Recurrence Plot Metrics Calculator

Formula used

Build delay-embedded vectors: v(i) = [x(i), x(i+τ), …, x(i+(m−1)τ)]. Compute pairwise distances d(i,j) using the chosen metric. A recurrence occurs when d(i,j) ≤ ε, excluding indices within the Theiler window.

• RR = recurrence points / possible points.
• DET = points forming diagonal lines (≥ lmin) / recurrence points.
• L = average diagonal line length; Lmax = maximum diagonal length; DIV = 1/Lmax.
• ENTR = Shannon entropy of diagonal-length distribution (natural log).
• LAM = points forming vertical lines (≥ vmin) / recurrence points.
• TT = average vertical line length; Vmax = maximum vertical length.
• Trend estimates slope of recurrence density across diagonals.

How to use this calculator

1. Paste your time-series values (comma or newline separated).
2. Choose m and tau to reconstruct state vectors.
3. Select normalization and a distance metric appropriate for your data.
4. Pick a threshold mode: set ε directly or target a recurrence rate.
5. Set lmin and vmin to filter short line artifacts.
6. Press Calculate Metrics. Results appear above the form.
7. Use the download buttons to export CSV or PDF reports.

Example data table

Try target RR when you want comparable densities across different signals.

Notes and limitations

• Large inputs can be slow due to the distance matrix cost.
• Normalization can strongly affect distance-based thresholds.
• Trend is a relative indicator, not an absolute diagnostic.
• For noisy series, consider increasing lmin and vmin.

Professional article

1) Recurrence analysis in physics

Many physical systems evolve on attractors that are not obvious in raw measurements. Recurrence analysis examines when a trajectory returns near a previous state, helping characterize oscillations, intermittency, and regime shifts in laboratory, geophysical, and space-plasma data.

2) Phase-space reconstruction choices

The calculator reconstructs state vectors with embedding dimension m and delay τ. In practice, start with m=2–4 and τ near the first decorrelation time. Too-small m can mix states, while too-large m requires more samples to stay reliable.

3) Normalization and distance metric

Because thresholds use distances, scaling matters. Z-score normalization supports fair comparisons across sensors with different amplitudes. Euclidean distance is common, Manhattan can be robust to spikes, and Chebyshev highlights worst-component deviations in multicomponent signals.

4) Threshold selection with ε or target RR

A fixed ε is suitable when units and noise levels are stable. Targeting a recurrence rate can standardize density across datasets. Many studies begin with RR in the 1–10% range; for example, RR=0.05 often yields readable structures without saturating the matrix.

5) Recurrence rate and sampling effects

RR is the fraction of recurrent pairs after excluding the Theiler window. Increasing sampling frequency can raise apparent RR by creating near-duplicates, so a modest Theiler window (often 1–5 samples) reduces trivial matches and improves physical interpretability.

6) Determinism and diagonal structures

Diagonal lines represent similar evolution over time. DET reports the share of recurrence points belonging to diagonal lines of length at least lmin. With clean periodic motion, DET can approach 1. With stochastic forcing, diagonals fragment and DET drops, especially when lmin is increased.

7) Laminarity, trapping time, and entropy

Vertical lines indicate slow change or laminar phases. LAM and TT increase with intermittency, sticking, or plateau-like segments in the signal. ENTR summarizes the diversity of diagonal lengths: broader distributions raise entropy, while narrow, regular patterns reduce it.

8) Reporting workflow for reproducible studies

A practical workflow is to document m, τ, metric, normalization, Theiler window, threshold mode, and line filters (lmin, vmin). Export CSV for batch comparison and PDF for reports. When comparing conditions, keep parameters fixed and interpret changes in RR, DET, LAM, and TT alongside domain knowledge and uncertainty sources.

For experimental datasets, run sensitivity checks: vary ε or target RR by ±20% and confirm qualitative trends persist. Use the exports to track parameter sets, then compare metrics across temperatures, forcing levels, or boundary conditions.

FAQs

1) What is a recurrence plot in simple terms?

It is a matrix marking when two reconstructed states are close under a chosen distance and threshold. Repeating patterns form lines and textures that reflect periodicity, chaos, or laminar phases.

2) How do I pick embedding dimension and delay?

Start with m=2–4 and τ near a decorrelation time or first minimum of mutual information. Increase m only if you have enough samples and results remain stable across small parameter changes.

3) Should I use fixed ε or target recurrence rate?

Use fixed ε when units and noise are consistent. Use target RR to standardize density across datasets. RR around 0.01–0.10 is common; adjust until the plot is not too sparse or saturated.

4) What does DET tell me physically?

Higher DET means more diagonal structure, indicating more predictable or deterministic evolution over short windows. Lower DET suggests randomness, strong noise, or rapidly changing dynamics breaking diagonal continuity.

5) Why do I need a Theiler window?

Adjacent time indices are naturally similar, especially with high sampling. Theiler exclusion removes near-diagonal pairs, reducing autocorrelation bias so RR and line-based metrics reflect genuine recurrences rather than trivial repeats.

6) What do LAM and TT indicate?

They measure vertical-line structure. Higher LAM and longer TT suggest laminar behavior, intermittency, or trapping near quasi-stationary states. They are useful for detecting pauses and regime sticking in physical signals.

7) How many samples do I need for stable metrics?

More is better because pair counts scale with length. As a rule, ensure N is large compared with (m−1)τ. If metrics change strongly with small parameter tweaks, increase samples or reduce m and τ.