Regression Analysis

Regression analysis is the cornerstone of predictive analytics, enabling us to model relationships between dependent and independent variables to predict continuous outcomes. In automotive applications, regression models power everything from pricing strategies to risk assessment.

Mathematical Foundation

Regression analysis seeks to find the optimal function that maps input features to continuous target values:

\boxed{\mathbf{Y = X\beta + \epsilon}}

Where:

$\mathbf{Y}$ is the $(n \times 1)$ vector of target values
$\mathbf{X}$ is the $(n \times p)$ feature matrix
$\boldsymbol{\beta}$ is the $(p \times 1)$ parameter vector
$\boldsymbol{\epsilon}$ is the $(n \times 1)$ error term with $E[\epsilon] = 0$

Linear Regression

Mathematical Formulation

For simple linear regression with one predictor:

y_i = \beta_0 + \beta_1 x_i + \epsilon_i

For multiple linear regression:

y_i = \beta_0 + \sum_{j=1}^{p} \beta_j x_{ij} + \epsilon_i

Parameter Estimation

The optimal parameters are found using the Ordinary Least Squares (OLS) method:

\boldsymbol{\hat{\beta}} = (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{Y}

The cost function being minimized is:

J(\boldsymbol{\beta}) = \frac{1}{2n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2

Automotive Example: Vehicle Price Prediction

Business Context: An auto dealership wants to predict used car prices based on vehicle characteristics.

Model Specification:

\text{Price} = \beta_0 + \beta_1 \times \text{Age} + \beta_2 \times \text{Mileage} + \beta_3 \times \text{Engine Size} + \epsilon

Sample Data Analysis:

Dataset: 10,000 used vehicle transactions
Target Variable: Sale price ( $Y$ )
Features: Vehicle age, mileage, engine size, brand rating

Mathematical Implementation:

\begin{bmatrix} \text{Price}_1 \\ \text{Price}_2 \\ \vdots \\ \text{Price}_n \end{bmatrix} = \begin{bmatrix} 1 & \text{Age}_1 & \text{Mileage}_1 & \text{EngineSize}_1 \\ 1 & \text{Age}_2 & \text{Mileage}_2 & \text{EngineSize}_2 \\ \vdots & \vdots & \vdots & \vdots \\ 1 & \text{Age}_n & \text{Mileage}_n & \text{EngineSize}_n \end{bmatrix} \begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3 \end{bmatrix} + \begin{bmatrix} \epsilon_1 \\ \epsilon_2 \\ \vdots \\ \epsilon_n \end{bmatrix}

Example Results:

$\hat{\beta}_0 = 45,000$ (base vehicle value)
$\hat{\beta}_1 = -2,500$ (depreciation per year)
$\hat{\beta}_2 = -0.15$ (price reduction per mile)
$\hat{\beta}_3 = 3,200$ (premium per liter of engine displacement)

Business Interpretation: A 3-year-old vehicle with 30,000 miles and 2.0L engine would be priced at:

\hat{y} = 45,000 + (-2,500)(3) + (-0.15)(30,000) + (3,200)(2.0) = \$39,400

Logistic Regression

Mathematical Formulation

Logistic regression predicts binary outcomes using the logistic function:

P(Y = 1|X) = \frac{1}{1 + e^{-(\beta_0 + \sum_{j=1}^{p} \beta_j x_j)}}

The logit transformation linearizes the relationship:

\log\left(\frac{P(Y=1|X)}{1-P(Y=1|X)}\right) = \beta_0 + \sum_{j=1}^{p} \beta_j x_j

Maximum Likelihood Estimation

Parameters are estimated by maximizing the likelihood function:

L(\boldsymbol{\beta}) = \prod_{i=1}^{n} P(Y=1|X_i)^{y_i} \times (1-P(Y=1|X_i))^{1-y_i}

The log-likelihood to be maximized:

\ell(\boldsymbol{\beta}) = \sum_{i=1}^{n} [y_i \log(p_i) + (1-y_i)\log(1-p_i)]

Automotive Example: Loan Default Prediction

Business Context: An auto finance company needs to assess the probability of loan default for potential borrowers.

Model Specification:

\log\left(\frac{P(\text{Default})}{1-P(\text{Default})}\right) = \beta_0 + \beta_1 \times \text{Credit Score} + \beta_2 \times \text{DTI Ratio} + \beta_3 \times \text{Loan Amount}

Sample Dataset:

Target Variable: Default (1) or No Default (0)
Features: Credit score, debt-to-income ratio, loan amount, employment history
Sample Size: 50,000 auto loans

Mathematical Results:

$\hat{\beta}_0 = 8.5$ (baseline log-odds)
$\hat{\beta}_1 = -0.012$ (credit score coefficient)
$\hat{\beta}_2 = 2.8$ (debt-to-income coefficient)
$\hat{\beta}_3 = 0.000015$ (loan amount coefficient)

Probability Calculation Example: For a borrower with credit score 720, DTI ratio 0.35, loan amount $25,000:

\text{Logit} = 8.5 + (-0.012)(720) + (2.8)(0.35) + (0.000015)(25,000) = 1.805

P(\text{Default}) = \frac{1}{1 + e^{-1.805}} = \frac{1}{1 + 0.165} = 0.858 = 85.8\%

Business Decision: High default probability suggests loan rejection or higher interest rate.

Polynomial Regression

Mathematical Foundation

Polynomial regression captures non-linear relationships by adding polynomial terms:

y = \beta_0 + \beta_1 x + \beta_2 x^2 + \beta_3 x^3 + \ldots + \beta_d x^d + \epsilon

For multiple features with interaction terms:

y = \beta_0 + \sum_{i=1}^{p} \beta_i x_i + \sum_{i=1}^{p} \sum_{j=i}^{p} \beta_{ij} x_i x_j + \sum_{i=1}^{p} \beta_{ii} x_i^2 + \epsilon

Automotive Example: Fuel Efficiency Modeling

Business Context: An automotive manufacturer wants to model fuel efficiency as a function of engine parameters.

Model Specification:

\text{MPG} = \beta_0 + \beta_1 \times \text{Engine Size} + \beta_2 \times \text{Engine Size}^2 + \beta_3 \times \text{Weight} + \beta_4 \times \text{Horsepower} + \epsilon

Mathematical Interpretation:

Linear term ( $\beta_1$ ): Base effect of engine size
Quadratic term ( $\beta_2$ ): Diminishing returns or accelerating effects
The parabolic relationship captures optimal engine size for fuel efficiency

Sample Results:

$\hat{\beta}_0 = 45.2$ (baseline MPG)
$\hat{\beta}_1 = -8.5$ (linear engine size effect)
$\hat{\beta}_2 = 1.2$ (quadratic engine size effect)
$\hat{\beta}_3 = -0.004$ (weight penalty)
$\hat{\beta}_4 = -0.02$ (horsepower penalty)

Optimal Engine Size Calculation: Taking the derivative and setting to zero:

\frac{\partial \text{MPG}}{\partial \text{Engine Size}} = -8.5 + 2(1.2) \times \text{Engine Size} = 0

\text{Optimal Engine Size} = \frac{8.5}{2.4} = 3.54 \text{ liters}

Regularized Regression

Ridge Regression (L2 Regularization)

Ridge regression adds a penalty term to prevent overfitting:

J(\boldsymbol{\beta}) = \frac{1}{2n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2 + \lambda \sum_{j=1}^{p} \beta_j^2

The closed-form solution:

\boldsymbol{\hat{\beta}_{ridge}} = (\mathbf{X}^T\mathbf{X} + \lambda\mathbf{I})^{-1}\mathbf{X}^T\mathbf{Y}

Lasso Regression (L1 Regularization)

Lasso regression performs feature selection through sparsity:

J(\boldsymbol{\beta}) = \frac{1}{2n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2 + \lambda \sum_{j=1}^{p} |\beta_j|

Automotive Example: Customer Lifetime Value Prediction

Business Context: An automotive marketing department has 200+ customer features and needs to predict customer lifetime value while avoiding overfitting.

Ridge Regression Application:

Features: Demographics, purchase history, service records, digital engagement
Challenge: High dimensionality with potential multicollinearity
Solution: Ridge regression with cross-validation to select optimal $\lambda$

Cross-Validation for Hyperparameter Selection:

\lambda^* = \arg\min_{\lambda} \frac{1}{K} \sum_{k=1}^{K} MSE_k(\lambda)

Where $MSE_k(\lambda)$ is the mean squared error on the k-th fold.

Business Impact:

Feature Stability: Ridge regression provides stable coefficients
Generalization: Better performance on new customer data
Interpretability: Regularization highlights most important customer characteristics

Model Evaluation Metrics

Regression Metrics

Mean Squared Error (MSE):

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

Root Mean Squared Error (RMSE):

RMSE = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2}

Mean Absolute Error (MAE):

MAE = \frac{1}{n} \sum_{i=1}^{n} |y_i - \hat{y}_i|

R-squared (Coefficient of Determination):

R^2 = 1 - \frac{SS_{res}}{SS_{tot}} = 1 - \frac{\sum_{i=1}^{n} (y_i - \hat{y}_i)^2}{\sum_{i=1}^{n} (y_i - \bar{y})^2}

Classification Metrics

Accuracy:

\text{Accuracy} = \frac{TP + TN}{TP + TN + FP + FN}

Precision:

\text{Precision} = \frac{TP}{TP + FP}

Recall (Sensitivity):

\text{Recall} = \frac{TP}{TP + FN}

F1-Score:

F_1 = \frac{2 \times \text{Precision} \times \text{Recall}}{\text{Precision} + \text{Recall}}

AUC-ROC: Area under the Receiver Operating Characteristic curve

Automotive Industry Implementation

Auto Finance Applications

1. Credit Risk Scoring

Model: Logistic regression for default probability
Features: Credit history, income, employment, vehicle type
Business Value: Reduced loan losses, optimized pricing

2. Lease Residual Value Prediction

Model: Multiple linear regression with polynomial terms
Features: Make, model, year, mileage, market conditions
Business Value: Accurate lease pricing, reduced residual risk

Auto Marketing Applications

1. Customer Lifetime Value

Model: Ridge regression with 150+ features
Features: Demographics, purchase history, service patterns
Business Value: Targeted marketing, resource allocation

2. Lead Conversion Prediction

Model: Logistic regression with interaction terms
Features: Digital behavior, demographics, vehicle interest
Business Value: Improved sales efficiency, better lead nurturing

Auto Sales Applications

1. Inventory Optimization

Model: Multiple regression with seasonal terms
Features: Historical sales, market trends, economic indicators
Business Value: Reduced carrying costs, improved availability

2. Dynamic Pricing

Model: Polynomial regression with interaction effects
Features: Competitor pricing, inventory levels, demand signals
Business Value: Optimized margins, competitive positioning

Regression analysis provides the mathematical foundation for data-driven decision making across the automotive industry, enabling organizations to quantify relationships, make accurate predictions, and optimize business outcomes through systematic analytical approaches.

Overview Time Series Analysis