Polynomial Regression ( From Scratch using Python )
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.
# 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() :
X = np.array( [ [1], [2], [3], [4], [5], [6], [7] ] )
Y = np.array( [ 45000, 50000, 60000, 80000, 110000, 150000, 200000 ] )
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:
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 \cdot 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) = w_1 x_1 + w_2 x_2 + \cdots + w_n x_n 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.
# 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:
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.
