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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

Statistical Analysis Methods

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.


  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.
  3. Hubbard, Bayarri. 2003. P-Values are not Error Probabilities.


  • 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. (
  • 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. (

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.


Last modified 11 years ago Last modified on 02/18/13 22:49:03

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