Implement fluctuation complexity

[kahaaga]

What's this?

The fluctuation complexity characterizes the deviations of the information content of individual states from some summary statistic (information measure) computed for the same distribution. I've here generalized the original measure, which uses Shannon entropy, to be compatible with any InformationMeasure.

Code changes

• Updated changelog and version.
• A bibtex entry for the original reference.
• The FluctuationComplexity type, which can be used with information in combination with any estimation method, just like the other entropies & friends.
• A documentation entry.
• Analytical tests for the case of using Shannon entropy as the summary statistic.
• Note: There's no information_maximum method for now, because I couldn't see any of generally computing the maximum value without some finicky additions to the codebase, which I don't want to deal with now.

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Codecov Report

Attention: Patch coverage is 90.00000% with 1 line in your changes missing coverage. Please review.

Project coverage is 91.97%. Comparing base (faa492a) to head (f1464dd). Report is 18 commits behind head on main.

Files	Patch %	Lines
...tion_measure_definitions/fluctuation_complexity.jl	90.00%	1 Missing :warning:

Additional details and impacted files

```diff @@ Coverage Diff @@ ## main #409 +/- ## ========================================== + Coverage 89.29% 91.97% +2.67% ========================================== Files 79 86 +7 Lines 2271 2468 +197 ========================================== + Hits 2028 2270 +242 + Misses 243 198 -45 ```

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[Datseris]

Is this really an Information measure or a ComplexityEstimator? Because it sounds like the latter. What axiomatic definition of an information measure does this satisfy? It isn't an entropy, or extropy, as far as I can tell.

[kahaaga]

Is this really an Information measure or a ComplexityEstimator? Because it sounds like the latter. What axiomatic definition of an information measure does this satisfy? It isn't an entropy, or extropy, as far as I can tell.

It is a functional of a PMF, so it is an information measure according to our definition.

We do not impose and demands on the axiomatic foundation of a measure in our API for it to quality as an information measure. Only that it is a functional of a PMF. Entropies and entropies are examples of information, measures, but not the only ones. The source of this method themselves call it a "information complexity measure":

I think we should stick with this simple distinction: non-probabilities-based or probabilities-based. Upstream, I've also named all measures that are functions of probabilities "information measures" (KL divergence, relative entropy, mutual info, etc) too, and they are computed using the information function, just with more input datasets.

[kahaaga]

To add: this is an information measure in our framework because it is something that can be estimated using any outcome space, any probabilities estimator and any generic information measure estimator (as a consequence of being a probabilities functional):

information(Jackknife(FluctuationComplexity()), BayesianRegularization(), OrdinalPatterns(), x) 

whereas things like SampleEntropy cannot be estimated in this way.

[Datseris]

okay, sounds good. distinction based on "functional of PMF or PDF" is fine for me.

[kahaaga]

okay, sounds good. distinction based on "functional of PMF or PDF" is fine for me.

Good, then we're on the same page 👍