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GLOSSARY OF TERMS USED IN TIME SERIES ANALYSIS
OF CARDIOVASCULAR DATA
APPROXIMATE ENTROPY

Measure of the regularity or predictability of time series.

To compute the approximate entropy, ApEn, of a time series x(n), n=1,2,...N, first the series of vectors of length m, v(n)=[x(n), x(n+1),...x(n+m-1)]T is derived from the signal samples x(n).  The distance D(i,j) between two vectors v(i) and v(j) is defined as the maximum difference in the scalar components of v(i) and v(j).

Then Nm,r(i), i.e., the number of vectors j (with jN-m+1) such that the distance between the vectors v(j) and the generic vector v(i) (with  iN-m+1) is lower than r,  D(i,j)r, is computed. The index r is a fixed parameter which set the "tolerance" of the comparison.
Let's now define Cm,r(i), the probability to find a vector which differs from v(i) less than the distance r, as:

Cm,r(i)=Nm,r(i)/(N-m+1)

and
i.e., the logarithmic average over all the vectors of the Cm,r(i) probability.

ApEn is given by:

Thus ApEn of a time series x(n) measures the logarithmic likelihood that runs of patterns of length m that are close to each other will remain close in the next incremental comparisons, m+1. A greater likelihood of remaining close (high regularity) produces smaller ApEn values, and, vice-versa, low regularity produces higher ApEn values. The two parameters, m and r, must be fixed to compute approximate entropy. The values m = 2 and r between 10% and 25% of the standard deviation of the data sets x(n) are recommended (Pincus et al, 1994). Heart-rate approximate entropy has been found to decrease with age and to be higher in women than in men (Ryan et al., 1994; Pikkujämsä et al, 1999).
An extension of approximate entropy to quantify the joint irregularity between two time series, the cross-entropy (Cross-ApEn), has been also proposed (Pincus et al., 1996).

Shortcomings of ApEn statistics are discussed in (Richman and Moorman, 2000), where a related complexity measure, sample-entropy (SampEn), is introduced and compared with ApEn.

Links
Approximate Entropy (ApEn) by G.B. Moody
Source Code in Matlab by D.Kaplan

References:
Richman JS, Moorman JR (2000) Physiological time-series analysis using approximate entropy and sample entropy. Am J Physiol
Pikkujamsa SM et al (1999) Cardiac interbeat interval dynamics from childhood to senescence : comparison of conventional and new measures based on fractals and chaos theory. Circulation FREE
Pincus SM et al (1996)Older males secrete luteinizing hormone and testosterone more irregularly, and jointly more asynchronously, than younger males Proc. Natl. Acad. Sci. FREE
Pincus SM, Goldberger AL. (1994) Physiological time-series analysis: what does regularity quantify? Am. J. Physiol.
Ryan SM et al (1994) Gender- and age-related differences in heart rate dynamics: are women more complex than men? J Am Coll Cardiol.
Pincus SM.(1994) Quantification of evolution from order to randomness in practical time series analysis. Methods Enzymol.

(PC 21-08-2001)

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