Comprehensive Computational Statistics Cheatsheet: Methods, Algorithms & Applications

Introduction to Computational Statistics

Computational statistics combines statistical theory with computational methods to solve complex statistical problems that are intractable through analytical means. It enables data scientists to implement simulation techniques, perform intensive calculations, apply resampling methods, and develop algorithmic approaches to statistical inference and modeling.

Core Concepts and Principles

Fundamental Elements

• Computational Inference: Using algorithms to estimate parameters and quantify uncertainty
• Simulation: Generating random samples from probability distributions
• Resampling: Creating new samples from existing data for inference
• Numerical Optimization: Finding parameter values that maximize/minimize objective functions
• Algorithm Efficiency: Balancing computational speed and statistical accuracy
• Numerical Stability: Ensuring calculations remain accurate despite finite precision

Statistical Computing Foundations

• Random Number Generation: Uniform, non-uniform, pseudorandom sequences
• Numerical Integration: Approximating integrals for probability calculations
• Linear Algebra Operations: Matrix manipulations for multivariate methods
• Function Optimization: Finding maxima/minima of likelihood functions
• Parallel Computing: Distributing statistical calculations across processors

Simulation Methods

Random Variable Generation

Basic Implementation (Box-Muller for Normal RVs)

def box_muller():
    u1, u2 = np.random.uniform(0, 1, 2)
    z1 = np.sqrt(-2 * np.log(u1)) * np.cos(2 * np.pi * u2)
    z2 = np.sqrt(-2 * np.log(u1)) * np.sin(2 * np.pi * u2)
    return z1, z2

Monte Carlo Methods

• Basic Monte Carlo Integration: Estimating integrals via random sampling
• Importance Sampling: Sampling from alternative distribution to reduce variance
• Stratified Sampling: Dividing sample space into strata for better coverage
• Quasi-Monte Carlo: Using low-discrepancy sequences for more uniform coverage
• Sequential Monte Carlo: Particle filtering for dynamic systems

Markov Chain Monte Carlo (MCMC)

Metropolis-Hastings Implementation

def metropolis_hastings(target_pdf, proposal_sampler, initial_state, n_samples):
    samples = [initial_state]
    current = initial_state
    for i in range(n_samples):
        proposed = proposal_sampler(current)
        acceptance_ratio = min(1, target_pdf(proposed) / target_pdf(current))
        if np.random.uniform() < acceptance_ratio:
            current = proposed
        samples.append(current)
    return samples

Resampling Methods

Bootstrap Techniques

• Standard Bootstrap: Resampling with replacement for parameter uncertainty
• Parametric Bootstrap: Resampling from fitted distribution
• Block Bootstrap: For dependent data (time series)
• Wild Bootstrap: For heteroscedastic errors
• Bootstrap Confidence Intervals: Percentile, BCa, t-bootstrap

Cross-Validation Methods

Computational Inference Methods

Maximum Likelihood Estimation

• Newton-Raphson Method: Second-order convergence using Hessian
• Scoring Method: Using expected Hessian
• EM Algorithm: For latent variable models and missing data
• Stochastic Gradient Methods: For large datasets

Bayesian Computation

• Laplace Approximation: Normal approximation to posterior
• Variational Inference: Approximating posterior with simpler distributions
• Approximate Bayesian Computation: Simulation-based inference without likelihood
• Integrated Nested Laplace Approximation: For latent Gaussian models

Optimization Algorithms

Derivative-Based Methods

Derivative-Free Methods

• Nelder-Mead Simplex: Robust but slow convergence
• Simulated Annealing: Global optimization with probabilistic jumps
• Genetic Algorithms: Evolution-inspired search
• Particle Swarm: Swarm intelligence for global optimization
• Bayesian Optimization: Sample-efficient for expensive functions

L-BFGS Implementation Example

from scipy.optimize import minimize

def negative_log_likelihood(params, data):
    # Compute the negative log-likelihood
    return -log_likelihood(params, data)

result = minimize(negative_log_likelihood, initial_guess, 
                  args=(data,), method='L-BFGS-B',
                  bounds=parameter_bounds)

Machine Learning in Statistics

Supervised Learning Methods

• Linear/Logistic Regression: Statistical foundations of simple models
• Regularization: Ridge, Lasso, Elastic Net for model complexity control
• Decision Trees: CART, C4.5, conditional inference trees
• Ensemble Methods: Random Forests, Boosting, Stacking
• Support Vector Machines: Maximum margin classification

Unsupervised Learning

Model Selection and Evaluation

• Information Criteria: AIC, BIC, DIC
• Cross-Validation Metrics: MSE, MAE, RMSE, R², AUC
• Regularization Paths: Solution paths for regularization parameters
• Bayesian Model Selection: Bayes factors, posterior model probabilities

Specialized Techniques

Time Series Analysis

• ARIMA/SARIMA: Box-Jenkins methodology
• State Space Models: Kalman filtering and smoothing
• Structural Time Series: Decomposition approaches
• GARCH Models: Volatility modeling
• Spectral Analysis: Frequency domain methods

Spatial Statistics

• Kriging: Gaussian process regression for spatial data
• Spatial Point Processes: Modeling point patterns
• Geographically Weighted Regression: Locally varying coefficients
• Spatial Autoregressive Models: Accounting for spatial dependence

Big Data Methods

• Stochastic Gradient Descent: Online learning for large datasets
• Random Projections: Dimension reduction via Johnson-Lindenstrauss
• Subsampling Approaches: Bag of Little Bootstraps, divide-and-conquer
• Streaming Algorithms: One-pass methods for data streams
• Distributed Computing: MapReduce, Spark for parallel statistics

Comparison of Statistical Software

Common Challenges and Solutions

Computational Efficiency

• Challenge: Long computation times for complex models
• Solutions:
  • Vectorize operations instead of loops
  • Use appropriate data structures (sparse matrices)
  • Implement parallel processing
  • Consider GPU acceleration
  • Use compiled languages for core functions

Convergence Issues

• Challenge: Optimization algorithms fail to converge
• Solutions:
  • Try multiple starting points
  • Use robust optimization methods
  • Reparameterize model
  • Check for identifiability issues
  • Monitor convergence diagnostics

Numerical Stability

• Challenge: Overflow, underflow, or ill-conditioning
• Solutions:
  • Work in log-scale for likelihoods
  • Use QR decomposition instead of direct matrix inversion
  • Apply regularization to ill-conditioned problems
  • Use stable algorithms for specific problems
  • Implement checks for edge cases

Scalability

• Challenge: Methods don’t scale to big data
• Solutions:
  • Use streaming algorithms
  • Implement distributed computing frameworks
  • Apply dimension reduction techniques
  • Consider subsampling approaches
  • Use online learning methods

Best Practices

Statistical Computing

• Write modular, reusable code
• Validate implemented methods against known results
• Use appropriate precision for numerical calculations
• Set random seeds for reproducibility
• Document statistical assumptions and methods

Simulation Studies

• Design simulations to answer specific questions
• Choose realistic parameter values
• Include sufficient replications
• Report Monte Carlo error
• Visualize simulation results effectively

Model Development

• Check model assumptions
• Perform sensitivity analysis
• Compare multiple models and methods
• Use diagnostics to assess fit
• Document limitations and scope

Reproducibility

• Use version control for code
• Document computational environment
• Share code and data when possible
• Use literate programming (R Markdown, Jupyter)
• Validate results with multiple approaches

Resources for Further Learning

Textbooks

• “Computational Statistics” by Gentle
• “Monte Carlo Statistical Methods” by Robert and Casella
• “Bootstrap Methods and Their Application” by Davison and Hinkley
• “Numerical Methods of Statistics” by Monahan
• “Statistical Computing with R” by Rizzo

Online Courses

• Coursera: “Bayesian Statistics” by Duke University
• edX: “Statistical Learning” by Stanford
• DataCamp: “Statistical Simulation in Python”
• Udemy: “Bayesian Machine Learning in Python: A/B Testing”

Software Documentation

• R Documentation and CRAN Task Views
• Python StatsModels and SciPy Documentation
• Stan User’s Guide
• TensorFlow Probability Tutorials

Communities and Forums

• Cross Validated (stats.stackexchange.com)
• R-bloggers
• PyData community
• Statistical Computing section of ASA

Journals

• Journal of Computational and Graphical Statistics
• Journal of Statistical Software
• Statistics and Computing
• Computational Statistics & Data Analysis

This cheatsheet provides a comprehensive overview of computational statistics methods, but the field is vast. For specific applications, deeper exploration of specialized resources may be necessary.