USER'S GUIDE - Biosignal Analysis and Medical Imaging Group
USER'S GUIDE - Biosignal Analysis and Medical Imaging Group USER'S GUIDE - Biosignal Analysis and Medical Imaging Group
3.3. Nonlinear methods 26Now the value of Cj m(r) will be between 0 and 1. Next, the values of Cm j (r) are averagedto yieldN−m+1C m 1 ∑(r) =Cj m (r) (3.13)N − m +1and the sample entropy is obtained asj=1SampEn(m, r, N) =ln(C m (r)/C m+1 (r)). (3.14)The values selected for the embedding dimension m and for the tolerance parameter rin the software are the same as those for the approximate entropy calculation. Both ApEnand SampEn are estimates for the negative natural logarithm of the conditional probabilitythat a data of length N, having repeated itself within a tolerance r for m points, will alsorepeat itself for m + 1 points. SampEn was designed to reduce the bias of ApEn and has acloser agreement with the theory for data with known probabilistic content [20].3.3.4 Detrended fluctuation analysisDetrended fluctuation analysis (DFA) measures the correlation within the signal. The correlationis extracted for different time scales as follows [36]. First, the RR interval time seriesis integratedk∑y(k) = (RR j − RR), k =1,...,N (3.15)j=1where RR is the average RR interval. Next, the integrated series is divided into segments ofequal length n. Within each segment, a least squares line is fitted into the data. Let y n (k)denote these regression lines. Next the integrated series y(k) is detrended by subtractingthe local trend within each segment and the root-mean-square fluctuation of this integratedand detrended time series is calculated byF (n) = √ 1 N∑(y(k) − y n (k))N2 . (3.16)k=1This computation is repeated over different segment lengths to yield the index F (n) asafunction of segment length n. Typically F (n) increases with segment length. A linear relationshipon a double log graph indicates presence of fractal scaling and the fluctuations canbe characterized by scaling exponent α (the slope of the regression line relating log F (n) tolog n. Different values of α indicate the followingα =1.5: Brown noise (integral of white noise)1
3.3. Nonlinear methods 27−0.6−0.8log F(n)−1−1.2−1.4α 1α 2−1.6−1.80.6 0.8 1 1.2 1.4 1.6 1.8log nFigure 3.2: Detrended fluctuation analysis. A double log plot of the index F (n) as a functionof segment length n. α 1 and α 2 are the short term and long term fluctuation slopes,respectively.3.3.5 Correlation dimensionAnother method for measuring the complexity or strangeness of the time series is the correlationdimension which was proposed in [13]. The correlation dimension is expected to giveinformation on the minimum number of dynamic variables needed to model the underlyingsystem and it can be obtained as follows.Similarly as in the calculation of approximate and sample entropies, form length mvectors u ju j =(RR j , RR j+1 ,...,RR j+m−1 ), j =1, 2,...,N − m + 1 (3.17)and calculate the number of vectors u k for which d(u j ,u k ) ≤ r, thatisCj m (r) = nbr of { ∣u k d(uj ,u k ) ≤ r }N − m +1∀ k (3.18)where the distance function d(u j ,u k ) is now defined as∑d(u j ,u k )= √ m (u j (l) − u k (l)) 2 . (3.19)l=1Next, an average of the term C m j(r) istakenC m (r) =N−m+11 ∑N − m +1j=1C m j(r) (3.20)Kubios HRV Analysisversion 2.0 betaBiosignal Analysis and Medical Imaging GroupDepartment of PhysicsUniversity of Kuopio, FINLAND
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3.3. Nonlinear methods 26Now the value of Cj m(r) will be between 0 <strong>and</strong> 1. Next, the values of Cm j (r) are averagedto yieldN−m+1C m 1 ∑(r) =Cj m (r) (3.13)N − m +1<strong>and</strong> the sample entropy is obtained asj=1SampEn(m, r, N) =ln(C m (r)/C m+1 (r)). (3.14)The values selected for the embedding dimension m <strong>and</strong> for the tolerance parameter rin the software are the same as those for the approximate entropy calculation. Both ApEn<strong>and</strong> SampEn are estimates for the negative natural logarithm of the conditional probabilitythat a data of length N, having repeated itself within a tolerance r for m points, will alsorepeat itself for m + 1 points. SampEn was designed to reduce the bias of ApEn <strong>and</strong> has acloser agreement with the theory for data with known probabilistic content [20].3.3.4 Detrended fluctuation analysisDetrended fluctuation analysis (DFA) measures the correlation within the signal. The correlationis extracted for different time scales as follows [36]. First, the RR interval time seriesis integratedk∑y(k) = (RR j − RR), k =1,...,N (3.15)j=1where RR is the average RR interval. Next, the integrated series is divided into segments ofequal length n. Within each segment, a least squares line is fitted into the data. Let y n (k)denote these regression lines. Next the integrated series y(k) is detrended by subtractingthe local trend within each segment <strong>and</strong> the root-mean-square fluctuation of this integrated<strong>and</strong> detrended time series is calculated byF (n) = √ 1 N∑(y(k) − y n (k))N2 . (3.16)k=1This computation is repeated over different segment lengths to yield the index F (n) asafunction of segment length n. Typically F (n) increases with segment length. A linear relationshipon a double log graph indicates presence of fractal scaling <strong>and</strong> the fluctuations canbe characterized by scaling exponent α (the slope of the regression line relating log F (n) tolog n. Different values of α indicate the followingα =1.5: Brown noise (integral of white noise)1
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