Understanding Regression Algorithms in Machine Learning

Regression is a fundamental type of predictive modeling technique that estimates the relationships among variables. It’s widely used to understand and predict continuous outcomes.

Types of Regression Algorithms

1. Linear Regression
2. Polynomial Regression
3. Logistic Regression

1. Linear Regression

Linear regression is a simple approach to modeling the relationship between a dependent variable yy and one or more independent variables XX. The relationship is modeled using a linear equation:

y=β0+β1X1+β2X2+⋯+βnXn+ϵ

Where:

• β0\beta_0 is the intercept.
• β1,β2,…,βn\beta_1, \beta_2, \ldots, \beta_n are the coefficients.
• ϵ\epsilon is the error term.

Mathematical Explanation

To estimate the coefficients, we use the Ordinary Least Squares (OLS) method, which minimizes the sum of squared residuals:

min⁡∑i=1N(yi−(β0+β1xi1+⋯+βnxin))2\min \sum_{i=1}^{N} (y_i — (\beta_0 + \beta_1 x_{i1} + \cdots + \beta_n x_{in}))²

Python Code for Linear Regression

import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression

# Generate sample data
np.random.seed(0)
X = 2 * np.random.rand(100, 1)
y = 4 + 3 * X + np.random.randn(100, 1)

# Create and fit the model
model = LinearRegression()
model.fit(X, y)

# Predictions
X_new = np.array([[0], [2]])
y_predict = model.predict(X_new)

# Plot the results
plt.scatter(X, y)
plt.plot(X_new, y_predict, color='red', linewidth=2)
plt.xlabel('Independent variable')
plt.ylabel('Dependent variable')
plt.title('Linear Regression')
plt.show()

The graph generated by the above code illustrates the linear relationship between the dependent and independent variables, with the red line representing the regression line.

2. Polynomial Regression

Polynomial regression models the relationship as an nth degree polynomial. For example, a second-degree polynomial:

y=β0+β1X+β2X2+ϵy = \beta_0 + \beta_1 X + \beta_2 X² + \epsilon

3. Logistic Regression

Although named regression, logistic regression is used for classification problems. It models the probability that a given input belongs to a certain class.

Conclusion

Regression algorithms are powerful tools in predictive modeling. Understanding the mathematical foundations and implementing these algorithms in Python can provide valuable insights into the data and improve decision-making.