Assessing the convolutedness of multivariate physiological time series

A feature of time-series variability that may reveal underlying complex dynamics is the degree of "convolutedness". For multivariate series of m components, convolutedness can be defined as the propensity of the trail of the time-series samples to fill the m-dimensional space. This work proposes different convolutedness indices and compare them on synthesized and real physiological signals. The indices are based on length L and planar extension d of the trail in m dimensions. The classical ones are: the L/d ratio, and the Mandelbrot\´s fractal dimension (FD) of a curve: FD_M =log(L)/log(d). In this work we also consider a correction of the Katz\´s estimator of FD_M, i.e., FD_KC =log(N)/(log(N)+log(d/L)), with N the number of samples; and FD_MC, an estimator of FD_M based on FD_KC calculated over a shorter running window N_w<;N appropriately selected to reduce estimation bias. Synthesized fractional Brownian motions indicated that all the indices increase with FD, but differ for other aspects, namely the dependence on N; the capacity to estimate FD; or to distinguish between true bivariate and degenerate bivariate time series. Application on real multivariate recordings of muscular activity before and after exercise-induced fatigue suggests that these indices can be profitably used to identify complex changes in the dynamics of physiological signals.