Embedded Dimension and Time Series Length. Practical Influence on Permutation Entropy and Its Applications

Abstract

Permutation Entropy (PE) is a time series complexity measure commonly used in a variety of contexts, with medicine being the prime example. In its general form, it requires three input parameters for its calculation: time series length N, embedded dimension m, and embedded delay τ . Inappropriate choices of these parameters may potentially lead to incorrect interpretations. However, there are no specific guidelines for an optimal selection of N, m, or τ , only general recommendations such as N > > m ! , τ = 1 , or m = 3 , … , 7 . This paper deals specifically with the study of the practical implications of N > > m ! , since long time series are often not available, or non-stationary, and other preliminary results suggest that low N values do not necessarily invalidate PE usefulness. Our study analyses the PE variation as a function of the series length N and embedded dimension m in the context of a diverse experimental set, both synthetic (random, spikes, or logistic model time series) and real–world (climatology, seismic, financial, or biomedical time series), and the classification performance achieved with varying N and m. The results seem to indicate that shorter lengths than those suggested by N > > m ! are sufficient for a stable PE calculation, and even very short time series can be robustly classified based on PE measurements before the stability point is reached. This may be due to the fact that there are forbidden patterns in chaotic time series, not all the patterns are equally informative, and differences among classes are already apparent at very short lengths.

Algorithmics, Possibilities and Limits of Ordinal Pattern Based Entropies

“…In this context, also aspects of data simulation are touched. In a certain sense, this paper is complementary to the recent paper [8] of Cuesta-Frau et. al.…”

“…The problem in the statement above is what sufficiently large means. First of all, in the case of existence of many different ordinal patterns in a system, N must be extremely large in order to realize them all in a time series (for practical recommendations relating N and d, see e.g., [8,9]). Even if arbitrarily long time series would be possible, there would be serious problems to reach the KSE.…”

“…First of all, if a system provides a large variety of ordinal patterns or ordinal pattern words with sufficiently large d or m, then a time series must be extremely long to represent the variety. This naturally bounds useful orders d and word lengths m (see e.g., [8,9] for further information). Even in the case that extremely long series and large d and m would be accessible, complexity estimation can be limited.…”

Permutation Entropy: Enhancing Discriminating Power by Using Relative Frequencies Vector of Ordinal Patterns Instead of Their Shannon Entropy

“…At the opposite end of the spectrum of randomness, Figure 1 b shows the histogram of a sinusoidal time series, with a more polarised distribution of motifs, some of them forbidden (zero probability) [ 22 ]. Figure 1 c,d shows the histograms of other time series with a different degree of determinism, a Logistic (coefficient 3.5) and a Lorenz (parameters 10, 8/3 and 28) time series, respectively [ 23 , 24 ]. When computing the associated entropy to the relative frequencies shown in Figure 1 , the single value obtained will be arguably very different in each case.…”

“…In a general sense, we consider a forbidden pattern with an ordinal pattern with a relative frequency of 0 and a forbidden transition , the impossibility to generate an ordinal pattern j if the previous one was an ordinal pattern i . Forbidden patterns have already demonstrated their usefulness to detect determinism in time series [ 27 , 28 ], and have even been used for classification purposes already [ 22 , 23 ], as additional distinguishing features. Specifically, the present study will use forbidden transitions as a tool to generate synthetic time series including forbidden patterns or certain histogram distributions.…”