Linear Regression from Scratch

Hello World, This is Saumya, and I am here to help you understand and implement Linear Regression from scratch without any libraries. Here, I will implement this code in Python, but you can implement the algorithm in any other programming language of your choice just by basically developing 4-5 simple functions.

So now, first of all, what exactly is Linear Regression?

As discussed in my blog on Machine Learning, Linear Regression is used to identify linear relationships between the input features x_(i) and the output labels of the training set y_(i) and thus form a function F(x_(i),θ), which would help in predicting future values.

This function, is called hypothesis and is usually denoted by h(x⁽ⁱ⁾_,θ). Note that, x(lowercase) is used to denote a single training example as a whole, where as we use X _(i,j) is used to point the j^th feature for the i^th training example. But confusing?? Let's simplify it!!

As shown, to show the whole feature set for a single example, we use x₍₁₎. We can also point to the first feature in third example in training set as X_(3,1) or x₁⁽³⁾.

For simplicity of understanding, let's assume, there is only one feature in our dataset of 10 examples. Let's plot it on a graph.

So, let's say, the equation of the line we are supposed to fit the curve is h(x⁽ⁱ⁾_,θ)= θ₀ + x₁θ₁.

Or for simplicity, let's say we have one more feature x₀ which is always equal to 1 so that we can rewrite h_θ (x⁽ⁱ⁾)= θ₀ x₀+θ₁ x₁.

Or for multiple features, we can write it as h(x⁽ⁱ⁾_,θ)= θ₀ x₀+ θ₁ x₁+ θ₂ x₂+ θ₃x₃…… θ_n x_n for n number of features.

Therefore, in short, h_θ(x⁽ⁱ⁾)= _i=1ⁿ∑  (θ_i*x_i).

Now, the code for our hypothesis will be as follows:

def HypoThe(Theta,xi):
    if(len(Theta)==len(xi)):
        sum=0
        for i in range(len(xi)):
            sum+=(Theta[i]*xi[i])
        return sum
    else:
        return False

Now, since we have defined a mathematical model for prediction, the next problem we face will be finding the rest set of parameters Theta(θ).

To find Theta, let's break it into basic steps:

1. Assume a pair of Theta.

2. Compute the prediction error.

3. Assume again so that it minimizes the Prediction Error.

Sounds complicated? Well it's quite simple actually. Suppose we have selected some particular values for parameter θ. Now, we will devise a new cost function, which basically calculates the prediction error for θ.

J(θ)= _i=1^m∑( h_θ(x⁽ⁱ⁾)-y⁽ⁱ⁾)²/ (2*m)

What is means is that, we predict the output using the h_θ(x⁽ⁱ⁾) function, and subtract it from actual label y⁽ⁱ⁾, and then square it and add it for all training examples(1….m). We average it out using (2*m). Why did we use 2*m instead of m, we'll find it out soon. Let’s just say, it's for the simplicity, and also because we are dealing with a mean-squared computed error function which we are supposed to minimize.

Although, cost function is useful only for debugging purposes, and not in actual implementation, we'll still code it out.

def RegCostFunc(Theta,X,Y):
    sum1=0
    for i in range(len(X)):
        sum1+=((HypoThe(Theta, X[i])-Y[i])**2)
    J=sum1
    return J/(2*len(X))

Now, what we are supposed to do next is find those value for Theta for which J(θ) reaches it minimum. If you are familiar with calculus, what we're supposed to do is to find a the minima for the J(θ) function. To solve this optimization problem, we'll be using the gradient descent algorithm.

So what exactly does it do?

Let's write down the algorithm first.

repeat until convergence: {

for all values of θ­ simultaneously update,

 θ_j= θ_j - α * (Ə J(θ)/ Əθ_j)

}

Let's understand this equation first.

(Ə J(θ)/ Əθ_j) means that take partial derivative of J(θ) w.r.t θ_j while rest of the θ are constant.

Therefore

Ə J(θ)/ Əθ_j= Ə (  _i=1^m∑( h_θ(x⁽ⁱ⁾)-y⁽ⁱ⁾)²/ (2*m) )  / Əθ_j.

If you solve this using calculus, we find that,

Ə J(θ)/ Əθ_j=  _i=1^m∑( h_θ(x⁽ⁱ⁾)-y⁽ⁱ⁾)² * x_j⁽ⁱ⁾/ m

This basically means, while keeping the rest constant, decrease the value of θ_j in the direction of the slope of the function J(θ).

α is the learning rate by which, we reduce the value of the θ_j. If α is too large, the function will never converge actually.

Note that, do not update θ_j continuously, but update the whole θ simultaneously.

i.e. after one complete loop.

Rewriting the Algorithm, we get.

repeat until convergence: {

for all values of θ­ simultaneously update,

 θ_j= θ_j - α * ( _i=1^m∑( h_θ(x⁽ⁱ⁾)-y⁽ⁱ⁾)² * x_j⁽ⁱ⁾/ m )

}

Let's code this now,

def GradTerm(X,Y,Theta,i):
    sum1=0
    for j in range(len(X)):
        sum1+=((HypoThe(Theta,X[j])-Y[j])*X[j][i])
    return sum1

Note that, here, I have divided the Gradient Descent Algorithm from the Gradient Term above, for the simplicity of the understanding of our code.

def GradDesc(Theta,alpha,Xfeature,Ylabels):
    Theta_=[]
    for i in range(0,len(Theta)):
        Theta_.append(Theta(i)-(alpha* GradTerm(Xfeature,Ylabels,Theta,i)/len(Xfeature)))
    return Theta_

Now we are done with everything required. So how to Run all these functions for proper functioning?

Basically, here is how we'll proceed:

1. We'll call linear regression and provide it with a Features and Labels.

2. We'll add feature x₀ = 1 to every example in data set.

3. We'll initialize Theta=[0]*len(Features[0]). (i.e.  number of parameters should be equal to number of features )

4. We'll select a particular value of alpha and no. of  iterations, and run gradient descent algorithm to minimize the parameters θ.

Make sure to keep alpha as small as possible in the order of 0.000001 or less and the number of iterations to be around 100,000-1,000,000 for better results.

def LinearRegression(Xfeature,Ylabels,alpha,iterations):
    if len(Xfeature)!=len(Ylabels):
        print("Missing Data");
        return False
    else:   
        for i in range(len(Xfeature)):
            Xfeature[i].insert(0,1)
        Theta=[0]*len(Xfeature[0])
        for i in range(iterations):
            print("\nIteration Number ",i)
            print(Theta)
            Theta=GradDesc(Theta, alpha, Xfeature, Ylabels)
        print(Theta)
        return Theta            

Now, I know you can easily implement it using scikit-learn library, but then, it's just a black-box module for machine learning. You'll never understand what is happening inside of it. However, if you have studied machine learning in detail, once, you'll have better understanding in using the sci-kit library and tweak the values accordingly for better results.

Below is the Complete code for this program. make sure to import math and sys for it. I am not a professional at coding, you might find some unwanted piece of lines in the code, which you can optimize yourself though.

That's it from this blog, if there are any suggestions, or corrections, feel free to mention in the comment section. Also if you have any doubts, feel free to ask.

References:-

- Machine Learning by Andrew Ng, Coursera.org (among the best MOOCs).