Multiple Linear Regression

Multiple Linear Regression is an extension of linear regression where we predict the value of a dependent variable (Y) based on multiple independent variables (X_1, X_2, X_3, ....). It models the relationship between the dependent variable and multiple predictors using the equation:

Y = b_0 + b_1X_1 + b_2X_2 + ... + b_nX_n + ϵ(epsilon)

Where:

• (Y): Dependent variable
• (X_1, X_2, ...., X_n): Independent variables
• (b_0): Intercept (value of (Y) when all (X_i = 0))
• (b_1, b_2, ...., b_n): Coefficients representing the effect of each (X_i) on (Y)
• ϵ: Error term (difference between actual and predicted (Y))

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Example: Predicting House Prices

Suppose we want to predict house prices based on three features: size of the house (X_1), number of bedrooms (X_2), and distance to the city center (X_3).

Data

House Size (sq ft) ((X_1))	Bedrooms ((X_2))	Distance to City (miles) ((X_3))	Price ((Y))
1500	3	5	400,000
2000	4	3	500,000
2500	4	2	600,000
3000	5	1	700,000

Regression Equation

Using multiple linear regression, we fit the following equation:

Y = b_0 + b_1.X_1 + b_2.X_2 + b_3.X_3

Assuming the computed coefficients are:

• (b_0 = 50,000) (intercept)
• (b_1 = 100) (per square foot)
• (b_2 = 10,000) (per bedroom)
• (b_3 = -20,000) (per mile from city)

The equation becomes:

Y = 50,000 + 100X_1 + 10,000X_2 - 20,000X_3

Prediction

For a house of size 2200 sq ft, 3 bedrooms, and 4 miles from the city:

Y = 50,000 + 100(2200) + 10,000(3) - 20,000(4)

Y = 50,000 + 220,000 + 30,000 - 80,000 = 220,000

The predicted price is $220,000.

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Advantages of Multiple Linear Regression

1. Captures Relationships: It identifies the influence of multiple factors on a dependent variable.
2. Predictive Power: It enhances prediction accuracy by incorporating multiple variables.
3. Quantifies Impact: Provides insight into the effect of each independent variable.

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Limitations

1. Multicollinearity: When predictors are highly correlated, it can distort coefficient estimates.
2. Overfitting: Too many predictors relative to the data size can reduce model generalizability.
3. Linearity Assumption: Assumes a linear relationship between predictors and the outcome.

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Multiple linear regression is a cornerstone of statistical modeling and machine learning, widely used in fields like economics, healthcare, and marketing for predictive analysis.

code to predict house prices

# Importing necessary libraries
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression

# Creating the dataset
data = {
    "House Size (sq ft)": [1500, 2000, 2500, 3000],
    "Bedrooms": [3, 4, 4, 5],
    "Distance to City (miles)": [5, 3, 2, 1],
    "Price": [400000, 500000, 600000, 700000]
}

# Convert to DataFrame
df = pd.DataFrame(data)

# Features (X) and Target (Y)
X = df[["House Size (sq ft)", "Bedrooms", "Distance to City (miles)"]]
Y = df["Price"]

# Linear regression model
model = LinearRegression()
model.fit(X, Y)

# Coefficients and Intercept
b0 = model.intercept_
coefficients = model.coef_

# Predict prices based on the model
df["Predicted Price"] = model.predict(X)

# Plot actual vs predicted prices
plt.figure(figsize=(8, 6))
plt.scatter(df["Price"], df["Predicted Price"], color='blue', label='Data points')
plt.plot(df["Price"], df["Price"], color='red', linestyle='--', label='Perfect Prediction Line')
plt.title("Actual vs Predicted Prices")
plt.xlabel("Actual Price")
plt.ylabel("Predicted Price")
plt.legend()
plt.grid(True)
plt.show()

# Display coefficients and intercept
b0, coefficients