Logistic Regression – Let’s Classify Things..!!

In my post on Categorising Deep Seas of ML, I introduced you to problems of Classification (a subcategory of Supervised Learning).

But wait..we are talking Logistic “Regression”. Blame history for it, but the only thing common between Logistic Regression and Regression is the word itself.

Logistic Regression Intuition

Consider a problem where you have to find the probability of a student being studious one or goofy one.
What can we do?
Data..key to every solution..yeah

So, we grade our test students on certain basis. Lets just say we rate them out of 10 on following points – study hrs/day(x1), attention in class(x2), interaction in class(x3), behaviour with peers(x4)..well for sake of simplicity lets just take these 4 features.

Representing this in a matrix

feature_matrix

Now, you might be thinking – Hey, I have seen probability in my mathematics class and I know it always lies in range [0, 1].

Hold your horse my friend. We are getting to that very step.

Activation Functions

The world of ML has taken many inputs from the field of Mathematics and perhaps this very part of activation function is taken entirely from the latter field.

A function used to transform the activation level of a unit(neuron) into an output signal is called an Activation Function.

Well, we’re going a little off from our topic here. But you can think of activation function as a function which provides us with the probability of our test being positive. In this case, it gives us the probability of a student being studious.

The function we’ll be using here is the Sigmoid function.

sigmoid

Lets just have a look at the graph of the function.

sigmoid_graph

‘z’ on x-axis vs. sigmoid(z) on y-axis

The graph clearly depicts that for any value of z, sigmoid function will return us a value between 0 and 1….MIND == BLOWN.. (“==” because for Programmers “=” != “==”).

So what’s left?
The only thing left for us to do is to define a mapping from our test data to z in sigmoid(z) and “minimize the error” in the mapping to get the best result.

“Minimize the error”..hmm..we have done something similar in Linear Regression too. Gradient Descent is the key.

So what’s our hypothesis? It will be nothing but

hypothesis

We’ll calculate these coefficients by minimizing our cost function.

The very basic step of Gradient descent is to find a Cost Function. I know all these functions are getting on your nerve so lets just depict these by using a flow chart. Stick with me and we’ll make it easy.

flow_chart_logistic

Flow Chart Depicting Logistic Regression.

Our Cost function here will be :

cost_function

Don’t worry you don’t have to memorise it. But lets just understand how this Cost function is implemented. Consider for our test data when y = 1 then our cost function is -log(h(theta)). The graph for the same is:

negative_log(y)

h(theta) on x-axis vs. -log(h(theta)) on y-axis

This shows that as the value of calculated hypothesis goes from 0 to 1(required value) our cost function decreases.

Now, consider when y = 0 then our cost function is -log(1-h(theta)). The graph for same is:

negative_log(1-y)

h(theta) on x-axis vs. -log(1-h(theta)) on y-axis

This shows that as the value of the calculated hypothesis goes from 1 to 0(required value) our cost function decreases. Pretty much what required.

Now, as we are familiar with our cost function lets just remember how Gradient Descent works.

batch_gradient_descent

Simultaneously,  update for every Theta. Alpha being the Learning Rate.

Hmmmm..partial differentiation and our apparent hide and seek with it..Let me make your task easy.

cost_func_differentiation

So, now we know how to get our coefficients tuned and how to run our gradient descent.

What about making predictions?
Well, that’s easy! A student with higher probability of being a studious one is of course more studious. But how will I compute it?. Deciding a threshold is upto you. For me.. a student with a probability greater than or equal to 0.5 works just fine. I am a little lenient I know. 😉

So, now you have it. Every tiny detail of logistic regression.

Now I’ve a task for you. I’ll be providing you with a dataset and you have to apply logistic regression on your own. No worries though, my next post will explain my way of logistic regression on the same dataset.

Explanation of dataset: The provided dataset contains 4 columns, namely – ‘admit’, ‘rank’, ‘gpa’ and ‘gre’. When given the ‘rank’ of the college then the ‘admit’ shows whether the person is provided with the admit to the college(1) or not(0) provided he has a corresponding ‘gpa’ and ‘gre’ scores. Your task is to find a mapping from ‘rank’, ‘gre’ and ‘gpa’ to ‘admit’ so as to find whether a person will be admitted to college or not.

Linear Regression – Giving Best Fit Line

In my last post, I gave you an intuition of types of Machine Learning problems. Here, we’ll be undertaking the problem of regression(or real values fitting) which is a sub-category of Supervised Machine Learning.

But first things first!
We’ll be using Python for implementation along with its open source libraries like – NumPy, Pandas, Matplotlib and Scikit-learn. So, knowledge of basic python syntax is a must. At the end of the post, there is a list of all the resources to help you out.

So, here comes the time of a deep dive…!

What is Linear Regression?

Linear Regression is the problem of fitting the best possible line to the data. Woooaahh!!

Let us go through it step by step…

Why..?
Because we (Programmers) break down our work into smaller, simpler pieces
Step 1: Collect data
Step 2: Find a mapping from input data to labels, which gives minimum error… Magic of Mathematics!!
Step 3: Use this mapping to find label to new or unseen data-sets.

Wait! Don’t leave I know its MATHS, but we’ll make it easy.

How to find the Line of Best Fit?

A line that will fit the data in the best possible manner will be the line that has the minimum error. So, here is our objective – to minimize the error between the Actual Labels and Calculated Labels. And in order to accomplish it, we’ll be using one of the widely used algorithms in the world of ML – Gradient Descent.

So lets just dive right into the coding part:

Step 1: Collecting Data

For this example, we’ll be using NumPy to read data from an excel file – “data.csv”. The file contains data comparing the amount of hours a student studied and how much he or she scored.


def run():
 #Step 1 - Collect the data
 points = np.genfromtxt('data.csv', delimiter = ',')

Step 2: Finding The mapping

Our goal is to find a mapping between the hours studied and marks obtained, i.e.

y = mx + b

where,
x = amount of hours studied
y = marks obtained
m = slope of the line
b = y intercept

Well, this step has further sub-steps..

Step 2.1: Computing Error

To Compute Error at each iteration, we’ll be using the cost function(or objective function):

Slide1


def compute_error_for_line_given_points(b, m, points):
 total_error = 0
 for ip in range(len(points)):
  x = points[ip, 0]
  y = points[ip, 1]
  total_error += (y - (m * x + b )) ** 2
 return total_error / float(len(points))

Looking at the mapping, it is clear that the only possible parts we can play around with to reduce the error are- m(the slope of the line) and b(the y intercept).

Step 2.2: Setting Hyper-parameters

In ML, we have certain parameters to tune in our model over the data we are analysing. These parameters are known as Hyper-parameters and they define certain terms that make sure our function(or model) fits data on a particular rate and in a particular way. We have a whole bunch of hyper-parameters with their own importance. More on this will be in a future post.

In our case, the Hyper-parameters that we must take into consideration are the Learning Rate, initial value of ‘b’, initial value of ‘m’ and number of iterations. Learning Rate defines how fast should our model converge. By converge, it means how fast model reaches the global optimum(in this case global minimum).


def run():
 #Step 1 - Collect the data
 points = np.genfromtxt('data.csv', delimiter = ',')

 #Step 2 - define hyperparameters
 learning_rate = 0.0001
 initial_b = 0
 initial_m = 0
 num_iterations = 1000

Well, then why isn’t Learning Rate big like a million?

The main reason being there must be a balance. If learning rate is too big, our error function might not converge(decrease), but if its too small, our error function might take too long to converge.

Step 2.3: Perform Gradient Descent

In this step, we will modify the value of y intercept ‘b’ and slope of line ‘m’ to decrease the value of the error (or objective function).

To do this, we will subtract the partial differentiation of our objective function with respect to b and m from former values of b and m respectively.


#Computes new values of b and m for a given iteration of gradient_descent_runner
def step_gradient(points, b_current, m_current, learning_rate):
 b_gradient = 0
 m_gradient = 0
 N = float(len(points))
 for i in range(len(points)):
  x = points[i, 0]
  y = points[i, 1]
  b_gradient += -(2/N) * (y - ((m_current * x) + b_current))
  m_gradient += -(2/N) * x * (y - ((m_current * x) + b_current))
 new_b = b_current - learning_rate * b_gradient
 new_m = m_current - learning_rate * m_gradient
 return [new_b, new_m]

#Finds the best fit line to the data.
def gradient_descent_runner(points, initial_b, initial_m, learning_rate, num_iterations):
 b = initial_b
 m = initial_m
 for i in range(num_iterations):
  b, m = step_gradient(points, b, m, learning_rate)
 return [b, m]

 

Step 3: Visualizing

Here, we will use Matplotlib – an open source library to plot graph of our mapping over the scatter plot of data.


def run():
 #Step 1 - Collect the data
 points = np.genfromtxt('data.csv', delimiter = ',')

 #Step 2 - define hyperparameters
 learning_rate = 0.0001
 initial_b = 0
 initial_m = 0
 num_iterations = 1000

 #Step 3 - train our model
 print('Starting gradient descent at b = {0}, m = {1}, error = {2}'
 .format(initial_b, initial_m, compute_error_for_line_given_points(initial_b, initial_m, points)))
 b, m = gradient_descent_runner(points, initial_b, initial_m, learning_rate, num_iterations)

 print('Ending gradient descent at Iteration = {0} b = {1}, m = {2}, error = {3}'
 .format(num_iterations, b, m, compute_error_for_line_given_points(b, m, points)))

 #Step 4 - Visualization
 x = [ix[0] for ix in points]
 y = [iy[1] for iy in points]
 y_predict = [m * ix + b for ix in x]

 plt.figure(0)
 plt.scatter(x, y)
 plt.plot(x, y_predict)
 plt.show()

 

output

Output : Graph Displaying the Best Fit line over the Data

To see a visual comparison between decrease in error and line fitting – go to link.

GitHub link to Source Code

Resources for further reading:

Python tutorial
NumPy tutorial
Matplotlib tutorial

PS: To get a good grip on the concept of Linear Regression visit this link.

Categorising Deep Seas of Machine Learning…

In my last post, I gave you a teaser to what “Machine Learning” is. Taking things up a notch, let us dive into the deep seas of Machine Learning(ML).

Deep ML seas(don’t confuse with Deep Learning) have variations, curves, biasness, data and its diversity. There are different types of ML algorithms to tackle these stuffs.

Broadly, these ML algorithms can be divided into 3 categories:

Supervised Learning – To tackle the data which have a known answer.
Unsupervised Learning – To tackle the data which have no known answer.
Reinforcement Learning – To tackle the data based on final outcome.

Supervised Learning

The game of the labeled data resides in this colony of ML..

The word labeled data signifies that for a given values of a set of input variables(x) we will have a given value of output variable(y; labels). The algorithms belonging to this category generally gives you a mapping

y = f(x)

which gives the best possible relation between input and output variables.

Think of the algorithm being a student supervised by a teacher. Teacher, being aware of the correct answers, corrects his student(algorithm) on each iteration till the student(algorithm) gives an acceptable level of performance.

Going further down the lane, we can divide supervised learning problems in two sub-categories:

Classification: A problem where output variable is a category or a class, such as “Cancer positive” or “Cancer negative”.
Regression: A problem where output variable is a real value, such as “rupees” or “weight”.

We’ll further look into these deep corners of supervised learning in upcoming posts.

Unsupervised Learning

Have cluttered, unlabeled data – we’ve you covered..!!

Unsupervised Learning problems are where you have input data but no corresponding output.

The algorithms covering this domain have a goal to model the underlying data into meaningful structures in order to learn more about the data gathered.

Consider these algorithms as those innovative students who don’t have a supervisor(teacher) to guide them. They are left on their own with the data to fiddle with it and give it a structure as they like. “…Mischievous!!”

Unsupervised Learning problems can further be divided into two sub-categories:

Clustering: A problem to find the inherent groupings in the data, such as grouping customers by purchasing behavior.
Association: A problem to discover association between large portions of data, such as people that buy X tend to buy Y.

We’ll cover these sub-categories in the future posts, tackling one at a time.

Reinforcement Learning

Greed for reward is the key..!!

In Reinforcement Learning problems, a software agent adapts in an environment so as to maximise its rewards.

To understand it, consider teaching a dog a new trick – there is no way to tell a dog “what to do!”, but you can reward or punish it depending on it being right or wrong.

These algorithms can, in a similar manner, train computers to perform complex tasks, such as playing chess. In Reinforcement Learning problems, if the modelling of problem is well handled, some algorithms can even converge to global optimum (that is, to the ideal behavior that maximises reward).

While we are talking of Greed..let me give you a greedy motivation to give your best in this field and keep going.

A breakthrough in machine learning would be worth ten Microsoft’s

-Bill Gates