Approximate entropy

In statistics, an approximate entropy (ApEn) is a technique used to quantify the amount of regularity and the unpredictability of fluctuations over time-series data.

[1] For example, consider two series of data: Moment statistics, such as mean and variance, will not distinguish between these two series.

Nor will rank order statistics distinguish between these series.

Yet series A is perfectly regular: knowing a term has the value of 1 enables one to predict with certainty that the next term will have the value of 0.

In contrast, series B is randomly valued: knowing a term has the value of 1 gives no insight into what value the next term will have.

Regularity was originally measured by exact regularity statistics, which has mainly centered on various entropy measures.

[1] However, accurate entropy calculation requires vast amounts of data, and the results will be greatly influenced by system noise,[2] therefore it is not practical to apply these methods to experimental data.

ApEn was first proposed (under a different name) by A. Cohen and I. Procaccia,[3] as an approximate algorithm to compute an exact regularity statistic, Kolmogorov–Sinai entropy, and later popularized by Steve M. Pincus.

ApEn was initially used to analyze chaotic dynamics and medical data, such as heart rate,[1] and later spread its applications in finance,[4] physiology,[5] human factors engineering,[6] and climate sciences.

[7] A comprehensive step-by-step tutorial with an explanation of the theoretical foundations of Approximate Entropy is available.

[8] The algorithm is: An implementation on Physionet,[9] which is based on Pincus,[2] use

While a concern for artificially constructed examples, it is usually not a concern in practice.

samples of heart rate equally spaced in time: Note the sequence is periodic with a period of 3.

can be varied without affecting the result).

Form a sequence of vectors: Distance is calculated repeatedly as follows.

, so Similarly, The result is a total of 17 terms

is Note in Step 4,

, and the total number is 16.

At the end of these calculations, we have Then we repeat the above steps for

First form a sequence of vectors: By calculating distances between vector

, we find the vectors satisfying the filtering level have the following characteristic: Therefore, At the end of these calculations, we have Finally, The value is very small, so it implies the sequence is regular and predictable, which is consistent with the observation.

The presence of repetitive patterns of fluctuation in a time series renders it more predictable than a time series in which such patterns are absent.

ApEn reflects the likelihood that similar patterns of observations will not be followed by additional similar observations.

[10] A time series containing many repetitive patterns has a relatively small ApEn; a less predictable process has a higher ApEn.

The advantages of ApEn include:[2] The ApEn algorithm counts each sequence as matching itself to avoid the occurrence of

This step might introduce bias in ApEn, which causes ApEn to have two poor properties in practice:[11] ApEn has been applied to classify electroencephalography (EEG) in psychiatric diseases, such as schizophrenia,[12] epilepsy,[13] and addiction.