Poincaré Plot

// 01 — The chart

What it looks like

// 02 — Definition

What is a Poincaré plot?

// 03 — When to use

When a Poincaré plot is the right call

Reach for a Poincaré plot whenever the question is about how the next sample relates to the current one. In cardiology that is the canonical short-term HRV question; in dynamics it is a quick visual stationarity check.

// 04 — When not to use

When a Poincaré plot is the wrong call

The chart only makes sense when the data is genuinely a time series with a stable lag. Outside that, simpler charts almost always communicate better.

// 05 — Data requirements

What your data needs to look like

Before building the chart, your dataset needs to fit a specific shape. Use this checklist to confirm yours does.

// 06 — Anatomy

Parts of a Poincaré plot

Five elements make up every Poincaré plot: two equal-scale axes, the line of identity, the point cloud, and the SD1 / SD2 ellipse drawn over it.

// 07 — Step-by-step

Step-by-step: how to build a good Poincaré plot

Eight steps that work regardless of tool. The artifact filter and the SD1 / SD2 ellipse are the two non-trivial moves; the rest is mechanical once the data is in shape.

1. 1

   Confirm the series is genuinely a time series

   Order matters. If the rows are unordered or the time gaps are irregular, the lag transform is meaningless. Sort by time and either verify constant sampling or document the gap distribution before plotting.
2. 2

   Choose the lag

   The default lag is 1 — plot x(n) against x(n+1). Larger lags expose periodicity (lag = period gives a tight diagonal). Stick with lag 1 for the canonical ‘Poincaré plot’ unless you have a reason to deviate.
3. 3

   Filter or annotate non-stationary segments

   Drift, regime changes, and ectopic beats produce off-diagonal clusters that swamp the variability story. Either filter them or color them differently so readers can see what’s signal vs artifact.
4. 4

   Draw the line of identity

   Add a 45-degree dashed line where x = y. Without it, the cloud has no anchor and SD1 / SD2 mean nothing geometrically.
5. 5

   Compute SD1 and SD2

   SD1 = std(x − y) / sqrt(2) and SD2 = std(x + y) / sqrt(2). They project the point cloud onto the perpendicular and parallel directions of the identity line.
6. 6

   Overlay the SD1 / SD2 ellipse

   Draw an ellipse centered at the cloud mean with semi-axes equal to SD1 (perpendicular to identity) and SD2 (along identity), rotated 45 degrees. This is the classical Poincaré ellipse.
7. 7

   Use partial transparency for overplotting

   Even moderate-sized series produce overplotting. Use alpha around 0.3–0.5 so density becomes visible without losing the underlying scatter.
8. 8

   Title the takeaway

   Replace “Poincaré plot of RR intervals” with “Tight elliptical cloud, SD1 = 38 ms, SD2 = 92 ms — healthy heart rate variability” so the chart announces the conclusion.

// 08 — Real-world examples

Where you’ll see Poincaré plots used

Cardiology is the headline application but Poincaré geometry shows up wherever a single time series needs a quick visual summary of short-term variability.

// 09 — Variations

Variants of the Poincaré plot

The basic chart has a few common variants — lag choice, density encoding, and segment overlay — each suited to a slightly different question.

// 10 — Comparisons

How it compares

Poincaré plots sit at the intersection of scatter plots and time-series visualization. The most useful comparisons are with general scatter plots, the broader lag-plot family, and recurrence plots from nonlinear dynamics.

// 11 — Common mistakes

Common Poincaré plot mistakes

The chart’s simplicity is deceptive. Most failures come from forgetting that order matters, that artifacts dominate, or that the geometry depends on equal axis scales.

// 12 — Accessibility

Accessibility checklist

Run through this list before publishing. The chart should still communicate its message to readers using assistive technology, color-blind users, keyboard navigation, and reduced-motion settings.

• ✓

  Color contrast for points and reference lines

  WCAG 1.4.3

  The point cloud should reach 3:1 contrast against the chart background; the line of identity, ellipse outline, and any annotations should reach 4.5:1 so they remain visible at thumbnail size.
• ✓

  Don’t encode meaning by color alone

  WCAG 1.4.1

  When you split the cloud (e.g., normal vs. ectopic beats), use both color and a marker shape so color-vision-deficient readers can still distinguish the groups.
• ✓

  Provide a meaningful text alternative

  WCAG 1.1.1

  An accessible name should describe the takeaway: “Tight elliptical cloud along the identity line; SD1 = 38 ms, SD2 = 92 ms, indicating healthy short- and long-term heart rate variability.”
• ✓

  Expose the underlying time series

  WCAG 1.3.1

  Offer a downloadable CSV of the original RR intervals plus the computed SD1 / SD2. The Poincaré representation is essentially impossible to experience without sight.
• ✓

  Label SD1 and SD2 explicitly

  WCAG 3.3.2

  Caption the chart with both the values and the units (typically ms). Numbers without labels are meaningless to anyone unfamiliar with HRV conventions.
• ✓

  Respect prefers-reduced-motion

  WCAG 2.3.3

  Avoid animated unfolding of the cloud unless the animation is gated behind a prefers-reduced-motion: no-preference media query.
• ✓

  Resizable and zoomable

  WCAG 1.4.4

  Use a responsive viewBox so the chart remains legible at 200% browser zoom. Make sure the ellipse outline and identity line stay visible at every container width.

// 13 — Best practices

Design and craft tips

A short list of dos and don’ts that consistently separate publication-quality Poincaré plots from the demo-grade ones produced by default in most scripting environments.

// 14 — Alternatives

Related and alternative charts

If a Poincaré plot isn’t quite what you need, these adjacent time-series visualizations cover similar ground.

// 15 — Tool instructions

How to build it in your tool of choice

Recipes for the libraries and clinical tools that ship a Poincaré implementation. RHRV and Kubios are the closest things to canonical references.

// 16 — Code examples

Working code in the most common stacks

Three runnable snippets that produce equivalent Poincaré plots in Python, R, and JavaScript. Each computes SD1 and SD2 alongside the visual.

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Ellipse

rng = np.random.default_rng(42)
n = 600

# Simulated RR intervals (ms): mean ~880 with realistic short-term variability.
rr = 880 + np.cumsum(rng.normal(0, 6, n))           # slow drift
rr += rng.normal(0, 18, n)                          # short-term variability

x = rr[:-1]
y = rr[1:]

mu_x, mu_y = x.mean(), y.mean()
sd1 = np.std(x - y, ddof=1) / np.sqrt(2)
sd2 = np.std(x + y, ddof=1) / np.sqrt(2)

fig, ax = plt.subplots(figsize=(6.5, 6.5))
ax.scatter(x, y, s=12, alpha=0.4, color="#c94a2e")

lo, hi = rr.min() - 30, rr.max() + 30
ax.plot([lo, hi], [lo, hi], "--", color="#6b6b67", alpha=0.5, lw=1)

ellipse = Ellipse(
    (mu_x, mu_y),
    width=2 * sd2, height=2 * sd1,
    angle=45,
    fill=False, edgecolor="#1a1a18", lw=1.2, linestyle=":",
)
ax.add_patch(ellipse)

ax.set_xlim(lo, hi); ax.set_ylim(lo, hi); ax.set_aspect("equal")
ax.set_xlabel("RR(n) [ms]")
ax.set_ylabel("RR(n+1) [ms]")
ax.set_title(
    f"Healthy HRV: SD1 = {sd1:.0f} ms, SD2 = {sd2:.0f} ms, "
    f"ratio = {sd1 / sd2:.2f}",
    loc="left",
)
ax.spines[["top", "right"]].set_visible(False)
plt.tight_layout()
plt.savefig("poincare.png", dpi=200, bbox_inches="tight")
plt.show()

// 17 — FAQs

Frequently asked questions

// 18 — References

References and further reading

Primary sources, official library documentation, and the international HRV standards cited throughout this guide.

• Brennan, M., Palaniswami, M., Kamen, P. — Do existing measures of Poincaré plot geometry reflect non-linear features of heart rate variability? IEEE Trans. Biomed. Eng. (2001)Primary source
  The classical paper deriving SD1 and SD2 as projections onto the identity line and discussing what Poincaré geometry actually captures.
  https://ieeexplore.ieee.org/document/959330
• Tulppo, M.P., Mäkikallio, T.H., Takala, T.E., Seppänen, T., Huikuri, H.V. — Quantitative beat-to-beat analysis of heart rate dynamics during exercise (1996)Primary source
  One of the foundational HRV papers introducing SD1 / SD2 in a clinical exercise-physiology context.
  https://journals.physiology.org/doi/10.1152/ajpheart.1996.271.1.H244
• Wikipedia — Poincaré plotReference
  Encyclopedia entry covering construction, SD1 / SD2, and applications across cardiology, signal processing, and dynamical systems.
  https://en.wikipedia.org/wiki/Poincar%C3%A9_plot
• RHRV documentationDocs
  Official documentation for the R package most widely used to compute Poincaré metrics and render the chart.
  https://rhrv.r-forge.r-project.org/
• Kubios HRV — user guideDocs
  Documentation for the Kubios HRV analysis software, the de-facto clinical tool that ships Poincaré plots out of the box.
  https://www.kubios.com/
• Task Force of ESC and NASPE — Heart rate variability: standards of measurement, physiological interpretation, and clinical use (1996)Standard
  International HRV standards document. Defines the conventions Poincaré plots adhere to in clinical practice.
  https://academic.oup.com/eurheartj/article/17/3/354/485572
• Goldberger, A.L. et al. — PhysioBank, PhysioToolkit, and PhysioNet (2000)Dataset
  The open repository of physiological signals (including RR-interval datasets) used to teach and test Poincaré plot pipelines.
  https://www.physionet.org/
• WAI — Complex Images: Charts and GraphsAccessibility
  Web Accessibility Initiative guidance on text alternatives, long descriptions, and data tables for complex scientific charts.
  https://www.w3.org/WAI/tutorials/images/complex/
• Wikipedia — Henri PoincaréReference
  Background on Henri Poincaré and his foundational work on dynamical systems, after whom the chart is named.
  https://en.wikipedia.org/wiki/Henri_Poincar%C3%A9