Kalauzi Aleksandar, Bojić Tijana, Rakić Ljubisav
Department for Life Sciences, Institute for Multidisciplinary Research, University of Belgrade Kneza Viseslava 1, 11000 Belgrade Serbia.
Nonlinear Biomed Phys. 2009 Aug 14;3(1):8. doi: 10.1186/1753-4631-3-8.
Nonlinear methods provide a direct way of estimating complexity of one-dimensional sampled signals through calculation of Higuchi's fractal dimension (1<FD<2). In most cases the signal is treated as being characterized by one value of FD and consequently analyzed as one epoch or, if divided into more epochs, often only mean and standard deviation of epoch FD are calculated. If its complexity variation (or running fractal dimension), FD(t), is to be extracted, a moving window (epoch) approach is needed. However, due to low-pass filtering properties of moving windows, short epochs are preferred. Since Higuchi's method is based on consecutive reduction of signal sampling frequency, it is not suitable for estimating FD of very short epochs (N < 100 samples).
In this work we propose a new and simple way to estimate FD for N < 100 by introducing 'normalized length density' of a signal epoch,where yn(i) represents the ith signal sample after amplitude normalization. The actual calculation of signal FD is based on construction of a monotonic calibration curve, FD = f(NLD), on a set of Weierstrass functions, for which FD values are given theoretically. The two existing methods, Higuchi's and consecutive differences, applied simultaneously on signals with constant FD (white noise and Brownian motion), showed that standard deviation of calculated window FD (FDw) increased sharply as the epoch became shorter. However, in case of the new NLD method a considerably lower scattering was obtained, especially for N < 30, at the expense of some lower accuracy in calculating average FDw. Consequently, more accurate reconstruction of FD waveforms was obtained when synthetic signals were analyzed, containig short alternating epochs of two or three different FD values. Additionally, scatter plots of FDw of an occipital human EEG signal for 10 sample epochs demontrated that Higuchi's estimations for some epochs exceeded the theoretical FD limits, while NLD-derived values did not.
The presented approach was more accurate than the existing two methods in FD(t) extraction for very short epochs and could be used in physiological signals when FD is expected to change abruptly, such as short phasic phenomena or transient artefacts, as well as in other fields of science.
非线性方法提供了一种通过计算 Higuchi 分形维数（1<FD<2）来直接估计一维采样信号复杂度的方法。在大多数情况下，信号被视为由一个 FD 值表征，因此作为一个时段进行分析，或者如果划分为更多时段，通常只计算时段 FD 的均值和标准差。如果要提取其复杂度变化（或运行分形维数）FD(t)，则需要采用移动窗口（时段）方法。然而，由于移动窗口的低通滤波特性，较短的时段更受青睐。由于 Higuchi 方法基于信号采样频率的连续降低，它不适用于估计非常短的时段（N < 100 个样本）的 FD。
在这项工作中，我们提出了一种新的简单方法来估计 N < 100 时的 FD，即引入信号时段的“归一化长度密度”，其中 yn(i) 表示幅度归一化后的第 i 个信号样本。信号 FD 的实际计算基于在一组 Weierstrass 函数上构建单调校准曲线 FD = f(NLD)，其 FD 值是理论给定的。将现有的两种方法 Higuchi 方法和连续差分法同时应用于具有恒定 FD 的信号（白噪声和布朗运动），结果表明，随着时段变短，计算得到的窗口 FD（FDw）的标准差急剧增加。然而，在新的 NLD 方法中，得到的散射要低得多，特别是对于 N < 30 的情况，代价是计算平均 FDw 时精度略有降低。因此，在分析包含两个或三个不同 FD 值的短交替时段的合成信号时，能够更准确地重建 FD 波形。此外，对 人类枕叶脑电信号的 10 个样本时段的 FDw 散点图表明，Higuchi 方法对某些时段的估计超出了理论 FD 极限，而 NLD 方法得到的值则没有。
在非常短的时段的 FD(t) 提取中，本文提出的方法比现有的两种方法更准确，可用于预期 FD 会突然变化的生理信号，如短相位现象或瞬态伪迹，以及其他科学领域。
