Chapter 3 Introduction to Computational Statistics

Computational statistics is a field that combines statistical methods with computational techniques to analyze data, make predictions, and solve complex problems that traditional analytical methods struggle with. It bridges the gap between theoretical statistics and practical data analysis by leveraging the power of computing.

Motivation for Computational Statistics

  1. Complex Dataset
    • Modern data is high-dimensional, large-scale, and often messy. Traditional statistical methods, which rely on closed-form solutions, struggle with such data. Computational statistics provides scalable solutions for big data.
  2. Optimization in Statistics
    • Many statistical techniques involve optimization (e.g., Maximum Likelihood Estimation, LASSO regression). Computational methods like gradient descent and EM algorithms allow us to estimate parameters efficiently.
  3. Monte Carlo Methods and Simulation
    • Many real-world problems cannot be solved analytically, especially in Bayesian inference, finance, and genetics. Computational statistics enables the use of Monte Carlo simulations, which approximate solutions by repeated random sampling.
  4. Resampling Techniques
    • Bootstrapping and permutation tests provide non-parametric alternatives when classical statistical assumptions (such as normality) do not hold. These methods rely heavily on computational power.
  5. Machine Learning and AI
    • Many machine learning methods, such as decision trees, neural networks, and clustering algorithms, are fundamentally statistical but rely on computational techniques to scale and optimize performance.

We mention the topics that can be considered part of computational statistics to help you understand the difference between these and the more traditional methods of statistics. The following table gives an excellent comparison of the two areas (Wegman, 1988).

Aspect Traditional Statistics Computational Statistics
Sample Size Small to moderate sample size Large to very large sample size
Observations and Datasets Independent, identically distributed data sets Nonhomogeneous datasets
Predictors One or low dimensional High Dimensional
Computation Manually Computational Computationally Intensive
Tractability Mathematically Tractable Numerically Tractable
Assumptions

Strong unverifiable assumptions

  • Relationships (linearity, additivity)
  • Error Structures (normality)

Weak or no assumptions

  • Relationships (nonlinearity)
  • Error structures (distribution free)
Inference Statistical Inference Structural Inference
Algorithms Predominantly closed-form algorithms Iterative algorithms are possible
Statistical Property

Statistical Optimality

  • Minimum Variance
  • Highest Likelihood

Statistical Robustness

  • Out-of-sample prediction accuracy
  • Robust to outliers

Computational statistics is essential in modern data analysis, enabling statisticians to tackle problems that traditional methods cannot handle efficiently. As data grows in size and complexity, computational techniques continue to expand the scope of statistical applications in science, business, and technology.

In this Chapter, we will cover core topics in Statistical Computing with mix of theory and application.

The next chapter will then cover results of these methods in computational statistics, mainly on machine learning algorithms.