Increase accuracy in Averager

[Peter]

It's been discussed on ​help-gsl how to best calculate mean and variance. Our approach in Averager using variable n, sum_x and sum_xx is likely the fastest because the update only involves three additions. Unfortunately, it is not the most accurate and variance may in some cases become negative which yields invalid standard deviation. Knuth suggests using variables n, mean and M2 where the latter is the centered sum squared. The update rules (including weighted case) can be found ​here. Only downside is that it is slightly slower as they involve a couple of divisions and multiplications. Any opinions whether this is worthwhile to implement?

The Averager classes are heavily used in, e.g., the classifiers and thus this change would influence not only the Averager classes. Speed versus accuracy. The fact that GSL uses the accurate method make lean towards that, but I'm not sure.

[Jari Häkkinen]

I think the accurate way is better. When it is time to optimize speed then we could create a faster class.

[Peter]

OK, let's include this in milestone:"yat 0.8"

[Peter]

(In [2558]) refs #671. Follow Knuth's algorithm to calculate average and variance online

[Peter]

(In [2559]) minor speed-up. refs #671

[Peter]

(In [2561]) refs #671. Correct so mean is NaN when there is no data

[Peter]

(In [2562]) refs #671. Reimplement AveragerWeighted?

[Peter]

Just for the record, here's a paper on the subject: ​http://www.janinebennett.org/index_files/ParallelStatisticsAlgorithms.pdf

[Peter]

(In [2565]) refs #671

[Peter]

(In [2568]) refs #671

[Peter]

(In [2569]) refs #671 optimization

[Peter]

(In [3039]) avoid building static when static is disabled which might indicate static archives are not available on system. refs #671