Statistics

1. Regression Analysis
   1. Linear Regression
      1. Simple Linear Regression
         1. Definition and Purpose
         2. Mathematical Representation
            1. Two-variable Equation: y = b0 + b1x + e
            2. Parameters: b0 (intercept), b1 (slope), e (error term)
         3. Data Fitting
            1. Least Squares Method
            2. Interpretation of Coefficients
         4. Visualization
            1. Scatter Plots with Regression Line
         5. Applications and Examples
      2. Multiple Linear Regression
         1. Definition and Purpose
         2. Mathematical Representation
            1. Multi-variable Equation: y = b0 + b1x1 + b2x2 + ... + bnxn + e
         3. Importance of Multicollinearity
            1. Detection (Variance Inflation Factor)
            2. Remedies (Ridge Regression, Variable Selection)
         4. Assumptions
            1. Linearity
            2. Independence
            3. Homoscedasticity (constant variance of errors)
            4. Normality of error terms
         5. Diagnostics
            1. Residual Analysis
            2. Cook's Distance for Influential Points
            3. Durbin-Watson Test for Autocorrelation
         6. Model Interpretation
            1. Coefficient Significance
            2. R-Squared and Adjusted R-Squared
         7. Visualization
            1. Residual Plots
            2. Partial Regression Plots
   2. Non-linear Regression
      1. Introduction and Use Cases
      2. Models
         1. Polynomial Regression
         2. Logistic Function for S-curve Fitting
      3. Estimation Methods
         1. Iterative Approximation (e.g., Newton-Raphson)
         2. Convergence Criteria
      4. Assumptions and Diagnostics
         1. Error Term Behavior
         2. Model Fit Analysis
   3. Logistic Regression
      1. Purpose and Applications
         1. Binary Outcome Variables
         2. Odds and Log Odds
      2. Mathematical Representation
         1. Logit Function: log(p/(1-p)) = b0 + b1x1 + b2x2 + ... + bnxn
      3. Extensions
         1. Multinomial Logistic Regression
         2. Ordinal Logistic Regression
      4. Model Evaluation
         1. Confusion Matrix
         2. ROC Curve and AUC
      5. Assumptions
         1. Linearity of Logits
         2. Independence of Errors
         3. Absence of Multicollinearity
   4. Correlation and Causation
      1. Correlation Coefficient (Pearson's r)
         1. Strength and Direction of Linear Relationship
         2. Testing Significance
      2. Limitations of Correlation
         1. Correlation does not imply causation
         2. Spurious Correlations
      3. Causation Analysis
         1. Granger Causality
         2. Structural Equation Modeling (SEM)
   5. Model Selection Techniques
      1. Criteria for Model Selection
         1. Akaike Information Criterion (AIC)
         2. Bayesian Information Criterion (BIC)
      2. Stepwise Regression
         1. Forward Selection
         2. Backward Elimination
         3. Bidirectional Elimination
      3. Regularization Methods
         1. Lasso Regression
         2. Ridge Regression
      4. Cross-Validation
         1. k-Fold Cross-Validation
         2. Leave-One-Out Cross-Validation (LOOCV)
      5. Considerations for Overfitting
         1. Balancing Complexity and Generalization