Linear Regression

Linear regression is finding the linear relationship between two variables (x,y). Where x is the input variable and y is the result output and follows the equation y = mx + b where m is the gradient and b is the intercept. To summarise, we have a training data which shows the relationship between the X and Y values, and from the training set we will draw the relationship and then use this to predict Y values for unknown X values. I will use gradient descent algorithm instead of the regression formula as it has a better optimization.

The gradient descent algorithm is finding the minimum of a diffrentiable function. We apply the gradient descent algorithm by calculaing the partial derivative of the slope and the intercept and then update the slope and intercept in the gradient descent formula after each iteration.

From highschool we have learned that to find the minima we need to calculate its derivative. And that is exactly what we are doing here We are finding the minima of the intercept and slope so the loss function that we are using here which is mean squared error function is (ideally) 0.

The loss equation looks like this:

while the derivative in respect to m and c is

Ideally we would split 80% of the data to train the function i.e use the algorithim on this data and use the rest 20% of the data to test it, however, for this program i am using all the data to train the function.

NOTE : We square the equation because we are trying to find the distance between the actual,correct point and the line that we have drawn to fit the points.

``````import numpy as np
import pandas as pd #a library that allows us to read CSV files and other table-like data structures
import matplotlib.pyplot as plt
``````
``````plt.rcParams['figure.figsize'] = (12.0, 9.0)

# Preprocessing Input data
data
``````

``````X = data.iloc[:, 1]#gets all the values in the 2nd column
Y = data.iloc[:, 8]#gets all the values in the 8th column
plt.scatter(X, Y)
plt.xlabel('GRE score')
plt.ylabel('Chance of getting into university %')
plt.show()
``````

``````m = 0
c = 0

L = 0.0000001  #small learning rate
i = 100

n = float(len(X))

for i in range(i):
Y_pred = m*X + c  # The current predicted value of Y
D_m = (-2/n) * sum(X * (Y - Y_pred))  # Derivative wrt m
D_c = (-2/n) * sum(Y - Y_pred)  # Derivative wrt c
m = m - L * D_m  # Update m
c = c - L * D_c  # Update c

``````

For this program we have just guessed the learning rate and havent normalized it so the line will not accurately predict and have best fit line

``````Y_pred = m * X + c
y_new = m * 338 + c
y_test1 = m * 292 + c
#here we are testing with two values and plotting the output as a red dot

plt.plot(338,y_new,'ro') #red dot
plt.plot(292,y_test1,'ro')

plt.scatter(X, Y)
plt.plot([min(X), max(X)], [min(Y_pred), max(Y_pred)], color='red')  # regression line
plt.show()
``````