Unit 2964: Statistical Modeling Comprehensive Outline

1. Introduction to Statistical Modeling

• Definition and scope of statistical modeling
• Importance in data analysis and decision making
• Types of statistical models
  • Descriptive vs. inferential models
  • Parametric vs. non-parametric models
  • Linear vs. nonlinear models

2. Probability Distributions

• Introduction to probability distributions
• Normal distribution
  • Properties and applications
  • Standard normal distribution and Z-scores
• Binomial distribution
  • Parameters and interpretation
  • Applications in modeling binary outcomes
• Poisson distribution
  • Use for modeling count data
  • Characteristics and examples

3. Simple Linear Regression

• Concept of predictor (independent) and response (dependent) variables
• Model formulation: Y = β0 + β1X + ε
• Assumptions of simple linear regression
  • Linearity, independence, homoscedasticity, normality
• Estimation of parameters using least squares
• Interpretation of coefficients
• Model diagnostics and residual analysis
• Application examples

4. Multiple Regression Analysis

• Extension from simple to multiple predictors
• Model formulation: Y = β0 + β1X1 + β2X2 + ... + βpXp + ε
• Assumptions and multicollinearity considerations
• Model building strategies
• Interpretation of coefficients in multiple regression
• Model diagnostics and validation techniques
• Practical applications and case studies

5. Model Selection and Validation

• Importance of selecting appropriate models
• Techniques for model selection
  • Akaike Information Criterion (AIC)
  • Bayesian Information Criterion (BIC)
  • Cross-validation methods
• Overfitting and underfitting concepts
• Validation approaches
  • Train-test split
  • k-fold cross-validation
• Model performance metrics

6. Logistic Regression

• When to use logistic regression (categorical response variables)
• Binary logistic regression
  • Model formulation and link function
  • Odds, odds ratios, and interpretation of coefficients
• Multinomial logistic regression
• Model assumptions and diagnostics
• Applications in classification problems

7. Time Series Analysis

• Characteristics of time series data
  • Trends, seasonality, cycles, and noise
• Stationarity and differencing
• Autoregressive Integrated Moving Average (ARIMA) models
  • Components: AR, I, and MA
  • Model identification and parameter estimation
• Seasonal ARIMA (SARIMA)
• Forecasting techniques and accuracy assessment

8. Bayesian Modeling

• Fundamentals of Bayesian statistics
  • Prior, likelihood, posterior distributions
• Comparison with frequentist approaches
• Markov Chain Monte Carlo (MCMC) methods
  • Basics of MCMC and sampling techniques
• Bayesian model comparison and selection
• Applications and examples

9. Machine Learning in Statistical Modeling

• Introduction to machine learning concepts
• Decision trees
  • Structure and splitting criteria
• Random forests
  • Ensemble methods and feature importance
• Support vector machines (SVM)
  • Kernel functions and margin maximization
• Neural networks
  • Architecture and learning process
• Application of these algorithms in predictive modeling and classification
• Advantages and limitations compared to classical statistical models