Is RPDE the same as sample entropy?

No. They use different constructions and count different relationships in a time series.

Does a value of one mean 100% randomness?

No. It means maximum histogram entropy under the selected normalization, not a universal randomness probability.

Can RPDE be compared across recordings?

Only with careful attention to sampling, preprocessing, parameters and estimation uncertainty; sharing a numerical range does not make all calculations directly comparable.

AI Glossary / Mathematics and optimization / Advanced

Recurrence period density entropy

Definition

Recurrence period density entropy, or RPDE, is a time-series measure that summarizes how spread out the observed return times are when a signal's reconstructed state revisits a nearby region. Concentrated return times suggest regular repetition; a broader distribution has greater entropy. The measure was developed for analyzing recurrence and aperiodicity in signals, including research on sustained voice recordings.

What it means

A signal can repeat approximately without reproducing exactly the same sample values. RPDE examines recurrence in a reconstructed state space, where each point combines a sample with delayed samples. It then measures uncertainty in the time needed to return near an earlier state. This differs from calculating entropy of the signal's amplitude histogram or power spectrum. A normalized RPDE value is a feature of the signal and analysis choices, not a probability that a person has a disorder or that a machine has failed.

How it works

Select an embedding dimension, delay, neighborhood radius and recurrence-time range. Construct delay-coordinate vectors from the time series. For a reference point, follow the trajectory until it leaves the radius-defined neighborhood, then identify its first return. Measure the recurrence interval from the reference time to that return, and repeat across eligible points. Normalize the recurrence-count histogram into probabilities P(T). Compute H = −Σ P(T) ln P(T), taking zero-probability terms as zero. The original normalized form divides by ln(Tmax), the maximum entropy for the Tmax possible integer-time bins. Parameters, signal scaling, record length and treatment of absent or truncated returns affect the result and must be documented.

Three practical examples

A speech-research workflow can use RPDE as one feature describing recurrence irregularity in carefully recorded sustained sounds, alongside other measures and independent assessment.
A synthetic-signal study can compare a periodic oscillator with perturbed or noise-contaminated versions under identical analysis settings.
A condition-monitoring experiment can investigate whether return-time dispersion adds useful information about rotating-equipment signals. This is a proposed feature-validation use, not evidence that RPDE alone identifies a specific fault or is suitable for every signal type.

Think of it this way

Think of timing how long it takes a walker to return near a marked location after leaving it. If the return interval is nearly always the same, the schedule is predictable; if intervals vary widely, it is less predictable. The analogy captures recurrence-time uncertainty. In RPDE the location is a delay-coordinate state, not a physical point, and both the neighborhood definition and sampling clock influence what counts as a return.

Worked example

Suppose an illustrative analysis uses integer recurrence-time bins 1 through 4 and records counts [0, 4, 2, 2]. There are eight recorded returns, so P = [0, 0.5, 0.25, 0.25]. The entropy is −0.5 ln(0.5) − 0.25 ln(0.25) − 0.25 ln(0.25) = 1.5 ln(2), approximately 1.0397 nats. Divide by ln(4) = 2 ln(2) to obtain normalized RPDE 0.75. If all eight returns instead occur at lag 3, P = [0, 0, 1, 0] and the entropy is zero. If the counts are [2, 2, 2, 2], the entropy is ln(4) and the normalized value is one. The calculation uses the full declared four-bin range, including the empty bin; normalizing by the three occupied bins would produce a different statistic. These are invented recurrence histograms for explaining the formula, not measured voices or a demonstration that every random signal produces a uniform histogram.

Illustrative case study

In a fictional project, Alder Acoustics evaluates recurrence features on a collection of synthetic and consented research recordings. The analysts find that changing recording duration and neighborhood radius changes RPDE values, so they standardize those settings and test sensitivity before comparing groups. They keep feature development separate from the final evaluation recordings and avoid interpreting one number as a diagnosis. The case illustrates reproducibility and evaluation design without inventing clinical performance or claiming that the feature generalizes beyond the tested recording conditions.

Common misconceptions and mistakes

High RPDE does not prove chaos, randomness, disease or mechanical damage. Several different dynamics and measurement artifacts can produce dispersed return times. Low entropy means concentration in the measured recurrence distribution, not that every aspect of the signal is simple. Remaining inside the neighborhood on consecutive samples is not the same as leaving and returning; failing to handle this can distort the lag histogram. The maximum entropy occurs for a uniform distribution over the declared bins, but real noise need not satisfy the special assumptions used to approximate that distribution. Normalization does not remove every dependence on sampling or parameter selection.

Business use

Use RPDE as a defined feature in a validated signal-analysis workflow. Preserve sample rate, preprocessing, amplitude normalization, embedding settings and return-search limits. Check whether enough eligible returns exist to support a stable estimate. Compare repeated measurements and synthetic controls so that apparent group differences are not driven by recording artifacts or analysis choices. When the intended application changes, reevaluate the feature's usefulness rather than carrying over a threshold from unrelated research. For decision reports, describe the score as recurrence-time dispersion and state the evidence linking it to the specific operational outcome being studied.

Check your understanding

All recorded recurrence intervals fall in one bin, with several other declared bins empty. What is the entropy?

Answer: Zero, because the occupied bin has probability one and contributes −1 ln(1) = 0. This statement concerns the recorded return-time distribution, not a complete characterization of the signal.

Related terms

recurrence plot
delay embedding
Shannon entropy
aperiodicity
return time
nonlinear time-series analysis

Learn next

calculate the entropy of the three example histograms, then analyze a synthetic sinusoid with a fixed sample rate and settings. Add controlled noise and vary the neighborhood radius separately. Record how both histogram shape and normalized entropy change before interpreting the score.

Fixed sample

Sources

Sources: Max A. Little and colleagues, Exploiting Nonlinear Recurrence and Fractal Scaling Properties for Voice Disorder Detection, recurrence-analysis section and equations 5–12, https://arxiv.org/pdf/0707.0086 ; published article, https://doi.org/10.1186/1475-925X-6-23 . The four-bin calculation and fictional research workflow are original illustrations; no clinical decision threshold is asserted.

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