Polynomial Regression ( From Scratch using Python )

Prerequisites 

1. Linear Regression
2. Gradient Descent

Introduction

Linear Regression finds the correlation between the dependent variable ( or target variable ) and independent variables ( or features ). In short, it is a linear model to fit the data linearly. But it fails to fit and catch the pattern in non-linear data. 

Let’s first apply Linear Regression on non-linear data to understand the need for Polynomial Regression. The Linear Regression model used in this article is imported from sklearn. You can refer to the separate article for the implementation of the Linear Regression model from scratch.

Python3

# Importing libraries

import numpy as np

import pandas as pd

from sklearn.model_selection import train_test_split

import matplotlib.pyplot as plt

from sklearn.linear_model import LinearRegression

# driver code

def main() :

    # Create dataset

    X = np.array( [ [1], [2], [3], [4], [5], [6], [7] ] )

    Y = np.array( [ 45000, 50000, 60000, 80000, 110000, 150000, 200000 ] )

    # Model training

    model = LinearRegression()

    model.fit( X, Y )

    # Prediction

    Y_pred = model.predict( X )

    # Visualization

    plt.scatter( X, Y, color = 'blue' )

    plt.plot( X, Y_pred, color = 'orange' )

    plt.title( 'X vs Y' )

    plt.xlabel( 'X' )

    plt.ylabel( 'Y' )

    plt.show()


if __name__ == "__main__" :

    main()

Output : 

Visualization

As shown in the output visualization, Linear Regression even failed to fit the training data well ( or failed to decode the pattern in the Y with respect to X ). Because its hypothetical function is linear in nature and Y is a non-linear function of X in the data. 

For univariate linear regression : 

h( x ) = w * x

here,  x is the feature vector.
and w is the weight vector.

 This problem is also called as underfitting. To overcome the underfitting, we introduce new features vectors just by adding power to the original feature vector.

For univariate polynomial regression : 
 
h( x ) = w1x + w2x2  + .... + wnxn 
 
here, w is the weight vector. 
where x2  is the derived feature from x. 

After transforming the original X into their higher degree terms, it will make our hypothetical function able to fit the non-linear data. Here is the implementation of the Polynomial Regression model from scratch and validation of the model on a dummy dataset.

Python

# Importing libraries

import numpy as np

import math

import matplotlib.pyplot as plt

# Univariate Polynomial Regression

class PolynomailRegression() :

    def __init__( self, degree, learning_rate, iterations ) :

        self.degree = degree

        self.learning_rate = learning_rate

        self.iterations = iterations

    # function to transform X

    def transform( self, X ) :

        # initialize X_transform

        X_transform = np.ones( ( self.m, 1 ) )

        j = 0

        for j in range( self.degree + 1 ) :

            if j != 0 :

                x_pow = np.power( X, j )

                # append x_pow to X_transform

                X_transform = np.append( X_transform, x_pow.reshape( -1, 1 ), axis = 1 )

        return X_transform

    # function to normalize X_transform

    def normalize( self, X ) :

        X[:, 1:] = ( X[:, 1:] - np.mean( X[:, 1:], axis = 0 ) ) / np.std( X[:, 1:], axis = 0 )

        return X

    # model training

    def fit( self, X, Y ) :

        self.X = X

        self.Y = Y

        self.m, self.n = self.X.shape

        # weight initialization

        self.W = np.zeros( self.degree + 1 )

        # transform X for polynomial  h( x ) = w0 * x^0 + w1 * x^1 + w2 * x^2 + ........+ wn * x^n

        X_transform = self.transform( self.X )

        # normalize X_transform

        X_normalize = self.normalize( X_transform )

        # gradient descent learning

        for i in range( self.iterations ) :

            h = self.predict( self.X )

            error = h - self.Y

            # update weights

            self.W = self.W - self.learning_rate * ( 1 / self.m ) * np.dot( X_normalize.T, error )

        return self

    # predict

    def predict( self, X ) :

        # transform X for polynomial  h( x ) = w0 * x^0 + w1 * x^1 + w2 * x^2 + ........+ wn * x^n

        X_transform = self.transform( X )

        X_normalize = self.normalize( X_transform )

        return np.dot( X_transform, self.W )


# Driver code

def main() :

    # Create dataset

    X = np.array( [ [1], [2], [3], [4], [5], [6], [7] ] )

    Y = np.array( [ 45000, 50000, 60000, 80000, 110000, 150000, 200000 ] )

    # model training

    model = PolynomailRegression( degree = 2, learning_rate = 0.01, iterations = 500 )

    model.fit( X, Y )

    # Prediction on training set

    Y_pred = model.predict( X )

    # Visualization

    plt.scatter( X, Y, color = 'blue' )

    plt.plot( X, Y_pred, color = 'orange' )

    plt.title( 'X vs Y' )

    plt.xlabel( 'X' )

    plt.ylabel( 'Y' )

    plt.show()


if __name__ == "__main__" :

    main()

Output : 

Visualization

We also normalized the X before feeding into the model just to avoid gradient vanishing and exploding problems.

Output visualization showed Polynomial Regression fit the non-linear data by generating a curve.