Approximate entropy

Approximate entropy is a measure of how regular a series of heartbeat intervals is: the average log-likelihood that runs of beats which are close for $m$ beats stay close for $m+1$. It has no unit.

Why this measure exists

Steven Pincus proposed it in 1991 as a way to put a number on the complexity of a system from a relatively short, noisy record, a thousand points or so, at a time when the fashionable tool, the correlation dimension, needed far more data and could be fooled into announcing chaos. Applied to heart rhythm it ranks a steady, repeating series low and an unpredictable one high. Richman and Moorman showed in 2000 that it has two flaws: each run is counted as matching itself, which guarantees some matches and biases the result towards regularity, and the bias changes with the length of the record, so values from recordings of different lengths are inconsistent. Their sample entropy sample entropy Sample entropy How unpredictable the heartbeat series is: minus the log of the chance that runs of beats that match also match one beat longer. Near 0 is a repeating pattern; 2 or more is close to random. More about Sample entropy was built to remove both, and it is the one to prefer.

The formula

$$\mathrm{ApEn}(m, r) = \Phi^{m}(r) - \Phi^{m+1}(r),\qquad \Phi^{m}(r) = \frac{1}{N-m+1}\sum_{i}\ln C_i^{m}(r)$$

For every run of $m$ beats, count the share $C_i^m(r)$ of all runs (itself included) that lie within $r$ of it, take the log, and average those logs to get $\Phi^m$. Do the same for runs one beat longer, and subtract. $m$ is usually 2 and $r$ is 0.2 times the standard deviation of the intervals.

Worked example

How predictable is the next beat?

Five minutes of the teaching recording

First 64 intervals matches over m and over m + 1

SampEn

1.67

ApEn

1.02

Tolerance r

5.7 ms

m matches

13

m+1 matches

1

Rested, m = 2, r = 0.20 × SD = 5.7 ms. The highlighted run of 2 beats matches 13 other runs within 5.7 ms; 1 of them still matches when both runs are one beat longer. Counted over every run in the recording, sample entropy is 1.67: the breathing rhythm makes the next beat partly predictable.

Periodic Rested Random

Run length m 2 beats Tolerance r 0.20 × SD

The controls work when JavaScript is on.

A line chart of 64 heartbeat intervals in order. A highlighted box marks a template run of m beats with a tolerance band of plus or minus r milliseconds around each of its beats; lighter boxes mark other runs of m beats that match the template within the band, and darker boxes those that still match when both runs are one beat longer. Readouts give sample entropy, approximate entropy, the tolerance in milliseconds, and the two match counts, followed by a sentence explaining them. Two sliders set m (1 to 4 beats) and r (0.1 to 0.6 times the standard deviation); three buttons choose a periodic series, the rested recording or random beats.

Five minutes of the teaching recording give an approximate entropy of 1.02 next to a sample entropy of 1.67, from the same $m = 2$ and $r$ = 5.7 ms. The gap between the two numbers is the self-match bias: the random preset scores 1.10 by ApEn against 2.35 by SampEn. The picture shows one template run, the band of tolerance around it, and the runs that match it.

What moves it

The same things as sample entropy: regular breathing lowers it, irregular breathing, artifacts artifacts Artifact An error in a heartbeat recording, such as a missed, extra or mistimed beat, often caused by movement or poor sensor contact. Artifacts can badly distort HRV numbers. More about Artifact and noise raise it, and $m$, $r$ and the record length all shift it. Because of the self-match bias it sits lower than sample entropy on the same data and falls further as the recording gets shorter.

When it misleads

Whenever recordings of different lengths are compared, and whenever it is read as a pure property of the heart rather than of the record, the parameters and the program. Shaffer and Ginsberg describe large values as low predictability and describe sample entropy as the less biased measure.

Where you will see it

Kubios HRV prints ApEn beside SampEn under its nonlinear measures, both with $m = 2$ and $r = 0.2 \times \mathrm{SDNN}$ by default. Older research papers report it; newer ones use sample entropy. Wearables do not show it.