7.4 Advanced regression models

Advanced regression models expand on basic linear regression, offering tools to capture complex relationships in data. These techniques include polynomial regression, interaction terms, and feature engineering, allowing for more accurate modeling of non-linear patterns and variable interactions.

Implementing these models involves careful consideration of model complexity, interpretation, and potential overfitting. Techniques like cross-validation and regularization help balance model flexibility with generalizability, ensuring robust predictive performance on new data.

Polynomial Regression and Interaction Terms

Non-linear Relationships in Regression

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• Polynomial regression extends linear regression by including higher-order terms of independent variables to capture non-linear relationships
• Order of polynomial regression model determined by highest power of independent variable (quadratic for 2nd order, cubic for 3rd order)
• Interaction terms capture combined effect of two or more independent variables on dependent variable, beyond individual effects
• Create interaction terms by multiplying two or more independent variables together
• Allows modeling of complex relationships between variables
• Polynomial regression can lead to overfitting if order of polynomial too high
  • Results in model fitting noise rather than underlying relationship
• Interpretation of polynomial and interaction terms requires careful consideration of coefficients and statistical significance
• Visualization techniques (partial dependence plots) aid in understanding effects of polynomial and interaction terms

Examples and Applications

• Quadratic relationship example: housing prices vs. square footage
  • Price increases with size but at decreasing rate
• Cubic relationship example: crop yield vs. fertilizer amount
  • Yield increases, plateaus, then decreases with excessive fertilizer
• Interaction term example: effect of temperature on ice cream sales moderated by humidity
• Partial dependence plot example: visualizing non-linear relationship between age and income in salary prediction model
• Overfitting example: using 10th degree polynomial to model simple quadratic relationship
  • Results in perfect fit to training data but poor generalization

Implementing Polynomial Regression Models

Model Implementation and Evaluation

• Implement polynomial regression by creating new features from original independent variables (x^2, x^3)
• Use Sklearn's PolynomialFeatures class to generate polynomial and interaction features automatically
• Compare R-squared values and other model fit statistics between linear and polynomial models to assess improvement
• Examine residual plots for polynomial regression models
  • Should show random scatter around zero, indicating captured non-linear patterns
• Apply cross-validation techniques to select appropriate polynomial degree and avoid overfitting
• Use regularization methods (ridge or lasso regression) to control multicollinearity and overfitting in polynomial models

Interpretation and Examples

• Interpret polynomial regression coefficients by considering combined effect of all terms containing particular variable
• Example: Interpreting quadratic model for housing prices
  • Positive linear term: price increases with size
  • Negative quadratic term: rate of increase slows for larger houses
• Example: Cubic model for crop yield vs. fertilizer
  • Positive linear and quadratic terms: yield increases rapidly at first
  • Negative cubic term: yield plateaus and eventually decreases with excessive fertilizer
• Example: Cross-validation for polynomial degree selection
  • Compare mean squared error across different polynomial degrees
  • Choose degree with lowest cross-validated error

Feature Engineering for Regression

Feature Creation and Transformation

• Feature engineering creates new features or transforms existing ones to better capture underlying relationships
• Apply techniques such as binning continuous variables, creating interaction terms, and mathematical transformations (log, square root)
• Utilize domain knowledge to guide creation of meaningful and interpretable features
• Create time-based features (seasonality indicators, lag variables) to improve performance of regression models on time series data
• Example: Transform skewed income data using log transformation to normalize distribution
• Example: Create binary feature for weekday/weekend to capture different patterns in daily sales data

Feature Selection and Evaluation

• Apply feature selection methods (recursive feature elimination, LASSO) to identify most important engineered features
• Use dimensionality reduction techniques (Principal Component Analysis) to handle multicollinearity introduced by feature engineering
• Assess impact of feature engineering on model performance using cross-validation and comparison of evaluation metrics
• Example: Use LASSO regression to automatically select relevant polynomial terms in complex model
• Example: Apply PCA to reduce dimensionality of dataset with many interaction terms, preserving most important information

Advanced Regression Techniques

Stepwise Regression and Model Selection

• Stepwise regression iteratively adds or removes predictors based on statistical significance
• Three common approaches: forward selection, backward elimination, and bidirectional elimination
• Forward selection starts with no variables and adds most significant predictor at each step
• Backward elimination starts with all variables and removes least significant predictor at each step
• Bidirectional elimination combines forward and backward approaches
• Example: Using stepwise regression to select best subset of predictors for customer churn model
• Example: Comparing AIC (Akaike Information Criterion) values to determine optimal model in stepwise process

Generalized Linear Models and Regularization

• Generalized Linear Models (GLMs) extend linear regression to handle response variables with non-normal distributions
• Link function in GLMs transforms expected value of response variable to allow linear relationship with predictors
• Common GLM types: logistic regression (binary outcomes), Poisson regression (count data)
• Regularized regression techniques (LASSO, Ridge, Elastic Net) prevent overfitting
• LASSO (L1 regularization) performs feature selection by shrinking some coefficients to zero
• Ridge regression (L2 regularization) shrinks all coefficients towards zero, but not exactly to zero
• Elastic Net combines L1 and L2 regularization
• Example: Using logistic regression to predict probability of customer default based on credit history
• Example: Applying Poisson regression to model number of customer support tickets per day