New estimator: recurrence measure of conditional dependence

[gabriel-ferr]

Describe the feature you'd like to have
Probably we can estimate recurrence measure of conditional dependence (RMCD) using RMA.

Cite scientific papers related to the feature/algorithm

Recurrence measure of conditional dependence and applications, Phys. Rev. E 95, 052206 (2017), doi: 10.1103/PhysRevE.95.052206

Detecting causality in coupled systems using recurrence analysis and maximum entropy principle, Phys. A 697, 131707 (2026), doi: 10.1016/j.physa.2026.131707

If possible, sketch out an implementation strategy
Maybe, can we use the Associations.jl here?

[Datseris]

Maybe, can we use the Associations.jl here?

I would argue it should be the other way around. This new measure should go into Associations.jl instead (and you can refernece it in the documentation here).

[gabriel-ferr]

Fine. As soon as possible, I'll fork Associations.jl and try to implement it. I'll probably also need to make some changes to RecurrenceMicrostatesAnalysis.jl to support distributions computed from a JRP, as well as allow the computation for more than two time series (I'll open another issue for it later).

[Datseris]

Hm, yes it is not clear where the measure should go actually. Since the measure is computed from a single JRP, but the JRP itself is the product of two timeseries. In JuliaDynamics we made the design decision that all measures that are computed from a single timeseries (univariate) go into ComplexityMeasures.jl and those that need 2 or more timeseries go into Associations.jl.

[gabriel-ferr]

No, we need to compute more than one JRP. This measure is defined as

$\mathcal{I}_{RMCD}(X,Y|Z)=\frac{1}{N} \sum_i p_i^{X,Y,Z} \log\left(\frac{p_i^{X,Y|Z}}{p_i^{X|Z}p_i^{Y|Z}}\right),$

where $p_i^{X,Y,\ldots}$ is the recurrence rate of column (or row) $i$ of the JRP constructed from $X, Y, \ldots$, and

$p_i^{X|Y}=\frac{p_i^{X,Y}}{p_i^Y}.$

It captures the influence of $X$ on $Y$ while excluding the influence of $Z$. Technically, we can also exclude the influence of multiple lags by computing $\mathcal{I}_{RMCD}(X^{\tau},Y|Y^{\tau},Y^{\tau-1},Y^{\tau+1},...)$, but this requires calculating even more JRPs. With RMA, we can estimate the recurrence rate of each column using only $1\times 1$ microstates, so I think implementing this should be very simple 🙂. It is similar to construct a RMA distribution for each column of the RP.

Then, we need to compute the JRPs for $(X,Y,Z)$, $(X,Z)$, and $(Y,Z)$ in the simplest case. Using RMA, this means estimating the recurrence rate of each column by restricting the sampling of the microstate distribution to that column in each JRP.

Just a note, from the paper:

Whenever $\tau=\tau'$, RMCD incorporates the recurrence behavior into the transfer entropy definition, i.e., it captures the influence of the past of $X^\tau$ into the present of $Y$ by excluding any contemporaneous self-influence of $Y^\tau$.