# 5. Computational Statistics Lectures

- Author:
Raymond Bisdorff, Emeritus Professor of Applied Mathematics and Computer Science

- Copyright:
Bisdorff © 2013-2023

## 5.1. Introduction

From 2007 to 2011 the Algorithmic Decision Theory COST Action IC0602, coordinated by Alexis Tsoukiàs, gathered researchers coming from different fields such as *Decision Theory*, *Discrete Mathematics*, *Theoretical Computer Science* and *Artificial Intelligence* in order to improve decision support in the presence of **massive data bases**, **combinatorial structures**, **partial** and/or **uncertain information** and **distributed**, possibly **interoperating decision makers**.

A positive result a.o. of this COST action was the organisation from 2010 to 2020 of a Semester Course on Computational Statistics at the *University of Luxembourg* in the context of its **Master in Information and Computer Science**.

Below are gathered 2x2 reduced copies of the presentation slides for 8 Lectures from the Winter Semester 2019.

## 5.2. Lectures

- L1. Generating random numbers for simulations
On numbers “chosen at random”, computer generated random numbers and multiple recursive random number generators over

*F2*Recommendations and traps to watch for with home brewed generators

Combining random number generators and testing randomness

- L2. Introduction to statistical computing
On generating simulation data with Python and exploring simulation data with

*gretl*. Getting started with*R*. Introducing*R*objects: vectors, matrices, lists, data frames. Reading CSV data files into data framesDoing linear algebra in

*R*. Constructing matrix objects, matrix operations and inversion, solving linear systems, eigen-values and -vectors, singular value decomposition, Choleski and QR decompositionsPrincipal component analysis (PCA) and discrete Markov chain simulating

- L3. Continuous Random Variables
Probability distributions in

*R*-core, simulating a continuous uniform random distribution, the spectral test for random number generatorsSimulating random variables by a continuous inverse transform, standard exponential law based generators

The Gaussian random variables, important properties, simulating Gaussian random variables

- L4. Simulating from Discrete Random Variables
Simulating Bernoulli and Binomial random variables. The CLT for binomial distributions

Simulating a Poisson random variable and Poisson processes with exponential time intervals

Simulating Gamma variables, integer alpha parameter and the sum rule for Gamma Variables

- L5. Simulating from arbitrary empirical random distributions
Single pass estimation of arbitrary quantiles: computing sample quantiles, quantiles via selecting algorithms, tracking the

*M*-largest element in a single passComputing quantiles from binned data: equally binned observation data, linear integration formulas, regular binned data quantiles

Incremental quantiles estimation with the IQ-agent: using the IQ-agent for Monte-Carlo simulations

- L6. Two distributions, are they of the same kind?
Methodology: comparing statistical distributions, methodological approach and statistical tests

Comparing histograms: Chi-square test against a known distribution, comparing two binned data sets, testing uniform randomness

Comparing continuous distributions with the Kolmogorov-Smirnov test

- L7. On Averaging
The benefit from averaging: the law of large numbers, estimating distribution parameters, how to reduce noise

Convergence of the averaging: Convergence of the mean for a standard Gaussian, and if there are outliers? Non convergence of a Cauchy mean

Comparing two empiric means: robustness of the

*t*statistic, estimating*t*statistics, Monte Carlo simulation of the*H0*rejection

- L8. Accept-Reject Simulation Methods
Classical Monte Carlo Integration: principles and applications

Accept-reject simulation methods

Accept-reject simulation applications: pi estimation, Box-Muller transform, Ratio-Of-Uniforms method