Fractal dimension of electroencephalographic time series and underlying brain processes

 Fractal dimension has been proposed as a useful measure for the characterization of electrophysiological time series. This paper investigates what the pointwise dimension of electroencephalographic (EEG) time series can reveal about underlying neuronal generators. The following theoretical assumptions concerning brain function were made (i) within the cortex, strongly coupled neural assemblies exist which oscillate at certain frequencies when they are active, (ii) several such assemblies can oscillate at a time, and (iii) activity flow between assemblies is minimal. If these assumptions are made, cortical activity can be considered as the weighted sum of a finite number of oscillations (plus noise). It is shown that the correlation dimension of finite time series generated by multiple oscillators increases monotonically with the number of oscillators. Furthermore, it is shown that a reliable estimate of the pointwise dimension of the raw EEG signal can be calculated from a time series as short as a few seconds. These results indicate that (i) The pointwise dimension of the EEG allows conclusions regarding the number of independently oscillating networks in the cortex, and (ii) a reliable estimate of the pointwise dimension of the EEG is possible on the basis of short raw signals. Received: 1 September 1994/Accepted in revised form: 16 May 1995     

Monitoring anaerobic sequential batch reactors via fractal analysis of pH time series

Efficient monitoring and control schemes are mandatory in the current operation of biological wastewater treatment plants because they must accomplish more demanding environmental policies. This fact is of particular interest in anaerobic digestion processes where the availability of accurate, inexpensive, and suitable sensors for the on‐line monitoring of key process variables remains an open problem nowadays. In particular, this problem is more challenging when dealing with batch processes where the monitoring strategy has to be performed in finite time, which limits the application of current advanced monitoring schemes as those based in the proposal of nonlinear observers (i.e., software sensors). In this article, a fractal time series analysis of pH fluctuations in an anaerobic sequential batch reactor (AnSBR) used for the treatment of tequila vinasses is presented. Results indicated that conventional on‐line pH measurements can be correlated with off‐line determined key process variables, such as COD, VFA and biogas production via some fractality indexes. Biotechnol. Bioeng. 2013; 110: 2131–2139. © 2013 Wiley Periodicals, Inc.     

Characterization of melanophore morphology by fractal dimension analysis

Fractal or focal dimension (FD) analysis is a valuable tool to identify physiologic stimuli at the cellular and tissue levels that allows for quantification of cell perimeter complexity. The FD analysis was determined on fluorescence images of caffeine- or epinephrine-treated (or untreated control) killifish Fundulus heteroclitus (Linneaus) melanophores in culture. Cell perimeters were indicated by rhodamine-phalloidin labeling of cortical microfilaments using box-counting FD analysis. Caffeine-treated melanophores displayed dispersed melanosomes in cells with less serrated edges and reduced FD and complexity. Complexity in epinephrine-treated cells was significantly higher than the caffeine-treated cells or in the control. Cytoarchitectural variability of the cell perimeter is expected because cells change shape when cued with agents. Epinephrine-treated melanophores demonstrated aggregated melanosomes in cells with more serrated edges, significantly higher FD and thus complexity. Melanophores not treated with caffeine or epinephrine produced variable distributions of melanosomes and resulted in cells with variably serrated edges and intermediate FD with a larger SE of the regression and greater range of complexity. Dispersion of melanosomes occurs with rearrangements of the cytoskeleton to accommodate centrifugal distribution of melanosomes throughout the cell and to the periphery. The loading of melanosomes onto cortical microfilaments may provide a less complex cell contour, with the even distribution of the cytoskeleton and melanosomes. Aggregation of melanosomes occurs with rearrangements of the cytoskeleton to accommodate centripetal distribution of melanosomes. The aggregation of melanosomes may contribute to centripetal retraction of the cytoskeleton and plasma membrane. The FD analysis is, therefore, a convenient method to measure contrasting morphologic changes within stimulated cells.     

Fractal dimension of birds population sizes time series

Information about fractal dimension is collected so that it can be applied to time series interpreting Hurst coefficient. The population size of a species is modelled as a dynamic system. The Hurst coefficient is calculated for these times series. A computer programme has been elaborated to compute the Hurst exponent of time series using the algorithms of range increment, second order moment increment and local second order moment increment. It has been applied to time series of birds' populations.     

Fractal dimension and histogram method: Algorithm and some preliminary results of noise-like time series analysis

In the present work a methodological background for the histogram method of time series analysis is developed. Connection between shapes of smoothed histograms constructed on the basis of short segments of time series of fluctuations and the fractal dimension of the segments is studied. It is shown that the fractal dimension possesses all main properties of the histogram method. Based on it a further development of fractal dimension determination algorithm is proposed. This algorithm allows more precision determination of the fractal dimension by using the “all possible combination” method. The application of the method to noise-like time series analysis leads to results, which could be obtained earlier only by means of the histogram method based on human expert comparisons of histograms shapes.     

沙冬青灌丛地的土壤颗粒大小分形维数空间变异性分析

通过土壤颗粒大小分形维数对土壤质地定量分析表明,土壤颗粒大小分形维数与沙粒含量（＞0.1mm）呈显著的线性负相关,与粘粉粒含量（＜0.05mm）呈显著的线性正相关关系（p＜0.0001）,沙粒和粘粉粒含量每增加1﹪,分形维数则降低或升高0.022个单位,沙粒增加导致分形维数的降低和粘粉粒增加导致分形维数升高幅度一致,所以土壤颗粒分形维数可以作为评价土地沙质荒漠化程度的定量指标之一.荒漠地带以沙冬青为优势种的沙生植被地段,地表土壤颗粒粗粒化和50～70cm土层土壤颗粒的细粒化,为该区有限降水的深层渗漏提供了基质,为沙冬青的存活提供了保障,一定程度上支持了一些专家提出的概念模型,验证了粗质土壤质地支持以灌木为主的群落的假说.但从空间变异性分析来看,荒漠区沙冬青群落因地表物质的吹蚀和堆积过程频繁以及灌丛沙堆发育程度引起的分枝形态差异,导致了土壤颗粒含量空间变异尺度并未集中在冠幅范围,并不支持灌丛在沙漠生态系统对降尘等细粒物质拦截所形成的＂沃岛＂作用.正是因为以沙冬青为优势种的沙生植被地段土壤颗粒组成的高度空间异质性,并且空间变异的范围并不局限于灌木冠幅范围和冠幅间的裸地,没有为其它植物种的入侵创造土壤基质条件,才使得沙冬青群落在该区稳定存在,为荒漠残遗植物种--沙冬青的迁地保护和干旱沙区植被恢复过程中合理地利用土壤资源,以避免营林失败等问题提供了理论依据.     

TSAN: A package for time series analysis

Many biomedical data are in the form of time series. Analyses of these datainclude: (1) search for any biorhytm; (2) test of homogeneity of several time series; (3) assessment of stationarity; (4) test of normality of the time histogram; (5) evaluation of dependence between data points. In this paper we present a subroutine package called TSAN. It is developed to accomplish these tasks. Computational methods, as well as flowcharts, for these subroutines are described. Two sample runs are demostrated.     

Cluster analysis of respiratory time series

We have investigated the respiratory control system with the hypothesis that, although many variables such as minute ventilation (V _I), tidal volume (V _T),breathing period (T _T),inspiratory duration (T _I),and exspiratory duration (T _E),may be observed, the controller functions more simply by manipulating only 2 or 3 of these. Thus, if tidal volume is the only independent variable, T _Ibeing determined by the off-switch threshold, these variables should have very similar time courses. Anesthetized dogs were subjected to CO₂ breathing and carotid sinus perfusion to stimulate both chemoreceptors. The time series of the variables V _I, V _T, T_T, T_Eand T _Ias well as P _{A CO 2}were determined on a breath by breath basis. Derived characteristics of these time series were compared using Cluster Analysis and the latent dimensionality of respiratory control determined by Factor Analysis. The characteristics of the time series clustered into 4 groups: magnitude (of the response), speed, variability and relative change. One respiratory factor accounted for 86% of the variance for the variability characteristics, 2 factors for magnitude (91%) and relative change (85%) and 3 factors for speed (89%). The respiratory variables were analysed for each of the 4 groups of characteristics with the following results: V _Tand T _I clustered together only for the magnitude and relative change characteristics where as T _Tand T _Eclustered closely for all four characteristics. One latent factor was associated with the [T _T-T_E]group and the other usually with P _{A CO 2}.Supported by USPHS 5t01 01919-05, NIH HL 12564 and GM 07033     

Wavelet analysis of ecological time series

Wavelet analysis is a powerful tool that is already in use throughout science and engineering. The versatility and attractiveness of the wavelet approach lie in its decomposition properties, principally its time-scale localization. It is especially relevant to the analysis of non-stationary systems, i.e., systems with short-lived transient components, like those observed in ecological systems. Here, we review the basic properties of the wavelet approach for time-series analysis from an ecological perspective. Wavelet decomposition offers several advantages that are discussed in this paper and illustrated by appropriate synthetic and ecological examples. Wavelet analysis is notably free from the assumption of stationarity that makes most methods unsuitable for many ecological time series. Wavelet analysis also permits analysis of the relationships between two signals, and it is especially appropriate for following gradual change in forcing by exogenous variables.     

Scale dependency and fractal dimension of rarity

The rarity of species in a country is usually determined by counting the number of grid cells occupied by those species on a geographical observation grid. In this paper, we present a measure of rarity that is less sensitive to the shape and size of a country. We demonstrate that the distribution of species on a national grid is not monofractal. Consequently, rarity figures cannot be scaled down to a finer grid merely using scale-area plots. We propose a downscaling method that takes into account the non-monofractal distribution of species. Rarity figures have often been published on a scale comprising a limited number of rarity classes. This article finally provides an insight into the degree of accuracy of such classes.     

Autoregressive time series analysis via representatives

In any realization of an autoregressive time series there exist a few observations having a noticeable feature: they express the useful properties of the time series and, therefore, they represent the entire process. Such representative observations (or, simply, representatives) can be determined by an optimization procedure, provided that the absolute value criterion is used instead of the customary least squares. To achieve this, a special kind of optimization operator (optimator) which generate the parameters of the time series is considered. The concepts of strong and weak similarity of the time series are defined in terms of the representatives and sufficient conditions for both strong and weak similarity are derived. It is shown that there exists a subclass of strongly similar processes, say X, such that ordinary addition is a binary operation in X. An analogous result is shown to hold for weakly similar autoregressive processes. Some examples illustrating these results are given.     

Evolutionary factor analysis of replicated time series

Summary In this article, we develop a novel method that explains the dynamic structure of multi-channel electroencephalograms (EEGs) recorded from several trials in a motor-visual task experiment. Preliminary analyses of our data suggest two statistical challenges. First, the variance at each channel and cross-covariance between each pair of channels evolve over time. Moreover, the cross-covariance profiles display a common structure across all pairs, and these features consistently appear across all trials. In the light of these features, we develop a novel evolutionary factor model (EFM) for multi-channel EEG data that systematically integrates information across replicated trials and allows for smoothly time-varying factor loadings. The individual EEGs series share common features across trials, thus, suggesting the need to pool information across trials, which motivates the use of the EFM for replicated time series. We explain the common co-movements of EEG signals through the existence of a small number of common factors. These latent factors are primarily responsible for processing the visual-motor task which, through the loadings, drive the behavior of the signals observed at different channels. The estimation of the time-varying loadings is based on the spectral decomposition of the estimated time-varying covariance matrix.     

Wavelet-based time series analysis of circadian rhythms

Analysis of circadian oscillations that exhibit variability in period or amplitude can be accomplished through wavelet transforms. Wavelet-based methods can also be used quite effectively to remove trend and noise from time series and to assess the strength of rhythms in different frequency bands, for example, ultradian versus circadian components in an activity record. In this article, we describe how to apply discrete and continuous wavelet transforms to time series of circadian rhythms, illustrated with novel analyses of 2 case studies involving mouse wheel-running activity and oscillations in PER2::LUC bioluminescence from SCN explants.