The study in is devoted to distinguishing white noise from noisy deterministic time series. Since PE conception, the length N of a time series under analysis using PE has been recommended to be significantly greater than the number of possible order permutations, given by the factorial of the embedded dimension m, that is, m ! 5 m !. ![]() However, as far as we know, there is no study that quantifies the effect of N and its relationship with m on PE applications. Their conclusion was that PE performed best for m = 3, and τ = 2, 3, and proposed to combine those two cases in a single index. The authors explored the effect of m = 3–7 and τ = 1–5 on anaesthetic depth assessment, based on the electroencephalogram. PE parameters have been addressed in works such as in. ![]() Another important conclusion of, strongly related to the present work, is that length has an almost negligible effect on the ability of the entropy measurements to classify records. Although this work acknowledges the extreme difficulty of studying the effect of up to 6 degrees of freedom, and the need for more studies, they were able to conclude that length N should be at least 200 samples for r = 0.2 σ. FuzzyEn and FuzzyMEn are apparently quite insensitive to r values, whereas ApEn exhibits the flip–flop effect (depending on r, the entropy values of two signals under comparison may swap order ). These methods require from 3 up to 6 parameters. The study in, addresses the problem of parameter configuration for ApEn, SampEn, Fuzzy (FuzzyEn), and Fuzzy Measure (FuzzyMEn) entropies in the framework of heart rate variability. The research into this parameter has been extended to other entropy statistics. They also noticed that longer series can have a detrimental effect due to non-stationarities and drifts, and therefore these issues should always be checked in advance. According to their results, SampEn is more stable than ApEn, and the required minimum length should be at least 200 samples. In, an analysis of ApEn and SampEn performance with changing parameters, using short length spatio–temporal gait time series was researched. For SampEn, works such as have focused on optimizing the input parameters for a specific field of application, the estimation of atrial fibrillation organisation. The authors also proposed a method to reduce the computational cost of this approach. For example, ref proposed the computation of all the ApEn results with the tolerance threshold varying from 0 to 1 in order to find its maximum, which leads to a more correct complexity assessment. Since the first widely used methods, such as Approximate Entropy (ApEn), or Sample Entropy (SampEn), the characterization of this influence has become a topic of intense research. If the selected values do not match the intended purpose or application, the results can be completely meaningless. The influence of input parameters on the performance of entropy statistics is a well known issue. 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. 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. ![]() ![]() 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. 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. 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. Inappropriate choices of these parameters may potentially lead to incorrect interpretations. In its general form, it requires three input parameters for its calculation: time series length N, embedded dimension m, and embedded delay τ. Permutation Entropy (PE) is a time series complexity measure commonly used in a variety of contexts, with medicine being the prime example.
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