Approximate Entropy Analysis of Electroencephalogram

Chinese Jourmal of Biomedical Engineering (English Edition) Volume 20 Number 1, March 2011Approximate Entropy Analysis of ElectroencephalogramYOU Rong -yiDepartment of Physics, Jimei Universit, Xiamen 361021, ChinaAbstract. Based on the time- -delayed embedding method of phase spacereconstruction, a new method to compute the approximate entropy (ApEn) ofelectroencephalogram (EEG) is proposed. The computational results show that there aresignificant differences between epileptic EEC and normal EEG in the approximateentropy with the variance of embedding dimension. This conclusion is helpful to analyzethe dynamical behavior of diferent EEGs by entropy.Key words: approximate entropy (ApEn); electroencephalogram (EEG); phasespaceINTRODUCTIONThe human brain is a complex nonlinear dynamical system", and EEG is a kind of complex timesequence recorded from scalp by many electrodes, it is usually used as the clinical reference ofencephalopathy treatment. However, it is difficult to analyze exactly the characteristics of EEG due tohe complexity and diversity of EEG itself. After the nonlinearity has been found in human EEG,various nonlinear dynamical methods have become the popular methods for the analysis of EECG-), forexample, the computation and analysis of nonlinear invariants of EEG such as correlation dimension,Lyapunov exponent, approximate entropy and the like. However, one of the indispensable processes forthe solution of these invariants is the phase space reconstruction. The purpose of phase spacereconstruction is to construct the system state by its history and view it in a higher-dimensional space. .A popular method used for phase space reconstruction is known as the time -delayed embeddingmethod proposed by Packard and Takens et al459. It is usually that the extermal characteristics of manynonlinear systerms is only represented by a time sequence of a certain single variable. The time -delayed embedding method can extract the nonlinear dynamical characteristics from the system byreconstructing the single time sequence of system. In the reconstructed space, each system invariantscan be resrored or reversed. In this paper, a new method to compute the ApEn of EEG is proposedbased on the phase space reconstruction of EEG time sequence. The computational results of 200中国煤化工CLC number: R 318.6 Document code:A Article ID: 1004 -0552(201 1)01 -0019-05M出CNMHGGrant sponsor.Natural Sriernce Foundation of Fujan Province of China; grant number: 2Corresponding author: YOU Rong -yi. E mail:yyou@jmu.edu.cnReceivred 25 July 2010; revised 10 March 201 1- 19-CHINESE J. BIOMED. ENG. V0IL.20 NO.1, MAR.2011samples show that there are significant differences between epileptic EEG and normal EEG in ApEnwith the variance of embedding dimension. This conclusion is helpful to analyze the dynamicalbehavior of different EEG by entropy.PHASE SPACE RECONSTRUCTIONAssume a continuous-time dynamical system is expressed as dx(t)dt = F(x(), x()∈R", it becomesa discrete- time system after disretization. f;R" -→R", x ([+1)= f(), t= 0,1,2,.. Let x(t+m) =f" (x()and y() a component of system state [) and h:R^- →R, so as to y() = h()). Let y"() = (r(t+m -1), y(t+m-2),.. y()). If U is a compact manifold, then form≥2n + 1, ]"(x()) ="[()=(h (o-1 (())，h(fm2(x()),) ., h()) is a embedding from U to J" (U).Generally, there exist a function g:Rm -→R", makey"(t+1)=g (r"() = ( (t+m), y(+m-1,".*.(+1)). Take noteof y"(t+1)= Jr*(x(t+1))= (f(f:()), s0 wehave r"(f(x(t)) = g ()). Under the hypothetical condition of J”is a homeomorphism, both f and gare topological isomorphic. Therefore, the dynamical characteristics off can be reserved in g. It isobvious that some dynamical invariants of the original system can be computed and analyzed byobserving a component y(t) of system state x(t).Let X denote a matrix X=xXxX*-,XJ], each row of X denotes a vector of the embedding space,i.e. X; represents the system state at discrete time point i. For a time series x={x, 2的. x} withlength of L, the phase space will be R" after time-delayed reconstruction, then the dynamical state ofsystem in phase space is determined by Equation(1):X=(x, xim " Xi0-m))(1wherei= 1,2,3..",N, T denotes the time lag, m denotes the embedding dimension, and the vectornumber IN=L -(m- -1)r, that is, there are N state points in the embedding space.APPROXIMATE ENTROPY OF EEGApEn is a positive number proposed firstly by Pincus 网in 1991, which is usually used to measurethe complexity of time sequence. In the present work, we used the method of the referencel, the ApEnof 200 samples of epileptic EEG were normal EEG were computed with the length of data in the rangeof 500-1,000. The computational results show that the ApEn is less related to the length of sampledata. The ApEn of epileptic EEG is approximately varied in the range of 0.75 -0.85 with a averagevalue of 0.8, and the ApEn of normal EEG is varied in the range of 0.80- -0.95 with a average value of0.9. In other words, there is a small difference between epileptic EEG and normal EEG in ApEn.Obviously, from the ApEn we can see that both the epileptic EEG and normal EEG are quite complex,which are diffcult to distinguish from each other. Based on the previous method, in this paper, wepresent a new method to compute the ApEn of EEG, which determine the similarity of different systemstates by relative Euclidian distance and standard deviation of sequence, so that the ApEn of EEG isdetermined in a new way.中国煤化工The computational process of ApEn can be described(1) Give the certain value of embedding dimension m;YHCNM H. G,M=1s, r=1 or2),then reconstruct EEG time sequrence by Equation (1).- 20-Chinese Jourmal of Biomedical Engineering (English Fdition) Volume 20 Number 1, March 2011(2) Using Equation (1), compute all relative Euclidian distances dy among an arbitrary vector Xand others: dy= Dg/m(k, j =1,..N, j≠k), where(2)D.-=+VXx.-xj(3) Give a threshold value of t(t=0.2std(x), where std(x) is the standard deviation of EEG sequence),for each vector Xk, counting the number of d;