function(output_vector)
        
        output_vector = np.dot(self.who, 
                               output_vector)
        output_vector = activation_function(output_vector)
    
        return output_vector
    
    def evaluate(self, data, labels):
        corrects, wrongs = 0, 0
        for i in range(len(data)):
            res = self.run(data[i])
            res_max = res.argmax()
            if res_max == labels[i]:
                corrects += 1
            else:
                wrongs += 1
        return corrects, wrongs
            

    
epochs = 10

ANN = NeuralNetwork(no_of_in_nodes = image_pixels, 
                               no_of_out_nodes = 10, 
                               no_of_hidden_nodes = 100,
                               learning_rate = 0.15)
    
    
 
weights = ANN.train(train_imgs, 
                    train_labels_one_hot, 
                    epochs=epochs, 
                    intermediate_results=True)
        
**********
cm = ANN.confusion_matrix(train_imgs, train_labels)
        
print(ANN.run(train_imgs[i]))
[[7.43484529e-02]
 [3.66837230e-08]
 [2.25297794e-06]
 [6.17959264e-08]
 [7.28907405e-07]
 [1.52789856e-05]
 [2.23938658e-07]
 [9.98247522e-05]
 [9.99993490e-01]
 [1.75621871e-05]]
cm = list(cm.items())
print(sorted(cm))
[((0.0, 0), 5897), ((0.0, 3), 1), ((0.0, 4), 1), ((0.0, 5), 3), ((0.0, 6), 4), ((0.0, 7), 1), ((0.0, 8), 7), ((0.0, 9), 9), ((1.0, 0), 1), ((1.0, 1), 6678), ((1.0, 2), 12), ((1.0, 3), 11), ((1.0, 4), 13), ((1.0, 5), 5), ((1.0, 6), 1), ((1.0, 7), 4), ((1.0, 8), 5), ((1.0, 9), 12), ((2.0, 0), 36), ((2.0, 1), 8), ((2.0, 2), 5836), ((2.0, 3), 13), ((2.0, 4), 12), ((2.0, 5), 3), ((2.0, 6), 6), ((2.0, 7), 18), ((2.0, 8), 15), ((2.0, 9), 11), ((3.0, 0), 37), ((3.0, 1), 6), ((3.0, 2), 40), ((3.0, 3), 5936), ((3.0, 5), 30), ((3.0, 6), 2), ((3.0, 7), 16), ((3.0, 8), 25), ((3.0, 9), 39), ((4.0, 0), 8), ((4.0, 1), 3), ((4.0, 2), 4), ((4.0, 3), 2), ((4.0, 4), 5707), ((4.0, 5), 2), ((4.0, 6), 7), ((4.0, 7), 2), ((4.0, 8), 3), ((4.0, 9), 104), ((5.0, 0), 16), ((5.0, 1), 3), ((5.0, 2), 2), ((5.0, 3), 14), ((5.0, 4), 5), ((5.0, 5), 5339), ((5.0, 6), 11), ((5.0, 7), 3), ((5.0, 8), 6), ((5.0, 9), 22), ((6.0, 0), 33), ((6.0, 1), 5), ((6.0, 2), 5), ((6.0, 3), 1), ((6.0, 4), 9), ((6.0, 5), 29), ((6.0, 6), 5830), ((6.0, 8), 6), ((7.0, 0), 7), ((7.0, 1), 18), ((7.0, 2), 30), ((7.0, 3), 2), ((7.0, 4), 22), ((7.0, 5), 4), ((7.0, 7), 6068), ((7.0, 8), 7), ((7.0, 9), 107), ((8.0, 0), 23), ((8.0, 1), 38), ((8.0, 2), 13), ((8.0, 3), 17), ((8.0, 4), 16), ((8.0, 5), 32), ((8.0, 6), 13), ((8.0, 7), 1), ((8.0, 8), 5620), ((8.0, 9), 78), ((9.0, 0), 16), ((9.0, 1), 3), ((9.0, 2), 1), ((9.0, 3), 7), ((9.0, 4), 16), ((9.0, 5), 5), ((9.0, 6), 1), ((9.0, 7), 14), ((9.0, 8), 5), ((9.0, 9), 5881)]
for i in range(epochs):  
    print("epoch: ", i)
    ANN.wih = weights[i][0]
    ANN.who = weights[i][1]
   
    corrects, wrongs = ANN.evaluate(train_imgs, train_labels)
    print("accuracy train: ", corrects / ( corrects + wrongs))
    corrects, wrongs = ANN.evaluate(test_imgs, test_labels)
    print("accuracy: test", corrects / ( corrects + wrongs))
epoch:  0
accuracy train:  0.9419666666666666
accuracy: test 0.9416
epoch:  1
accuracy train:  0.9600833333333333
accuracy: test 0.9557
epoch:  2
accuracy train:  0.96535
accuracy: test 0.9606
epoch:  3
accuracy train:  0.9685833333333334
accuracy: test 0.9605
epoch:  4
accuracy train:  0.97385
accuracy: test 0.9654
epoch:  5
accuracy train:  0.97355
accuracy: test 0.9649
epoch:  6
accuracy train:  0.9767833333333333
accuracy: test 0.9678
epoch:  7
accuracy train:  0.9798
accuracy: test 0.9692
epoch:  8
accuracy train:  0.9777166666666667
accuracy: test 0.9663
epoch:  9
accuracy train:  0.9798666666666667
accuracy: test 0.9682



With Bias Nodes

import numpy as np

@np.vectorize
def sigmoid(x):
    return 1 / (1 + np.e ** -x)
activation_function = sigmoid

from scipy.stats import truncnorm

def truncated_normal(mean=0, sd=1, low=0, upp=10):
    return truncnorm((low - mean) / sd, 
                     (upp - mean) / sd, 
                     loc=mean, 
                     scale=sd)


class NeuralNetwork:
        
    
    def __init__(self, 
                 no_of_in_nodes, 
                 no_of_out_nodes, 
                 no_of_hidden_nodes,
                 learning_rate,
                 bias=None
                ):  

        self.no_of_in_nodes = no_of_in_nodes
        self.no_of_out_nodes = no_of_out_nodes      
        self.no_of_hidden_nodes = no_of_hidden_nodes     
        self.learning_rate = learning_rate 
        self.bias = bias
        self.create_weight_matrices()
    
        
    
    def create_weight_matrices(self):
        """ 
        A method to initialize the weight 
        matrices of the neural network with 
        optional bias nodes
        """
        
        bias_node = 1 if self.bias else 0
        
        rad = 1 / np.sqrt(self.no_of_in_nodes + bias_node)
        X = truncated_normal(mean=0, 
                             sd=1, 
                             low=-rad, 
                             upp=rad)
        self.wih = X.rvs((self.no_of_hidden_nodes, 
                          self.no_of_in_nodes + bias_node))

        rad = 1 / np.sqrt(self.no_of_hidden_nodes + bias_node)
        X = truncated_normal(mean=0, sd=1, low=-rad, upp=rad)
        self.who = X.rvs((self.no_of_out_nodes, 
                          self.no_of_hidden_nodes + bias_node))
        
        
        
    def train(self, input_vector, target_vector):
        """ 
        input_vector and target_vector can 
        be tuple, list or ndarray
        """
        
        bias_node = 1 if self.bias else 0
        if self.bias:
            # adding bias node to the end of the inpuy_vector
            input_vector = np.concatenate((input_vector, 
                                           [self.bias]) )
                                    
            
        input_vector = np.array(input_vector, ndmin=2).T
        target_vector = np.array(target_vector, ndmin=2).T

        
        output_vector1 = np.dot(self.wih, 
                                input_vector)
        output_hidden = activation_function(output_vector1)
        
        if self.bias:
            output_hidden = np.concatenate((output_hidden, 
                                            [[self.bias]]) )
        
        
        output_vector2 = np.dot(self.who, 
                                output_hidden)
        output_network = activation_function(output_vector2)
        
        output_errors = target_vector - output_network
        # update the weights:
        tmp = output_errors * output_network * (1.0 - output_network)     
        tmp = self.learning_rate  * np.dot(tmp, output_hidden.T)
        self.who += tmp


        # calculate hidden errors:
        hidden_errors = np.dot(self.who.T, 
                               output_errors)
        # update the weights:
        tmp = hidden_errors * output_hidden * (1.0 - output_hidden)
        if self.bias:
            x = np.dot(tmp, input_vector.T)[:-1,:]     
        else:
            x = np.dot(tmp, input_vector.T)
        self.wih += self.learning_rate * x
        
       
    
    def run(self, input_vector):
        """
        input_vector can be tuple, list or ndarray
        """
        
        if self.bias:
            # adding bias node to the end of the inpuy_vector
            input_vector = np.concatenate((input_vector, [1]) )
        input_vector = np.array(input_vector, ndmin=2).T

        output_vector = np.dot(self.wih, 
                               input_vector)
        output_vector = activation_function(output_vector)
        
        if self.bias:
            output_vector = np.concatenate( (output_vector, 
                                             [[1]]) )
            

        output_vector = np.dot(self.who, 
                               output_vector)
        output_vector = activation_function(output_vector)
        return output_vector
    
    def evaluate(self, data, labels):
        corrects, wrongs = 0, 0
        for i in range(len(data)):
            res = self.run(data[i])
            res_max = res.argmax()
            if res_max == labels[i]:
                corrects += 1
            else:
                wrongs += 1
        return corrects, wrongs
    
ANN = NeuralNetwork(no_of_in_nodes=image_pixels, 
                    no_of_out_nodes=10, 
                    no_of_hidden_nodes=200,
                    learning_rate=0.1,
                    bias=None)
    
    
for i in range(len(train_imgs)):
    ANN.train(train_imgs[i], train_labels_one_hot[i])
for i in range(20):
    res = ANN.run(test_imgs[i])
    print(test_labels[i], np.argmax(res), np.max(res))
[7.] 7 0.9993819859155288
[2.] 2 0.9672200549506809
[1.] 1 0.9995092558845281
[0.] 0 0.9854240675765327
[4.] 4 0.9919590660928057
[1.] 1 0.9994948509796988
[4.] 4 0.9985819713073246
[9.] 9 0.998669443193103
[5.] 6 0.2028693078785877
[9.] 9 0.9973215949202802
[0.] 0 0.9859700151014654
[6.] 6 0.9381796046018899
[9.] 9 0.9985601946879261
[0.] 0 0.9956192009922595
[1.] 1 0.9994045488656444
[5.] 5 0.9569624440783905
[9.] 9 0.9984763761796004
[7.] 7 0.9978166154278303
[3.] 3 0.9853957301739942
[4.] 4 0.9996234543869396
corrects, wrongs = ANN.evaluate(train_imgs, train_labels)
print("accuracy train: ", corrects / ( corrects + wrongs))
corrects, wrongs = ANN.evaluate(test_imgs, test_labels)
print("accuracy: test", corrects / ( corrects + wrongs))
accuracy train:  0.9501
accuracy: test 0.9512



Version with Bias and Epochs:

import numpy as np

@np.vectorize
def sigmoid(x):
    return 1 / (1 + np.e ** -x)
activation_function = sigmoid

from scipy.stats import truncnorm

def truncated_normal(mean=0, sd=1, low=0, upp=10):
    return truncnorm((low - mean) / sd,
                     (upp - mean) / sd,
                     loc=mean,
                     scale=sd)


class NeuralNetwork:
 
    def __init__(self, 
                 no_of_in_nodes, 
                 no_of_out_nodes, 
                 no_of_hidden_nodes,
                 learning_rate,
                 bias=None
                ):  

        self.no_of_in_nodes = no_of_in_nodes
        self.no_of_out_nodes = no_of_out_nodes
        
        self.no_of_hidden_nodes = no_of_hidden_nodes
            
        self.learning_rate = learning_rate 
        self.bias = bias
        self.create_weight_matrices()
    
        
    
    def create_weight_matrices(self):
        """ 
        A method to initialize the weight matrices 
        of the neural network with optional 
        bias nodes"""
        
        bias_node = 1 if self.bias else 0
        
        rad = 1 / np.sqrt(self.no_of_in_nodes + bias_node)
        X = truncated_normal(mean=0, sd=1, low=-rad, upp=rad)
        self.wih = X.rvs((self.no_of_hidden_nodes, 
                          self.no_of_in_nodes + bias_node))

        rad = 1 / np.sqrt(self.no_of_hidden_nodes + bias_node)
        X = truncated_normal(mean=0, 
                             sd=1, 
                             low=-rad, 
                             upp=rad)
        self.who = X.rvs((self.no_of_out_nodes, 
                          self.no_of_hidden_nodes + bias_node))
        
 
    def train_single(self, input_vector, target_vector):
        """
        input_vector and target_vector can be tuple, 
        list or ndarray
        """

        bias_node = 1 if self.bias else 0
        if self.bias:
            # adding bias node to the end of the inpuy_vector
            input_vector = np.concatenate( (input_vector, 
                                            [self.bias]) )
        
        output_vectors = []
        input_vector = np.array(input_vector, ndmin=2).T
        target_vector = np.array(target_vector, ndmin=2).T

        
        output_vector1 = np.dot(self.wih, 
                                input_vector)
        output_hidden = activation_function(output_vector1)
        
        if self.bias:
            output_hidden = np.concatenate((output_hidden, 
                                            [[self.bias]]) )

        
        output_vector2 = np.dot(self.who, 
                                output_hidden)
        output_network = activation_function(output_vector2)
        
        output_errors = target_vector - output_network
        # update the weights:
        tmp = output_errors * output_network * (1.0 - output_network)          
        tmp = self.learning_rate  * np.dot(tmp, 
                                           output_hidden.T) 
        self.who += tmp

        
        # calculate hidden errors:
        hidden_errors = np.dot(self.who.T, 
                               output_errors)
        # update the weights:
        tmp = hidden_errors * output_hidden * (1.0 - output_hidden)
        if self.bias:
            x = np.dot(tmp, input_vector.T)[:-1,:] 
        else:
            x = np.dot(tmp, input_vector.T)
        self.wih += self.learning_rate * x
        

    def train(self, data_array, 
              labels_one_hot_array,
              epochs=1,
              intermediate_results=False):
        intermediate_weights = []
        for epoch in range(epochs):  
            for i in range(len(data_array)):
                self.train_single(data_array[i], 
                                  labels_one_hot_array[i])
            if intermediate_results:
                intermediate_weights.append((self.wih.copy(), 
                                             self.who.copy()))
        return intermediate_weights      
        

        
    
    def run(self, input_vector):
        # input_vector can be tuple, list or ndarray
        
        if self.bias:
            # adding bias node to the end of the inpuy_vector
            input_vector = np.concatenate( (input_vector, 
                                            [self.bias]) )
        input_vector = np.array(input_vector, ndmin=2).T

        output_vector = np.dot(self.wih, 
                               input_vector)
        output_vector = activation_function(output_vector)
        
        if self.bias:
            output_vector = np.concatenate( (output_vector, 
                                             [[self.bias]]) )
            

        output_vector = np.dot(self.who, 
                               output_vector)
        output_vector = activation_function(output_vector)
    
        return output_vector
    
    
    def evaluate(self, data, labels):
        corrects, wrongs = 0, 0
        for i in range(len(data)):
            res = self.run(data[i])
            res_max = res.argmax()
            if res_max == labels[i]:
                corrects += 1
            else:
                wrongs += 1
        return corrects, wrongs
            
    
epochs = 12

network = NeuralNetwork(no_of_in_nodes=image_pixels, 
                        no_of_out_nodes=10, 
                        no_of_hidden_nodes=100,
                        learning_rate=0.1,
                        bias=None)

weights = network.train(train_imgs, 
                        train_labels_one_hot, 
                        epochs=epochs, 
                        intermediate_results=True) 
for epoch in range(epochs):  
    print("epoch: ", epoch)
    network.wih = weights[epoch][0]
    network.who = weights[epoch][1]
    corrects, wrongs = network.evaluate(train_imgs, 
                                        train_labels)
    print("accuracy train: ", corrects / ( corrects + wrongs))                   
    corrects, wrongs = network.evaluate(test_imgs, 
                                        test_labels)
    print("accuracy test: ", corrects / ( corrects + wrongs)) 
    
epoch:  0
accuracy train:  0.94595
accuracy test:  0.9455
epoch:  1
accuracy train:  0.9622
accuracy test:  0.9574
epoch:  2
accuracy train:  0.96945
accuracy test:  0.9615
epoch:  3
accuracy train:  0.9747666666666667
accuracy test:  0.9643
epoch:  4
accuracy train:  0.97855
accuracy test:  0.9662
epoch:  5
accuracy train:  0.9800666666666666
accuracy test:  0.9663
epoch:  6
accuracy train:  0.9814166666666667
accuracy test:  0.967
epoch:  7
accuracy train:  0.9828333333333333
accuracy test:  0.9671
epoch:  8
accuracy train:  0.9847
accuracy test:  0.9697
epoch:  9
accuracy train:  0.9858833333333333
accuracy test:  0.9703
epoch:  10
accuracy train:  0.9867
accuracy test:  0.9702
epoch:  11
accuracy train:  0.9880666666666666
accuracy test:  0.9702
epochs = 12

with open("nist_tests.csv", "w") as fh_out:  
    for hidden_nodes in [20, 50, 100, 150]:
        for learning_rate in [0.05, 0.1, 0.2]:
            for bias in [None, 0.5]:
                network = NeuralNetwork(no_of_in_nodes=image_pixels, 
                                       no_of_out_nodes=10, 
                                       no_of_hidden_nodes=hidden_nodes,
                                       learning_rate=learning_rate,
                                       bias=bias)
                weights = network.train(train_imgs, 
                                       train_labels_one_hot, 
                                       epochs=epochs, 
                                       intermediate_results=True) 
                for epoch in range(epochs):  
                    print("*", end="")
                    network.wih = weights[epoch][0]
                    network.who = weights[epoch][1]
                    train_corrects, train_wrongs = network.evaluate(train_imgs, 
                                                                    train_labels)
                    
                    test_corrects, test_wrongs = network.evaluate(test_imgs, 
                                                                  test_labels)
                    outstr = str(hidden_nodes) + " " + str(learning_rate) + " " + str(bias) 
                    outstr += " " + str(epoch) + " "
                    outstr += str(train_corrects / (train_corrects + train_wrongs)) + " "
                    outstr += str(train_wrongs / (train_corrects + train_wrongs)) + " "
                    outstr += str(test_corrects / (test_corrects + test_wrongs)) + " "
                    outstr += str(test_wrongs / (test_corrects + test_wrongs)) 
                    
                    fh_out.write(outstr + "\n" )
                    fh_out.flush()
************************************************************************************************************************************************************************************************************************************************************************************************

The file nist_tests_20_50_100_120_150.csv contains the results from a run of the previous program.

Networks with multiple hidden layers

We will write a new neural network class, in which we can define an arbitrary number of hidden layers. The code is also improved, because the weight matrices are now build inside of a loop instead redundant code:

In [ ]:
import numpy as np
from scipy.special import expit as activation_function
from scipy.stats import truncnorm

def truncated_normal(mean=0, sd=1, low=0, upp=10):
    return truncnorm((low - mean) / sd, 
                     (upp - mean) / sd, 
                     loc=mean, 
                     scale=sd)


class NeuralNetwork:
        
    
    def __init__(self, 
                 network_structure, # ie. [input_nodes, hidden1_nodes, ... , hidden_n_nodes, output_nodes]
                 learning_rate,
                 bias=None
                ):  

        self.structure = network_structure
        self.learning_rate = learning_rate 
        self.bias = bias
        self.create_weight_matrices()
    
    
    def create_weight_matrices(self):
        
        bias_node = 1 if self.bias else 0
        self.weights_matrices = []
        
        layer_index = 1
        no_of_layers = len(self.structure)
        while layer_index < no_of_layers:
            nodes_in = self.structure[layer_index-1]
            nodes_out = self.structure[layer_index]
            n = (nodes_in + bias_node) * nodes_out
            rad = 1 / np.sqrt(nodes_in)
            X = truncated_normal(mean=2, 
                                 sd=1, 
                                 low=-rad, 
                                 upp=rad)
            wm = X.rvs(n).reshape((nodes_out, nodes_in + bias_node))
            self.weights_matrices.append(wm)
            layer_index += 1

        
        
    def train(self, input_vector, target_vector):
        """
        input_vector and target_vector can be tuple, 
        list or ndarray
        """                              

        no_of_layers = len(self.structure)
        input_vector = np.array(input_vector, ndmin=2).T
        layer_index = 0
        # The output/input vectors of the various layers:
        res_vectors = [input_vector]
        while layer_index < no_of_layers - 1:
            in_vector = res_vectors[-1]
            if self.bias:
                # adding bias node to the end of the 'input'_vector
                in_vector = np.concatenate( (in_vector, 
                                             [[self.bias]]) )
                res_vectors[-1] = in_vector
            x = np.dot(self.weights_matrices[layer_index], 
                       in_vector)
            out_vector = activation_function(x)
            # the output of one layer is the input of the next one:
            res_vectors.append(out_vector)    
            layer_index += 1
        
        layer_index = no_of_layers - 1
        target_vector = np.array(target_vector, ndmin=2).T
         # The input vectors to the various layers
        output_errors = target_vector - out_vector  
        while layer_index > 0:
            out_vector = res_vectors[layer_index]
            in_vector = res_vectors[layer_index-1]

            if self.bias and not layer_index==(no_of_layers-1):
                out_vector = out_vector[:-1,:].copy()

            tmp = output_errors * out_vector * (1.0 - out_vector)     
            tmp = np.dot(tmp, in_vector.T)
            
            #if self.bias:
            #    tmp = tmp[:-1,:] 
                
            self.weights_matrices[layer_index-1] += self.learning_rate * tmp
            
            output_errors = np.dot(self.weights_matrices[layer_index-1].T, 
                                   output_errors)
            if self.bias:
                output_errors = output_errors[:-1,:]
            layer_index -= 1
            
            
               
    
    def run(self, input_vector):
        # input_vector can be tuple, list or ndarray

        no_of_layers = len(self.structure)
        if self.bias:
            # adding bias node to the end of the inpuy_vector
            input_vector = np.concatenate( (input_vector, 
                                            [self.bias]) )
        in_vector = np.array(input_vector, ndmin=2).T

        layer_index = 1
        # The input vectors to the various layers
        while layer_index < no_of_layers:
            x = np.dot(self.weights_matrices[layer_index-1], 
                       in_vector)
            out_vector = activation_function(x)
            
            # input vector for next layer
            in_vector = out_vector
            if self.bias:
                in_vector = np.concatenate( (in_vector, 
                                             [[self.bias]]) )            
            
            layer_index += 1
  
    
        return out_vector
    
    def evaluate(self, data, labels):
        corrects, wrongs = 0, 0
        for i in range(len(data)):
            res = self.run(data[i])
            res_max = res.argmax()
            if res_max == labels[i]:
                corrects += 1
            else:
                wrongs += 1
        return corrects, wrongs
    
    
In [ ]:
ANN = NeuralNetwork(network_structure=[image_pixels, 50, 50, 10],
                               learning_rate=0.1,
                               bias=None)
    
    
for i in range(len(train_imgs)):
    ANN.train(train_imgs[i], train_labels_one_hot[i])
In [ ]:
corrects, wrongs = ANN.evaluate(train_imgs, train_labels)
print("accuracy train: ", corrects / ( corrects + wrongs))
corrects, wrongs = ANN.evaluate(test_imgs, test_labels)
print("accuracy: test", corrects / ( corrects + wrongs))

Networks with multiple hidden layers and Epochs

In [ ]:
import numpy as np
from scipy.special import expit as activation_function
from scipy.stats import truncnorm

def truncated_normal(mean=0, sd=1, low=0, upp=10):
    return truncnorm((low - mean) / sd,
                     (upp - mean) / sd, 
                     loc=mean, 
                     scale=sd)


class NeuralNetwork:
        
    
    def __init__(self, 
                 network_structure, # ie. [input_nodes, hidden1_nodes, ... , hidden_n_nodes, output_nodes]
                 learning_rate,
                 bias=None
                ):  

        self.structure = network_structure
        self.learning_rate = learning_rate 
        self.bias = bias
        self.create_weight_matrices()

    
    
    def create_weight_matrices(self):
        X = truncated_normal(mean=2, sd=1, low=-0.5, upp=0.5)
        
        bias_node = 1 if self.bias else 0
        self.weights_matrices = []    
        layer_index = 1
        no_of_layers = len(self.structure)
        while layer_index < no_of_layers:
            nodes_in = self.structure[layer_index-1]
            nodes_out = self.structure[layer_index]
            n = (nodes_in + bias_node) * nodes_out
            rad = 1 / np.sqrt(nodes_in)
            X = truncated_normal(mean=2, sd=1, low=-rad, upp=rad)
            wm = X.rvs(n).reshape((nodes_out, nodes_in + bias_node))
            self.weights_matrices.append(wm)
            layer_index += 1

        
        
    def train_single(self, input_vector, target_vector):
        # input_vector and target_vector can be tuple, list or ndarray
                                       
        no_of_layers = len(self.structure)        
        input_vector = np.array(input_vector, ndmin=2).T

        layer_index = 0
        # The output/input vectors of the various layers:
        res_vectors = [input_vector]          
        while layer_index < no_of_layers - 1:
            in_vector = res_vectors[-1]
            if self.bias:
                # adding bias node to the end of the 'input'_vector
                in_vector = np.concatenate( (in_vector, 
                                             [[self.bias]]) )
                res_vectors[-1] = in_vector
            x = np.dot(self.weights_matrices[layer_index], in_vector)
            out_vector = activation_function(x)
            res_vectors.append(out_vector)   
            layer_index += 1
        
        layer_index = no_of_layers - 1
        target_vector = np.array(target_vector, ndmin=2).T
         # The input vectors to the various layers
        output_errors = target_vector - out_vector  
        while layer_index > 0:
            out_vector = res_vectors[layer_index]
            in_vector = res_vectors[layer_index-1]

            if self.bias and not layer_index==(no_of_layers-1):
                out_vector = out_vector[:-1,:].copy()

            tmp = output_errors * out_vector * (1.0 - out_vector)     
            tmp = np.dot(tmp, in_vector.T)
            
            #if self.bias:
            #    tmp = tmp[:-1,:] 
                
            self.weights_matrices[layer_index-1] += self.learning_rate * tmp
            
            output_errors = np.dot(self.weights_matrices[layer_index-1].T, 
                                   output_errors)
            if self.bias:
                output_errors = output_errors[:-1,:]
            layer_index -= 1
            

       

    def train(self, data_array, 
              labels_one_hot_array,
              epochs=1,
              intermediate_results=False):
        intermediate_weights = []
        for epoch in range(epochs):  
            for i in range(len(data_array)):
                self.train_single(data_array[i], labels_one_hot_array[i])
            if intermediate_results:
                intermediate_weights.append((self.wih.copy(), 
                                             self.who.copy()))
        return intermediate_weights      
        

               
    
    def run(self, input_vector):
        # input_vector can be tuple, list or ndarray

        no_of_layers = len(self.structure)
        if self.bias:
            # adding bias node to the end of the inpuy_vector
            input_vector = np.concatenate( (input_vector, [self.bias]) )
        in_vector = np.array(input_vector, ndmin=2).T

        layer_index = 1
        # The input vectors to the various layers
        while layer_index < no_of_layers:
            x = np.dot(self.weights_matrices[layer_index-1], 
                       in_vector)
            out_vector = activation_function(x)
            
            # input vector for next layer
            in_vector = out_vector
            if self.bias:
                in_vector = np.concatenate( (in_vector, 
                                             [[self.bias]]) )            
            
            layer_index += 1
  
    
        return out_vector
    
    def evaluate(self, data, labels):
        corrects, wrongs = 0, 0
        for i in range(len(data)):
            res = self.run(data[i])
            res_max = res.argmax()
            if res_max == labels[i]:
                corrects += 1
            else:
                wrongs += 1
        return corrects, wrongs
    
    
In [ ]:
epochs = 3

ANN = NeuralNetwork(network_structure=[image_pixels, 80, 80, 10],
                               learning_rate=0.01,
                               bias=None)
    
    
ANN.train(train_imgs, train_labels_one_hot, epochs=epochs)
In [ ]:
corrects, wrongs = ANN.evaluate(train_imgs, train_labels)
print("accuracy train: ", corrects / ( corrects + wrongs))
corrects, wrongs = ANN.evaluate(test_imgs, test_labels)
print("accuracy: test", corrects / ( corrects + wrongs))

Footnotes


1 Wan, Li; Matthew Zeiler; Sixin Zhang; Yann LeCun; Rob Fergus (2013). Regularization of Neural Network using DropConnect. International Conference on Machine Learning(ICML).