Recurrence Period Density Entropy (RPDE)

• RPDE is a nonparametric measure that quantifies dynamic complexity by embedding time series and analyzing recurrence time distributions.
• It computes the normalized Shannon entropy of the empirical recurrence period density, distinguishing periodic, chaotic, and noise-like behaviors.
• RPDE is applied in biomedical contexts, aiding in the detection of anomalies in voice signals and cardiac dynamics by contrasting healthy and pathological states.

Recurrence Period Density Entropy (RPDE) is a nonparametric, time series–based quantifier of dynamic complexity, originally introduced to objectively characterize aperiodicity and irregularity in signals exhibiting nonstationary or nonlinear dynamics. It is constructed by embedding the scalar time series in a reconstructed phase space, extracting all first recurrence times to a local neighborhood (the “recurrence period”), constructing their empirical probability density function, and then measuring the normalized Shannon entropy of this distribution. RPDE has been demonstrated to have diagnostic utility in biomedical applications, notably for distinguishing pathological from normal voice signals and differentiating healthy from pathological cardiac dynamics by quantifying the degree of deterministic versus stochastic structure in the phase-space trajectories (0707.0086, Mukherjee et al., 2015).

1. Formal Definition and Computational Workflow

Given a scalar time series $x(n)$, $n = 1, \ldots, N$, RPDE is computed via the following steps:

• Phase-Space Embedding:

Each point is embedded as a delay vector in $\mathbb{R}^m$:

$$\mathbf{x}_n = [x(n),\, x(n-\tau),\, \ldots,\, x(n-(m-1)\tau)]^T$$

where $m$ is the embedding dimension and $\tau$ the embedding delay.

• Recurrence Times Extraction:

For each $\mathbf{x}_n$, identify the smallest $j>0$ such that

$\|\mathbf{x}_n - \mathbf{x}_{n+j}\| \leq r$

where $r$ is the recurrence threshold and $\|\cdot\|$ denotes Euclidean norm. The resulting $j$ is the recurrence time $T$.

• Recurrence-Period Density:

The histogram $R(T)$ of observed recurrence times is normalized:

$$P(T) = \frac{R(T)}{\sum_{i=1}^{T_{\max}} R(i)}\,, \qquad \sum_{T=1}^{T_{\max}} P(T) = 1$$

with $T_{\max}$ a finite cutoff, typically chosen to encompass all relevant recurrence intervals.

• Entropy Calculation:

The Shannon entropy of $P(T)$ is computed:

$H = -\sum_{T=1}^{T_{\max}} P(T)\,\ln P(T)$

Normalization yields

$H_{\rm norm} = \frac{H}{\ln T_{\max}}$

with $H_{\rm norm} \in [0,1]$ by construction.

Low $H_{\rm norm}$ ($\approx0$) is indicative of nearly periodic or regular dynamics, while high $H_{\rm norm}$ ($\approx1$) signals highly aperiodic, noise-like, or stochastic behavior (0707.0086, Mukherjee et al., 2015).

2. Parameter Selection and Practical Considerations

The robustness of RPDE depends critically on selecting appropriate embedding parameters $(m, \tau, r)$. Recommended procedures include:

• Time Delay ($\tau$):

Select via the first local minimum of the Average Mutual Information (AMI):

$$\text{AMI}(\tau) = \sum_{i} \Pr(x(i), x(i+\tau)) \cdot \log \bigg( \frac{\Pr(x(i), x(i+\tau))}{\Pr(x(i))\Pr(x(i+\tau))} \bigg)$$

• Embedding Dimension ($m$):

Use False Nearest Neighbors (FNN), increasing $m$ until the fraction of false neighbors drops to near zero.

• Recurrence Threshold ($r$):

Typically set as a small fraction (e.g., $0.1$) of the time series standard deviation; values in $[0.02, 0.5]$ have been used upon normalization to $[-1,1]$.

• Binning ($T_{\max}$):

Chosen to capture all physiologically or physically relevant recurrence times, e.g., $T_{\max} = 1000$.

Values found effective include $m=4$, $\tau=35$, $r=0.12$ for voice signals (0707.0086); for ECG, $m\approx3$–$9$ and $\tau\approx12$–$30$ depending on health status (Mukherjee et al., 2015).

3. Interpretative Framework

RPDE provides a scalar representation of time series complexity and regularity. Two reference cases are:

• Perfectly periodic signals:

$P(T)$ is a Kronecker delta, $H_{\rm norm}\rightarrow0$.

• Purely stochastic/i.i.d. noise:

$P(T)$ is nearly uniform, $H_{\rm norm}\rightarrow1$.

Intermediate values reflect varying degrees of chaoticity. Empirical studies demonstrate that regular/periodic (e.g., healthy heart or normal voice) signals yield significantly lower $H_{\rm norm}$ than pathological or noise-corrupted signals (0707.0086, Mukherjee et al., 2015).

The following table illustrates typical interpretative ranges:

$H_{\rm norm}$ Range	Signal Type	Example Condition
$0 \lesssim H_{\rm norm} \lesssim 0.3$	Periodic / Regular	Type I voice, healthy ECG
$0.4 \lesssim H_{\rm norm} \lesssim 0.8$	Chaotic / Complex	Type II voice, chaotic heart
$0.8 \lesssim H_{\rm norm} \leq 1.0$	Noise-Like / Irregular	Disordered voice, pathological ECG

In practical application, thresholds can be defined by empirical distributions, such as $H_{\rm norm} \geq 0.82$ for healthy heart vs. $H_{\rm norm} \leq 0.52$ for congestive heart failure (Mukherjee et al., 2015).

4. Windowed and Temporal Analysis

To accommodate nonstationary or evolving systems, RPDE can be adapted to a windowed approach:

• The time series $x(1...N)$ is divided into contiguous, nonoverlapping windows of length $L$.
• RPDE is computed independently within each window, yielding $H_{\rm norm}^{(w)}$ for window $w$.
• The mean or trajectory of $H_{\rm norm}^{(w)}$ characterizes temporal variation in complexity.

For biomedical signals such as ECG, window length $L=10^4$ samples is employed, and $W=1000$ windows encompass the record. This averaging procedure preserves discrimination between classes (e.g., normal vs. pathological) while enabling time-resolved assessment (Mukherjee et al., 2015).

5. Biomedical and Applied Usage

RPDE has been validated in several biomedical contexts:

• Voice Pathology Detection:

On a dataset of $\approx 760$ sustained vowels, RPDE (paired with detrended fluctuation analysis in a quadratic discriminant) achieved $91.8\%\pm2.0\%$ overall classification accuracy, with $95.4\%\pm3.2\%$ true positive and $91.5\%\pm2.3\%$ true negative rates, surpassing traditional metrics (jitter, shimmer, HNR) (0707.0086). RPDE values are robust to non-Gaussian noise and highly nonlinear oscillatory phenomena characteristic of disordered vocal fold vibration.

• Cardiac Complexity Analysis:

In ECG applications, windowed RPDE reveals lower complexity (lower $\langle H_{\rm norm} \rangle$) in congestive heart failure compared to healthy heart dynamics. Empirically, $\langle H_{\rm norm} \rangle \geq 0.82$ in healthy subjects and $\langle H_{\rm norm} \rangle \leq 0.52$ in heart failure, with a discriminative gap of $d\approx0.30$ (Mukherjee et al., 2015).

A plausible implication is that RPDE, by measuring average entropic complexity over attractor reconstructions, is sensitive to the shift from deterministic to stochastic dynamical regimes in physiological systems.

6. Limitations and Theoretical Distinction

RPDE is to be distinguished from universal typical-signal entropy estimators based on recurrence/waiting time statistics in symbolic dynamics, as in the context of nonequilibrium statistical mechanics and information theory (Cristadoro et al., 2022). In those works, estimators rely on asymptotic scaling of integer-valued recurrence times and their logarithmic rates, not on constructing return-time density functions or their associated Shannon entropies. Thus RPDE is not directly related to the Shannon entropy rate $s(P)$ via such return-time scaling limits; it instead operates as an instantaneous, empirical entropy of distributional recurrence statistics. No directly comparable bounds or inequalities relating RPDE to $s(P)$ are established.

RPDE’s effective domain remains continuous-valued or real-world signals requiring nonlinear dynamical analysis, rather than discrete-alphabet processes or symbolic shifts (0707.0086, Mukherjee et al., 2015, Cristadoro et al., 2022).

7. Summary Table: Core Steps and Distinctions

Step	Continuous-State RPDE (0707.0086, Mukherjee et al., 2015)	Symbolic Recurrence Time Estimators (Cristadoro et al., 2022)
Embedding	Required (phase-space reconstruction)	Not used
Recurrence Stat	Empirical recurrence time distribution / $P(T)$	First return time $R_n$ (integer sequence)
Entropic Measure	Shannon entropy of $P(T)$ ("RPDE")	Asymptotic logarithmic scaling ($\log R_n/n$)
Domain	Real-valued signals, biomedical/chaotic time series	Symbolic sequences, ergodic processes

The combination of phase-space topology and statistical entropy distinguishes RPDE as a diagnostic and analytic tool in nonlinear signal analysis—particularly in biomedical applications where quantifying the complexity of oscillatory and irregular physiological signals is essential.