Linear Regression Algorithm From Scratch In Python
What is Linear Regression?
A linear regression is one of the easiest statistical models in machine learning. It is used to show the linear relationship between a dependent variable and one or more independent variables.
Before we drill down to linear regression in depth, let me just give you a quick overview of what is a regression as Linear Regression is one of a type of Regression algorithm
What is Regression?
Regression analysis is a form of predictive modeling technique which investigates the relationship between a dependent and independent variable
Types of Regression
- Linear Regression
- Logistic Regression
- Polynomial Regression
- Stepwise Regression
Linear Regression vs Logistic Regression
Where is Linear Regression Used?
1. Evaluating Trends and Sales Estimates
Linear regressions can be used in business to evaluate trends and make estimates or forecasts.
For example, if a company’s sales have increased steadily every month for the past few years, conducting a linear analysis on the sales data with monthly sales on the y-axis and time on the x-axis would produce a line that that depicts the upward trend in sales. After creating the trend line, the company could use the slope of the line to forecast sales in future months.
2. Analyzing the Impact of Price Changes
Linear regression can also be used to analyze the effect of pricing on consumer behavior.
For example, if a company changes the price on a certain product several times, it can record the quantity it sells for each price level and then performs a linear regression with quantity sold as the dependent variable and price as the explanatory variable. The result would be a line that depicts the extent to which consumers reduce their consumption of the product as prices increase, which could help guide future pricing decisions.
3. Assessing Risk
Linear regression can be used to analyze risk.
For example
A health insurance company might conduct a linear regression plotting number of claims per customer against age and discover that older customers tend to make more health insurance claims. The results of such an analysis might guide important business decisions made to account for risk.
How do Linear Regression Algorithm works?
Least Square Method — Finding the best fit line
Least squares is a statistical method used to determine the best fit line or the regression line by minimizing the sum of squares created by a mathematical function. The “square” here refers to squaring the distance between a data point and the regression line. The line with the minimum value of the sum of square is the best-fit regression line.
Regression Line, y = mx+c where,
y = Dependent Variable
x= Independent Variable ; c = y-Intercept
Least Square Method — Implementation using Python
For the implementation part, I will be using a dataset consisting of head size and brain weight of different people.
# Importing Necessary Libraries
%matplotlib inline
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
plt.rcParams['figure.figsize'] = (20.0, 10.0)
# Reading Data
data = pd.read_csv('headbrain.csv')
print(data.shape)
data.head()
# Collecting X and Y
X = data['Head Size(cm^3)'].values
Y = data['Brain Weight(grams)'].valuesIn order to find the value of m and c, you first need to calculate the mean of X and Y
# Mean X and Y
mean_x = np.mean(X)
mean_y = np.mean(Y)
# Total number of values
n = len(X)
# Using the formula to calculate m and c
numer = 0
denom = 0
for i in range(n):
numer += (X[i] - mean_x) * (Y[i] - mean_y)
denom += (X[i] - mean_x) ** 2
m = numer / denom
c = mean_y - (m * mean_x)
# Print coefficients
print(m, c)The value of m and c from above will be added to this equation
BrainWeight = c + m ∗ HeadSize
Plotting Linear Regression Line
Now that we have the equation of the line. So for each actual value of x, we will find the predicted values of y. Once we get the points we can plot them over and create the Linear Regression Line.
# Plotting Values and Regression Line
max_x = np.max(X) + 100
min_x = np.min(X) - 100
# Calculating line values x and y
x = np.linspace(min_x, max_x, 1000)
y = c + m * x
# Ploting Line
plt.plot(x, y, color='#52b920', label='Regression Line')
# Ploting Scatter Points
plt.scatter(X, Y, c='#ef4423', label='Scatter Plot')
plt.xlabel('Head Size in cm3')
plt.ylabel('Brain Weight in grams')
plt.legend()
plt.show()R Square Method — Goodness of Fit
R–squared value is the statistical measure to show how close the data are to the fitted regression line
y = actual value
y ̅ = mean value of y
yp = predicted value of y
R-squared does not indicate whether a regression model is adequate. You can have a low R-squared value for a good model, or high R-squared value for a model that does not fit the data!
#ss_t is the total sum of squares and ss_r is the total sum of squares of residuals(relate them to the formula).
ss_t = 0
ss_r = 0
for i in range(m):
y_pred = c + m * X[i]
ss_t += (Y[i] — mean_y) ** 2
ss_r += (Y[i] — y_pred) ** 2
r2 = 1 — (ss_r/ss_t)
print(r2)Linear Regression — Implementation using scikit learn
If you have reached up here, I assume now you have a good understanding of Linear Regression Algorithm using Least Square Method. Now its time that I tell you about how you can simplify things and implement the same model using a Machine Learning Library called scikit-learn
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
# Cannot use Rank 1 matrix in scikit learn
X = X.reshape((m, 1))
# Creating Model
reg = LinearRegression()
# Fitting training data
reg = reg.fit(X, Y)
# Y Prediction
Y_pred = reg.predict(X)
# Calculating R2 Score
r2_score = reg.score(X, Y)
print(r2_score)If you wish to check out more articles on the market’s most trending technologies like Artificial Intelligence, DevOps, Ethical Hacking, then you can refer to Edureka’s official site.
Do look out for other articles in this series which will explain the various other aspects of Python and Data Science.
2. Python Programming Language
6. Scikit Learn Machine Learning
11. PyGame Tutorial
12. OpenCV Tutorial
14. PyCharm Tutorial
17. Python Regex
18. Loops in Python
19. Python Projects
21. Arrays in Python
22. Sets in Python
24. Python Interview Questions
25. Java vs Python
26. How To Become A Python Developer?
29. What is Socket Programming in Python
30. Python Database Connection
31. Golang vs Python
Originally published at www.edureka.co on September 6, 2018.
