7.1 Linear regression

Linear regression is a powerful statistical tool for modeling relationships between variables. It forms the foundation of many advanced machine learning techniques, allowing us to predict outcomes and understand the impact of different factors on a target variable.

This section explores the key concepts of linear regression, including model assumptions, coefficient interpretation, and evaluation metrics. We'll dive into the mathematics behind the method and discuss practical applications across various fields, equipping you with essential skills for data analysis and prediction.

Linear Regression Fundamentals

Model Concept and Key Assumptions

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• Linear regression models relationship between dependent variable and one or more independent variables by fitting linear equation to observed data
• Fundamental assumption exists linear relationship between dependent variable and independent variables
• Homoscedasticity requires variance of residual errors remain constant across all levels of independent variables
• Independence assumption necessitates observations remain independent of each other (particularly important for time series data)
• Multicollinearity refers to high correlations between independent variables leading to unstable and unreliable coefficient estimates
• Normality of residuals assumes residual errors follow normal distribution for valid statistical inference
• Absence of influential outliers prevents extreme data points from disproportionately affecting regression line and coefficient estimates

Mathematical Representation

• General form of multiple linear regression equation expressed as: $Y = β0 + β1X1 + β2X2 + ... + βnXn + ε$
  • Y represents dependent variable
  • X's denote independent variables
  • β's signify coefficients
  • ε indicates error term
• Ordinary least squares (OLS) method commonly estimates regression coefficients by minimizing sum of squared residuals
• Standardized coefficients (beta coefficients) enable comparison of relative importance among independent variables measured on different scales

Interpreting Regression Coefficients

Coefficient Interpretation

• Intercept (β0) represents expected value of Y when all independent variables equal zero (may lack meaningful interpretation in some real-world contexts)
• Slope coefficients (β1, β2, ..., βn) indicate change in Y for one-unit increase in corresponding X, holding all other variables constant
• Coefficient sign reveals direction of relationship between independent and dependent variables
• Coefficient magnitude demonstrates strength of relationship between variables
• Confidence intervals for coefficients provide range of plausible values and indicate precision of estimates

Advanced Interpretation Techniques

• Standardized coefficients allow comparison of predictor importance across different scales
• Interaction terms capture complex relationships between independent variables and their combined effect on dependent variable
• Polynomial terms model non-linear relationships within linear regression framework
• Regularization techniques (Ridge and LASSO regression) prevent overfitting and improve model generalization, especially in high-dimensional datasets

Model Fit and Prediction

Goodness of Fit Measures

• ###r-squared_0### (coefficient of determination) measures proportion of variance in dependent variable predictable from independent variable(s), ranging from 0 to 1
• Adjusted R-squared accounts for number of predictors, penalizing addition of variables not improving model's explanatory power
• F-statistic tests overall significance of regression model, comparing it to model with no predictors
• Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) used for model selection, balancing goodness of fit with model complexity

Predictive Performance Evaluation

• Root Mean Square Error (RMSE) quantifies standard deviation of residuals, measuring model's prediction error in original units of dependent variable
• Mean Absolute Error (MAE) represents average absolute difference between predicted and actual values, less sensitive to outliers than RMSE
• Cross-validation techniques (k-fold cross-validation) assess model generalization to unseen data by partitioning dataset into training and testing subsets
• Diagnostic plots (residual plots, Q-Q plots) validate model assumptions and identify potential issues (heteroscedasticity, non-linearity)

Linear Regression Applications

Data Preprocessing and Feature Selection

• Data preprocessing crucial for optimal model performance
  • Handle missing values
  • Encode categorical variables
  • Scale numerical features
• Feature selection techniques identify most relevant predictors
  • Forward selection
  • Backward elimination
  • LASSO regression

Real-World Implementation

• Apply linear regression to various domains (economics, healthcare, marketing)
• Interpret regression results considering practical significance alongside statistical significance
• Use polynomial regression to model non-linear relationships within linear regression framework
• Implement regularization techniques (Ridge, LASSO) to prevent overfitting in high-dimensional datasets
• Employ interaction terms to capture complex relationships between independent variables