ComplexityMeasures.jl

ComplexityMeasures.jl

ComplexityMeasures.jl is a Julia-based software for calculating 1000s of various kinds of probabilities, entropies, and other so-called complexity measures from a single-variable input datasets. For relational measures across many input datasets see its extension CausalityTools.jl. If you are a user of other programming languages (Python, R, MATLAB, ...), you can still use ComplexityMeasures.jl due to Julia's interoperability. For example, for Python use juliacall.

A careful comparison with alternative widely used software shows that ComplexityMeasures.jl outclasses the alternatives in several objective aspects of comparison, such as computational performance, overall amount of measures, reliability, and extendability. See the associated publication for more details.

The key features that it provides can be summarized as:

• A rigorous framework for extracting probabilities from data, based on the mathematical formulation of probability spaces.
• Several (12+) outcome spaces, i.e., ways to discretize data into probabilities.
• Several estimators for estimating probabilities given an outcome space, which correct theoretically known estimation biases.
• Several definitions of information measures, such as various flavours of entropies (Shannon, Tsallis, Curado...), extropies, and other complexity measures, that are used in the context of nonlinear dynamics, nonlinear timeseries analysis, and complex systems.
• Several discrete and continuous (differential) estimators for entropies, which correct theoretically known estimation biases.
• An extendable interface and well thought out API accompanied by dedicated developer documentation. This makes it trivial to define new outcome spaces, or new estimators for probabilities, information measures, or complexity measures and integrate them with everything else in the software without boilerplate code.

ComplexityMeasures.jl can be used as a standalone package, or as part of other projects in the JuliaDynamics organization, such as DynamicalSystems.jl or CausalityTools.jl.

To install it, run import Pkg; Pkg.add("ComplexityMeasures").

All further information is provided in the documentation, which you can either find online or build locally by running the docs/make.jl file.

Previously, this package was called Entropies.jl.

StateSpaceSet{D, T} <: AbstractStateSpaceSet{D,T}

A dedicated interface for sets in a state space. It is an ordered container of equally-sized points of length D. Each point is represented by SVector{D, T}. The data are a standard Julia Vector{SVector}, and can be obtained with vec(ssset::StateSpaceSet). Typically the order of points in the set is the time direction, but it doesn't have to be.

When indexed with 1 index, StateSpaceSet is like a vector of points. When indexed with 2 indices it behaves like a matrix that has each of the columns be the timeseries of each of the variables. When iterated over, it iterates over its contained points. See description of indexing below for more.

StateSpaceSet also supports almost all sensible vector operations like append!, push!, hcat, eachrow, among others.

Description of indexing

In the following let i, j be integers, typeof(X) <: AbstractStateSpaceSet and v1, v2 be <: AbstractVector{Int} (v1, v2 could also be ranges, and for performance benefits make v2 an SVector{Int}).

• X[i] == X[i, :] gives the ith point (returns an SVector)
• X[v1] == X[v1, :], returns a StateSpaceSet with the points in those indices.
• X[:, j] gives the jth variable timeseries (or collection), as Vector
• X[v1, v2], X[:, v2] returns a StateSpaceSet with the appropriate entries (first indices being "time"/point index, while second being variables)
• X[i, j] value of the jth variable, at the ith timepoint

Use Matrix(ssset) or StateSpaceSet(matrix) to convert. It is assumed that each column of the matrix is one variable. If you have various timeseries vectors x, y, z, ... pass them like StateSpaceSet(x, y, z, ...). You can use columns(dataset) to obtain the reverse, i.e. all columns of the dataset in a tuple.