Multiple Linear Regression: A Solved Numerical Example

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Scenario: Restaurant Revenue Prediction

The Objective: Predict a restaurant's weekly revenue (£'s) based on seating capacity and average customer rating.

Step 1: The Historical Data & Target Point

When we have more than one independent variable (feature), we use Multiple Linear Regression to find a hyper-plane of best fit. We are trying to predict the Weekly_Revenue_hundreds using the given features.

Data Point	Seating_Capacity	Avg_Rating	Weekly_Revenue_hundreds
P1	35	4	14
P2	40	3.8	15
P3	25	3.2	10
Target	45	4.1	?

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Step 2: Extract X and Y Matrices

First, we convert our tabular data into matrices. Notice how we add a column of 1s to the very beginning of the X Matrix. This acts as a placeholder for our Y-intercept $b_0$

Formula:

X = \begin{bmatrix} 1 & x_{11} & x_{12} \\ 1 & x_{21} & x_{22} \\ \dots \end{bmatrix}, \quad Y = \begin{bmatrix} y_1 \\ y_2 \\ \dots \end{bmatrix}

Matrix

X

(with 1s)

1354

1403.80

1253.20

Matrix

Y

Step 3: Calculate X-Transpose and Multiply

We flip the rows and columns of Matrix $X$ to create its Transpose ( $X^T$ ). Then, we multiply $X^T$ by the original $X$ matrix to create a square matrix.

Calculate Transpose (

X^T

)

X^T

111

354025

43.803.20

Multiply (

X^T \cdot X

)

(X^T \cdot X)

310011

1003450372

1137240.68

Step 4: Find Inverse and Multiply

Because there is no "division" in matrix math, we find the Inverse of our square matrix. Multiplying by an inverse is the mathematical equivalent of dividing!

Find Inverse

(X^T \cdot X)^{-1}

(X^T \cdot X)^{-1}

54.500.67-20.83

0.670.03-0.44

-20.83-0.449.72

Multiply Inverse by

X^T

(X^T \cdot X)^{-1} \cdot X^T

-5.502.004.50

-0.100.13-0.03

2.50-1.67-0.83

Step 5: Solve for B (Weights & Intercept)

Finally, we multiply our accumulated matrix by the original Matrix $Y$ . This yields the Beta Matrix ( $B$ ), which contains our intercept ( $b_0$ ) and the optimal weights ( $b_1, b_2, dots$ ) for our features.

Formula:

B = [(X^T \cdot X)^{-1} \cdot X^T] \cdot Y

Matrix

B

-2.00

0.27

1.67

Intercept ( $b_0$ ) : -2.00

Weight for $X_{1}$ ( $b_{1}$ ) : 0.27

Weight for $X_{2}$ ( $b_{2}$ ) : 1.67

Step 6: Final Equation & Prediction

We extract those weights and build our Multiple Linear Regression equation. Then we plug in our new Target Point to calculate the final prediction!

Formula:

Y = b_0 + b_1X_1 + b_2X_2 + \dots + b_nX_n

Calculating the Target Prediction

Line: Y = -2.0000 + (0.2667 * 45) + (1.6667 * 4.1)

Y = 16.83

Final Takeaway

Based on the interactions between all selected features, the model predicts an estimated result of 16.83.