Recurrence Quantification Analysis

The recurrence microstate analysis allows us to estimate values of typical RQA measures, such as determinism and laminarity, with a good precision, and defines some novel quantifiers. We will demonstate in this page how compute these quantifiers using a uniform distribution as input.

julia> using RecurrenceMicrostatesAnalysis, Distributions        # Generate our data =D
julia> data = rand(Uniform(0, 1), 3000);
julia> th, s = find_parameters(data, 3)(0.2712540022468859, 5.684788996949903)
julia> dist = distribution(data, th, 3);

Recurrence Entropy

To compute the recurrence entropy, it is possible to use the rentropy function that receives a recurrence motif probability distribution.

julia> entr = rentropy(dist)5.681533363657713

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Note that the entropy computed is also returned by the find_parameters function.

The rentropy function has also a keyword argument that can be used to ignore some motifs.

julia> entr = rentropy(dist; ignore_motifs = [1, 512])5.549076236101606

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Note that the kword ignore_motifs uses as index the notation of Julia, beginning in 1, instead 0. So, the motif 0 is identified by the number 1.

Recurrence Rate

The recurrence rate (RR) can be computed using a similar method to the recurrence entropy.

julia> rr = rrate(dist)0.4625747910794636

Since the recurrence rate is an estimated measure, it has a small error, how you can check in the following graphic, that displays the relative error between the RR computed by RecurrenceMicrostatesAnalysis.jl and the standard approach. Relative error of recurrence rate compared to the value computed using the standard method. Panel (f) provides an overview of the error distributions presented in panels (a)-(e).

Determinism

The determinism (DET) can be computed using a recurrence motifs probability distribution and the recurrence rate. It is important to note that it can be done using two motif constrained shapes: :square or :diagonal.

julia> det = determinism(rr, dist)0.7106182486784647
julia> det = determinism(rr, distribution(data, th, 3; shape = :diagonal))0.7120854334475366

Similar to RR, the determinism is a quantifier estimated using recurrence microstates analysis, so it has a small error that is demonstrated in the following figure. Relative error of determinism compared to the value computed using the standard method.

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(i) is the uniform distribution, (ii) is the Lorenz system, (iii) is the Logistic map, (iv) is the Rössler system, and (v) is the Bernoulli shifted generalized.

Warning

Determinism (DET) just can be computed using :square or :diagonal shapes.

We implement an way to do it without the need to compute the recurrence distributions.

julia> det = determinism(data, th)(0.7120381138764347, 0.46239059486723033)

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When we estimate DET directly using this function overload, the library will automatically use a diagonal motif constrained shape.

Laminarity

The laminarity (LAM) can be computed with a method similar to determinism (DET). It is important to note that it can be done using two motif constrained shapes: :square or :line.

julia> lam = laminarity(rr, dist)0.7300716755838004
julia> lam = laminarity(rr, distribution(data, th, 3; shape = :line))0.7310626266081572

In the same way, laminarity has a small error associated to it estimation. You can check it in the next figure. Relative error of laminarity compared to the value computed using the standard method.

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(i) is the uniform distribution, (ii) is the Lorenz system, (iii) is the Logistic map, (iv) is the Rössler system, and (v) is the Bernoulli shifted generalized.

Warning

Laminarity (LAM) just can be computed using :square or :line shapes.

We implement an way to do it without the need to compute the recurrence distributions.

julia> lam = laminarity(data, th)(0.7295071316160557, 0.4623579587598279)

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When we estimate LAM directly using this function overload, the library will automatically use a line motif constrained shape.

Disorder

Disorder is a very powerful quantifier introduced by Flauzino et al.[10] in 2025. The computation of this quantifier is very simple using RecurrenceMicrostatesAnalysis.jl:

julia> microstate_size = 33
julia> Ξ = disorder(data, microstate_size)0.9996368268697874

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