Simple Linear Regression Using California Housing Dataset (using scikit-learn)

 

Experiment 

Simple Linear Regression Using California Housing Dataset (scikit-learn)

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Aim

To implement Simple Linear Regression using one feature (AveRooms) from the California Housing dataset  to predict house prices using the top 150 records.

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Objectives

• Load dataset from a scikit-learn

• Select a single feature(AveRooms) and target variable ( MedHouseVal)

• Build a regression model using scikit-learn

• Compute MSE and R² using buitin functions

• Visualize regression line

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🛠️ Tools Required

• Python

• scikit-learn

• NumPy

• Pandas

• Matplotlib

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📖 Theory

🔹 Simple Linear Regression

$$y = b_0 + b_1 x$$

Where:

• $x$: Independent variable (AveRooms)

• $y$: Dependent variable (MedHouseVal)

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🔹 Regression Coefficients

$$b_1 = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2}$$$$b_0 = \bar{y} - b_1 \bar{x}$$

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🔹 Mean Squared Error (MSE)

$$MSE = \frac{1}{n} \sum (y_i - \hat{y}_i)^2$$

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🔹 R-squared (R²)

$$R^2 = 1 - \frac{SS_{res}}{SS_{tot}}$$

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📋 Procedure

1. Load dataset from scikit-learn

2. Select:

   • Feature: AveRooms

   • Target: MedHouseVal

3. Use first 150 rows

4. Compute regression coefficients

5. Predict values

6. Compute MSE and R²

7. Plot regression line

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💻 Program

import numpy as np

import matplotlib.pyplot as plt

from sklearn.datasets import fetch_california_housing

from sklearn.linear_model import LinearRegression

from sklearn.metrics import mean_squared_error, r2_score

# -----------------------------

# Load dataset directly

# -----------------------------

housing = fetch_california_housing()

X = housing.data[:, housing.feature_names.index('AveRooms')].reshape(-1, 1)

y = housing.target.reshape(-1, 1)

# Take first 150 samples (same as your original)

X = X[:150]

y = y[:150]

# -----------------------------

# Visualization (Scatter Plot)

# -----------------------------

plt.scatter(X, y, s=20)

# -----------------------------

# Train model

# -----------------------------

model = LinearRegression()

model.fit(X, y)

# -----------------------------

# Coefficients

# -----------------------------

b1 = model.coef_[0][0]

b0 = model.intercept_[0]

print("Slope (b1):", b1)

print("Intercept (b0):", b0)

# -----------------------------

# Predictions

# -----------------------------

y_pred = model.predict(X)

# -----------------------------

# Metrics

# -----------------------------

mse = mean_squared_error(y, y_pred)

r2 = r2_score(y, y_pred)

print("MSE:", mse)

print("R²:", r2)

# -----------------------------

# Plot regression line

# -----------------------------

plt.plot(X, y_pred)

plt.xlabel("Average Rooms")

plt.ylabel("Median House Value")

plt.title("Simple Linear Regression (150 Samples)")

plt.show()

 Output

• Scatter plot of data

• Regression line

• Computed values:

• Slope (b₁)

• Intercept (b₀)

• MSE

• R²

Slope (b1): 0.3072787169173846 Intercept (b0): 0.5097916053072971 MSE: 0.8823553195768612 R²: 0.17260066984923983

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Result

The linear regression model was successfully implemented using the AveRooms feature.
Model performance was evaluated using MSE and R².

• The model establishes a linear relationship between AveRooms and MedHouseVal

• Performance depends on how strongly the feature influences the target

• Using only one feature may limit accuracy