Lempel–Ziv complexity measure

[Datseris]

New complexity measure:

https://en.wikipedia.org/wiki/Lempel%E2%80%93Ziv_complexity

The wiki website also has an implementation. This is really easy to implement as far as I can tell from the algorithm section at the end.

[kahaaga]

If we're including the LZ-complexity here, perhaps we should also do the same for the compression complexity (part of open PR in CausalityTools.jl)?

If so, I can open a PR that moves basic compression complexity part here.

[Datseris]

here's the distinction I propose:

• if a measure has one input data argument, it is here
• if it has two or more, then it can be interpreted as a relational measure and goes into causalitytools.

Waht do you think?

[kahaaga]

Since the LZ-complexity operates on binary sequences, we should also consider integration with the probabilities estimators. Any estimator that internally encodes the input to integers can be used to convert a raw (multivariate) timeseries into a binary sequence. We just have to restrict the number of encodes symbols to 2

[Datseris]

We just have to restrict the number of encodes symbols to 2

I am not sure what this means. No encoding that we have at the moment can do this. THey all have more than 2 symbols (integers)

[kahaaga]

I am not sure what this means. No encoding that we have at the moment can do this. THey all have more than 2 symbols (integers)

Some of the estimators, through keyword arguments, can result in an outcome space with cardinality 2. Any sequence of such outcomes can be interpreted as a binary sequence. For example, one could use SymbolicPermutation(m=2) to convert a real-valued time series into a binary sequence. This is of course not the case for all estimators. Perhaps it's best to let the user encode themselves.

[kahaaga]

here's the distinction I propose:

• if a measure has one input data argument, it is here
• if it has two or more, then it can be interpreted as a relational measure and goes into causalitytools.

Waht do you think?

Yes, that makes sense.

The compression complexity causality algorithm uses two concepts:

1. The effort-to-compress compression complexity, which in the PR I've implemented as compression_complexity(x, EffortToCompress()), where x can be both univariate and multivariate. With the ComplexityMeasures.jl 2.X API, this becomes complexity(EffortToCompress(), x), which is parallel to e.g. complexity(LempelZiv(), x).

2. The joint effort-to-compress compression complexity. This has two inputs, i.e. complexity(EffortToCompress(), x, y), and can, as you say, therefore be considered as some sort of association measure.

I think the former belongs here, and the latter belongs in CausalityTools. Agree, @Datseris?

[Datseris]

Perhaps it's best to let the user encode themselves.

I think this is the best. You can add a note to the docstring about this in the docstring.

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I think the former belongs here, and the latter belongs in CausalityTools. Agree, @Datseris?

Yeap, CausalityTools.jl can simply extent the method.