Prerequisites
- Linear Regression
- 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 librariesimport numpy as npimport pandas as pdfrom sklearn.model_selection import train_test_splitimport matplotlib.pyplot as pltfrom sklearn.linear_model import LinearRegression# driver codedef 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 librariesimport numpy as npimport mathimport matplotlib.pyplot as plt# Univariate Polynomial Regressionclass 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.
