# Understanding Neural Style Transfer with Tensorflow and Keras

Neural Style Transfer (NST)  is a fascinating area of Deep Learning and Convolutional Neural Networks. NST is an interesting technique, in which the style from an image, known as the ‘style image’ is transferred to another image ‘content image’ and we get a third a image which is a generated image which has the content of the original image and the style of another image.

NST can be used to reimagine how famous painters like Van Gogh, Claude Monet or a Picasso would have visualised a scenery or architecture. NST uses Convolutional Neural Networks (CNNs) to achieve this artistic style transfer from one image to another. NST was originally implemented by Gati et al., in their paper Neural Algorithm of Artistic Style. Convolutional Neural Networks have been very successful in image classification image recognition et cetera. CNN networks have also been have also generated very interesting pictures using Neural Style Transfer which will be shown in this post. An interesting aspect of CNN’s is that the first couple of layers in the CNN capture basic features of the image like edges and  pixel values. But as we go deeper into the CNN, the network captures higher level features of the input image.

To get started with Neural Style transfer  we will be using the VGG19 pre-trained network. The VGG19 CNN is a compact pre-trained your network which can be used for performing the NST. However, we could have also used Resnet or InceptionV3 networks for this purpose but these are very large networks. The idea of using a network trained on a different task and applying it to a new task is called transfer learning.

What needs to be done to transfer the style from one of the image to another image. This brings us to the question – What is ‘style’? What is it that distinguishes Van Gogh’s painting or Picasso’s cubist art. Convolutional Neural Networks capture basic features in the lower layers and much more complex features in the deeper layers.  Style can be computed by taking the correlation of the feature maps in a layer L. This is my interpretation of how style is captured.  Since style  is intrinsic to  the image, it  implies that the style feature would exist across all the filters in a layer. Hence, to pick up this style we would need to get the correlation of the filters across channels of a lawyer. This is computed mathematically, using the Gram matrix which calculates the correlation of the activation of a the filter by the style image and generated image

To transfer the style from one image to the content image we need to do two parallel operations while doing forward propagation
– Compute the content loss between the source image and the generated image
– Compute the style loss between the style image and the generated image
– Finally we need to compute the total loss

In order to get transfer the style from the ‘style’ image to the ‘content ‘image resulting in a  ‘generated’  image  the total loss has to be minimised. Therefore backward propagation with gradient descent  is done to minimise the total loss comprising of the content and style loss.

Initially we make the Generated Image ‘G’ the same as the source image ‘S’

The content loss at layer ‘l’

$L_{content} = 1/2 \sum_{i}^{j} ( F^{l}_{i,j} - P^{l}_{i,j})^{2}$

where $F^{l}_{i,j}$ and $P^{l}_{i,j}$ represent the activations at layer ‘l’ in a filter i, at position ‘j’. The intuition is that the activations will be same for similar source and generated image. We need to minimise the content loss so that the generated stylized image is as close to the original image as possible. An intermediate layer of VGG19 block5_conv2 is used

The Style layers that are are used are

style_layers = [‘block1_conv1’,
‘block2_conv1’,
‘block3_conv1’,
‘block4_conv1’,
‘block5_conv1’]
To compute the Style Loss the Gram matrix needs to be computed. The Gram Matrix is computed by unrolling the filters as shown below (source: Convolutional Neural Networks by Prof Andrew Ng, Coursera). The result is a matrix of size $n_{c}$ x $n_{c}$ where $n_{c}$ is the number of channels
The above diagram shows the filters of height $n_{H}$ and width $n_{W}$ with $n_{C}$ channels
The contribution of layer ‘l’ to style loss is given by
$L^{'}_{style} = \frac{\sum_{i}^{j} (G^{2}_{i,j} - A^l{i,j})^2}{4N^{2}_{l}M^{2}_{l}}$
where $G_{i,j}$  and $A_{i,j}$ are the Gram matrices of the style and generated images respectively. By minimising the distance in the gram matrices of the style and generated image we can ensure that generated image is a stylized version of the original image similar to the style image
The total loss is given by
$L_{total} = \alpha L_{content} + \beta L_{style}$
Back propagation with gradient descent works to minimise the content loss between the source and generated image, while the style loss tries to minimise the discrepancies in the style of the style image and generated image. Running through forward and backpropagation through several epochs successfully transfers the style from the style image to the source image.
You can check the Notebook at Neural Style Transfer

Note: The code in this notebook is largely based on the Neural Style Transfer tutorial from Tensorflow, though I may have taken some changes from other blogs. I also made a few changes to the code in this tutorial, like removing the scaling factor, or the class definition (Personally, I belong to the old school (C language) and am not much in love with the ‘self.”..All references are included below

Note: Here is a interesting thought. Could we do a Neural Style Transfer in music? Imagine Carlos Santana playing ‘Hotel California’ or Brian May style in ‘Another brick in the wall’. While our first reaction would be that it may not sound good as we are used to style of these songs, we may be surprised by a possible style transfer. This is definitely music to the ears!

Here are few runs from this

## A) Run 1

1. Neural Style Transfer – a) Content Image – My portrait.  b) Style Image – Wassily Kadinsky Oil on canvas, 1913, Vassily Kadinsky’s composition

2. Result of Neural Style Transfer

2) Run 2

a) Content Image – Portrait of my parents b) Style Image –  Vincent Van Gogh’s ,Starry Night Oil on canvas 1889

2. Result of Neural Style Transfer

## Run 3

1.  Content Image – Caesar 2 (Masai Mara- 20 Jun 2018).  Style Image – The Great Wave at Kanagawa – Katsushika Hokosai, 1826-1833

2. Result of Neural Style Transfer

## Run 4

1.   Content Image – Junagarh Fort , Rajasthan   Sep 2016              b) Style Image – Le Pont Japonais by Claude Monet, Oil on canvas, 1920

2. Result of Neural Style Transfer

Neural Style Transfer is a very ingenious idea which shows that we can segregate the style of a painting and transfer to another image.

### References

1. A Neural Algorithm of Artistic Style, Leon A. Gatys, Alexander S. Ecker, Matthias Bethge
2. Neural style transfer
3. Neural Style Transfer: Creating Art with Deep Learning using tf.keras and eager execution
4. Convolutional Neural Network, DeepLearning.AI Specialization, Prof Andrew Ng
5. Intuitive Guide to Neural Style Transfer

To see all posts click Index of posts

# The mechanics of Convolutional Neural Networks in Tensorflow and Keras

Convolutional Neural Networks (CNNs), have been very popular in the last decade or so. CNNs have been used in multiple applications like image recognition, image classification, facial recognition, neural style transfer etc. CNN’s have been extremely successful in handling these kind of problems. How do they work? What makes them so successful? What is the principle behind CNN’s ?

Note: this post is based on two Coursera courses I did, namely namely Deep Learning specialisation by Prof Andrew Ng and Tensorflow Specialisation by  Laurence Moroney.

In this post I show you how CNN’s work. To understand how CNNs work, we need to understand the concept behind machine learning algorithms. If you take a simple machine learning algorithm in which you are trying to do multi-class classification using softmax or binary classification with the sigmoid function, for a set of for a set of input features against a target variable we need to create an objective function of the input features versus the target variable. Then we need to minimise this objective function, while performing gradient descent, such that the cost  is the lowest. This will give the set of weights for the different variables in the objective function.

The central problem in ML algorithms is to do feature selection, i.e.  we need to find the set of features that actually influence the target.  There are various methods for doing features selection – best fit, forward fit, backward fit, ridge and lasso regression. All these methods try to pick out the predictors that influence the output most, by making the weights of the other features close to zero. Please look at my post – Practical Machine Learning in R and Python – Part 3, where I show you the different methods for doing features selection.

In image classification or Image recognition we need to find the important features in the image. How do we do that? Many years back, have played around with OpenCV.  While working with OpenCV I came across are numerous filters like the Sobel ,the Laplacian, Canny, Gaussian filter et cetera which can be used to identify key features of the image. For example the Canny filter feature can be used for edge detection, Gaussian for smoothing, Sobel for determining the derivative and we have other filters for detecting vertical or horizontal edges. Take a look at my post Computer Vision: Ramblings on derivatives, histograms and contours So for handling images we need to apply these filters to pick  out the key features of the image namely the edges and other features. So rather than using the entire image’s pixels against the target class we can pick out the features from the image and use that as predictors of the target output.

Note: that in Convolutional Neural Network, fixed filter values like the those shown above  are not used directly. Rather the filter values are learned through back propagation and gradient descent as shown below.

In CNNs the filter values are considered to be weights which are then learned and updated in each forward/backward propagation cycle much like the way a fully connected Deep Learning Network learns the weights of the network.

Here is a short derivation of the most important parts of how a CNNs work

The convolution of a filter F with the input X can be represented as.

Convolving we get

This the forward propagation as it passes through a non-linear function like Relu

To go through back propagation we need to compute the $\partial L$  at every node of Convolutional Neural network

The loss with respect to the output is $\partial L/\partial O$. $\partial O/\partial X$ & $\partial O/\partial F$ are the local derivatives

We need these local derivatives because we can learn the filter values using gradient descent

where $\alpha$ is the learning rate. Also $\partial L/\partial X$ is the loss which is back propagated to the previous layers. You can see the detailed derivation of back propagation in my post Deep Learning from first principles in Python, R and Octave – Part 3 in a L-layer, multi-unit Deep Learning network.

In the fully connected layers the weights associated with each connection is computed in every cycle of forward and backward propagation using gradient descent. Similarly, the filter values are also computed and updated in each forward and backward propagation cycle. This is done so as to minimize the loss at the output layer.

By using the chain rule and simplifying the back propagation for the Convolutional layers we get these 2 equations. The first equation is used to learn the filter values and the second is used pass the loss to layers before

(for the detailed derivation see Convolutions and Backpropagations

An important aspect of performing convolutions is to reduce the size of  the flattened image that is passed into the fully connected DL network. Successively convolving with 2D filters and doing a max pooling helps to reduce the size of the features that we can use for learning the images. Convolutions also enable a sparsity of connections  as you can see in the diagram below. In the LeNet-5 Convolution Neural Network of Yann Le Cunn, successive convolutions reduce the image size from 28 x 28=784 to 120 flattened values.

Here is an interesting Deep Learning problem. Convolutions help in picking out important features of images and help in image classification/ detection. What would be its equivalent if we wanted to identify the Carnatic ragam of a song? A Carnatic ragam is roughly similar to Western scales (major, natural, melodic, blues) with all its modes Lydian, Aeolion, Phyrgian etc. Except in the case of the ragams, it is more nuanced, complex and involved. Personally, I can rarely identify a ragam on which a carnatic song is based (I am tone deaf when it comes to identifying ragams). I have come to understand that each Carnatic ragam has its own character, which is made up of several melodic phrases which are unique to that flavor of a ragam. What operation like convolution would be needed so that we can pick out these unique phrases in a Carnatic ragam? Of course, we would need to use it in Recurrent Neural Networks with LSTMs as a song is a time sequence of notes to identify sequences. Nevertheless, if there was some operation with which we can pick up the distinct, unique phrases from a song and then run it through a classifier, maybe we would be able to identify the ragam of the song.

Below I implement 3 simple CNN using the Dogs vs Cats Dataset from Kaggle. The first CNN uses regular Convolutions a Fully connected network to classify the images. The second approach uses Image Augmentation. For some reason, I did not get a better performance with Image Augumentation. Thirdly I use the pre-trained Inception v3 network.

# 1. Basic Convolutional Neural Network in Tensorflow & Keras

You can view the Colab notebook here – Cats_vs_dogs_1.ipynb

Here some important parts of the notebook

## Create CNN Model

• Use 3 Convolution + Max pooling layers with 32,64 and 128 filters respectively
• Flatten the data
• Have 2 Fully connected layers with 128, 512 neurons with relu activation
• Use sigmoid for binary classification
In [0]:
model = tf.keras.models.Sequential([
tf.keras.layers.Conv2D(32,(3,3),activation='relu',input_shape=(150,150,3)),
tf.keras.layers.MaxPooling2D(2,2),
tf.keras.layers.Conv2D(64,(3,3),activation='relu'),
tf.keras.layers.MaxPooling2D(2,2),
tf.keras.layers.Conv2D(128,(3,3),activation='relu'),
tf.keras.layers.MaxPooling2D(2,2),
tf.keras.layers.Flatten(),
tf.keras.layers.Dense(128,activation='relu'),
tf.keras.layers.Dense(512,activation='relu'),
tf.keras.layers.Dense(1,activation='sigmoid')
])


## Print model summary

In [13]:
model.summary()

Model: "sequential"
_________________________________________________________________
Layer (type)                 Output Shape              Param #
=================================================================
conv2d (Conv2D)              (None, 148, 148, 32)      896
_________________________________________________________________
max_pooling2d (MaxPooling2D) (None, 74, 74, 32)        0
_________________________________________________________________
conv2d_1 (Conv2D)            (None, 72, 72, 64)        18496
_________________________________________________________________
max_pooling2d_1 (MaxPooling2 (None, 36, 36, 64)        0
_________________________________________________________________
conv2d_2 (Conv2D)            (None, 34, 34, 128)       73856
_________________________________________________________________
max_pooling2d_2 (MaxPooling2 (None, 17, 17, 128)       0
_________________________________________________________________
flatten (Flatten)            (None, 36992)             0
_________________________________________________________________
dense (Dense)                (None, 128)               4735104
_________________________________________________________________
dense_1 (Dense)              (None, 512)               66048
_________________________________________________________________
dense_2 (Dense)              (None, 1)                 513
=================================================================
Total params: 4,894,913
Trainable params: 4,894,913
Non-trainable params: 0
_________________________________________________________________


## Use the Adam Optimizer with binary cross entropy

model.compile(optimizer='adam',
loss='binary_crossentropy',
metrics=['accuracy'])


• Do Gradient Descent for 15 epochs
history=model.fit(train_generator,
validation_data=validation_generator,
steps_per_epoch=100,
epochs=15,
validation_steps=50,
verbose=2)

Epoch 1/15
100/100 - 13s - loss: 0.6821 - accuracy: 0.5425 - val_loss: 0.6484 - val_accuracy: 0.6131
Epoch 2/15
100/100 - 13s - loss: 0.6227 - accuracy: 0.6456 - val_loss: 0.6161 - val_accuracy: 0.6394
Epoch 3/15
100/100 - 13s - loss: 0.5975 - accuracy: 0.6719 - val_loss: 0.5558 - val_accuracy: 0.7206
Epoch 4/15
100/100 - 13s - loss: 0.5480 - accuracy: 0.7241 - val_loss: 0.5431 - val_accuracy: 0.7138
Epoch 5/15
100/100 - 13s - loss: 0.5182 - accuracy: 0.7447 - val_loss: 0.4839 - val_accuracy: 0.7606
Epoch 6/15
100/100 - 13s - loss: 0.4773 - accuracy: 0.7781 - val_loss: 0.5029 - val_accuracy: 0.7506
Epoch 7/15
100/100 - 13s - loss: 0.4466 - accuracy: 0.7972 - val_loss: 0.4573 - val_accuracy: 0.7912
Epoch 8/15
100/100 - 13s - loss: 0.4395 - accuracy: 0.7997 - val_loss: 0.4252 - val_accuracy: 0.8119
Epoch 9/15
100/100 - 13s - loss: 0.4314 - accuracy: 0.8019 - val_loss: 0.4931 - val_accuracy: 0.7481
Epoch 10/15
100/100 - 13s - loss: 0.4309 - accuracy: 0.7969 - val_loss: 0.4203 - val_accuracy: 0.8109
Epoch 11/15
100/100 - 13s - loss: 0.4329 - accuracy: 0.7916 - val_loss: 0.4189 - val_accuracy: 0.8069
Epoch 12/15
100/100 - 13s - loss: 0.4248 - accuracy: 0.8050 - val_loss: 0.4476 - val_accuracy: 0.7925
Epoch 13/15
100/100 - 13s - loss: 0.3868 - accuracy: 0.8306 - val_loss: 0.3900 - val_accuracy: 0.8236
Epoch 14/15
100/100 - 13s - loss: 0.3710 - accuracy: 0.8328 - val_loss: 0.4520 - val_accuracy: 0.7900
Epoch 15/15
100/100 - 13s - loss: 0.3654 - accuracy: 0.8353 - val_loss: 0.3999 - val_accuracy: 0.8100

## Plot results

• Plot training and validation accuracy

• Plot training and validation loss

#-----------------------------------------------------------
# Retrieve a list of list results on training and test data
# sets for each training epoch
#-----------------------------------------------------------
acc      = history.history[     'accuracy' ]
val_acc  = history.history[ 'val_accuracy' ]
loss     = history.history[    'loss' ]
val_loss = history.history['val_loss' ]

epochs   = range(len(acc)) # Get number of epochs

#------------------------------------------------
# Plot training and validation accuracy per epoch
#------------------------------------------------
plt.plot  ( epochs,     acc,label="training accuracy" )
plt.plot  ( epochs, val_acc, label='validation acuracy' )
plt.title ('Training and validation accuracy')
plt.legend()

plt.figure()

#------------------------------------------------
# Plot training and validation loss per epoch
#------------------------------------------------
plt.plot  ( epochs,     loss , label="training loss")
plt.plot  ( epochs, val_loss,label="validation loss" )
plt.title ('Training and validation loss'   )
plt.legend()



# 2. CNN with Image Augmentation

You can check the Cats_vs_Dogs_2.ipynb

Including the important parts of this implementation below

## Use Image Augumentation

Use Image Augumentation to improve performance

• Use the same model parameters as before
• Perform the following image augmentation
• width, height shift
• shear and zoom

import tensorflow as tf
from tensorflow import keras
from tensorflow.keras.optimizers import RMSprop
from tensorflow.keras.preprocessing.image import ImageDataGenerator
model = tf.keras.models.Sequential([
tf.keras.layers.Conv2D(32,(3,3),activation='relu',input_shape=(150,150,3)),
tf.keras.layers.MaxPooling2D(2,2),
tf.keras.layers.Conv2D(64,(3,3),activation='relu'),
tf.keras.layers.MaxPooling2D(2,2),
tf.keras.layers.Conv2D(128,(3,3),activation='relu'),
tf.keras.layers.MaxPooling2D(2,2),
tf.keras.layers.Flatten(),
tf.keras.layers.Dense(128,activation='relu'),
tf.keras.layers.Dense(512,activation='relu'),
tf.keras.layers.Dense(1,activation='sigmoid')
])

train_datagen = ImageDataGenerator(
rescale=1./255,
#rotation_range=90,
width_shift_range=0.2,
height_shift_range=0.2,
shear_range=0.2,
zoom_range=0.2)
#horizontal_flip=True,
#fill_mode='nearest')

validation_datagen = ImageDataGenerator(rescale=1./255)
#
train_generator = train_datagen.flow_from_directory(train_dir,
batch_size=32,
class_mode='binary',
target_size=(150, 150))
# --------------------
# Flow validation images in batches of 20 using test_datagen generator
# --------------------
validation_generator =  validation_datagen.flow_from_directory(validation_dir,
batch_size=32,
class_mode  = 'binary',
target_size = (150, 150))

loss='binary_crossentropy',
metrics=['accuracy'])

Found 20000 images belonging to 2 classes.
Found 5000 images belonging to 2 classes.


history=model.fit(train_generator,
validation_data=validation_generator,
steps_per_epoch=100,
epochs=15,
validation_steps=50,
verbose=2)

Epoch 1/15
100/100 - 27s - loss: 0.5716 - accuracy: 0.6922 - val_loss: 0.4843 - val_accuracy: 0.7744
Epoch 2/15
100/100 - 27s - loss: 0.5575 - accuracy: 0.7084 - val_loss: 0.4683 - val_accuracy: 0.7750
Epoch 3/15
100/100 - 26s - loss: 0.5452 - accuracy: 0.7228 - val_loss: 0.4856 - val_accuracy: 0.7665
Epoch 4/15
100/100 - 27s - loss: 0.5294 - accuracy: 0.7347 - val_loss: 0.4654 - val_accuracy: 0.7812
Epoch 5/15
100/100 - 27s - loss: 0.5352 - accuracy: 0.7350 - val_loss: 0.4557 - val_accuracy: 0.7981
Epoch 6/15
100/100 - 26s - loss: 0.5136 - accuracy: 0.7453 - val_loss: 0.4964 - val_accuracy: 0.7621
Epoch 7/15
100/100 - 27s - loss: 0.5249 - accuracy: 0.7334 - val_loss: 0.4959 - val_accuracy: 0.7556
Epoch 8/15
100/100 - 26s - loss: 0.5035 - accuracy: 0.7497 - val_loss: 0.4555 - val_accuracy: 0.7969
Epoch 9/15
100/100 - 26s - loss: 0.5024 - accuracy: 0.7487 - val_loss: 0.4675 - val_accuracy: 0.7728
Epoch 10/15
100/100 - 27s - loss: 0.5015 - accuracy: 0.7500 - val_loss: 0.4276 - val_accuracy: 0.8075
Epoch 11/15
100/100 - 26s - loss: 0.5002 - accuracy: 0.7581 - val_loss: 0.4193 - val_accuracy: 0.8131
Epoch 12/15
100/100 - 27s - loss: 0.4733 - accuracy: 0.7706 - val_loss: 0.5209 - val_accuracy: 0.7398
Epoch 13/15
100/100 - 27s - loss: 0.4999 - accuracy: 0.7538 - val_loss: 0.4109 - val_accuracy: 0.8075
Epoch 14/15
100/100 - 27s - loss: 0.4550 - accuracy: 0.7859 - val_loss: 0.3770 - val_accuracy: 0.8288
Epoch 15/15
100/100 - 26s - loss: 0.4688 - accuracy: 0.7688 - val_loss: 0.4764 - val_accuracy: 0.7786


## Plot results

• Plot training and validation accuracy
• Plot training and validation loss
In [15]:
import matplotlib.pyplot as plt
#-----------------------------------------------------------
# Retrieve a list of list results on training and test data
# sets for each training epoch
#-----------------------------------------------------------
acc      = history.history[     'accuracy' ]
val_acc  = history.history[ 'val_accuracy' ]
loss     = history.history[    'loss' ]
val_loss = history.history['val_loss' ]

epochs   = range(len(acc)) # Get number of epochs

#------------------------------------------------
# Plot training and validation accuracy per epoch
#------------------------------------------------
plt.plot  ( epochs,     acc,label="training accuracy" )
plt.plot  ( epochs, val_acc, label='validation acuracy' )
plt.title ('Training and validation accuracy')
plt.legend()

plt.figure()

#------------------------------------------------
# Plot training and validation loss per epoch
#------------------------------------------------
plt.plot  ( epochs,     loss , label="training loss")
plt.plot  ( epochs, val_loss,label="validation loss" )
plt.title ('Training and validation loss'   )
plt.legend()



# Implementation using Inception Network V3

The implementation is in the Colab notebook Cats_vs_Dog_3.ipynb

This is implemented as below

## Use Inception V3

import os

from tensorflow.keras import layers
from tensorflow.keras import Model

from tensorflow.keras.applications.inception_v3 import InceptionV3
pre_trained_model = InceptionV3(input_shape = (150, 150, 3),
include_top = False,
weights = 'imagenet')

for layer in pre_trained_model.layers:
layer.trainable = False

# pre_trained_model.summary()

last_layer = pre_trained_model.get_layer('mixed7')
print('last layer output shape: ', last_layer.output_shape)
last_output = last_layer.output

Downloading data from https://storage.googleapis.com/tensorflow/keras-applications/inception_v3/inception_v3_weights_tf_dim_ordering_tf_kernels_notop.h5
87916544/87910968 [==============================] - 1s 0us/step
last layer output shape:  (None, 7, 7, 768)


## Use Layer 7 of Inception Network

• Use Image Augumentation
In [0]:
import tensorflow as tf
from tensorflow import keras
from tensorflow.keras.optimizers import RMSprop
from tensorflow.keras.preprocessing.image import ImageDataGenerator
# Flatten the output layer to 1 dimension
x = layers.Flatten()(last_output)
# Add a fully connected layer with 1,024 hidden units and ReLU activation
x = layers.Dense(1024, activation='relu')(x)
# Add a dropout rate of 0.2
x = layers.Dropout(0.2)(x)
# Add a final sigmoid layer for classification
x = layers.Dense  (1, activation='sigmoid')(x)

model = Model( pre_trained_model.input, x)
#train_datagen = ImageDataGenerator( rescale = 1.0/255. )
#validation_datagen = ImageDataGenerator( rescale = 1.0/255. )

train_datagen = ImageDataGenerator(
rescale=1./255,
#rotation_range=90,
width_shift_range=0.2,
height_shift_range=0.2,
shear_range=0.2,
zoom_range=0.2)
#horizontal_flip=True,
#fill_mode='nearest')

validation_datagen = ImageDataGenerator(rescale=1./255)
#
train_generator = train_datagen.flow_from_directory(train_dir,
batch_size=32,
class_mode='binary',
target_size=(150, 150))
# --------------------
# Flow validation images in batches of 20 using test_datagen generator
# --------------------
validation_generator =  validation_datagen.flow_from_directory(validation_dir,
batch_size=32,
class_mode  = 'binary',
target_size = (150, 150))

loss='binary_crossentropy',
metrics=['accuracy'])

Found 20000 images belonging to 2 classes.
Found 5000 images belonging to 2 classes.


## Fit model

history=model.fit(train_generator,
validation_data=validation_generator,
steps_per_epoch=100,
epochs=15,
validation_steps=50,
verbose=2)

Epoch 1/15
100/100 - 31s - loss: 0.5961 - accuracy: 0.8909 - val_loss: 0.1919 - val_accuracy: 0.9456
Epoch 2/15
100/100 - 30s - loss: 0.2002 - accuracy: 0.9259 - val_loss: 0.1025 - val_accuracy: 0.9550
Epoch 3/15
100/100 - 30s - loss: 0.1618 - accuracy: 0.9366 - val_loss: 0.0920 - val_accuracy: 0.9581
Epoch 4/15
100/100 - 29s - loss: 0.1442 - accuracy: 0.9381 - val_loss: 0.0960 - val_accuracy: 0.9600
Epoch 5/15
100/100 - 30s - loss: 0.1402 - accuracy: 0.9381 - val_loss: 0.0703 - val_accuracy: 0.9794
Epoch 6/15
100/100 - 30s - loss: 0.1437 - accuracy: 0.9413 - val_loss: 0.1090 - val_accuracy: 0.9531
Epoch 7/15
100/100 - 30s - loss: 0.1325 - accuracy: 0.9428 - val_loss: 0.0756 - val_accuracy: 0.9670
Epoch 8/15
100/100 - 29s - loss: 0.1341 - accuracy: 0.9491 - val_loss: 0.0625 - val_accuracy: 0.9737
Epoch 9/15
100/100 - 29s - loss: 0.1186 - accuracy: 0.9513 - val_loss: 0.0934 - val_accuracy: 0.9581
Epoch 10/15
100/100 - 29s - loss: 0.1171 - accuracy: 0.9513 - val_loss: 0.0642 - val_accuracy: 0.9727
Epoch 11/15
100/100 - 29s - loss: 0.1018 - accuracy: 0.9591 - val_loss: 0.0930 - val_accuracy: 0.9606
Epoch 12/15
100/100 - 29s - loss: 0.1190 - accuracy: 0.9541 - val_loss: 0.0737 - val_accuracy: 0.9719
Epoch 13/15
100/100 - 29s - loss: 0.1223 - accuracy: 0.9494 - val_loss: 0.0740 - val_accuracy: 0.9695
Epoch 14/15
100/100 - 29s - loss: 0.1158 - accuracy: 0.9516 - val_loss: 0.0659 - val_accuracy: 0.9744
Epoch 15/15
100/100 - 29s - loss: 0.1168 - accuracy: 0.9591 - val_loss: 0.0788 - val_accuracy: 0.9669


## Plot results

• Plot training and validation accuracy
• Plot training and validation loss
In [14]:
import matplotlib.pyplot as plt
#-----------------------------------------------------------
# Retrieve a list of list results on training and test data
# sets for each training epoch
#-----------------------------------------------------------
acc      = history.history[     'accuracy' ]
val_acc  = history.history[ 'val_accuracy' ]
loss     = history.history[    'loss' ]
val_loss = history.history['val_loss' ]

epochs   = range(len(acc)) # Get number of epochs

#------------------------------------------------
# Plot training and validation accuracy per epoch
#------------------------------------------------
plt.plot  ( epochs,     acc,label="training accuracy" )
plt.plot  ( epochs, val_acc, label='validation acuracy' )
plt.title ('Training and validation accuracy')
plt.legend()

plt.figure()

#------------------------------------------------
# Plot training and validation loss per epoch
#------------------------------------------------
plt.plot  ( epochs,     loss , label="training loss")
plt.plot  ( epochs, val_loss,label="validation loss" )
plt.title ('Training and validation loss'   )
plt.legend()

I intend to do some interesting stuff with Convolutional Neural Networks.

Watch this space!

To see all posts click Index of posts

# Getting started with Tensorflow, Keras in Python and R

The Pale Blue Dot

“From this distant vantage point, the Earth might not seem of any particular interest. But for us, it’s different. Consider again that dot. That’s here, that’s home, that’s us. On it everyone you love, everyone you know, everyone you ever heard of, every human being who ever was, lived out their lives. The aggregate of our joy and suffering, thousands of confident religions, ideologies, and economic doctrines, every hunter and forager, every hero and coward, every creator and destroyer of civilization, every king and peasant, every young couple in love, every mother and father, hopeful child, inventor and explorer, every teacher of morals, every corrupt politician, every “superstar,” every “supreme leader,” every saint and sinner in the history of our species lived there—on the mote of dust suspended in a sunbeam.”

Carl Sagan

Tensorflow and Keras are Deep Learning frameworks that really simplify a lot of things to the user. If you are familiar with Machine Learning and Deep Learning concepts then Tensorflow and Keras are really a playground to realize your ideas.  In this post I show how you can get started with Tensorflow in both Python and R

### Tensorflow in Python

For tensorflow in Python, I found Google’s Colab an ideal environment for running your Deep Learning code. This is an Google’s research project  where you can execute your code  on GPUs, TPUs etc

### Tensorflow in R (RStudio)

To execute tensorflow in R (RStudio) you need to install tensorflow and keras as shown below
In this post I show how to get started with Tensorflow and Keras in R.

# Install Tensorflow in RStudio
#install_tensorflow()
# Install Keras
#install_packages("keras")
library(tensorflow)
libary(keras)

This post takes 3 different Machine Learning problems and uses the
Tensorflow/Keras framework to solve it

Note:
You can view the Google Colab notebook at Tensorflow in Python
The RMarkdown file has been published at RPubs and can be accessed
at Getting started with Tensorflow in R

Checkout my book ‘Deep Learning from first principles: Second Edition – In vectorized Python, R and Octave’. My book starts with the implementation of a simple 2-layer Neural Network and works its way to a generic L-Layer Deep Learning Network, with all the bells and whistles. The derivations have been discussed in detail. The code has been extensively commented and included in its entirety in the Appendix sections. My book is available on Amazon as paperback ($14.99) and in kindle version($9.99/Rs449).

## 1. Multivariate regression with Tensorflow – Python

This code performs multivariate regression using Tensorflow and keras on the advent of Parkinson disease through sound recordings see Parkinson Speech Dataset with Multiple Types of Sound Recordings Data Set . The clinician’s motorUPDRS score has to be predicted from the set of features

In [0]:
# Import tensorflow
import tensorflow as tf
from tensorflow import keras

In [2]:
#Get the data rom the UCI Machine Learning repository
dataset = keras.utils.get_file("parkinsons_updrs.data", "https://archive.ics.uci.edu/ml/machine-learning-databases/parkinsons/telemonitoring/parkinsons_updrs.data")

Downloading data from https://archive.ics.uci.edu/ml/machine-learning-databases/parkinsons/telemonitoring/parkinsons_updrs.data
917504/911261 [==============================] - 0s 0us/step

In [3]:
# Read the CSV file
import pandas as pd
parkinsons = pd.read_csv(dataset, na_values = "?", comment='\t',
sep=",", skipinitialspace=True)
print(parkinsons.shape)
print(parkinsons.columns)
#Check if there are any NAs in the rows
parkinsons.isna().sum()

(5875, 22)
Index(['subject#', 'age', 'sex', 'test_time', 'motor_UPDRS', 'total_UPDRS',
'Jitter(%)', 'Jitter(Abs)', 'Jitter:RAP', 'Jitter:PPQ5', 'Jitter:DDP',
'Shimmer', 'Shimmer(dB)', 'Shimmer:APQ3', 'Shimmer:APQ5',
'Shimmer:APQ11', 'Shimmer:DDA', 'NHR', 'HNR', 'RPDE', 'DFA', 'PPE'],
dtype='object')

Out[3]:
subject#         0
age              0
sex              0
test_time        0
motor_UPDRS      0
total_UPDRS      0
Jitter(%)        0
Jitter(Abs)      0
Jitter:RAP       0
Jitter:PPQ5      0
Jitter:DDP       0
Shimmer          0
Shimmer(dB)      0
Shimmer:APQ3     0
Shimmer:APQ5     0
Shimmer:APQ11    0
Shimmer:DDA      0
NHR              0
HNR              0
RPDE             0
DFA              0
PPE              0
dtype: int64
Note: To see how to create dummy variables see my post Practical Machine Learning with R and Python – Part 2
In [4]:
# Drop the columns subject number as it is not relevant
parkinsons1=parkinsons.drop(['subject#'],axis=1)

# Create dummy variables for sex (M/F)
parkinsons2=pd.get_dummies(parkinsons1,columns=['sex'])

Out[4]
age test_time motor_UPDRS total_UPDRS Jitter(%) Jitter(Abs) Jitter:RAP Jitter:PPQ5 Jitter:DDP Shimmer Shimmer(dB) Shimmer:APQ3 Shimmer:APQ5 Shimmer:APQ11 Shimmer:DDA NHR HNR RPDE DFA PPE sex_0 sex_1
0 72 5.6431 28.199 34.398 0.00662 0.000034 0.00401 0.00317 0.01204 0.02565 0.230 0.01438 0.01309 0.01662 0.04314 0.014290 21.640 0.41888 0.54842 0.16006 1 0
1 72 12.6660 28.447 34.894 0.00300 0.000017 0.00132 0.00150 0.00395 0.02024 0.179 0.00994 0.01072 0.01689 0.02982 0.011112 27.183 0.43493 0.56477 0.10810 1 0
2 72 19.6810 28.695 35.389 0.00481 0.000025 0.00205 0.00208 0.00616 0.01675 0.181 0.00734 0.00844 0.01458 0.02202 0.020220 23.047 0.46222 0.54405 0.21014 1 0
3 72 25.6470 28.905 35.810 0.00528 0.000027 0.00191 0.00264 0.00573 0.02309 0.327 0.01106 0.01265 0.01963 0.03317 0.027837 24.445 0.48730 0.57794 0.33277 1 0
4 72 33.6420 29.187 36.375 0.00335 0.000020 0.00093 0.00130 0.00278 0.01703 0.176 0.00679 0.00929 0.01819 0.02036 0.011625 26.126 0.47188 0.56122 0.19361 1 0


# Create a training and test data set with 80%/20%
train_dataset = parkinsons2.sample(frac=0.8,random_state=0)
test_dataset = parkinsons2.drop(train_dataset.index)

# Select columns
train_dataset1= train_dataset[['age', 'test_time', 'Jitter(%)', 'Jitter(Abs)',
'Jitter:RAP', 'Jitter:PPQ5', 'Jitter:DDP', 'Shimmer', 'Shimmer(dB)',
'Shimmer:APQ3', 'Shimmer:APQ5', 'Shimmer:APQ11', 'Shimmer:DDA', 'NHR',
'HNR', 'RPDE', 'DFA', 'PPE', 'sex_0', 'sex_1']]
test_dataset1= test_dataset[['age','test_time', 'Jitter(%)', 'Jitter(Abs)',
'Jitter:RAP', 'Jitter:PPQ5', 'Jitter:DDP', 'Shimmer', 'Shimmer(dB)',
'Shimmer:APQ3', 'Shimmer:APQ5', 'Shimmer:APQ11', 'Shimmer:DDA', 'NHR',
'HNR', 'RPDE', 'DFA', 'PPE', 'sex_0', 'sex_1']]

In [7]:
# Generate the statistics of the columns for use in normalization of the data
train_stats = train_dataset1.describe()
train_stats = train_stats.transpose()
train_stats

Out[7]:
count mean std min 25% 50% 75% max
age 4700.0 64.792766 8.870401 36.000000 58.000000 65.000000 72.000000 85.000000
test_time 4700.0 93.399490 53.630411 -4.262500 46.852250 93.405000 139.367500 215.490000
Jitter(%) 4700.0 0.006136 0.005612 0.000830 0.003560 0.004900 0.006770 0.099990
Jitter(Abs) 4700.0 0.000044 0.000036 0.000002 0.000022 0.000034 0.000053 0.000396
Jitter:RAP 4700.0 0.002969 0.003089 0.000330 0.001570 0.002235 0.003260 0.057540
Jitter:PPQ5 4700.0 0.003271 0.003760 0.000430 0.001810 0.002480 0.003460 0.069560
Jitter:DDP 4700.0 0.008908 0.009267 0.000980 0.004710 0.006705 0.009790 0.172630
Shimmer 4700.0 0.033992 0.025922 0.003060 0.019020 0.027385 0.039810 0.268630
Shimmer(dB) 4700.0 0.310487 0.231016 0.026000 0.175000 0.251000 0.363250 2.107000
Shimmer:APQ3 4700.0 0.017125 0.013275 0.001610 0.009190 0.013615 0.020562 0.162670
Shimmer:APQ5 4700.0 0.020151 0.016848 0.001940 0.010750 0.015785 0.023733 0.167020
Shimmer:APQ11 4700.0 0.027508 0.020270 0.002490 0.015630 0.022685 0.032713 0.275460
Shimmer:DDA 4700.0 0.051375 0.039826 0.004840 0.027567 0.040845 0.061683 0.488020
NHR 4700.0 0.032116 0.060206 0.000304 0.010827 0.018403 0.031452 0.748260
HNR 4700.0 21.704631 4.288853 1.659000 19.447750 21.973000 24.445250 37.187000
RPDE 4700.0 0.542549 0.100212 0.151020 0.471235 0.543490 0.614335 0.966080
DFA 4700.0 0.653015 0.070446 0.514040 0.596470 0.643285 0.710618 0.865600
PPE 4700.0 0.219559 0.091506 0.021983 0.156470 0.205340 0.264017 0.731730
sex_0 4700.0 0.681489 0.465948 0.000000 0.000000 1.000000 1.000000 1.000000
sex_1 4700.0 0.318511 0.465948 0.000000 0.000000 0.000000 1.000000 1.000000
In [0]:
# Create the target variable
train_labels = train_dataset.pop('motor_UPDRS')
test_labels = test_dataset.pop('motor_UPDRS')

In [0]:
# Normalize the data by subtracting the mean and dividing by the standard deviation
def normalize(x):
return (x - train_stats['mean']) / train_stats['std']

# Create normalized training and test data
normalized_train_data = normalize(train_dataset1)
normalized_test_data = normalize(test_dataset1)

In [0]:
# Create a Deep Learning model with keras
model = tf.keras.Sequential([
keras.layers.Dense(6, activation=tf.nn.relu, input_shape=[len(train_dataset1.keys())]),
keras.layers.Dense(9, activation=tf.nn.relu),
keras.layers.Dense(6,activation=tf.nn.relu),
keras.layers.Dense(1)
])

# Use the Adam optimizer with a learning rate of 0.01

# Set the metrics required to be Mean Absolute Error and Mean Squared Error.For regression, the loss is mean_squared_error
model.compile(loss='mean_squared_error',
optimizer=optimizer,
metrics=['mean_absolute_error', 'mean_squared_error'])

In [0]:
# Create a model
history=model.fit(
normalized_train_data, train_labels,
epochs=1000, validation_data = (normalized_test_data,test_labels), verbose=0)

In [26]:
hist = pd.DataFrame(history.history)
hist['epoch'] = history.epoch
hist.tail()

Out[26]:
loss mean_absolute_error mean_squared_error val_loss val_mean_absolute_error val_mean_squared_error epoch
995 15.773989 2.936990 15.773988 16.980803 3.028168 16.980803 995
996 15.238623 2.873420 15.238622 17.458752 3.101033 17.458752 996
997 15.437594 2.895500 15.437593 16.926016 2.971508 16.926018 997
998 15.867891 2.943521 15.867892 16.950249 2.985036 16.950249 998
999 15.846878 2.938914 15.846880 17.095623 3.014504 17.095625 999
In [30]:
def plot_history(history):
hist = pd.DataFrame(history.history)
hist['epoch'] = history.epoch

plt.figure()
plt.xlabel('Epoch')
plt.ylabel('Mean Abs Error')
plt.plot(hist['epoch'], hist['mean_absolute_error'],
label='Train Error')
plt.plot(hist['epoch'], hist['val_mean_absolute_error'],
label = 'Val Error')
plt.ylim([2,5])
plt.legend()

plt.figure()
plt.xlabel('Epoch')
plt.ylabel('Mean Square Error ')
plt.plot(hist['epoch'], hist['mean_squared_error'],
label='Train Error')
plt.plot(hist['epoch'], hist['val_mean_squared_error'],
label = 'Val Error')
plt.ylim([10,40])
plt.legend()
plt.show()

plot_history(history)


### Observation

It can be seen that the mean absolute error is on an average about +/- 4.0. The validation error also is about the same. This can be reduced by playing around with the hyperparamaters and increasing the number of iterations

### 1a. Multivariate Regression in Tensorflow – R

# Install Tensorflow in RStudio
#install_tensorflow()
# Install Keras
#install_packages("keras")
library(tensorflow)
library(keras)

library(dplyr)
library(dummies)
## dummies-1.5.6 provided by Decision Patterns
library(tensorflow)
library(keras)

## Multivariate regression

This code performs multivariate regression using Tensorflow and keras on the advent of Parkinson disease through sound recordings see Parkinson Speech Dataset with Multiple Types of Sound Recordings Data Set. The clinician’s motorUPDRS score has to be predicted from the set of features.

# Download the Parkinson's data from UCI Machine Learning repository

# Set the column names
names(dataset) <- c("subject","age", "sex", "test_time","motor_UPDRS","total_UPDRS","Jitter","Jitter.Abs",
"Jitter.RAP","Jitter.PPQ5","Jitter.DDP","Shimmer", "Shimmer.dB", "Shimmer.APQ3",
"Shimmer.APQ5","Shimmer.APQ11","Shimmer.DDA", "NHR","HNR", "RPDE", "DFA","PPE")

# Remove the column 'subject' as it is not relevant to analysis
dataset1 <- subset(dataset, select = -c(subject))

# Make the column 'sex' as a factor for using dummies
dataset1$sex=as.factor(dataset1$sex)
# Add dummy variables for categorical cariable 'sex'
dataset2 <- dummy.data.frame(dataset1, sep = ".")
## Warning in model.matrix.default(~x - 1, model.frame(~x - 1), contrasts =
## FALSE): non-list contrasts argument ignored
dataset3 <- na.omit(dataset2)

### Split the data as training and test in 80/20

## Split data 80% training and 20% test
sample_size <- floor(0.8 * nrow(dataset3))

## set the seed to make your partition reproducible
set.seed(12)
train_index <- sample(seq_len(nrow(dataset3)), size = sample_size)

train_dataset <- dataset3[train_index, ]
test_dataset <- dataset3[-train_index, ]

train_data <- train_dataset %>% select(sex.0,sex.1,age, test_time,Jitter,Jitter.Abs,Jitter.PPQ5,Jitter.DDP,
Shimmer, Shimmer.dB,Shimmer.APQ3,Shimmer.APQ11,
Shimmer.DDA,NHR,HNR,RPDE,DFA,PPE)

train_labels <- select(train_dataset,motor_UPDRS)
test_data <- test_dataset %>% select(sex.0,sex.1,age, test_time,Jitter,Jitter.Abs,Jitter.PPQ5,Jitter.DDP,
Shimmer, Shimmer.dB,Shimmer.APQ3,Shimmer.APQ11,
Shimmer.DDA,NHR,HNR,RPDE,DFA,PPE)
test_labels <- select(test_dataset,motor_UPDRS)

## Normalize the data

 # Normalize the data by subtracting the mean and dividing by the standard deviation
normalize<-function(x) {
y<-(x - mean(x)) / sd(x)
return(y)
}

normalized_train_data <-apply(train_data,2,normalize)
# Convert to matrix
train_labels <- as.matrix(train_labels)
normalized_test_data <- apply(test_data,2,normalize)
test_labels <- as.matrix(test_labels)

### Create the Deep Learning Model

model <- keras_model_sequential()
model %>%
layer_dense(units = 6, activation = 'relu', input_shape = dim(normalized_train_data)[2]) %>%
layer_dense(units = 9, activation = 'relu') %>%
layer_dense(units = 6, activation = 'relu') %>%
layer_dense(units = 1)

# Set the metrics required to be Mean Absolute Error and Mean Squared Error.For regression, the loss is
# mean_squared_error
model %>% compile(
loss = 'mean_squared_error',
optimizer = optimizer_rmsprop(),
metrics = c('mean_absolute_error','mean_squared_error')
)

# Fit the model
# Use the test data for validation
history <- model %>% fit(
normalized_train_data, train_labels,
epochs = 30, batch_size = 128,
validation_data = list(normalized_test_data,test_labels)
)

### Plot mean squared error, mean absolute error and loss for training data and test data

plot(history)


Fig1

## 2. Binary classification in Tensorflow – Python

This is a simple binary classification problem from UCI Machine Learning repository and deals with data on Breast cancer from the Univ. of Wisconsin Breast Cancer Wisconsin (Diagnostic) Data Set bold text

In [31]:
import tensorflow as tf
from tensorflow import keras
import pandas as pd
# Read the data set from UCI ML site
dataset_path = keras.utils.get_file("breast-cancer-wisconsin.data", "https://archive.ics.uci.edu/ml/machine-learning-databases/breast-cancer-wisconsin/breast-cancer-wisconsin.data")
raw_dataset = pd.read_csv(dataset_path, sep=",", na_values = "?", skipinitialspace=True,)
dataset = raw_dataset.copy()

#Check for Null and drop
dataset.isna().sum()
dataset = dataset.dropna()
dataset.isna().sum()

# Set the column names
"barenuclei","chromatin","normalnucleoli","mitoses","class"]

Downloading data from https://archive.ics.uci.edu/ml/machine-learning-databases/breast-cancer-wisconsin/breast-cancer-wisconsin.data
24576/19889 [=====================================] - 0s 1us/step
id	thickness	cellsize	cellshape	adhesion	epicellsize	barenuclei	chromatin	normalnucleoli	mitoses	class
0	1002945	5	4	4	5	7	10.0	3	2	1	2
1	1015425	3	1	1	1	2	2.0	3	1	1	2
2	1016277	6	8	8	1	3	4.0	3	7	1	2
3	1017023	4	1	1	3	2	1.0	3	1	1	2
4	1017122	8	10	10	8	7	10.0	9	7	1	4
# Create a training/test set in the ratio 80/20
train_dataset = dataset.sample(frac=0.8,random_state=0)
test_dataset = dataset.drop(train_dataset.index)

# Set the training and test set
'epicellsize', 'barenuclei', 'chromatin', 'normalnucleoli','mitoses']]
'epicellsize', 'barenuclei', 'chromatin', 'normalnucleoli','mitoses']]

In [34]:
# Generate the stats for each column to be used for normalization
train_stats = train_dataset1.describe()
train_stats = train_stats.transpose()
train_stats

Out[34]:
count mean std min 25% 50% 75% max
thickness 546.0 4.430403 2.812768 1.0 2.0 4.0 6.0 10.0
cellsize 546.0 3.179487 3.083668 1.0 1.0 1.0 5.0 10.0
cellshape 546.0 3.225275 3.005588 1.0 1.0 1.0 5.0 10.0
adhesion 546.0 2.921245 2.937144 1.0 1.0 1.0 4.0 10.0
epicellsize 546.0 3.261905 2.252643 1.0 2.0 2.0 4.0 10.0
barenuclei 546.0 3.560440 3.651946 1.0 1.0 1.0 7.0 10.0
chromatin 546.0 3.483516 2.492687 1.0 2.0 3.0 5.0 10.0
normalnucleoli 546.0 2.875458 3.064305 1.0 1.0 1.0 4.0 10.0
mitoses 546.0 1.609890 1.736762 1.0 1.0 1.0 1.0 10.0
In [0]:
# Create target variables
train_labels = train_dataset.pop('class')
test_labels = test_dataset.pop('class')

In [0]:
# Set the target variables as 0 or 1
train_labels[train_labels==2] =0 # benign
train_labels[train_labels==4] =1 # malignant

test_labels[test_labels==2] =0 # benign
test_labels[test_labels==4] =1 # malignant

In [0]:
# Normalize by subtracting mean and dividing by standard deviation
def normalize(x):
return (x - train_stats['mean']) / train_stats['std']

# Convert columns to numeric
train_dataset1 = train_dataset1.apply(pd.to_numeric)
test_dataset1 = test_dataset1.apply(pd.to_numeric)

# Normalize
normalized_train_data = normalize(train_dataset1)
normalized_test_data = normalize(test_dataset1)

In [0]:
# Create a model
model = tf.keras.Sequential([
keras.layers.Dense(6, activation=tf.nn.relu, input_shape=[len(train_dataset1.keys())]),
keras.layers.Dense(9, activation=tf.nn.relu),
keras.layers.Dense(6,activation=tf.nn.relu),
keras.layers.Dense(1)
])

# Use the RMSProp optimizer
optimizer = tf.keras.optimizers.RMSprop(0.01)

# Since this is binary classification use binary_crossentropy
model.compile(loss='binary_crossentropy',
optimizer=optimizer,
metrics=['acc'])

# Fit a model
history=model.fit(
normalized_train_data, train_labels,
epochs=1000, validation_data=(normalized_test_data,test_labels), verbose=0)

In [55]:
hist = pd.DataFrame(history.history)
hist['epoch'] = history.epoch
hist.tail()

loss acc val_loss val_acc epoch
995 0.112499 0.992674 0.454739 0.970588 995
996 0.112499 0.992674 0.454739 0.970588 996
997 0.112499 0.992674 0.454739 0.970588 997
998 0.112499 0.992674 0.454739 0.970588 998
999 0.112499 0.992674 0.454739 0.970588 999
In [58]:
# Plot training and test accuracy
plt.plot(history.history['acc'])
plt.plot(history.history['val_acc'])
plt.title('model accuracy')
plt.ylabel('accuracy')
plt.xlabel('epoch')
plt.legend(['train', 'test'], loc='upper left')
plt.ylim([0.9,1])
plt.show()

# Plot training and test loss
plt.plot(history.history['loss'])
plt.plot(history.history['val_loss'])
plt.title('model loss')
plt.ylabel('loss')
plt.xlabel('epoch')
plt.legend(['train', 'test'], loc='upper left')
plt.ylim([0,0.5])
plt.show()



### 2a. Binary classification in Tensorflow -R

This is a simple binary classification problem from UCI Machine Learning repository and deals with data on Breast cancer from the Univ. of Wisconsin Breast Cancer Wisconsin (Diagnostic) Data Set

# Read the data for Breast cancer (Wisconsin)

# Rename the columns
"barenuclei","chromatin","normalnucleoli","mitoses","class")

# Remove the columns id and class
dataset1 <- subset(dataset, select = -c(id, class))
dataset2 <- na.omit(dataset1)

# Convert the column to numeric
dataset2$barenuclei <- as.numeric(dataset2$barenuclei)

## Normalize the data

train_data <-apply(dataset2,2,normalize)
train_labels <- as.matrix(select(dataset,class))

# Set the target variables as 0 or 1 as it binary classification
train_labels[train_labels==2,]=0
train_labels[train_labels==4,]=1

### Create the Deep Learning model

model <- keras_model_sequential()
model %>%
layer_dense(units = 6, activation = 'relu', input_shape = dim(train_data)[2]) %>%
layer_dense(units = 9, activation = 'relu') %>%
layer_dense(units = 6, activation = 'relu') %>%
layer_dense(units = 1)

# Since this is a binary classification we use binary cross entropy
model %>% compile(
loss = 'binary_crossentropy',
optimizer = optimizer_rmsprop(),
metrics = c('accuracy')  # Metrics is accuracy
)

### Fit the model. Use 20% of data for validation

history <- model %>% fit(
train_data, train_labels,
epochs = 30, batch_size = 128,
validation_split = 0.2
)

### Plot the accuracy and loss for training and validation data

plot(history)


### 3. MNIST in Tensorflow – Python

This takes the famous MNIST handwritten digits . It ca be seen that Tensorflow and Keras make short work of this famous problem of the late 1980s

# Download MNIST data
mnist=tf.keras.datasets.mnist
# Set training and test data and labels

print(training_images.shape)
print(test_images.shape)

(60000, 28, 28)
(10000, 28, 28)

In [61]:
# Plot a sample image from MNIST and show contents
import matplotlib.pyplot as plt
plt.imshow(training_images[1])
print(training_images[1])
[[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 159 253
159 50 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 48 238 252 252
252 237 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 54 227 253 252 239
233 252 57 6 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 10 60 224 252 253 252 202
84 252 253 122 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 163 252 252 252 253 252 252
96 189 253 167 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 51 238 253 253 190 114 253 228
47 79 255 168 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 48 238 252 252 179 12 75 121 21
0 0 253 243 50 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 38 165 253 233 208 84 0 0 0 0
0 0 253 252 165 0 0 0 0 0]
[ 0 0 0 0 0 0 0 7 178 252 240 71 19 28 0 0 0 0
0 0 253 252 195 0 0 0 0 0]
[ 0 0 0 0 0 0 0 57 252 252 63 0 0 0 0 0 0 0
0 0 253 252 195 0 0 0 0 0]
[ 0 0 0 0 0 0 0 198 253 190 0 0 0 0 0 0 0 0
0 0 255 253 196 0 0 0 0 0]
[ 0 0 0 0 0 0 76 246 252 112 0 0 0 0 0 0 0 0
0 0 253 252 148 0 0 0 0 0]
[ 0 0 0 0 0 0 85 252 230 25 0 0 0 0 0 0 0 0
7 135 253 186 12 0 0 0 0 0]
[ 0 0 0 0 0 0 85 252 223 0 0 0 0 0 0 0 0 7
131 252 225 71 0 0 0 0 0 0]
[ 0 0 0 0 0 0 85 252 145 0 0 0 0 0 0 0 48 165
252 173 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 86 253 225 0 0 0 0 0 0 114 238 253
162 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 85 252 249 146 48 29 85 178 225 253 223 167
56 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 85 252 252 252 229 215 252 252 252 196 130 0
0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 28 199 252 252 253 252 252 233 145 0 0 0
0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 25 128 252 253 252 141 37 0 0 0 0
0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0]]


# Normalize the images by dividing by 255.0
training_images = training_images/255.0
test_images = test_images/255.0

# Create a Sequential Keras model
model = tf.keras.models.Sequential([tf.keras.layers.Flatten(),
tf.keras.layers.Dense(1024,activation=tf.nn.relu),
tf.keras.layers.Dense(10,activation=tf.nn.softmax)])

In [68]:
history=model.fit(training_images,training_labels,validation_data=(test_images, test_labels), epochs=5, verbose=1)

Train on 60000 samples, validate on 10000 samples
Epoch 1/5
60000/60000 [==============================] - 17s 291us/sample - loss: 0.0020 - acc: 0.9999 - val_loss: 0.0719 - val_acc: 0.9810
Epoch 2/5
60000/60000 [==============================] - 17s 284us/sample - loss: 0.0021 - acc: 0.9998 - val_loss: 0.0705 - val_acc: 0.9821
Epoch 3/5
60000/60000 [==============================] - 17s 286us/sample - loss: 0.0017 - acc: 0.9999 - val_loss: 0.0729 - val_acc: 0.9805
Epoch 4/5
60000/60000 [==============================] - 17s 284us/sample - loss: 0.0014 - acc: 0.9999 - val_loss: 0.0762 - val_acc: 0.9804
Epoch 5/5
60000/60000 [==============================] - 17s 280us/sample - loss: 0.0015 - acc: 0.9999 - val_loss: 0.0735 - val_acc: 0.9812

Fig 1

Fig 2

## MNIST in Tensorflow – R

The following code uses Tensorflow to learn MNIST’s handwritten digits ### Load MNIST data

mnist <- dataset_mnist()
x_train <- mnist$train$x
y_train <- mnist$train$y
x_test <- mnist$test$x
y_test <- mnist$test$y

### Reshape and rescale

# Reshape the array
x_train <- array_reshape(x_train, c(nrow(x_train), 784))
x_test <- array_reshape(x_test, c(nrow(x_test), 784))
# Rescale
x_train <- x_train / 255
x_test <- x_test / 255

### Convert out put to One Hot encoded format

y_train <- to_categorical(y_train, 10)
y_test <- to_categorical(y_test, 10)

### Fit the model

Use the softmax activation for recognizing 10 digits and categorical cross entropy for loss

model <- keras_model_sequential()
model %>%
layer_dense(units = 256, activation = 'relu', input_shape = c(784)) %>%
layer_dense(units = 128, activation = 'relu') %>%
layer_dense(units = 10, activation = 'softmax') # Use softmax

model %>% compile(
loss = 'categorical_crossentropy',
optimizer = optimizer_rmsprop(),
metrics = c('accuracy')
)

### Fit the model

Note: A smaller number of epochs has been used. For better performance increase number of epochs

history <- model %>% fit(
x_train, y_train,
epochs = 5, batch_size = 128,
validation_data = list(x_test,y_test)
)

### Plot the accuracy and loss for training and test data

plot(history)


Conclusion
This post shows how to use Tensorflow and Keras in both Python & R
Hope you have fun with Tensorflow!!

To see all posts click Index of posts

# Take 4+: Presentations on ‘Elements of Neural Networks and Deep Learning’ – Parts 1-8

“Lights, camera and … action – Take 4+!”

This post includes  a rework of all presentation of ‘Elements of Neural Networks and Deep  Learning Parts 1-8 ‘ since my earlier presentations had some missing parts, omissions and some occasional errors. So I have re-recorded all the presentations.
This series of presentation will do a deep-dive  into Deep Learning networks starting from the fundamentals. The equations required for performing learning in a L-layer Deep Learning network  are derived in detail, starting from the basics. Further, the presentations also discuss multi-class classification, regularization techniques, and gradient descent optimization methods in deep networks methods. Finally the presentations also touch on how  Deep Learning Networks can be tuned.

The corresponding implementations are available in vectorized R, Python and Octave are available in my book ‘Deep Learning from first principles:Second edition- In vectorized Python, R and Octave

1. Elements of Neural Networks and Deep Learning – Part 1
This presentation introduces Neural Networks and Deep Learning. A look at history of Neural Networks, Perceptrons and why Deep Learning networks are required and concluding with a simple toy examples of a Neural Network and how they compute. This part also includes a small digression on the basics of Machine Learning and how the algorithm learns from a data set

2. Elements of Neural Networks and Deep Learning – Part 2
This presentation takes logistic regression as an example and creates an equivalent 2 layer Neural network. The presentation also takes a look at forward & backward propagation and how the cost is minimized using gradient descent

The implementation of the discussed 2 layer Neural Network in vectorized R, Python and Octave are available in my post ‘Deep Learning from first principles in Python, R and Octave – Part 1‘

3. Elements of Neural Networks and Deep Learning – Part 3
This 3rd part, discusses a primitive neural network with an input layer, output layer and a hidden layer. The neural network uses tanh activation in the hidden layer and a sigmoid activation in the output layer. The equations for forward and backward propagation are derived.

To see the implementations for the above discussed video see my post ‘Deep Learning from first principles in Python, R and Octave – Part 2

4. Elements of Neural Network and Deep Learning – Part 4
This presentation is a continuation of my 3rd presentation in which I derived the equations for a simple 3 layer Neural Network with 1 hidden layer. In this video presentation, I discuss step-by-step the derivations for a L-Layer, multi-unit Deep Learning Network, with any activation function g(z)

The implementations of L-Layer, multi-unit Deep Learning Network in vectorized R, Python and Octave are available in my post Deep Learning from first principles in Python, R and Octave – Part 3

5. Elements of Neural Network and Deep Learning – Part 5
This presentation discusses multi-class classification using the Softmax function. The detailed derivation for the Jacobian of the Softmax is discussed, and subsequently the derivative of cross-entropy loss is also discussed in detail. Finally the final set of equations for a Neural Network with multi-class classification is derived.

The corresponding implementations in vectorized R, Python and Octave are available in the following posts
a. Deep Learning from first principles in Python, R and Octave – Part 4
b. Deep Learning from first principles in Python, R and Octave – Part 5

6. Elements of Neural Networks and Deep Learning – Part 6
This part discusses initialization methods specifically like He and Xavier. The presentation also focuses on how to prevent over-fitting using regularization. Lastly the dropout method of regularization is also discussed

The corresponding implementations in vectorized R, Python and Octave of the above discussed methods are available in my post Deep Learning from first principles in Python, R and Octave – Part 6

7. Elements of Neural Networks and Deep Learning – Part 7
This presentation introduces exponentially weighted moving average and shows how this is used in different approaches to gradient descent optimization. The key techniques discussed are learning rate decay, momentum method, rmsprop and adam.

The equivalent implementations of the gradient descent optimization techniques in R, Python and Octave can be seen in my post Deep Learning from first principles in Python, R and Octave – Part 7

8. Elements of Neural Networks and Deep Learning – Part 8
This last part touches on the method to adopt while tuning hyper-parameters in Deep Learning networks

Checkout my book ‘Deep Learning from first principles: Second Edition – In vectorized Python, R and Octave’. My book starts with the implementation of a simple 2-layer Neural Network and works its way to a generic L-Layer Deep Learning Network, with all the bells and whistles. The derivations have been discussed in detail. The code has been extensively commented and included in its entirety in the Appendix sections. My book is available on Amazon as paperback ($18.99) and in kindle version($9.99/Rs449).

This concludes this series of presentations on “Elements of Neural Networks and Deep Learning’

To see all posts click Index of posts

# My presentations on ‘Elements of Neural Networks & Deep Learning’ -Parts 6,7,8

This is the final set of presentations in my series ‘Elements of Neural Networks and Deep Learning’. This set follows the earlier 2 sets of presentations namely
1. My presentations on ‘Elements of Neural Networks & Deep Learning’ -Part1,2,3
2. My presentations on ‘Elements of Neural Networks & Deep Learning’ -Parts 4,5

In this final set of presentations I discuss initialization methods, regularization techniques including dropout. Next I also discuss gradient descent optimization methods like momentum, rmsprop, adam etc. Lastly, I briefly also touch on hyper-parameter tuning approaches. The corresponding implementations are available in vectorized R, Python and Octave are available in my book ‘Deep Learning from first principles:Second edition- In vectorized Python, R and Octave

1. Elements of Neural Networks and Deep Learning – Part 6
This part discusses initialization methods specifically like He and Xavier. The presentation also focuses on how to prevent over-fitting using regularization. Lastly the dropout method of regularization is also discusses

The corresponding implementations in vectorized R, Python and Octave of the above discussed methods are available in my post Deep Learning from first principles in Python, R and Octave – Part 6

2. Elements of Neural Networks and Deep Learning – Part 7
This presentation introduces exponentially weighted moving average and shows how this is used in different approaches to gradient descent optimization. The key techniques discussed are learning rate decay, momentum method, rmsprop and adam.

The equivalent implementations of the gradient descent optimization techniques in R, Python and Octave can be seen in my post Deep Learning from first principles in Python, R and Octave – Part 7

3. Elements of Neural Networks and Deep Learning – Part 8
This last part touches upon hyper-parameter tuning in Deep Learning networks

This concludes this series of presentations on “Elements of Neural Networks and Deep Learning’

Important note: Do check out my later version of these videos at Take 4+: Presentations on ‘Elements of Neural Networks and Deep Learning’ – Parts 1-8 . These have more content and also include some corrections. Check it out!

Checkout my book ‘Deep Learning from first principles: Second Edition – In vectorized Python, R and Octave’. My book starts with the implementation of a simple 2-layer Neural Network and works its way to a generic L-Layer Deep Learning Network, with all the bells and whistles. The derivations have been discussed in detail. The code has been extensively commented and included in its entirety in the Appendix sections. My book is available on Amazon as paperback ($18.99) and and in kindle version($9.99/Rs449).

To see all posts click Index of posts

# My presentations on ‘Elements of Neural Networks & Deep Learning’ -Parts 4,5

This is the next set of presentations on “Elements of Neural Networks and Deep Learning”.  In the 4th presentation I discuss and derive the generalized equations for a multi-unit, multi-layer Deep Learning network.  The 5th presentation derives the equations for a Deep Learning network when performing multi-class classification along with the derivations for cross-entropy loss. The corresponding implementations are available in vectorized R, Python and Octave are available in my book ‘Deep Learning from first principles:Second edition- In vectorized Python, R and Octave

Important note: Do check out my later version of these videos at Take 4+: Presentations on ‘Elements of Neural Networks and Deep Learning’ – Parts 1-8 . These have more content and also include some corrections. Check it out!

1. Elements of Neural Network and Deep Learning – Part 4
This presentation is a continuation of my 3rd presentation in which I derived the equations for a simple 3 layer Neural Network with 1 hidden layer. In this video presentation, I discuss step-by-step the derivations for a L-Layer, multi-unit Deep Learning Network, with any activation function g(z)

The implementations of L-Layer, multi-unit Deep Learning Network in vectorized R, Python and Octave are available in my post Deep Learning from first principles in Python, R and Octave – Part 3

2. Elements of Neural Network and Deep Learning – Part 5
This presentation discusses multi-class classification using the Softmax function. The detailed derivation for the Jacobian of the Softmax is discussed, and subsequently the derivative of cross-entropy loss is also discussed in detail. Finally the final set of equations for a Neural Network with multi-class classification is derived.

The corresponding implementations in vectorized R, Python and Octave are available in the following posts
a. Deep Learning from first principles in Python, R and Octave – Part 4
b. Deep Learning from first principles in Python, R and Octave – Part 5

To be continued. Watch this space!

Checkout my book ‘Deep Learning from first principles: Second Edition – In vectorized Python, R and Octave’. My book starts with the implementation of a simple 2-layer Neural Network and works its way to a generic L-Layer Deep Learning Network, with all the bells and whistles. The derivations have been discussed in detail. The code has been extensively commented and included in its entirety in the Appendix sections. My book is available on Amazon as paperback ($18.99) and in kindle version($9.99/Rs449).

To see all posts click Index of Posts

# My presentations on ‘Elements of Neural Networks & Deep Learning’ -Part1,2,3

I will be uploading a series of presentations on ‘Elements of Neural Networks and Deep Learning’. In these video presentations I discuss the derivations of L -Layer Deep Learning Networks, starting from the basics. The corresponding implementations are available in vectorized R, Python and Octave are available in my book ‘Deep Learning from first principles:Second edition- In vectorized Python, R and Octave

1. Elements of Neural Networks and Deep Learning – Part 1
This presentation introduces Neural Networks and Deep Learning. A look at history of Neural Networks, Perceptrons and why Deep Learning networks are required and concluding with a simple toy examples of a Neural Network and how they compute

2. Elements of Neural Networks and Deep Learning – Part 2
This presentation takes logistic regression as an example and creates an equivalent 2 layer Neural network. The presentation also takes a look at forward & backward propagation and how the cost is minimized using gradient descent

The implementation of the discussed 2 layer Neural Network in vectorized R, Python and Octave are available in my post ‘Deep Learning from first principles in Python, R and Octave – Part 1

3. Elements of Neural Networks and Deep Learning – Part 3
This 3rd part, discusses a primitive neural network with an input layer, output layer and a hidden layer. The neural network uses tanh activation in the hidden layer and a sigmoid activation in the output layer. The equations for forward and backward propagation are derived.

To see the implementations for the above discussed video see my post ‘Deep Learning from first principles in Python, R and Octave – Part 2

Important note: Do check out my later version of these videos at Take 4+: Presentations on ‘Elements of Neural Networks and Deep Learning’ – Parts 1-8 . These have more content and also include some corrections. Check it out!

To be continued. Watch this space!

Checkout my book ‘Deep Learning from first principles: Second Edition – In vectorized Python, R and Octave’. My book starts with the implementation of a simple 2-layer Neural Network and works its way to a generic L-Layer Deep Learning Network, with all the bells and whistles. The derivations have been discussed in detail. The code has been extensively commented and included in its entirety in the Appendix sections. My book is available on Amazon as paperback ($18.99) and in kindle version($9.99/Rs449).

To see all posts click Index of posts

# My book ‘Deep Learning from first principles:Second Edition’ now on Amazon

The second edition of my book ‘Deep Learning from first principles:Second Edition- In vectorized Python, R and Octave’, is now available on Amazon, in both paperback ($18.99) and kindle ($9.99/Rs449/-)  versions. Since this book is almost 70% code, all functions, and code snippets have been formatted to use the fixed-width font ‘Lucida Console’. In addition line numbers have been added to all code snippets. This makes the code more organized and much more readable. I have also fixed typos in the book

The book includes the following chapters

Table of Contents
Preface 4
Introduction 6
1. Logistic Regression as a Neural Network 8
2. Implementing a simple Neural Network 23
3. Building a L- Layer Deep Learning Network 48
4. Deep Learning network with the Softmax 85
5. MNIST classification with Softmax 103
6. Initialization, regularization in Deep Learning 121
7. Gradient Descent Optimization techniques 167
8. Gradient Check in Deep Learning 197
1. Appendix A 214
2. Appendix 1 – Logistic Regression as a Neural Network 220
3. Appendix 2 - Implementing a simple Neural Network 227
4. Appendix 3 - Building a L- Layer Deep Learning Network 240
5. Appendix 4 - Deep Learning network with the Softmax 259
6. Appendix 5 - MNIST classification with Softmax 269
7. Appendix 6 - Initialization, regularization in Deep Learning 302
8. Appendix 7 - Gradient Descent Optimization techniques 344
9. Appendix 8 – Gradient Check 405
References 475

To see posts click Index of Posts

# Deep Learning from first principles in Python, R and Octave – Part 3

“Once upon a time, I, Chuang Tzu, dreamt I was a butterfly, fluttering hither and thither, to all intents and purposes a butterfly. I was conscious only of following my fancies as a butterfly, and was unconscious of my individuality as a man. Suddenly, I awoke, and there I lay, myself again. Now I do not know whether I was then a man dreaming I was a butterfly, or whether I am now a butterfly dreaming that I am a man.”
from The Brain: The Story of you – David Eagleman

“Thought is a great big vector of neural activity”
Prof Geoffrey Hinton

# Introduction

This is the third part in my series on Deep Learning from first principles in Python, R and Octave. In the first part Deep Learning from first principles in Python, R and Octave-Part 1, I implemented logistic regression as a 2 layer neural network. The 2nd part Deep Learning from first principles in Python, R and Octave-Part 2, dealt with the implementation of 3 layer Neural Networks with 1 hidden layer to perform classification tasks, where the 2 classes cannot be separated by a linear boundary. In this third part, I implement a multi-layer, Deep Learning (DL) network of arbitrary depth (any number of hidden layers) and arbitrary height (any number of activation units in each hidden layer). The implementations of these Deep Learning networks, in all the 3 parts, are based on vectorized versions in Python, R and Octave. The implementation in the 3rd part is for a L-layer Deep Netwwork, but without any regularization, early stopping, momentum or learning rate adaptation techniques. However even the barebones multi-layer DL, is a handful and has enough hyperparameters to fine-tune and adjust.

Checkout my book ‘Deep Learning from first principles: Second Edition – In vectorized Python, R and Octave’. My book starts with the implementation of a simple 2-layer Neural Network and works its way to a generic L-Layer Deep Learning Network, with all the bells and whistles. The derivations have been discussed in detail. The code has been extensively commented and included in its entirety in the Appendix sections. My book is available on Amazon as paperback ($18.99) and in kindle version($9.99/Rs449).

The implementation of the vectorized L-layer Deep Learning network in Python, R and Octave were both exhausting, and exacting!! Keeping track of the indices, layer number and matrix dimensions required quite bit of focus. While the implementation was demanding, it was also very exciting to get the code to work. The trick was to be able to shift gears between the slight quirkiness between the languages. Here are some of challenges I faced.

1. Python and Octave allow multiple return values to be unpacked in a single statement. With R, unpacking multiple return values from a list, requires the list returned, to be unpacked separately. I did see that there is a package gsubfn, which does this.  I hope this feature becomes a base R feature.
2. Python and R allow dissimilar elements to be saved and returned from functions using dictionaries or lists respectively. However there is no real equivalent in Octave. The closest I got to this functionality in Octave, was the ‘cell array’. But the cell array can be accessed only by the index, and not with the key as in a Python dictionary or R list. This makes things just a bit more difficult in Octave.
3. Python and Octave include implicit broadcasting. In R, broadcasting is not implicit, but R has a nifty function, the sweep(), with which we can broadcast either by columns or by rows
4. The closest equivalent of Python’s dictionary, or R’s list, in Octave is the cell array. However I had to manage separate cell arrays for weights and biases and during gradient descent and separate gradients dW and dB
5. In Python the rank-1 numpy arrays can be annoying at times. This issue is not present in R and Octave.

Though the number of lines of code for Deep Learning functions in Python, R and Octave are about ~350 apiece, they have been some of the most difficult code I have implemented. The current vectorized implementation supports the relu, sigmoid and tanh activation functions as of now. I will be adding other activation functions like the ‘leaky relu’, ‘softmax’ and others, to the implementation in the weeks to come.

While testing with different hyper-parameters namely i) the number of hidden layers, ii) the number of activation units in each layer, iii) the activation function and iv) the number iterations, I found the L-layer Deep Learning Network to be very sensitive to these hyper-parameters. It is not easy to tune the parameters. Adding more hidden layers, or more units per layer, does not help and mostly results in gradient descent getting stuck in some local minima. It does take a fair amount of trial and error and very close observation on how the DL network performs for logical changes. We then can zero in on the most the optimal solution. Feel free to download/fork my code from Github DeepLearning-Part 3 and play around with the hyper-parameters for your own problems.

#### Derivation of a Multi Layer Deep Learning Network

Note: A detailed discussion of the derivation below is available in my video presentation Neural Network 4
Lets take a simple 3 layer Neural network with 3 hidden layers and an output layer

In the forward propagation cycle the equations are

$Z_{1} = W_{1}A_{0} +b_{1}$  and  $A_{1} = g(Z_{1})$
$Z_{2} = W_{2}A_{1} +b_{2}$  and  $A_{2} = g(Z_{2})$
$Z_{3} = W_{3}A_{2} +b_{3}$  and $A_{3} = g(Z_{3})$

The loss function is given by
$L = -(ylogA3 + (1-y)log(1-A3))$
and $dL/dA3 = -(Y/A_{3} + (1-Y)/(1-A_{3}))$

For a binary classification the output activation function is the sigmoid function given by
$A_{3} = 1/(1+ e^{-Z3})$. It can be shown that
$dA_{3}/dZ_{3} = A_{3}(1-A_3)$ see equation 2 in Part 1

$\partial L/\partial Z_{3} = \partial L/\partial A_{3}* \partial A_{3}/\partial Z_{3} = A3-Y$ see equation (f) in  Part 1
and since
$\partial L/\partial A_{2} = \partial L/\partial Z_{3} * \partial Z_{3}/\partial A_{2} = (A_{3} -Y) * W_{3}$ because $\partial Z_{3}/\partial A_{2} = W_{3}$ -(1a)
and $\partial L/\partial Z_{2} =\partial L/\partial A_{2} * \partial A_{2}/\partial Z_{2} = (A_{3} -Y) * W_{3} *g'(Z_{2})$ -(1b)
$\partial L/\partial W_{2} = \partial L/\partial Z_{2} * A_{1}$ -(1c)
since $\partial Z_{2}/\partial W_{2} = A_{1}$
and
$\partial L/\partial b_{2} = \partial L/\partial Z_{2}$ -(1d)
because
$\partial Z_{2}/\partial b_{2} =1$

Also

$\partial L/\partial A_{1} =\partial L/\partial Z_{2} * \partial Z_{2}/\partial A_{1} = \partial L/\partial Z_{2} * W_{2}$     – (2a)
$\partial L/\partial Z_{1} =\partial L/\partial A_{1} * \partial A_{1}/\partial Z_{1} = \partial L/\partial A_{1} * W_{2} *g'(Z_{1})$          – (2b)
$\partial L/\partial W_{1} = \partial L/\partial Z_{1} * A_{0}$ – (2c)
$\partial L/\partial b_{1} = \partial L/\partial Z_{1}$ – (2d)

Inspecting the above equations (1a – 1d & 2a-2d), our ‘Uber deep, bottomless’ brain  can easily discern the pattern in these equations. The equation for any layer ‘l’ is of the form
$Z_{l} = W_{l}A_{l-1} +b_{l}$     and  $A_{l} = g(Z_{l})$
The equation for the backward propagation have the general form
$\partial L/\partial A_{l} = \partial L/\partial Z_{l+1} * W^{l+1}$
$\partial L/\partial Z_{l}=\partial L/\partial A_{l} *g'(Z_{l})$
$\partial L/\partial W_{l} =\partial L/\partial Z_{l} *A^{l-1}$
$\partial L/\partial b_{l} =\partial L/\partial Z_{l}$

Some other important results The derivatives of the activation functions in the implemented Deep Learning network
g(z) = sigmoid(z) = $1/(1+e^{-z})$ = a g’(z) = a(1-a) – See Part 1
g(z) = tanh(z) = a g’(z) = $1 - a^{2}$
g(z) = relu(z) = z  when z>0 and 0 when z 0 and 0 when z <= 0
While it appears that there is a discontinuity for the derivative at 0 the small value at the discontinuity does not present a problem

The implementation of the multi layer vectorized Deep Learning Network for Python, R and Octave is included below. For all these implementations, initially I create the size and configuration of the the Deep Learning network with the layer dimennsions So for example layersDimension Vector ‘V’ of length L indicating ‘L’ layers where

V (in Python)= $[v_{0}, v_{1}, v_{2}$, … $v_{L-1}]$
V (in R)= $c(v_{1}, v_{2}, v_{3}$ , … $v_{L})$
V (in Octave)= [ $v_{1} v_{2} v_{3}$$v_{L}]$

In all of these implementations the first element is the number of input features to the Deep Learning network and the last element is always a ‘sigmoid’ activation function since all the problems deal with binary classification.

The number of elements between the first and the last element are the number of hidden layers and the magnitude of each $v_{i}$ is the number of activation units in each hidden layer, which is specified while actually executing the Deep Learning network using the function L_Layer_DeepModel(), in all the implementations Python, R and Octave

## 1a. Classification with Multi layer Deep Learning Network – Relu activation(Python)

In the code below a 4 layer Neural Network is trained to generate a non-linear boundary between the classes. In the code below the ‘Relu’ Activation function is used. The number of activation units in each layer is 9. The cost vs iterations is plotted in addition to the decision boundary. Further the accuracy, precision, recall and F1 score are also computed

import os
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.colors
import sklearn.linear_model

from sklearn.model_selection import train_test_split
from sklearn.datasets import make_classification, make_blobs
from matplotlib.colors import ListedColormap
import sklearn
import sklearn.datasets

#from DLfunctions import plot_decision_boundary
execfile("./DLfunctions34.py") #
os.chdir("C:\\software\\DeepLearning-Posts\\part3")

# Create clusters of 2 classes
X1, Y1 = make_blobs(n_samples = 400, n_features = 2, centers = 9,
cluster_std = 1.3, random_state = 4)
#Create 2 classes
Y1=Y1.reshape(400,1)
Y1 = Y1 % 2
X2=X1.T
Y2=Y1.T
# Set the dimensions of DL Network
#  Below we have
#  2 - 2 input features
#  9,9 - 2 hidden layers with 9 activation units per layer and
#  1 - 1 sigmoid activation unit in the output layer as this is a binary classification
# The activation in the hidden layer is the 'relu' specified in L_Layer_DeepModel

layersDimensions = [2, 9, 9,1] #  4-layer model
parameters = L_Layer_DeepModel(X2, Y2, layersDimensions,hiddenActivationFunc='relu', learning_rate = 0.3,num_iterations = 2500, fig="fig1.png")
#Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), X2,Y2,str(0.3),"fig2.png")

# Compute the confusion matrix
yhat = predict(parameters,X2)
from sklearn.metrics import confusion_matrix
a=confusion_matrix(Y2.T,yhat.T)
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score
print('Accuracy: {:.2f}'.format(accuracy_score(Y2.T, yhat.T)))
print('Precision: {:.2f}'.format(precision_score(Y2.T, yhat.T)))
print('Recall: {:.2f}'.format(recall_score(Y2.T, yhat.T)))
print('F1: {:.2f}'.format(f1_score(Y2.T, yhat.T)))
## Accuracy: 0.90
## Precision: 0.91
## Recall: 0.87
## F1: 0.89

For more details on metrics like Accuracy, Recall, Precision etc. used in classification take a look at my post Practical Machine Learning with R and Python – Part 2. More details about these and other metrics besides implementation of the most common machine learning algorithms are available in my book My book ‘Practical Machine Learning with R and Python’ on Amazon

## 1b. Classification with Multi layer Deep Learning Network – Relu activation(R)

In the code below, binary classification is performed on the same data set as above using the Relu activation function. The DL network is same as above

library(ggplot2)
x <- z[,1:2]
y <- z[,3]
X1 <- t(x)
Y1 <- t(y)

# Set the dimensions of the Deep Learning network
# No of input features =2, 2 hidden layers with 9 activation units and 1 output layer
layersDimensions = c(2, 9, 9,1)
# Execute the Deep Learning Neural Network
retvals = L_Layer_DeepModel(X1, Y1, layersDimensions,
hiddenActivationFunc='relu',
learningRate = 0.3,
numIterations = 5000,
print_cost = True)
library(ggplot2)
source("DLfunctions33.R")
# Get the computed costs
costs <- retvals[['costs']]
# Create a sequence of iterations
numIterations=5000
iterations <- seq(0,numIterations,by=1000)
df <-data.frame(iterations,costs)
# Plot the Costs vs number of iterations
ggplot(df,aes(x=iterations,y=costs)) + geom_point() +geom_line(color="blue") +
xlab('No of iterations') + ylab('Cost') + ggtitle("Cost vs No of iterations")

# Plot the decision boundary
plotDecisionBoundary(z,retvals,hiddenActivationFunc="relu",0.3)

library(caret)
# Predict the output for the data values
yhat <-predict(retvals$parameters,X1,hiddenActivationFunc="relu") yhat[yhat==FALSE]=0 yhat[yhat==TRUE]=1 # Compute the confusion matrix confusionMatrix(yhat,Y1) ## Confusion Matrix and Statistics ## ## Reference ## Prediction 0 1 ## 0 201 10 ## 1 21 168 ## ## Accuracy : 0.9225 ## 95% CI : (0.8918, 0.9467) ## No Information Rate : 0.555 ## P-Value [Acc > NIR] : < 2e-16 ## ## Kappa : 0.8441 ## Mcnemar's Test P-Value : 0.07249 ## ## Sensitivity : 0.9054 ## Specificity : 0.9438 ## Pos Pred Value : 0.9526 ## Neg Pred Value : 0.8889 ## Prevalence : 0.5550 ## Detection Rate : 0.5025 ## Detection Prevalence : 0.5275 ## Balanced Accuracy : 0.9246 ## ## 'Positive' Class : 0 ##  ## 1c. Classification with Multi layer Deep Learning Network – Relu activation(Octave) Included below is the code for performing classification. Incidentally Octave does not seem to have implemented the confusion matrix, but confusionmat is available in Matlab. # Read the data data=csvread("data.csv"); X=data(:,1:2); Y=data(:,3); # Set layer dimensions layersDimensions = [2 9 7 1] #tanh=-0.5(ok), #relu=0.1 best! # Execute Deep Network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='relu', learningRate = 0.1, numIterations = 10000); plotCostVsIterations(10000,costs); plotDecisionBoundary(data,weights, biases,hiddenActivationFunc="tanh")  ## 2a. Classification with Multi layer Deep Learning Network – Tanh activation(Python) Below the Tanh activation function is used to perform the same classification. I found the Tanh activation required a simpler Neural Network of 3 layers. # Tanh activation import os import numpy as np import matplotlib.pyplot as plt import matplotlib.colors import sklearn.linear_model from sklearn.model_selection import train_test_split from sklearn.datasets import make_classification, make_blobs from matplotlib.colors import ListedColormap import sklearn import sklearn.datasets #from DLfunctions import plot_decision_boundary os.chdir("C:\\software\\DeepLearning-Posts\\part3") execfile("./DLfunctions34.py") # Create the dataset X1, Y1 = make_blobs(n_samples = 400, n_features = 2, centers = 9, cluster_std = 1.3, random_state = 4) #Create 2 classes Y1=Y1.reshape(400,1) Y1 = Y1 % 2 X2=X1.T Y2=Y1.T # Set the dimensions of the Neural Network layersDimensions = [2, 4, 1] # 3-layer model # Compute the DL network parameters = L_Layer_DeepModel(X2, Y2, layersDimensions, hiddenActivationFunc='tanh', learning_rate = .5,num_iterations = 2500,fig="fig3.png") #Plot the decision boundary plot_decision_boundary(lambda x: predict(parameters, x.T), X2,Y2,str(0.5),"fig4.png")  ## 2b. Classification with Multi layer Deep Learning Network – Tanh activation(R) R performs better with a Tanh activation than the Relu as can be seen below  #Set the dimensions of the Neural Network layersDimensions = c(2, 9, 9,1) library(ggplot2) # Read the data z <- as.matrix(read.csv("data.csv",header=FALSE)) x <- z[,1:2] y <- z[,3] X1 <- t(x) Y1 <- t(y) # Execute the Deep Model retvals = L_Layer_DeepModel(X1, Y1, layersDimensions, hiddenActivationFunc='tanh', learningRate = 0.3, numIterations = 5000, print_cost = True) # Get the costs costs <- retvals[['costs']] iterations <- seq(0,numIterations,by=1000) df <-data.frame(iterations,costs) # Plot Cost vs number of iterations ggplot(df,aes(x=iterations,y=costs)) + geom_point() +geom_line(color="blue") + xlab('No of iterations') + ylab('Cost') + ggtitle("Cost vs No of iterations") #Plot the decision boundary plotDecisionBoundary(z,retvals,hiddenActivationFunc="tanh",0.3) ## 2c. Classification with Multi layer Deep Learning Network – Tanh activation(Octave) The code below uses the Tanh activation in the hidden layers for Octave # Read the data data=csvread("data.csv"); X=data(:,1:2); Y=data(:,3); # Set layer dimensions layersDimensions = [2 9 7 1] #tanh=-0.5(ok), #relu=0.1 best! # Execute Deep Network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='tanh', learningRate = 0.1, numIterations = 10000); plotCostVsIterations(10000,costs); plotDecisionBoundary(data,weights, biases,hiddenActivationFunc="tanh")  ## 3. Bernoulli’s Lemniscate To make things more interesting, I create a 2D figure of the Bernoulli’s lemniscate to perform non-linear classification. The Lemniscate is given by the equation $(x^{2} + y^{2})^{2}$ = $2a^{2}*(x^{2}-y^{2})$ ## 3a. Classifying a lemniscate with Deep Learning Network – Relu activation(Python) import os import numpy as np import matplotlib.pyplot as plt os.chdir("C:\\software\\DeepLearning-Posts\\part3") execfile("./DLfunctions33.py") x1=np.random.uniform(0,10,2000).reshape(2000,1) x2=np.random.uniform(0,10,2000).reshape(2000,1) X=np.append(x1,x2,axis=1) X.shape # Create a subset of values where squared is <0,4. Perform ravel() to flatten this vector # Create the equation # (x^{2} + y^{2})^2 - 2a^2*(x^{2}-y^{2}) <= 0 a=np.power(np.power(X[:,0]-5,2) + np.power(X[:,1]-5,2),2) b=np.power(X[:,0]-5,2) - np.power(X[:,1]-5,2) c= a - (b*np.power(4,2)) <=0 Y=c.reshape(2000,1) # Create a scatter plot of the lemniscate plt.scatter(X[:,0], X[:,1], c=Y, marker= 'o', s=15,cmap="viridis") Z=np.append(X,Y,axis=1) plt.savefig("fig50.png",bbox_inches='tight') plt.clf() # Set the data for classification X2=X.T Y2=Y.T # These settings work the best # Set the Deep Learning layer dimensions for a Relu activation layersDimensions = [2,7,4,1] #Execute the DL network parameters = L_Layer_DeepModel(X2, Y2, layersDimensions, hiddenActivationFunc='relu', learning_rate = 0.5,num_iterations = 10000, fig="fig5.png") #Plot the decision boundary plot_decision_boundary(lambda x: predict(parameters, x.T), X2, Y2,str(2.2),"fig6.png") # Compute the Confusion matrix yhat = predict(parameters,X2) from sklearn.metrics import confusion_matrix a=confusion_matrix(Y2.T,yhat.T) from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score print('Accuracy: {:.2f}'.format(accuracy_score(Y2.T, yhat.T))) print('Precision: {:.2f}'.format(precision_score(Y2.T, yhat.T))) print('Recall: {:.2f}'.format(recall_score(Y2.T, yhat.T))) print('F1: {:.2f}'.format(f1_score(Y2.T, yhat.T))) ## Accuracy: 0.93 ## Precision: 0.77 ## Recall: 0.76 ## F1: 0.76 We could get better performance by tuning further. Do play around if you fork the code. Note:: The lemniscate data is saved as a CSV and then read in R and also in Octave. I do this instead of recreating the lemniscate shape ## 3b. Classifying a lemniscate with Deep Learning Network – Relu activation(R code) The R decision boundary for the Bernoulli’s lemniscate is shown below Z <- as.matrix(read.csv("lemniscate.csv",header=FALSE)) Z1=data.frame(Z) # Create a scatter plot of the lemniscate ggplot(Z1,aes(x=V1,y=V2,col=V3)) +geom_point() #Set the data for the DL network X=Z[,1:2] Y=Z[,3] X1=t(X) Y1=t(Y) # Set the layer dimensions for the tanh activation function layersDimensions = c(2,5,4,1) # Execute the Deep Learning network with Tanh activation retvals = L_Layer_DeepModel(X1, Y1, layersDimensions, hiddenActivationFunc='tanh', learningRate = 0.3, numIterations = 20000, print_cost = True) # Plot cost vs iteration costs <- retvals[['costs']] numIterations = 20000 iterations <- seq(0,numIterations,by=1000) df <-data.frame(iterations,costs) ggplot(df,aes(x=iterations,y=costs)) + geom_point() +geom_line(color="blue") + xlab('No of iterations') + ylab('Cost') + ggtitle("Cost vs No of iterations") #Plot the decision boundary plotDecisionBoundary(Z,retvals,hiddenActivationFunc="tanh",0.3) ## 3c. Classifying a lemniscate with Deep Learning Network – Relu activation(Octave code) Octave is used to generate the non-linear lemniscate boundary.  # Read the data data=csvread("lemniscate.csv"); X=data(:,1:2); Y=data(:,3); # Set the dimensions of the layers layersDimensions = [2 9 7 1] # Compute the DL network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='relu', learningRate = 0.20, numIterations = 10000); plotCostVsIterations(10000,costs); plotDecisionBoundary(data,weights, biases,hiddenActivationFunc="relu")  ## 4a. Binary Classification using MNIST – Python code Finally I perform a simple classification using the MNIST handwritten digits, which according to Prof Geoffrey Hinton is “the Drosophila of Deep Learning”. The Python code for reading the MNIST data is taken from Alex Kesling’s github link MNIST. In the Python code below, I perform a simple binary classification between the handwritten digit ‘5’ and ‘not 5’ which is all other digits. I will perform the proper classification of all digits using the Softmax classifier some time later. import os import numpy as np import matplotlib.pyplot as plt os.chdir("C:\\software\\DeepLearning-Posts\\part3") execfile("./DLfunctions34.py") execfile("./load_mnist.py") training=list(read(dataset='training',path="./mnist")) test=list(read(dataset='testing',path="./mnist")) lbls=[] pxls=[] print(len(training)) # Select the first 10000 training data and the labels for i in range(10000): l,p=training[i] lbls.append(l) pxls.append(p) labels= np.array(lbls) pixels=np.array(pxls) # Sey y=1 when labels == 5 and 0 otherwise y=(labels==5).reshape(-1,1) X=pixels.reshape(pixels.shape[0],-1) # Create the necessary feature and target variable X1=X.T Y1=y.T # Create the layer dimensions. The number of features are 28 x 28 = 784 since the 28 x 28 # pixels is flattened to single vector of length 784. layersDimensions=[784, 15,9,7,1] # Works very well parameters = L_Layer_DeepModel(X1, Y1, layersDimensions, hiddenActivationFunc='relu', learning_rate = 0.1,num_iterations = 1000, fig="fig7.png") # Test data lbls1=[] pxls1=[] for i in range(800): l,p=test[i] lbls1.append(l) pxls1.append(p) testLabels=np.array(lbls1) testData=np.array(pxls1) ytest=(testLabels==5).reshape(-1,1) Xtest=testData.reshape(testData.shape[0],-1) Xtest1=Xtest.T Ytest1=ytest.T yhat = predict(parameters,Xtest1) from sklearn.metrics import confusion_matrix a=confusion_matrix(Ytest1.T,yhat.T) from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score print('Accuracy: {:.2f}'.format(accuracy_score(Ytest1.T, yhat.T))) print('Precision: {:.2f}'.format(precision_score(Ytest1.T, yhat.T))) print('Recall: {:.2f}'.format(recall_score(Ytest1.T, yhat.T))) print('F1: {:.2f}'.format(f1_score(Ytest1.T, yhat.T))) probs=predict_proba(parameters,Xtest1) from sklearn.metrics import precision_recall_curve precision, recall, thresholds = precision_recall_curve(Ytest1.T, probs.T) closest_zero = np.argmin(np.abs(thresholds)) closest_zero_p = precision[closest_zero] closest_zero_r = recall[closest_zero] plt.xlim([0.0, 1.01]) plt.ylim([0.0, 1.01]) plt.plot(precision, recall, label='Precision-Recall Curve') plt.plot(closest_zero_p, closest_zero_r, 'o', markersize = 12, fillstyle = 'none', c='r', mew=3) plt.xlabel('Precision', fontsize=16) plt.ylabel('Recall', fontsize=16) plt.savefig("fig8.png",bbox_inches='tight')   ## Accuracy: 0.99 ## Precision: 0.96 ## Recall: 0.89 ## F1: 0.92 In addition to plotting the Cost vs Iterations, I also plot the Precision-Recall curve to show how the Precision and Recall, which are complementary to each other vary with respect to the other. To know more about Precision-Recall, please check my post Practical Machine Learning with R and Python – Part 4. Check out my compact and minimal book “Practical Machine Learning with R and Python:Second edition- Machine Learning in stereo” available in Amazon in paperback($10.99) and kindle($7.99) versions. My book includes implementations of key ML algorithms and associated measures and metrics. The book is ideal for anybody who is familiar with the concepts and would like a quick reference to the different ML algorithms that can be applied to problems and how to select the best model. Pick your copy today!! A physical copy of the book is much better than scrolling down a webpage. Personally, I tend to use my own book quite frequently to refer to R, Python constructs, subsetting, machine Learning function calls and the necessary parameters etc. It is useless to commit any of this to memory, and a physical copy of a book is much easier to thumb through for the relevant code snippet. Pick up your copy today! ## 4b. Binary Classification using MNIST – R code In the R code below the same binary classification of the digit ‘5’ and the ‘not 5’ is performed. The code to read and display the MNIST data is taken from Brendan O’ Connor’s github link at MNIST source("mnist.R") load_mnist() #show_digit(train$x[2,]
layersDimensions=c(784, 7,7,3,1) # Works at 1500
x <- t(train$x) # Choose only 5000 training data x2 <- x[,1:5000] y <-train$y
# Set labels for all digits that are 'not 5' to 0
y[y!=5] <- 0
# Set labels of digit 5 as 1
y[y==5] <- 1
# Set the data
y1 <- as.matrix(y)
y2 <- t(y1)
# Choose the 1st 5000 data
y3 <- y2[,1:5000]

#Execute the Deep Learning Model
retvals = L_Layer_DeepModel(x2, y3, layersDimensions,
hiddenActivationFunc='tanh',
learningRate = 0.3,
numIterations = 3000, print_cost = True)
# Plot cost vs iteration
costs <- retvals[['costs']]
numIterations = 3000
iterations <- seq(0,numIterations,by=1000)
df <-data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() +geom_line(color="blue") +
xlab('No of iterations') + ylab('Cost') + ggtitle("Cost vs No of iterations")

# Compute probability scores
scores <- computeScores(retvals$parameters, x2,hiddenActivationFunc='relu') a=y3==1 b=y3==0 # Compute probabilities of class 0 and class 1 class1=scores[a] class0=scores[b] # Plot ROC curve pr <-pr.curve(scores.class0=class1, scores.class1=class0, curve=T) plot(pr) The AUC curve hugs the top left corner and hence the performance of the classifier is quite good. ## 4c. Binary Classification using MNIST – Octave code This code to load MNIST data was taken from Daniel E blog. Precision recall curves are available in Matlab but are yet to be implemented in Octave’s statistics package.  load('./mnist/mnist.txt.gz'); % load the dataset # Subset the 'not 5' digits a=(trainY != 5); # Subset '5' b=(trainY == 5); #make a copy of trainY #Set 'not 5' as 0 and '5' as 1 y=trainY; y(a)=0; y(b)=1; X=trainX(1:5000,:); Y=y(1:5000); # Set the dimensions of layer layersDimensions=[784, 7,7,3,1]; # Compute the DL network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='relu', learningRate = 0.1, numIterations = 5000);  # Conclusion It was quite a challenge coding a Deep Learning Network in Python, R and Octave. The Deep Learning network implementation, in this post,is the base Deep Learning network, without any of the regularization methods included. Here are some key learning that I got while playing with different multi-layer networks on different problems a. Deep Learning Networks come with many levers, the hyper-parameters, – learning rate – activation unit – number of hidden layers – number of units per hidden layer – number of iterations while performing gradient descent b. Deep Networks are very sensitive. A change in any of the hyper-parameter makes it perform very differently c. Initially I thought adding more hidden layers, or more units per hidden layer will make the DL network better at learning. On the contrary, there is a performance degradation after the optimal DL configuration d. At a sub-optimal number of hidden layers or number of hidden units, gradient descent seems to get stuck at a local minima e. There were occasions when the cost came down, only to increase slowly as the number of iterations were increased. Probably early stopping would have helped. f. I also did come across situations of ‘exploding/vanishing gradient’, cost went to Inf/-Inf. Here I would think inclusion of ‘momentum method’ would have helped I intend to add the additional hyper-parameters of L1, L2 regularization, momentum method, early stopping etc. into the code in my future posts. Feel free to fork/clone the code from Github Deep Learning – Part 3, and take the DL network apart and play around with it. I will be continuing this series with more hyper-parameters to handle vanishing and exploding gradients, early stopping and regularization in the weeks to come. I also intend to add some more activation functions to this basic Multi-Layer Network. Hang around, there are more exciting things to come. Watch this space! To see all posts see Index of posts # Deep Learning from first principles in Python, R and Octave – Part 2 “What does the world outside your head really ‘look’ like? Not only is there no color, there’s also no sound: the compression and expansion of air is picked up by the ears, and turned into electrical signals. The brain then presents these signals to us as mellifluous tones and swishes and clatters and jangles. Reality is also odorless: there’s no such thing as smell outside our brains. Molecules floating through the air bind to receptors in our nose and are interpreted as different smells by our brain. The real world is not full of rich sensory events; instead, our brains light up the world with their own sensuality.” The Brain: The Story of You” by David Eagleman The world is Maya, illusory. The ultimate reality, the Brahman, is all-pervading and all-permeating, which is colourless, odourless, tasteless, nameless and formless Bhagavad Gita ## 1. Introduction This post is a follow-up post to my earlier post Deep Learning from first principles in Python, R and Octave-Part 1. In the first part, I implemented Logistic Regression, in vectorized Python,R and Octave, with a wannabe Neural Network (a Neural Network with no hidden layers). In this second part, I implement a regular, but somewhat primitive Neural Network (a Neural Network with just 1 hidden layer). The 2nd part implements classification of manually created datasets, where the different clusters of the 2 classes are not linearly separable. Neural Network perform really well in learning all sorts of non-linear boundaries between classes. Initially logistic regression is used perform the classification and the decision boundary is plotted. Vanilla logistic regression performs quite poorly. Using SVMs with a radial basis kernel would have performed much better in creating non-linear boundaries. To see R and Python implementations of SVMs take a look at my post Practical Machine Learning with R and Python – Part 4. Checkout my book ‘Deep Learning from first principles: Second Edition – In vectorized Python, R and Octave’. My book starts with the implementation of a simple 2-layer Neural Network and works its way to a generic L-Layer Deep Learning Network, with all the bells and whistles. The derivations have been discussed in detail. The code has been extensively commented and included in its entirety in the Appendix sections. My book is available on Amazon as paperback ($18.99) and in kindle version($9.99/Rs449). You may also like my companion book “Practical Machine Learning with R and Python:Second Edition- Machine Learning in stereo” available in Amazon in paperback($10.99) and Kindle($7.99/Rs449) versions. This book is ideal for a quick reference of the various ML functions and associated measurements in both R and Python which are essential to delve deep into Deep Learning. Take a look at my video presentation which discusses the below derivation step-by- step Elements of Neural Networks and Deep Learning – Part 3 You can clone and fork this R Markdown file along with the vectorized implementations of the 3 layer Neural Network for Python, R and Octave from Github DeepLearning-Part2 ### 2. The 3 layer Neural Network A simple representation of a 3 layer Neural Network (NN) with 1 hidden layer is shown below. In the above Neural Network, there are 2 input features at the input layer, 3 hidden units at the hidden layer and 1 output layer as it deals with binary classification. The activation unit at the hidden layer can be a tanh, sigmoid, relu etc. At the output layer the activation is a sigmoid to handle binary classification # Superscript indicates layer 1 $z_{11} = w_{11}^{1}x_{1} + w_{21}^{1}x_{2} + b_{1}$ $z_{12} = w_{12}^{1}x_{1} + w_{22}^{1}x_{2} + b_{1}$ $z_{13} = w_{13}^{1}x_{1} + w_{23}^{1}x_{2} + b_{1}$ Also $a_{11} = tanh(z_{11})$ $a_{12} = tanh(z_{12})$ $a_{13} = tanh(z_{13})$ # Superscript indicates layer 2 $z_{21} = w_{11}^{2}a_{11} + w_{21}^{2}a_{12} + w_{31}^{2}a_{13} + b_{2}$ $a_{21} = sigmoid(z21)$ Hence $Z1= \begin{pmatrix} z11\\ z12\\ z13 \end{pmatrix} =\begin{pmatrix} w_{11}^{1} & w_{21}^{1} \\ w_{12}^{1} & w_{22}^{1} \\ w_{13}^{1} & w_{23}^{1} \end{pmatrix} * \begin{pmatrix} x1\\ x2 \end{pmatrix} + b_{1}$ And $A1= \begin{pmatrix} a11\\ a12\\ a13 \end{pmatrix} = \begin{pmatrix} tanh(z11)\\ tanh(z12)\\ tanh(z13) \end{pmatrix}$ Similarly $Z2= z_{21} = \begin{pmatrix} w_{11}^{2} & w_{21}^{2} & w_{31}^{2} \end{pmatrix} *\begin{pmatrix} z_{11}\\ z_{12}\\ z_{13} \end{pmatrix} +b_{2}$ and $A2 = a_{21} = sigmoid(z_{21})$ These equations can be written as $Z1 = W1 * X + b1$ $A1 = tanh(Z1)$ $Z2 = W2 * A1 + b2$ $A2 = sigmoid(Z2)$ I) Some important results (a memory refresher!) $d/dx(e^{x}) = e^{x}$ and $d/dx(e^{-x}) = -e^{-x}$ -(a) and $sinhx = (e^{x} - e^{-x})/2$ and $coshx = (e^{x} + e^{-x})/2$ Using (a) we can shown that $d/dx(sinhx) = coshx$ and $d/dx(coshx) = sinhx$ (b) Now $d/dx(f(x)/g(x)) = (g(x)*d/dx(f(x)) - f(x)*d/dx(g(x)))/g(x)^{2}$ -(c) Since $tanhx =z= sinhx/coshx$ and using (b) we get $tanhx = (coshx*d/dx(sinhx) - sinhx*d/dx(coshx))/(cosh^{2})$ Using the values of the derivatives of sinhx and coshx from (b) above we get $d/dx(tanhx) = (coshx^{2} - sinhx{2})/coshx{2} = 1 - tanhx^{2}$ Since $tanhx =z$ $d/dx(tanhx) = 1 - tanhx^{2}= 1 - z^{2}$ -(d) II) Derivatives $L=-(Ylog(A2) + (1-Y)log(1-A2))$ $dL/dA2 = -(Y/A2 + (1-Y)/(1-A2))$ Since $A2 = sigmoid(Z2)$ therefore $dA2/dZ2 = A2(1-A2)$ see Part1 $Z2 = W2A1 +b2$ $dZ2/dW2 = A1$ $dZ2/db2 = 1$ $A1 = tanh(Z1)$ and $dA1/dZ1 = 1 - A1^{2}$ $Z1 = W1X + b1$ $dZ1/dW1 = X$ $dZ1/db1 = 1$ III) Back propagation Using the derivatives from II) we can derive the following results using Chain Rule $\partial L/\partial Z2 = \partial L/\partial A2 * \partial A2/\partial Z2$ $= -(Y/A2 + (1-Y)/(1-A2)) * A2(1-A2) = A2 - Y$ $\partial L/\partial W2 = \partial L/\partial A2 * \partial A2/\partial Z2 * \partial Z2/\partial W2$ $= (A2-Y) *A1$ -(A) $\partial L/\partial b2 = \partial L/\partial A2 * \partial A2/\partial Z2 * \partial Z2/\partial b2 = (A2-Y)$ -(B) $\partial L/\partial Z1 = \partial L/\partial A2 * \partial A2/\partial Z2 * \partial Z2/\partial A1 *\partial A1/\partial Z1 = (A2-Y) * W2 * (1-A1^{2})$ $\partial L/\partial W1 = \partial L/\partial A2 * \partial A2/\partial Z2 * \partial Z2/\partial A1 *\partial A1/\partial Z1 *\partial Z1/\partial W1$ $=(A2-Y) * W2 * (1-A1^{2}) * X$ -(C) $\partial L/\partial b1 = \partial L/\partial A2 * \partial A2/\partial Z2 * \partial Z2/\partial A1 *dA1/dZ1 *dZ1/db1$ $= (A2-Y) * W2 * (1-A1^{2})$ -(D) IV) Gradient Descent The key computations in the backward cycle are $W1 = W1-learningRate * \partial L/\partial W1$ – From (C) $b1 = b1-learningRate * \partial L/\partial b1$ – From (D) $W2 = W2-learningRate * \partial L/\partial W2$ – From (A) $b2 = b2-learningRate * \partial L/\partial b2$ – From (B) The weights and biases (W1,b1,W2,b2) are updated for each iteration thus minimizing the loss/cost. These derivations can be represented pictorially using the computation graph (from the book Deep Learning by Ian Goodfellow, Joshua Bengio and Aaron Courville) ### 3. Manually create a data set that is not lineary separable Initially I create a dataset with 2 classes which has around 9 clusters that cannot be separated by linear boundaries. Note: This data set is saved as data.csv and is used for the R and Octave Neural networks to see how they perform on the same dataset. import numpy as np import matplotlib.pyplot as plt import matplotlib.colors import sklearn.linear_model from sklearn.model_selection import train_test_split from sklearn.datasets import make_classification, make_blobs from matplotlib.colors import ListedColormap import sklearn import sklearn.datasets colors=['black','gold'] cmap = matplotlib.colors.ListedColormap(colors) X, y = make_blobs(n_samples = 400, n_features = 2, centers = 7, cluster_std = 1.3, random_state = 4) #Create 2 classes y=y.reshape(400,1) y = y % 2 #Plot the figure plt.figure() plt.title('Non-linearly separable classes') plt.scatter(X[:,0], X[:,1], c=y, marker= 'o', s=50,cmap=cmap) plt.savefig('fig1.png', bbox_inches='tight') ### 4. Logistic Regression On the above created dataset, classification with logistic regression is performed, and the decision boundary is plotted. It can be seen that logistic regression performs quite poorly import numpy as np import matplotlib.pyplot as plt import matplotlib.colors import sklearn.linear_model from sklearn.model_selection import train_test_split from sklearn.datasets import make_classification, make_blobs from matplotlib.colors import ListedColormap import sklearn import sklearn.datasets #from DLfunctions import plot_decision_boundary execfile("./DLfunctions.py") # Since import does not work in Rmd!!! colors=['black','gold'] cmap = matplotlib.colors.ListedColormap(colors) X, y = make_blobs(n_samples = 400, n_features = 2, centers = 7, cluster_std = 1.3, random_state = 4) #Create 2 classes y=y.reshape(400,1) y = y % 2 # Train the logistic regression classifier clf = sklearn.linear_model.LogisticRegressionCV(); clf.fit(X, y); # Plot the decision boundary for logistic regression plot_decision_boundary_n(lambda x: clf.predict(x), X.T, y.T,"fig2.png")  ### 5. The 3 layer Neural Network in Python (vectorized) The vectorized implementation is included below. Note that in the case of Python a learning rate of 0.5 and 3 hidden units performs very well. ## Random data set with 9 clusters import numpy as np import matplotlib import matplotlib.pyplot as plt import sklearn.linear_model import pandas as pd from sklearn.datasets import make_classification, make_blobs execfile("./DLfunctions.py") # Since import does not work in Rmd!!! X1, Y1 = make_blobs(n_samples = 400, n_features = 2, centers = 9, cluster_std = 1.3, random_state = 4) #Create 2 classes Y1=Y1.reshape(400,1) Y1 = Y1 % 2 X2=X1.T Y2=Y1.T #Perform gradient descent parameters,costs = computeNN(X2, Y2, numHidden = 4, learningRate=0.5, numIterations = 10000) plot_decision_boundary(lambda x: predict(parameters, x.T), X2, Y2,str(4),str(0.5),"fig3.png") ## Cost after iteration 0: 0.692669 ## Cost after iteration 1000: 0.246650 ## Cost after iteration 2000: 0.227801 ## Cost after iteration 3000: 0.226809 ## Cost after iteration 4000: 0.226518 ## Cost after iteration 5000: 0.226331 ## Cost after iteration 6000: 0.226194 ## Cost after iteration 7000: 0.226085 ## Cost after iteration 8000: 0.225994 ## Cost after iteration 9000: 0.225915 ### 6. The 3 layer Neural Network in R (vectorized) For this the dataset created by Python is saved to see how R performs on the same dataset. The vectorized implementation of a Neural Network was just a little more interesting as R does not have a similar package like ‘numpy’. While numpy handles broadcasting implicitly, in R I had to use the ‘sweep’ command to broadcast. The implementaion is included below. Note that since the initialization with random weights is slightly different, R performs best with a learning rate of 0.1 and with 6 hidden units source("DLfunctions2_1.R") z <- as.matrix(read.csv("data.csv",header=FALSE)) # x <- z[,1:2] y <- z[,3] x1 <- t(x) y1 <- t(y) #Perform gradient descent nn <-computeNN(x1, y1, 6, learningRate=0.1,numIterations=10000) # Good ## [1] 0.7075341 ## [1] 0.2606695 ## [1] 0.2198039 ## [1] 0.2091238 ## [1] 0.211146 ## [1] 0.2108461 ## [1] 0.2105351 ## [1] 0.210211 ## [1] 0.2099104 ## [1] 0.2096437 ## [1] 0.209409 plotDecisionBoundary(z,nn,6,0.1) ### 7. The 3 layer Neural Network in Octave (vectorized) This uses the same dataset that was generated using Python code. source("DL-function2.m") data=csvread("data.csv"); X=data(:,1:2); Y=data(:,3); # Make sure that the model parameters are correct. Take the transpose of X & Y #Perform gradient descent [W1,b1,W2,b2,costs]= computeNN(X', Y',4, learningRate=0.5, numIterations = 10000); ### 8a. Performance for different learning rates (Python) import numpy as np import matplotlib import matplotlib.pyplot as plt import sklearn.linear_model import pandas as pd from sklearn.datasets import make_classification, make_blobs execfile("./DLfunctions.py") # Since import does not work in Rmd!!! # Create data X1, Y1 = make_blobs(n_samples = 400, n_features = 2, centers = 9, cluster_std = 1.3, random_state = 4) #Create 2 classes Y1=Y1.reshape(400,1) Y1 = Y1 % 2 X2=X1.T Y2=Y1.T # Create a list of learning rates learningRate=[0.5,1.2,3.0] df=pd.DataFrame() #Compute costs for each learning rate for lr in learningRate: parameters,costs = computeNN(X2, Y2, numHidden = 4, learningRate=lr, numIterations = 10000) print(costs) df1=pd.DataFrame(costs) df=pd.concat([df,df1],axis=1) #Set the iterations iterations=[0,1000,2000,3000,4000,5000,6000,7000,8000,9000] #Create data frame #Set index df1=df.set_index([iterations]) df1.columns=[0.5,1.2,3.0] fig=df1.plot() fig=plt.title("Cost vs No of Iterations for different learning rates") plt.savefig('fig4.png', bbox_inches='tight') ### 8b. Performance for different hidden units (Python) import numpy as np import matplotlib import matplotlib.pyplot as plt import sklearn.linear_model import pandas as pd from sklearn.datasets import make_classification, make_blobs execfile("./DLfunctions.py") # Since import does not work in Rmd!!! #Create data set X1, Y1 = make_blobs(n_samples = 400, n_features = 2, centers = 9, cluster_std = 1.3, random_state = 4) #Create 2 classes Y1=Y1.reshape(400,1) Y1 = Y1 % 2 X2=X1.T Y2=Y1.T # Make a list of hidden unis numHidden=[3,5,7] df=pd.DataFrame() #Compute costs for different hidden units for numHid in numHidden: parameters,costs = computeNN(X2, Y2, numHidden = numHid, learningRate=1.2, numIterations = 10000) print(costs) df1=pd.DataFrame(costs) df=pd.concat([df,df1],axis=1) #Set the iterations iterations=[0,1000,2000,3000,4000,5000,6000,7000,8000,9000] #Set index df1=df.set_index([iterations]) df1.columns=[3,5,7] #Plot fig=df1.plot() fig=plt.title("Cost vs No of Iterations for different no of hidden units") plt.savefig('fig5.png', bbox_inches='tight') ### 9a. Performance for different learning rates (R) source("DLfunctions2_1.R") # Read data z <- as.matrix(read.csv("data.csv",header=FALSE)) # x <- z[,1:2] y <- z[,3] x1 <- t(x) y1 <- t(y) #Loop through learning rates and compute costs learningRate <-c(0.1,1.2,3.0) df <- NULL for(i in seq_along(learningRate)){ nn <- computeNN(x1, y1, 6, learningRate=learningRate[i],numIterations=10000) cost <- nn$costs
df <- cbind(df,cost)

}      

#Create dataframe
df <- data.frame(df)
iterations=seq(0,10000,by=1000)
df <- cbind(iterations,df)
names(df) <- c("iterations","0.5","1.2","3.0")
library(reshape2)
df1 <- melt(df,id="iterations")  # Melt the data
#Plot
ggplot(df1) + geom_line(aes(x=iterations,y=value,colour=variable),size=1)  +
xlab("Iterations") +
ylab('Cost') + ggtitle("Cost vs No iterations for  different learning rates")

### 9b. Performance  for different hidden units (R)

source("DLfunctions2_1.R")
# Loop through Num hidden units
numHidden <-c(4,6,9)
df <- NULL
for(i in seq_along(numHidden)){
nn <-  computeNN(x1, y1, numHidden[i], learningRate=0.1,numIterations=10000)
cost <- nn\$costs
df <- cbind(df,cost)

}      
df <- data.frame(df)
iterations=seq(0,10000,by=1000)
df <- cbind(iterations,df)
names(df) <- c("iterations","4","6","9")
library(reshape2)
# Melt
df1 <- melt(df,id="iterations")
# Plot
ggplot(df1) + geom_line(aes(x=iterations,y=value,colour=variable),size=1)  +
xlab("Iterations") +
ylab('Cost') + ggtitle("Cost vs No iterations for  different number of hidden units")

## 10a. Performance of the Neural Network for different learning rates (Octave)

source("DL-function2.m") plotLRCostVsIterations() print -djph figa.jpg

## 10b. Performance of the Neural Network for different number of hidden units (Octave)

source("DL-function2.m") plotHiddenCostVsIterations() print -djph figa.jpg

## 11. Turning the heat on the Neural Network

In this 2nd part I create a a central region of positives and and the outside region as negatives. The points are generated using the equation of a circle (x – a)^{2} + (y -b) ^{2} = R^{2} . How does the 3 layer Neural Network perform on this?  Here’s a look! Note: The same dataset is also used for R and Octave Neural Network constructions

## 12. Manually creating a circular central region

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.colors
import sklearn.linear_model

from sklearn.model_selection import train_test_split
from sklearn.datasets import make_classification, make_blobs
from matplotlib.colors import ListedColormap
import sklearn
import sklearn.datasets

colors=['black','gold']
cmap = matplotlib.colors.ListedColormap(colors)
x1=np.random.uniform(0,10,800).reshape(800,1)
x2=np.random.uniform(0,10,800).reshape(800,1)
X=np.append(x1,x2,axis=1)
X.shape
# Create (x-a)^2 + (y-b)^2 = R^2
# Create a subset of values where squared is <0,4. Perform ravel() to flatten this vector
a=(np.power(X[:,0]-5,2) + np.power(X[:,1]-5,2) <= 6).ravel()
Y=a.reshape(800,1)

cmap = matplotlib.colors.ListedColormap(colors)

plt.figure()
plt.title('Non-linearly separable classes')
plt.scatter(X[:,0], X[:,1], c=Y,
marker= 'o', s=15,cmap=cmap)
plt.savefig('fig6.png', bbox_inches='tight')

### 13a. Decision boundary with hidden units=4 and learning rate = 2.2 (Python)

With the above hyper parameters the decision boundary is triangular

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.colors
import sklearn.linear_model
execfile("./DLfunctions.py")
x1=np.random.uniform(0,10,800).reshape(800,1)
x2=np.random.uniform(0,10,800).reshape(800,1)
X=np.append(x1,x2,axis=1)
X.shape

# Create a subset of values where squared is <0,4. Perform ravel() to flatten this vector
a=(np.power(X[:,0]-5,2) + np.power(X[:,1]-5,2) <= 6).ravel()
Y=a.reshape(800,1)

X2=X.T
Y2=Y.T

parameters,costs = computeNN(X2, Y2, numHidden = 4, learningRate=2.2, numIterations = 10000)
plot_decision_boundary(lambda x: predict(parameters, x.T), X2, Y2,str(4),str(2.2),"fig7.png")

## Cost after iteration 0: 0.692836
## Cost after iteration 1000: 0.331052
## Cost after iteration 2000: 0.326428
## Cost after iteration 3000: 0.474887
## Cost after iteration 4000: 0.247989
## Cost after iteration 5000: 0.218009
## Cost after iteration 6000: 0.201034
## Cost after iteration 7000: 0.197030
## Cost after iteration 8000: 0.193507
## Cost after iteration 9000: 0.191949

### 13b. Decision boundary with hidden units=12 and learning rate = 2.2 (Python)

With the above hyper parameters the decision boundary is triangular

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.colors
import sklearn.linear_model
execfile("./DLfunctions.py")
x1=np.random.uniform(0,10,800).reshape(800,1)
x2=np.random.uniform(0,10,800).reshape(800,1)
X=np.append(x1,x2,axis=1)
X.shape

# Create a subset of values where squared is <0,4. Perform ravel() to flatten this vector
a=(np.power(X[:,0]-5,2) + np.power(X[:,1]-5,2) <= 6).ravel()
Y=a.reshape(800,1)

X2=X.T
Y2=Y.T

parameters,costs = computeNN(X2, Y2, numHidden = 12, learningRate=2.2, numIterations = 10000)
plot_decision_boundary(lambda x: predict(parameters, x.T), X2, Y2,str(12),str(2.2),"fig8.png")

## Cost after iteration 0: 0.693291
## Cost after iteration 1000: 0.383318
## Cost after iteration 2000: 0.298807
## Cost after iteration 3000: 0.251735
## Cost after iteration 4000: 0.177843
## Cost after iteration 5000: 0.130414
## Cost after iteration 6000: 0.152400
## Cost after iteration 7000: 0.065359
## Cost after iteration 8000: 0.050921
## Cost after iteration 9000: 0.039719

### 14a. Decision boundary with hidden units=9 and learning rate = 0.5 (R)

When the number of hidden units is 6 and the learning rate is 0,1, is also a triangular shape in R

source("DLfunctions2_1.R")
x <- z[,1:2]
y <- z[,3]
x1 <- t(x)
y1 <- t(y)
nn <-computeNN(x1, y1, 9, learningRate=0.5,numIterations=10000) # Triangular
## [1] 0.8398838
## [1] 0.3303621
## [1] 0.3127731
## [1] 0.3012791
## [1] 0.3305543
## [1] 0.3303964
## [1] 0.2334615
## [1] 0.1920771
## [1] 0.2341225
## [1] 0.2188118
## [1] 0.2082687
plotDecisionBoundary(z,nn,6,0.1)

### 14b. Decision boundary with hidden units=8 and learning rate = 0.1 (R)

source("DLfunctions2_1.R")
x <- z[,1:2]
y <- z[,3]
x1 <- t(x)
y1 <- t(y)
nn <-computeNN(x1, y1, 8, learningRate=0.1,numIterations=10000) # Hemisphere
## [1] 0.7273279
## [1] 0.3169335
## [1] 0.2378464
## [1] 0.1688635
## [1] 0.1368466
## [1] 0.120664
## [1] 0.111211
## [1] 0.1043362
## [1] 0.09800573
## [1] 0.09126161
## [1] 0.0840379
plotDecisionBoundary(z,nn,8,0.1)

### 15a. Decision boundary with hidden units=12 and learning rate = 1.5 (Octave)

source("DL-function2.m") data=csvread("data1.csv"); X=data(:,1:2); Y=data(:,3); # Make sure that the model parameters are correct. Take the transpose of X & Y [W1,b1,W2,b2,costs]= computeNN(X', Y',12, learningRate=1.5, numIterations = 10000); plotDecisionBoundary(data, W1,b1,W2,b2) print -djpg fige.jpg

Conclusion: This post implemented a 3 layer Neural Network to create non-linear boundaries while performing classification. Clearly the Neural Network performs very well when the number of hidden units and learning rate are varied.

To be continued…
Watch this space!!

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