Statistical Computing

  1. Numerical Methods for Statistics
    1. Numerical Linear Algebra
      1. Matrix Fundamentals
        1. Matrix Representation
          1. Matrix Properties
            1. Symmetric Matrices
              1. Positive Definite Matrices
                1. Orthogonal Matrices
                2. Basic Matrix Operations
                  1. Addition and Subtraction
                    1. Scalar Multiplication
                      1. Matrix Multiplication
                        1. Transposition
                      2. Matrix Decompositions
                        1. LU Decomposition
                          1. QR Decomposition
                            1. Gram-Schmidt Process
                              1. Householder Reflections
                              2. Cholesky Decomposition
                                1. Singular Value Decomposition
                                  1. Properties and Applications
                                    1. Truncated SVD
                                    2. Eigenvalue Decomposition
                                      1. Eigenvalues and Eigenvectors
                                        1. Spectral Decomposition
                                      2. Solving Linear Systems
                                        1. Direct Methods
                                          1. Gaussian Elimination
                                            1. LU Factorization
                                              1. Forward and Back Substitution
                                              2. Iterative Methods
                                                1. Jacobi Method
                                                  1. Gauss-Seidel Method
                                                    1. Conjugate Gradient Method
                                                  2. Applications in Statistics
                                                    1. Least Squares Problems
                                                      1. Normal Equations
                                                        1. QR Approach
                                                          1. SVD Approach
                                                          2. Principal Component Analysis
                                                            1. Linear Discriminant Analysis
                                                          3. Numerical Optimization
                                                            1. Optimization Fundamentals
                                                              1. Objective Functions
                                                                1. Constraints
                                                                  1. Local vs. Global Optima
                                                                    1. Convexity
                                                                    2. Univariate Optimization
                                                                      1. Golden Section Search
                                                                        1. Brent's Method
                                                                          1. Derivative-Based Methods
                                                                          2. Root-Finding Methods
                                                                            1. Bisection Method
                                                                              1. Newton-Raphson Method
                                                                                1. Secant Method
                                                                                  1. Fixed-Point Iteration
                                                                                  2. Multivariate Optimization
                                                                                    1. Unconstrained Optimization
                                                                                      1. Gradient Descent
                                                                                        1. Steepest Descent
                                                                                          1. Stochastic Gradient Descent
                                                                                            1. Adaptive Learning Rates
                                                                                            2. Newton's Method
                                                                                              1. Quasi-Newton Methods
                                                                                                1. BFGS Algorithm
                                                                                                  1. L-BFGS Algorithm
                                                                                                  2. Conjugate Gradient Methods
                                                                                                  3. Constrained Optimization
                                                                                                    1. Lagrange Multipliers
                                                                                                      1. KKT Conditions
                                                                                                        1. Penalty Methods
                                                                                                          1. Barrier Methods
                                                                                                        2. Statistical Applications
                                                                                                          1. Maximum Likelihood Estimation
                                                                                                            1. Maximum A Posteriori Estimation
                                                                                                              1. Expectation-Maximization Algorithm
                                                                                                                1. E-Step Implementation
                                                                                                                  1. M-Step Implementation
                                                                                                                    1. Convergence Criteria
                                                                                                                2. Numerical Integration
                                                                                                                  1. Deterministic Integration
                                                                                                                    1. Newton-Cotes Formulas
                                                                                                                      1. Trapezoidal Rule
                                                                                                                        1. Simpson's Rule
                                                                                                                          1. Composite Rules
                                                                                                                          2. Gaussian Quadrature
                                                                                                                            1. Gauss-Legendre Quadrature
                                                                                                                              1. Gauss-Hermite Quadrature
                                                                                                                                1. Adaptive Quadrature
                                                                                                                              2. Monte Carlo Integration
                                                                                                                                1. Basic Monte Carlo
                                                                                                                                  1. Importance Sampling
                                                                                                                                    1. Stratified Sampling
                                                                                                                                    2. Applications in Statistics
                                                                                                                                      1. Marginal Likelihood Computation
                                                                                                                                        1. Posterior Integration
                                                                                                                                          1. Normalizing Constants
                                                                                                                                        2. Interpolation and Approximation
                                                                                                                                          1. Polynomial Interpolation
                                                                                                                                            1. Lagrange Interpolation
                                                                                                                                              1. Newton's Divided Differences
                                                                                                                                              2. Spline Interpolation
                                                                                                                                                1. Linear Splines
                                                                                                                                                  1. Cubic Splines
                                                                                                                                                    1. B-Splines
                                                                                                                                                    2. Kernel Methods
                                                                                                                                                      1. Kernel Density Estimation
                                                                                                                                                        1. Kernel Regression

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                                                                                                                                                    3. Data Management and Manipulation

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                                                                                                                                                    5. Simulation and Resampling Methods