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3.3. Nonlinear methods 280−0.5−1log C m (r)−1.5−2D 2−2.5−3−1.4 −1.2 −1 −0.8 −0.6 −0.4log rFigure 3.3: Approximation of the correlation dimension D 2 from the (log r, log C m (r)) plot.which is the so-called correlation integral. The correlation dimension D 2 is defined as thelimit valuelog C m (r)D 2 (m) = lim lim. (3.21)r→0 N→∞ log rIn practice this limit value is approximated by the slope of the regression curve(log r, log C m (r)) [17]. The slope is calculated from the linear part of the log-log plot,see Fig. 3.3. The slope of the regression curves tend to saturate on the finite value of D 2when m is increased. In the software, a value of m = 10 was selected for the embedding.3.3.6 Recurrence plot analysisYet another approach, included in the software, for analyzing the complexity of the timeseries is the so-called recurrence plot (RP) analysis. In this approach, vectorsu j =(RR j , RR j+τ ,...,RR j+(m−1)τ ), j =1, 2,...,N − (m − 1)τ (3.22)where m is the embedding dimension and τ the embedding lag. The vectors u j then representthe RR interval time series as a trajectory in m dimensional space. A recurrence plot is asymmetrical [N − (m − 1)τ] × [N − (m − 1)τ] matrix of zeros and ones. The element inthe j’th row and k’th column of the RP matrix, i.e. RP(j,k), is 1 if the point u j on thetrajectory is close to point u k .ThatisRP(j, k) ={1, d(uj − u k ) ≤ r0, otherwise(3.23)where d(u j ,u k ) is the Euclidean distance given in (3.19) andr is a fixed threshold. Thestructure of the RP matrix usually shows short line segments of ones parallel to the maindiagonal. The lengths of these diagonal lines describe the duration of which the two pointsare close to each other. An example RP for HRV time series is presented in Fig. 3.4.Kubios HRV Analysisversion 2.0 betaBiosignal Analysis and Medical Imaging GroupDepartment of PhysicsUniversity of Kuopio, FINLAND