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

Presentation on ‘Machine Learning in plain English – Part 1’

This is the first part on my series ‘Machine Learning in plain English – Part 1’ in which I discuss the intuition behind different Machine Learning algorithms, metrics and the approaches etc. These presentations will not include tiresome math or laborious programming constructs, and will instead focus on just the concepts behind the Machine Learning algorithms.  This presentation discusses what Machine Learning is, Gradient Descent, linear, multi variate & polynomial regression, bias/variance, under fit, good fit and over fit and finally logistic regression etc.

It is hoped that these presentations will trigger sufficient interest in you, to explore this fascinating field further

To see actual implementations of the most widely used Machine Learning algorithms in R and Python, check out My book ‘Practical Machine Learning with R and Python’ on Amazon

To see all post see “Index of posts

Deep Learning from first principles in Python, R and Octave – Part 1

“You don’t perceive objects as they are. You perceive them as you are.”
“Your interpretation of physical objects has everything to do with the historical trajectory of your brain – and little to do with the objects themselves.”
“The brain generates its own reality, even before it receives information coming in from the eyes and the other senses. This is known as the internal model”

                          David Eagleman - The Brain: The Story of You

This is the first in the series of posts, I intend to write on Deep Learning. This post is inspired by the Deep Learning Specialization by Prof Andrew Ng on Coursera and Neural Networks for Machine Learning by Prof Geoffrey Hinton also on Coursera. In this post I implement Logistic regression with a 2 layer Neural Network i.e. a Neural Network that just has an input layer and an output layer and with no hidden layer.I am certain that any self-respecting Deep Learning/Neural Network would consider a Neural Network without hidden layers as no Neural Network at all!

This 2 layer network is implemented in Python, R and Octave languages. I have included Octave, into the mix, as Octave is a close cousin of Matlab. These implementations in Python, R and Octave are equivalent vectorized implementations. So, if you are familiar in any one of the languages, you should be able to look at the corresponding code in the other two. You can download this R Markdown file and Octave code from DeepLearning -Part 1

Check out my video presentation which discusses the derivations in detail
1. Elements of Neural Networks and Deep Le- Part 1
2. Elements of Neural Networks and Deep Learning – Part 2

To start with, Logistic Regression is performed using sklearn’s logistic regression package for the cancer data set also from sklearn. This is shown below

1. Logistic Regression

import numpy as np
import pandas as pd
import os
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LogisticRegression
from sklearn.datasets import make_classification, make_blobs

from sklearn.metrics import confusion_matrix
from matplotlib.colors import ListedColormap
(X_cancer, y_cancer) = load_breast_cancer(return_X_y = True)
X_train, X_test, y_train, y_test = train_test_split(X_cancer, y_cancer,
random_state = 0)
# Call the Logisitic Regression function
clf = LogisticRegression().fit(X_train, y_train)
print('Accuracy of Logistic regression classifier on training set: {:.2f}'
.format(clf.score(X_train, y_train)))
print('Accuracy of Logistic regression classifier on test set: {:.2f}'
.format(clf.score(X_test, y_test)))
## Accuracy of Logistic regression classifier on training set: 0.96
## Accuracy of Logistic regression classifier on test set: 0.96

To check on other classification algorithms, check my post Practical Machine Learning with R and Python – Part 2. 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).

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.

2. Logistic Regression as a 2 layer Neural Network

In the following section Logistic Regression is implemented as a 2 layer Neural Network in Python, R and Octave. The same cancer data set from sklearn will be used to train and test the Neural Network in Python, R and Octave. This can be represented diagrammatically as below The cancer data set has 30 input features, and the target variable ‘output’ is either 0 or 1. Hence the sigmoid activation function will be used in the output layer for classification.

This simple 2 layer Neural Network is shown below
At the input layer there are 30 features and the corresponding weights of these inputs which are initialized to small random values. $Z= w_{1}x_{1} +w_{2}x_{2} +..+ w_{30}x_{30} + b$
where ‘b’ is the bias term

The Activation function is the sigmoid function which is $a= 1/(1+e^{-z})$
The Loss, when the sigmoid function is used in the output layer, is given by $L=-(ylog(a) + (1-y)log(1-a))$ (1)

Forward propagation

In forward propagation cycle of the Neural Network the output Z and the output of activation function, the sigmoid function, is first computed. Then using the output ‘y’ for the given features, the ‘Loss’ is computed using equation (1) above.

Backward propagation

The backward propagation cycle determines how the ‘Loss’ is impacted for small variations from the previous layers upto the input layer. In other words, backward propagation computes the changes in the weights at the input layer, which will minimize the loss. Several cycles of gradient descent are performed in the path of steepest descent to find the local minima. In other words the set of weights and biases, at the input layer, which will result in the lowest loss is computed by gradient descent. The weights at the input layer are decreased by a parameter known as the ‘learning rate’. Too big a ‘learning rate’ can overshoot the local minima, and too small a ‘learning rate’ can take a long time to reach the local minima. This is done for ‘m’ training examples.

Chain rule of differentiation
Let y=f(u)
and u=g(x) then $\partial y/\partial x = \partial y/\partial u * \partial u/\partial x$

Derivative of sigmoid $\sigma=1/(1+e^{-z})$
Let $x= 1 + e^{-z}$  then $\sigma = 1/x$ $\partial \sigma/\partial x = -1/x^{2}$ $\partial x/\partial z = -e^{-z}$
Using the chain rule of differentiation we get $\partial \sigma/\partial z = \partial \sigma/\partial x * \partial x/\partial z$ $=-1/(1+e^{-z})^{2}* -e^{-z} = e^{-z}/(1+e^{-z})^{2}$
Therefore $\partial \sigma/\partial z = \sigma(1-\sigma)$        -(2)

The 3 equations for the 2 layer Neural Network representation of Logistic Regression are $L=-(y*log(a) + (1-y)*log(1-a))$      -(a) $a=1/(1+e^{-Z})$      -(b) $Z= w_{1}x_{1} +w_{2}x_{2} +...+ w_{30}x_{30} +b = Z = \sum_{i} w_{i}*x_{i} + b$ -(c)

The back propagation step requires the computation of $dL/dw_{i}$ and $dL/db_{i}$. In the case of regression it would be $dE/dw_{i}$ and $dE/db_{i}$ where dE is the Mean Squared Error function.
Computing the derivatives for back propagation we have $dL/da = -(y/a + (1-y)/(1-a))$          -(d)
because $d/dx(logx) = 1/x$
Also from equation (2) we get $da/dZ = a (1-a)$                                  – (e)
By chain rule $\partial L/\partial Z = \partial L/\partial a * \partial a/\partial Z$
therefore substituting the results of (d) & (e) we get $\partial L/\partial Z = -(y/a + (1-y)/(1-a)) * a(1-a) = a-y$         (f)
Finally $\partial L/\partial w_{i}= \partial L/\partial a * \partial a/\partial Z * \partial Z/\partial w_{i}$                                                           -(g) $\partial Z/\partial w_{i} = x_{i}$            – (h)
and from (f) we have $\partial L/\partial Z =a-y$
Therefore  (g) reduces to $\partial L/\partial w_{i} = x_{i}* (a-y)$ -(i)
Also $\partial L/\partial b = \partial L/\partial a * \partial a/\partial Z * \partial Z/\partial b$ -(j)
Since $\partial Z/\partial b = 1$ and using (f) in (j) $\partial L/\partial b = a-y$

The gradient computes the weights at the input layer and the corresponding bias by using the values
of $dw_{i}$ and $db$ $w_{i} := w_{i} -\alpha * dw_{i}$ $b := b -\alpha * db$
I found the computation graph representation in the book Deep Learning: Ian Goodfellow, Yoshua Bengio, Aaron Courville, very useful to visualize and also compute the backward propagation. For the 2 layer Neural Network of Logistic Regression the computation graph is shown below 3. Neural Network for Logistic Regression -Python code (vectorized)

import numpy as np
import pandas as pd
import os
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split

# Define the sigmoid function
def sigmoid(z):
a=1/(1+np.exp(-z))
return a

# Initialize
def initialize(dim):
w = np.zeros(dim).reshape(dim,1)
b = 0
return w

# Compute the loss
def computeLoss(numTraining,Y,A):
loss=-1/numTraining *np.sum(Y*np.log(A) + (1-Y)*(np.log(1-A)))
return(loss)

# Execute the forward propagation
def forwardPropagation(w,b,X,Y):
# Compute Z
Z=np.dot(w.T,X)+b
# Determine the number of training samples
numTraining=float(len(X))
# Compute the output of the sigmoid activation function
A=sigmoid(Z)
#Compute the loss
loss = computeLoss(numTraining,Y,A)
# Compute the gradients dZ, dw and db
dZ=A-Y
dw=1/numTraining*np.dot(X,dZ.T)
db=1/numTraining*np.sum(dZ)

# Return the results as a dictionary
"db": db}
loss = np.squeeze(loss)

def gradientDescent(w, b, X, Y, numIerations, learningRate):
losses=[]
idx =[]
# Iterate
for i in range(numIerations):
#Get the derivates
w = w-learningRate*dw
b = b-learningRate*db

# Store the loss
if i % 100 == 0:
idx.append(i)
losses.append(loss)
params = {"w": w,
"b": b}
"db": db}

# Predict the output for a training set
def predict(w,b,X):
size=X.shape
yPredicted=np.zeros((1,size))
Z=np.dot(w.T,X)
# Compute the sigmoid
A=sigmoid(Z)
for i in range(A.shape):
#If the value is > 0.5 then set as 1
if(A[i] > 0.5):
yPredicted[i]=1
else:
# Else set as 0
yPredicted[i]=0

return yPredicted

#Normalize the data
def normalize(x):
x_norm = None
x_norm = np.linalg.norm(x,axis=1,keepdims=True)
x= x/x_norm
return x

# Run the 2 layer Neural Network on the cancer data set

(X_cancer, y_cancer) = load_breast_cancer(return_X_y = True)
# Create train and test sets
X_train, X_test, y_train, y_test = train_test_split(X_cancer, y_cancer,
random_state = 0)
# Normalize the data for better performance
X_train1=normalize(X_train)

# Create weight vectors of zeros. The size is the number of features in the data set=30
w=np.zeros((X_train.shape,1))
#w=np.zeros((30,1))
b=0

#Normalize the training data so that gradient descent performs better
X_train1=normalize(X_train)
#Transpose X_train so that we have a matrix as (features, numSamples)
X_train2=X_train1.T

# Reshape to remove the rank 1 array and then transpose
y_train1=y_train.reshape(len(y_train),1)
y_train2=y_train1.T

# Run gradient descent for 4000 times and compute the weights
w = parameters["w"]
b = parameters["b"]

# Normalize X_test
X_test1=normalize(X_test)
#Transpose X_train so that we have a matrix as (features, numSamples)
X_test2=X_test1.T

#Reshape y_test
y_test1=y_test.reshape(len(y_test),1)
y_test2=y_test1.T

# Predict the values for
yPredictionTest = predict(w, b, X_test2)
yPredictionTrain = predict(w, b, X_train2)

# Print the accuracy
print("train accuracy: {} %".format(100 - np.mean(np.abs(yPredictionTrain - y_train2)) * 100))
print("test accuracy: {} %".format(100 - np.mean(np.abs(yPredictionTest - y_test)) * 100))

# Plot the Costs vs the number of iterations
fig1=plt.plot(idx,costs)
fig1=plt.title("Gradient descent-Cost vs No of iterations")
fig1=plt.xlabel("No of iterations")
fig1=plt.ylabel("Cost")
fig1.figure.savefig("fig1", bbox_inches='tight')
## train accuracy: 90.3755868545 %
## test accuracy: 89.5104895105 %

Note: It can be seen that the Accuracy on the training and test set is 90.37% and 89.51%. This is comparatively poorer than the 96% which the logistic regression of sklearn achieves! But this is mainly because of the absence of hidden layers which is the real power of neural networks.

4. Neural Network for Logistic Regression -R code (vectorized)

source("RFunctions-1.R")
# Define the sigmoid function
sigmoid <- function(z){
a <- 1/(1+ exp(-z))
a
}

# Compute the loss
computeLoss <- function(numTraining,Y,A){
loss <- -1/numTraining* sum(Y*log(A) + (1-Y)*log(1-A))
return(loss)
}

# Compute forward propagation
forwardPropagation <- function(w,b,X,Y){
# Compute Z
Z <- t(w) %*% X +b
#Set the number of samples
numTraining <- ncol(X)
# Compute the activation function
A=sigmoid(Z)

#Compute the loss
loss <- computeLoss(numTraining,Y,A)

# Compute the gradients dZ, dw and db
dZ<-A-Y
dw<-1/numTraining * X %*% t(dZ)
db<-1/numTraining*sum(dZ)

fwdProp <- list("loss" = loss, "dw" = dw, "db" = db)
return(fwdProp)
}

# Perform one cycle of Gradient descent
gradientDescent <- function(w, b, X, Y, numIerations, learningRate){
losses <- NULL
idx <- NULL
# Loop through the number of iterations
for(i in 1:numIerations){
fwdProp <-forwardPropagation(w,b,X,Y)
#Get the derivatives
dw <- fwdProp$dw db <- fwdProp$db
w = w-learningRate*dw
b = b-learningRate*db
l <- fwdProp$loss # Stoe the loss if(i %% 100 == 0){ idx <- c(idx,i) losses <- c(losses,l) } } # Return the weights and losses gradDescnt <- list("w"=w,"b"=b,"dw"=dw,"db"=db,"losses"=losses,"idx"=idx) return(gradDescnt) } # Compute the predicted value for input predict <- function(w,b,X){ m=dim(X) # Create a ector of 0's yPredicted=matrix(rep(0,m),nrow=1,ncol=m) Z <- t(w) %*% X +b # Compute sigmoid A=sigmoid(Z) for(i in 1:dim(A)){ # If A > 0.5 set value as 1 if(A[1,i] > 0.5) yPredicted[1,i]=1 else # Else set as 0 yPredicted[1,i]=0 } return(yPredicted) } # Normalize the matrix normalize <- function(x){ #Create the norm of the matrix.Perform the Frobenius norm of the matrix n<-as.matrix(sqrt(rowSums(x^2))) #Sweep by rows by norm. Note '1' in the function which performing on every row normalized<-sweep(x, 1, n, FUN="/") return(normalized) } # Run the 2 layer Neural Network on the cancer data set # Read the data (from sklearn) cancer <- read.csv("cancer.csv") # Rename the target variable names(cancer) <- c(seq(1,30),"output") # Split as training and test sets train_idx <- trainTestSplit(cancer,trainPercent=75,seed=5) train <- cancer[train_idx, ] test <- cancer[-train_idx, ] # Set the features X_train <-train[,1:30] y_train <- train[,31] X_test <- test[,1:30] y_test <- test[,31] # Create a matrix of 0's with the number of features w <-matrix(rep(0,dim(X_train))) b <-0 X_train1 <- normalize(X_train) X_train2=t(X_train1) # Reshape then transpose y_train1=as.matrix(y_train) y_train2=t(y_train1) # Perform gradient descent gradDescent= gradientDescent(w, b, X_train2, y_train2, numIerations=3000, learningRate=0.77) # Normalize X_test X_test1=normalize(X_test) #Transpose X_train so that we have a matrix as (features, numSamples) X_test2=t(X_test1) #Reshape y_test and take transpose y_test1=as.matrix(y_test) y_test2=t(y_test1) # Use the values of the weights generated from Gradient Descent yPredictionTest = predict(gradDescent$w, gradDescent$b, X_test2) yPredictionTrain = predict(gradDescent$w, gradDescent$b, X_train2) sprintf("Train accuracy: %f",(100 - mean(abs(yPredictionTrain - y_train2)) * 100)) ##  "Train accuracy: 90.845070" sprintf("test accuracy: %f",(100 - mean(abs(yPredictionTest - y_test)) * 100)) ##  "test accuracy: 87.323944" df <-data.frame(gradDescent$idx, gradDescent\$losses)
names(df) <- c("iterations","losses")
ggplot(df,aes(x=iterations,y=losses)) + geom_point() + geom_line(col="blue") +
ggtitle("Gradient Descent - Losses vs No of Iterations") +
xlab("No of iterations") + ylab("Losses") 4. Neural Network for Logistic Regression -Octave code (vectorized)

 1; # Define sigmoid function function a = sigmoid(z) a = 1 ./ (1+ exp(-z)); end # Compute the loss function loss=computeLoss(numtraining,Y,A) loss = -1/numtraining * sum((Y .* log(A)) + (1-Y) .* log(1-A)); end
 # Perform forward propagation function [loss,dw,db,dZ] = forwardPropagation(w,b,X,Y) % Compute Z Z = w' * X + b; numtraining = size(X)(1,2); # Compute sigmoid A = sigmoid(Z);
 #Compute loss. Note this is element wise product loss =computeLoss(numtraining,Y,A); # Compute the gradients dZ, dw and db dZ = A-Y; dw = 1/numtraining* X * dZ'; db =1/numtraining*sum(dZ);

end
 # Compute Gradient Descent function [w,b,dw,db,losses,index]=gradientDescent(w, b, X, Y, numIerations, learningRate) #Initialize losses and idx losses=[]; index=[]; # Loop through the number of iterations for i=1:numIerations, [loss,dw,db,dZ] = forwardPropagation(w,b,X,Y); # Perform Gradient descent w = w - learningRate*dw; b = b - learningRate*db; if(mod(i,100) ==0) # Append index and loss index = [index i]; losses = [losses loss]; endif

end
end
 # Determine the predicted value for dataset function yPredicted = predict(w,b,X) m = size(X)(1,2); yPredicted=zeros(1,m); # Compute Z Z = w' * X + b; # Compute sigmoid A = sigmoid(Z); for i=1:size(X)(1,2), # Set predicted as 1 if A > 0,5 if(A(1,i) >= 0.5) yPredicted(1,i)=1; else yPredicted(1,i)=0; endif end end
 # Normalize by dividing each value by the sum of squares function normalized = normalize(x) # Compute Frobenius norm. Square the elements, sum rows and then find square root a = sqrt(sum(x .^ 2,2)); # Perform element wise division normalized = x ./ a; end
 # Split into train and test sets function [X_train,y_train,X_test,y_test] = trainTestSplit(dataset,trainPercent) # Create a random index ix = randperm(length(dataset)); # Split into training trainSize = floor(trainPercent/100 * length(dataset)); train=dataset(ix(1:trainSize),:); # And test test=dataset(ix(trainSize+1:length(dataset)),:); X_train = train(:,1:30); y_train = train(:,31); X_test = test(:,1:30); y_test = test(:,31); end

 cancer=csvread("cancer.csv"); [X_train,y_train,X_test,y_test] = trainTestSplit(cancer,75); w=zeros(size(X_train)(1,2),1); b=0; X_train1=normalize(X_train); X_train2=X_train1'; y_train1=y_train'; [w1,b1,dw,db,losses,idx]=gradientDescent(w, b, X_train2, y_train1, numIerations=3000, learningRate=0.75); # Normalize X_test X_test1=normalize(X_test); #Transpose X_train so that we have a matrix as (features, numSamples) X_test2=X_test1'; y_test1=y_test'; # Use the values of the weights generated from Gradient Descent yPredictionTest = predict(w1, b1, X_test2); yPredictionTrain = predict(w1, b1, X_train2); 

 trainAccuracy=100-mean(abs(yPredictionTrain - y_train1))*100 testAccuracy=100- mean(abs(yPredictionTest - y_test1))*100 trainAccuracy = 90.845 testAccuracy = 89.510 graphics_toolkit('gnuplot') plot(idx,losses); title ('Gradient descent- Cost vs No of iterations'); xlabel ("No of iterations"); ylabel ("Cost"); Conclusion
This post starts with a simple 2 layer Neural Network implementation of Logistic Regression. Clearly the performance of this simple Neural Network is comparatively poor to the highly optimized sklearn’s Logistic Regression. This is because the above neural network did not have any hidden layers. Deep Learning & Neural Networks achieve extraordinary performance because of the presence of deep hidden layers

The Deep Learning journey has begun… Don’t miss the bus!
Stay tuned for more interesting posts in Deep Learning!!

To see all posts check Index of posts

Video presentation on Machine Learning, Data Science, NLP and Big Data – Part 2

Here is the 1st part of my video presentation on “Machine Learning, Data Science, NLP and Big Data – Part 2”

Video presentation on Machine Learning, Data Science, NLP and Big Data – Part 1

Here is the 1st part of my video presentation on “Machine Learning, Data Science, NLP and Big Data – Part 1”

Simplifying ML: Logistic regression – Part 2

Logistic regression is another class of Machine Learning algorithms which comes under supervised learning. In this regression technique we need to classify data. Take a look at my earlier post Simplifying Machine Learning algorithms – Part 1 I had discussed linear regression. For e.g if we had data on tumor sizes versus the fact that the tumor was benign or malignant, the question is whether given a tumor size we can predict whether this tumor would be benign or cancerous. So we need to have the ability to classify this data.

This is shown below It is obvious that a line with a certain slope could easily separate the two.

As another example we could have an algorithm that is able to automatically classify mail as either spam or not spam based on the subject line. So for e.g if the subject line had words like medicine, prize, lottery etc we could with a fair degree of probability classify this as spam.

However some classification problems could be far more complex.  We may need to classify another problem as shown below. From the above it can be seen that hypothesis function is second order equation which is either a circle or an ellipse.

In the case of logistic regression the hypothesis function should be able to switch between 2 values 0 or 1 almost like a transistor either being in cutoff or in saturation state.

In the case of logistic regression 0 <= hƟ <= 1

The hypothesis function uses function of the following form

g(z) = 1/(1 + e‑z)

and hƟ (x) = g(ƟTX) The function g(z) shown above has the characteristic required for logistic regression as it has the following shape

The function rapidly asymptotes at 1 when hƟ (x) >= 0.5 and  hƟ (x) asymptotes to 0 when hƟ (x) < 0.5

As in linear regression we can have hypothesis function be of an appropriate order. So for e.g. in the ellipse figure above one could choose a hypothesis function as follows

hƟ (x) = Ɵ0 + Ɵ1x12 + Ɵ2x22 + Ɵ3x1 +  Ɵ4x2

or

hƟ (x) = 1/(1 + e –(Ɵ0 + Ɵ1×12 + Ɵ2×22 + Ɵ3×1 +  Ɵ4×2))

We could choose the general form of a circle which is

f(x) = ax2 + by2 +2gx + 2hy + d

The cost function for logistic regression is given below

Cost(hƟ (x),y) = { -log(hƟ (x))             if y = 1

-log(1 – hƟ (x)))       if y = 0

In the case of regression there was a single cost function which could determine the error of the data against the predicted value.

The cost in the event of logistic regression is given as above as a set of 2 equations one for the case where the data is 1 and another for the case where the data is 0.

The reason for this is as follows. If we consider y =1 as a positive value, then when our hypothesis correctly predicts 1 then we have a ‘true positive’ however if we predict 0 when it should be 1 then we have a false negative. Similarly when the data is 0 and we predict a 1 then this is the case of a false positive and if we correctly predict 0 when it is 0 it is true negative.

Here is the reason as how the cost function

Cost(hƟ (x),y) = { -log(hƟ (x))             if y = 1

-log(1 – hƟ (x)))       if y = 0

Was arrived at. By definition the cost function gives the error between the predicted value and the data value.

The logic for determining the appropriate function is as follows

For y = 1

y=1 & hypothesis = 1 then cost = 0

y= 1 & hypothesis = 0 then cost = Infinity

Similarly for y = 0

y = 0 & hypotheses  = 0 then cost = 0

y = 0 & hypothesis = 1 then cost = Infinity

and the the functions above serve exactly this purpose as can be seen Hence the cost can be written as

J(Ɵ) = Cost(hƟ (x),y) = -y * log(hƟ (x))  – (1-y) * (log(1 – hƟ (x))

This is the same as the equation above

The same gradient descent algorithm can now be used to minimize the cost function

So we can iterate througj

Ɵj =   Ɵj – α δ/δ Ɵj J(Ɵ0, Ɵ1,… Ɵn)

This works out to a function that is similar to linear regression

Ɵj = Ɵj – α 1/m { Σ hƟ (xi) – yi} xj i

This will enable the machine to fairly accurately determine the parameters Ɵj for the features x and provide the hypothesis function.

This is based on the Coursera course on Machine Learning by Professor Andrew Ng. Highly recommended!!!

Simplifying Machine Learning algorithms – Part 1

Machine learning or the ability to use computers to predict values, classify data or identify patterns is truly a fascinating field. It is amazing how algorithms can come to conclusions on data. Detecting patterns is a inborn ability of the human mind. But our mind cannot handle large quantities of data with many features. It is here that machines have an edge over us.

This post is inspired by the Machine Learning course at Coursera conducted by Professor Andrew Ng of Stanford. The lectures are truly lucid and delivered with amazing clarity. In a series of post I will be trying to distil the meaning and motivation behind the algorithms that are part of machine learning.

There are 2 major types of learning

a)      Supervised learning b) Unsupervised learning

Supervised learning: In supervised learning we have to infer the relationship between input data and output values. The intention of supervised learning is determine the possible out for some random input once the relationship has been determined. Some examples of supervised learning are linear regression, logistic regression etc.

Unsupervised learning: In unsupervised learning the problem is to determine patterns and structure in unlabeled data. Some examples of unsupervised learning are K-Means clustering, hidden Markov models etc.

In this post I would like to take a look at Supervised Learning algorithms

Linear Regression

In regression problems we try to infer the relationship between a set of input parameters to an output value. Let us we have data for the number of rooms vs. price of the house as shown below Depending on the data we could either fit a straight line or use a linear fit. Alternatively we could fit a higher order curve to data.

The function that determines the relationship is also known as hypothesis function. This can be represented as follows for e.g a hypothesis function with a single feature

hƟ(x) = Ɵ1x+ Ɵ0

The above equation is the hypothesis function where Ɵ is the parameter and x is the feature

We could have a higher order hypothesis function as follows

hƟ(x) = Ɵ2x2+ Ɵ1x+Ɵ0

To evaluate whether the hypothesis function is able to map the input and related output accurately is known as the ‘cost function’.

The cost function can be represented as

J(Ɵ) = 1/2m Σ(hƟ (xi)  – y i)2

The cost function really calculates the ‘mean squared error’ of the actual data points (y) with the points on the hypothesis function (hƟ). Clearly higher the value of J(Ɵ) the greater is the error in predicting the output based on a set of input parameters. If we just took the error instead of the squared error then if there were data points on either side of the predicted line then the positive & negative errors could cancel out. Hence the approach is usually to take the mean of the squared error. The goal would be to minimize the error which will result in the best fit.

So the approach would be to choose values for the parameters Ɵi

The algorithm that is used for determining the values of the parameters that will result in the minimum error is gradient descent

The formula is

Ɵj := Ɵj – αd/d Ɵj J(Ɵ)

Where α is the learning rate

Gradient descent starts by picking a random value for Ɵi. Then the algorithm looks around to search for the next combination that will take us down fastest. By continuing this process the local minima is determined.

Gradient descent is based on the observation that if the multivariable function  is defined and differentiable in a neighborhood of a point , then  decreases fastest if one goes from  in the direction of the negative gradient. This is shown in the below diagram taken from Wikipedia. For e.g for a curve as shown below This how I think the gradient descent works. In the above diagram at point A the slope is +ve and taking the negative of the slope multiplied by the learning factor α and subtracting it from Ɵj will result in a value that is less than Ɵj. That is we move towards the minima or C. Similarly at point B the slope will be -ve. If we multiply by  – α then we will add to Ɵj. Hence we will move to the right or towards point C.

By applying the iterative process of gradient descent we can get the combination of parameter values for  Ɵ that will provide the best fit for the set of data points

The iterative process of gradient descent is applied to minimize the cost function which is function of the error in the current hypothesis

δ/δ J(Ɵ) = δ/ δ Ɵ * 1/2m Σ(hƟ (xi)  – y i)2

This process is applied iteratively to the below equation to arrive at the values of Ɵi

The formula is

Ɵj := Ɵj – αd/d Ɵj J(Ɵ)

to obtain the values for the best fit equation

hƟ(x) = Ɵ2xn+ Ɵ1xn-1+ …+  Ɵ0

Also read my post on Simplifying ML: Logistic regression – Part 2