Computational Statistics

Computational Statistics is a subfield of statistics that leverages the power of computing to solve complex analytical problems. It focuses on the development and application of algorithms for implementing statistical methods that are computationally intensive or analytically intractable, such as Monte Carlo simulations for approximating distributions, bootstrapping for estimating uncertainty, and Markov Chain Monte Carlo (MCMC) for Bayesian inference. This discipline is essential for handling massive datasets and applying sophisticated models, forming a critical bridge between statistical theory and practical data analysis in the modern era.

1. Foundations of Computational Statistics
   1. The Role of Computation in Statistics
      1. Bridging Theory and Practice
         1. Translating Statistical Theory into Algorithms
         2. Implementing Statistical Procedures Computationally
         3. Computational Complexity Considerations
      2. Solving Intractable Problems
         1. Approximating Analytical Solutions
         2. Simulation-Based Approaches
         3. Numerical Integration Methods
      3. Handling Large Datasets
         1. Data Storage and Access
         2. Memory Management
         3. Efficient Data Manipulation
         4. Scalability Considerations
         5. Streaming Data Processing
   2. Core Mathematical Prerequisites
      1. Numerical Linear Algebra
         1. Matrix Operations
            1. Matrix Multiplication
            2. Matrix Inversion
            3. Matrix Norms
            4. Condition Numbers
         2. Matrix Decompositions
            1. Singular Value Decomposition (SVD)
            2. QR Decomposition
            3. Cholesky Decomposition
            4. LU Decomposition
         3. Solving Systems of Linear Equations
            1. Direct Methods
            2. Iterative Methods
            3. Sparse Matrix Methods
         4. Eigenvalues and Eigenvectors
            1. Computation Methods
            2. Applications in Statistics
      2. Probability and Distribution Theory
         1. Common Probability Distributions
            1. Discrete Distributions
               1. Bernoulli Distribution
               2. Binomial Distribution
               3. Poisson Distribution
               4. Geometric Distribution
               5. Negative Binomial Distribution
            2. Continuous Distributions
               1. Normal Distribution
               2. Exponential Distribution
               3. Gamma Distribution
               4. Beta Distribution
               5. Uniform Distribution
               6. Student's t-Distribution
               7. Chi-Square Distribution
               8. F-Distribution
         2. Multivariate Distributions
            1. Multivariate Normal Distribution
            2. Wishart Distribution
            3. Dirichlet Distribution
         3. Transformations of Random Variables
            1. Change of Variables
            2. Jacobian Determinant
            3. Simulation of Transformed Variables
         4. Moments and Expectations
            1. Method of Moments
            2. Moment Generating Functions
            3. Characteristic Functions
      3. Numerical Analysis Fundamentals
         1. Floating Point Arithmetic
            1. Representation and Precision
            2. Rounding Errors
            3. Numerical Stability
         2. Root Finding
            1. Bisection Method
            2. Newton-Raphson Method
            3. Secant Method
         3. Numerical Integration
            1. Trapezoidal Rule
            2. Simpson's Rule
            3. Gaussian Quadrature
   3. Statistical Programming Environments
      1. R Programming
         1. Data Structures in R
            1. Vectors
            2. Matrices
            3. Data Frames
            4. Lists
         2. Statistical Functions and Packages
            1. Base R Functions
            2. CRAN Package System
            3. Package Development
         3. Visualization Tools
            1. Base Graphics
            2. ggplot2
            3. Interactive Visualizations
      2. Python for Statistics
         1. NumPy for Numerical Computation
            1. Array Operations
            2. Broadcasting
            3. Linear Algebra Functions
         2. SciPy for Scientific Computing
            1. Statistical Functions
            2. Optimization Routines
            3. Special Functions
         3. Pandas for Data Manipulation
            1. Data Structures
            2. Data Cleaning
            3. Groupby Operations
         4. Visualization Libraries
            1. Matplotlib
            2. Seaborn
            3. Plotly
         5. Statistical Modeling Libraries
            1. Statsmodels
            2. Scikit-learn
            3. PyMC
      3. Specialized Languages and Tools
         1. Julia for High-Performance Computing
            1. Performance Characteristics
            2. Statistical Packages
         2. Stan for Bayesian Modeling
            1. Stan Language Syntax
            2. Compilation and Sampling
         3. JAGS and BUGS for MCMC
            1. Model Specification
            2. Interface with R and Python
   4. Random Number Generation
      1. Pseudo-random Number Generators (PRNGs)
         1. Principles of PRNGs
            1. Deterministic Algorithms
            2. Period and Cycle Length
         2. Linear Congruential Generators
            1. Algorithm Structure
            2. Parameter Selection
         3. Mersenne Twister
            1. Algorithm Details
            2. Implementation Considerations
         4. Seeding and Reproducibility
            1. Seed Selection
            2. Reproducible Research Practices
      2. Testing Random Number Generators
         1. Empirical Tests
            1. Frequency Test
            2. Runs Test
            3. Autocorrelation Test
            4. Gap Test
         2. Theoretical Tests
            1. Periodicity Analysis
            2. Uniformity Assessment
            3. Independence Verification
         3. Test Suites
            1. Diehard Tests
            2. TestU01 Suite
      3. Generating from Non-uniform Distributions
         1. Inverse Transform Sampling
            1. Method Description
            2. Implementation for Common Distributions
         2. Rejection Sampling
            1. Basic Algorithm
            2. Efficiency Considerations
         3. Adaptive Rejection Sampling
            1. Log-Concave Distributions
            2. Envelope Construction
         4. Box-Muller Method for Normal Variates
            1. Transformation Approach
            2. Polar Method Variant
         5. Alias Method for Discrete Distributions
            1. Setup Phase
            2. Generation Phase
         6. Ziggurat Method
            1. Algorithm Structure
            2. Applications to Continuous Distributions