A page for collecting and discussing statistical analysis of metaheuristic optimization experiments.

In metaheuristic optimization experiments we measure the outcome of a stochastic process. From this measurements we hope to estimate the distribution mean of the outcome. The outcome usually is a certain quality or the time/iterations required to reach a certain quality. These outcomes are random variables with unknown distributions.

The goal in those experiments is to show that one process is able to achieve a better output than another process. There are several ways to show this:

• Visual analysis methods
• Statistical hypotheses tests for unequality of two means

Statistical Background

• ​Behrens-Fischer Problem

Statistical Analysis Methods

• Single comparison
  • ​t-test
  • ​Mann-Whitney U test
  • ​Welch's t-test
• Multiple comparison
  • ​ANOVA
  • ​Friedman test
  • ​Kruskal-Wallis

Visual Analysis Methods

• In statistical analysis visual methods are often preferred. They can encode a lot more data and provide a clearer picture for expert human interpretation. Several possibilities exist, some have already been mentioned:

Name	Example	Purpose
Bubble charts		Bubbles are fun!
Density plots / histograms		Comparing/Estimating the exact shapes of the underlying distributions
Boxplots		Compact and efficient comparison of distributions
Forest plots		Comparing confidence intervals makes it probably a little more obvious if a result would be significant enough
Q-Q plots		Estimating whether data fits a certain distribution

Possible Workflow / Methodology

1. Testing data for e.g. normal distributions, equalness of variances, etc. to decide if parametric or non-parametric tests to apply
2. In case of multiple comparisons perform ANOVA, Friedman or other tests otherwise perform single comparison t-test, Mann-Whitney, etc.
3. In case multiple comparisons are significant use pairwise comparisons with post hoc analysis adjustments

One problem with such a workflow is that the error propagates in every step. So if testing for normal distributions has a certain error and ANOVA has a certain error, the final error would be a combination of the two. We should also identify and exclude cases where the application of tests would probably not be valid.

Critique

1. Steven Goodman. 2008. A Dirty Dozen: Twelve P-Value Misconceptions. Seminars in Hematology Volume 45, Issue 3, July 2008, Pages 135–140
2. Jacob Cohen. 1994. The Earth is Round (p < 0.05). American Psycologist. ​http://ist-socrates.berkeley.edu/~maccoun/PP279_Cohen1.pdf
3. Hubbard, Bayarri. 2003. P-Values are not Error Probabilities. ​http://ftp.isds.duke.edu/WorkingPapers/03-26.pdf

References

• García, Fernández, Luengo, Herrera. 2010. Advanced nonparametric tests for multiple comparisons in the design of experiments in computational intelligence and data mining: Experimental analysis of power. Information Sciences 180, pp. 2044–2064. (​http://sci2s.ugr.es/sicidm/pdf/2010-Garcia-INS.pdf)
• Salvador García, Daniel Molina, Manuel Lozano, Francisco Herrera. 2009. A study on the use of non-parametric tests for analyzing the evolutionary algorithms’ behaviour: a case study on the CEC’2005 Special Session on Real Parameter Optimization. (​http://sci2s.ugr.es/publications/ficheros/2009-garcia-JH.pdf)

External Libraries

• ​meta numerics implements some statistical tests (e.g. ANOVA, Mann-Whitney U, ...) but is licensed under the MS PL which is incompatible with the GPL.
• ​Accord.NET / ​AForge.NET: From the Accord.NET website: "Accord.NET is a framework for scientific computing in .NET. The framework builds upon AForge.NET, an also popular framework for image processing, supplying new tools and libraries. Those libraries encompass a wide range of scientific computing applications, such as statistical data processing, machine learning, pattern recognition, including but not limited to, computer vision and computer audition. The framework offers a large number of probability distributions, hypothesis tests, kernel functions and support for most popular performance measurements techniques." These libraries actually provide a lot of stuff which we don't need. Gladly the math/statistics parts can be extracted very easily (they are own assemblies). Accord.NET implements OneWayAnova, TwoWayAnova(1,2,3), T-test, Mann-Whitney Wilcoxon, Kolmogorov-Smirnov and a lot more. And they are both licensed under LGPL so we don't have a problem with licensing.

Links

• ​STATService for comparison of metaheuristic results
• ​Post-hoc analysis
• ​Friedman test example
• ​Criticism of friedman test
• ​Statistical Tests Overview

Significant