The measures of sequence complexity for EEG studies
type: Item
instance of: scholarly article
title: The measures of sequence complexity for EEG studies
zbMATH Open document ID: 0818.92005
DOI: 10.1016/0960-0779(94)90125-2
author: Jinghua Xu
author: Ren Liu
author: Zengrong Liu
published in: Chaos, Solitons and Fractals
publication date: 1995-08-17T00:00:00Z
review text: The authors consider sequences of data which take a finite number of values. The sequences of data form points of a symbol space. They introduce two measures of complexity of sequences, which they call \(c_ 1\) and \(c_ 2\). The complexity measure \(c_ 1\) is related to the entropy of the shift map on the space of allowable data points. It measures the rate of growth of the number of words of length \(n\) in the first \(n \cdot 2^ n\) terms of the sequences. The second measure \(c_ 2\) measures the rate of growth of the number of words of length \(n\) which do not occur, but which have the property that the subword of length \(n - 1\) obtained by right truncation does occur again in the first \(n \cdot 2^ n\) terms of the sequence. The authors point out that for periodic systems \(c_ 1\) and \(c_ 2\) are 0; for `completely random systems' (e.g. the full 2-shift), \(c_ 1\) is non-zero, while \(c_ 2\) is 0. They contend that this means that the dynamics are `simple but not complex', and argue that \(c_ 2\) is the more genuine measure of complexity. They then estimate numerically the values of \(c_ 1\) and \(c_ 2\) as a function of parameter for the family of logistic maps \(x \mapsto \lambda x(1-x)\). The authors then apply the methods introduced to data coming from EEG studies of brain activity. They unfortunately introduce some probabilities (which come from averaging trials), but fail to describe the meaning of these. Using entropy theory, it can be seen that the quantities \(C_ 1\) and \(C_ 2\) which they define converge to 0 as the sequence length is taken to \(\infty\). As \(C_ 1\) and \(C_ 2\) are meant to be analogues of \(c_ 1\) and \(c_ 2\) above, this is unsatisfactory, and it would probably be more sensible to define \[ C_ 1 = n^{-1} \sum_ i p_{ai} \log_ 2 N_{ai} \quad \text{and}\quad C_ 2 = n^{-1} \sum_ i p_{ai} \log_ 2 N_{fi}. \] Some conclusions are drawn on the basis of the \(C_ 1\) and \(C_ 2\) values.
Mathematics Subject Classification ID: 92C20
Mathematics Subject Classification ID: 28D20
Mathematics Subject Classification ID: 92C50
Mathematics Subject Classification ID: 37E99
zbMATH DE Number: 710141
zbMATH Keywords: sequence complexity
zbMATH Keywords: completely random systems
zbMATH Keywords: measures of complexity of sequences
zbMATH Keywords: entropy
zbMATH Keywords: shift map
zbMATH Keywords: periodic systems
zbMATH Keywords: logistic maps
zbMATH Keywords: EEG
zbMATH Keywords: brain activity
MaRDI profile type: Publication
full work available at URL: 0960-0779(94)90125-2
OpenAlex ID: W1980178211
cites work: Computation theory of cellular automata
cites work: An analysis of the structure and complexity of nonlinear binary sequence generators
cites work: An algebraic measure of complexity
cites work: On the Complexity of Finite Sequences
description: scientific article; zbMATH DE number 710141