In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, an approximate entropy (ApEn) is a technique used to quantify the amount of regularity and the
unpredictability
Predictability is the degree to which a correct prediction or forecast of a system's state can be made, either qualitatively or quantitatively.
Predictability and causality
Causal determinism has a strong relationship with predictability. Perfe ...
of fluctuations over
time-series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Ex ...
data.
For example, consider two series of data:
: Series A: (0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...), which alternates 0 and 1.
: Series B: (0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, ...), which has either a value of 0 or 1, chosen randomly, each with probability 1/2.
Moment statistics, such as
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
and
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
, will not distinguish between these two series. Nor will
rank order
A ranking is a relationship between a set of items such that, for any two items, the first is either "ranked higher than", "ranked lower than" or "ranked equal to" the second.
In mathematics, this is known as a weak order or total preorder of ...
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.
However, accurate entropy calculation requires vast amounts of data, and the results will be greatly influenced by system noise,
[
] therefore it is not practical to apply these methods to experimental data. ApEn was developed by
Steve M. Pincus to handle these limitations by modifying an exact regularity statistic,
Kolmogorov–Sinai entropy
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special ...
. ApEn was initially developed to analyze medical data, such as heart rate,
and later spread its applications in
finance
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fina ...
,
[
] physiology
Physiology (; ) is the scientific study of functions and mechanisms in a living system. As a sub-discipline of biology, physiology focuses on how organisms, organ systems, individual organs, cells, and biomolecules carry out the chemical ...
,
[
] human factors engineering
Human factors and ergonomics (commonly referred to as human factors) is the application of psychological and physiological principles to the engineering and design of products, processes, and systems. Four primary goals of human factors learnin ...
,
[
] and climate sciences.
Algorithm
A comprehensive step-by-step tutorial with an explanation of the theoretical foundations of Approximate Entropy is available. The algorithm is:
; Step 1: Assume a time series of data
. These are
raw data values from measurements equally spaced in time.
; Step 2: Let
be a positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, with
, which represents the length of a run of data (essentially a
window
A window is an opening in a wall, door, roof, or vehicle that allows the exchange of light and may also allow the passage of sound and sometimes air. Modern windows are usually glazed or covered in some other transparent or translucent materia ...
).
Let
be a positive
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
, which specifies a filtering level.
Let
.
; Step 3: Define