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### Means, arithmetic and geometric (= additive and multiplicative)

Now that we've replaced our old additive scoring algorithm with a multiplicative one (see this post), the algorithm that decides whether the simulation has reached its equilibrium isn't working well. The post-doc suggests that this is because we need to track the changing score of the genome using a multiplicative mean rather than an additive mean.

The test for equilibrium works as follows: The USS-score of the genome is calculated each generation, and is used to calculate both a "recent" mean score (over the interval between status updates, usually specified as a percent of the elapsed cycles) and a "grand" mean score (over the entire run). Both means are calculated as simple averages (sum of the scores divided by the number of scores). Early in the run the grand mean is much smaller than the recent mean. Equilibrium conditions are satisfied when the % difference between these means becomes smaller than some pre-specified threshold.

With additive scoring this worked well regardless of the length of the genome. But with multiplicative scoring, a single base change can cause a dramatic change in the score (ten-fold or more, depending on the scoring matrix used), especially when using short genomes. The post-doc, who is much more statistically sophisticated than I am, says that when numbers differ by large values, their means should be calculated geometrically rather than arithmetically.

Wikipedia explains that arithmetic means are the typical 'average' values, and geometric means are calculated using products and roots rather than sums and division. I'll say that another way: a geometric mean is calculated by first calculating the product of all the n values (rather than their sum) and then taking the n-th root of this product. Luckily this can be easily done using logarithms.

The post-doc has modified the code to do this, but she's now gone off to Barcelona (!) for the Society for Molecular Biology and Evolution meeting. I haven't yet tried out her version, but checking whether it finds better equilibria will be much easier now that the runs go so much faster.