# Entropy and Statistical Complexity in Bioelectrical Signals: A Literature Review
## 3 Results
### Permutation Entropy  
Permutation entropy is based on information entropy [43] and was first introduced by [PERSON] and [PERSON] in 2002 [34; 38; 44]. This metric information is widely recognized for its effectiveness, simplicity [28; 42; 44; 45; 46], robustness [28; 35; 44; 45; 46], efficiency [32; 35; 41; 43], and computational speed [42; 44; 46]. Permutation entropy has been referred to by various names depending on the author's perspective. For instance, it has been called PE [28; 34; 42; 43; 44; 45; 47] or also known as PermEn (Permutation Entropy) [14; 35], Permutation Entropy (PerEn) [24], PmEn [46], PeEn [32], Dynamic Entropy [47], or the Shannon Entropy of the ordinal patterns distribution [13]. PE has been regarded as a technique [35; 44], a statistical tool [43], an information metric [42], or a simple approach [47]. It can examine and characterize various types of real-world data [32] or measure [32; 39; 43; 44; 46] the complexity (i) in sequences [46], and time series of appropriate length [14; 39; 43; 44; 47], particularly in neuronal [44] and/or nonlinear context [38; 46]; and (ii) in signals [41; 43; 46], distinguishing periodic, chaotic, and stochastic behaviors [24], as well as characterizing the dynamics of short, non-stationary, and nonlinear signals [44]. It has also been recognized as a tool for quantifying irregularity [24; 47]. Furthermore, PE is robust against artifacts [41], signal-to-noise ratio variations [43], dynamic noise [28], and general noise interference [44; 46]. Symbolic dynamic theory underpins the generation of PE [36]. A sequence of symbols is derived from the data by encoding cardinal relationships [36; 45]. This approach enables the mapping of a time series of real-valued data into a symbolic sequence [47]. It relies on the counting and analysis of ordinal patterns [5; 34; 39; 41; 43; 44; 47], ordinal structures [14], and/or permutation patterns [14; 24; 32], as well as the order relationships of amplitude values [24]. The relative frequency of ordinal patterns forms the basis for PE computation. This is achieved through (i) the organization of adjacent points to evaluate complexity [32] or the probability of patterns [43], and their identification based on increasing or decreasing values [13]; (ii) the comparison of consecutive values within the time series [45] or adjacent time-series values via partitions [35], and/or the analysis of subsequences derived from the time series [39]; or (iii) the encoding of temporal relationships [45] and the ordinal arrangement of the data [43]. Notably, this approach does not require prior knowledge of the underlying distribution [45] or the predictability inferred from the probability distribution [43]. In addition to the aforementioned considerations, two crucial parameters must be defined for the calculation of PE: the embedding dimension m and the time delay \(\tau\)[28; 41]. However, the latter has minimal impact on complexity analysis [28; 48]. Regarding the embedding dimension m, it must be carefully balanced, typically set to 1, although values between 3 and 7 are generally preferred [28]. Subsequently, PE is determined by calculating the entropy S using the estimated probabilities of the ordinal patterns [39]. PE is a measure that reflects the predictability of data series [36]. When PE has a high value, approaching 1, it indicates that the data series is unpredictable [28; 36] and irregular [28]. Conversely, a low PE value, nearing 0, suggests that the data series is highly predictable [36] or regular, with a greater occurrence of one or a few patterns [41; 44]. It has also been observed that PE tends to decrease at lower frequencies, as permutations related to oscillations become less frequent. In contrast, at higher frequencies, PE increases, reflecting greater complexity in the data series under study [36]. Permutation entropy offers considerable advantages over other commonly used entropy measures due to its low computational complexity [28; 35; 44], making it widely utilized and applicable across various contexts.