Linear Regression is a supervised learning algorithm which is both a statistical and a machine learning algorithm. It is used to predict the real-valued output y based on the given input value x. It depicts the relationship between the dependent variable y and the independent variables xi ( or features ). The hypothetical function used for prediction is represented by h( x ).
h( x ) = w * x + b
here, b is the bias.
x represents the feature vector
w represents the weight vector.
Linear regression with one variable is also called univariant linear regression. After initializing the weight vector, we can find the weight vector to best fit the model by ordinary least squares method or gradient descent learning.
Mathematical Intuition: The cost function (or loss function) is used to measure the performance of a machine learning model or quantifies the error between the expected values and the values predicted by our hypothetical function. The cost function for Linear Regression is represented by J.
Here, m is the total number of training examples in the dataset. y(i) represents the value of target variable for ith training example.
So, our objective is to minimize the cost function J (or improve the performance of our machine learning model). To do this, we have to find the weights at which J is minimum. One such algorithm which can be used to minimize any differentiable function is Gradient Descent. It is a first-order iterative optimizing algorithm that takes us to a minimum of a function.
Gradient descent:
Pseudo Code:
- Start with some w
- Keep changing w to reduce J( w ) until we hopefully end up at a minimum.
Algorithm:
repeat until convergence {
tmpi = wi - alpha * dwi
wi = tmpi
}
where alpha is the learning rate.
Implementation:
Dataset used in this implementation can be downloaded from link.
It has 2 columns — “YearsExperience” and “Salary” for 30 employees in a company. So in this, we will train a Linear Regression model to learn the correlation between the number of years of experience of each employee and their respective salary. Once the model is trained, we will be able to predict the salary of an employee on the basis of his years of experience.
Python3
# Importing libraries import numpy as np import pandas as pd from sklearn.model_selection import train_test_split import matplotlib.pyplot as plt # Linear Regression class LinearRegression() : def __init__( self, learning_rate, iterations ) : self.learning_rate = learning_rate self.iterations = iterations # Function for model training def fit( self, X, Y ) : # no_of_training_examples, no_of_features self.m, self.n = X.shape # weight initialization self.W = np.zeros( self.n ) self.b = 0 self.X = X self.Y = Y # gradient descent learning for i in range( self.iterations ) : self.update_weights() return self # Helper function to update weights in gradient descent def update_weights( self ) : Y_pred = self.predict( self.X ) # calculate gradients dW = - ( 2 * ( self.X.T ).dot( self.Y - Y_pred ) ) / self.m db = - 2 * np.sum( self.Y - Y_pred ) / self.m # update weights self.W = self.W - self.learning_rate * dW self.b = self.b - self.learning_rate * db return self # Hypothetical function h( x ) def predict( self, X ) : return X.dot( self.W ) + self.b # driver code def main() : # Importing dataset df = pd.read_csv( "salary_data.csv" ) X = df.iloc[:,:-1].values Y = df.iloc[:,1].values # Splitting dataset into train and test set X_train, X_test, Y_train, Y_test = train_test_split( X, Y, test_size = 1/3, random_state = 0 ) # Model training model = LinearRegression( iterations = 1000, learning_rate = 0.01 ) model.fit( X_train, Y_train ) # Prediction on test set Y_pred = model.predict( X_test ) print( "Predicted values ", np.round( Y_pred[:3], 2 ) ) print( "Real values ", Y_test[:3] ) print( "Trained W ", round( model.W[0], 2 ) ) print( "Trained b ", round( model.b, 2 ) ) # Visualization on test set plt.scatter( X_test, Y_test, color = 'blue' ) plt.plot( X_test, Y_pred, color = 'orange' ) plt.title( 'Salary vs Experience' ) plt.xlabel( 'Years of Experience' ) plt.ylabel( 'Salary' ) plt.show() if __name__ == "__main__" : main() |
Output:
Predicted values [ 40594.69 123305.18 65031.88] Real values [ 37731 122391 57081] Trained W 9398.92 Trained b 26496.31
Linear Regression Visualization
