Review of this book in Mathematical Reviews (2003)
Abstract:
We compare two algorithms for the numerical estimation of the correlation dimension from a finite set of vectors: the "classical" algorithm of Grassberger and Procaccia (GPA) and the recently proposed algorithm of Judd (JA). Data set size requirements and their relations to systematic and statistical errors of the estimates are investigated. It is demonstrated that correlation dimensions of the order of 6 can correctly be resolved on the basis of about 100,000 data points in the case of a continuous trajectory on a strange attractor; the minimum data set size is, however, noticeably dependent on the geometrical structure of the system from which the vectors were sampled.
Abstract:
We address the issue of testing for nonlinearity in time series from continuous dynamics and propose a quantitative measure for nonlinearity which is based on discrete parametric modelling. The well-known problems of modelling continuous dynamical systems by discrete models are addressed by a subsampling approach. This measure should preferably be combined with conventional surrogate data testing. The performance of the test is demonstrated by application to simulated, heavily noise-contaminated time series from high-dimensional Lorenz systems, and to experimental time series from a high-dimensional mode of Taylor-Couette flow. We also discuss the discrimination power of the test under surrogate data testing, when compared with other well-tried test statistics.
Abstract:
We investigate the structure of dynamical correlations on reconstructed attractors which were obtained by time-delay embedding of periodic, quasi-periodic and chaotic time series. Within the specific sampling of the invariant density by a finite number of vectors which results from embedding, we identify two separate levels of sampling, corresponding to two different types of dynamical correlations, each of which produces characteristic artifacts in correlation dimension estimation: the well-known trajectory bias and a characteristic oscillation due to periodic sampling. For the second artifact we propose random sampling as a new correction method which is shown to provide improved sampling and to reduce dynamical correlations more efficiently than it has been possible by the standard Theiler correction. For accurate numerical analysis of correlation dimension in a bootstrap framework both corrections should be combined. For tori and the Lorenz attractor we also show how to construct time-delay embeddings which are completely free of any dynamical correlations.
Abstract:
We use the theory of nonlinear dynamical systems to measure the complexity of currency markets by estimating the correlation dimension of the returns of the Dollar/Pound and Dollar/Yen daily exchange rates (the spot rates). We test the significance of the results by comparing them to correlation dimension estimates for surrogate time series, i.e. stochastic linear time series with the same power spectrum and amplitude distribution as given by the original data. We find discernible nonlinear structure in the returns of the Dollar/Pound daily rate.
Abstract:
In this paper we consider the dynamical inverse problem of EEG generation where a specific dynamics for the electrical current distribution is assumed. By casting this problem into a state space representation and assuming a specific class of parametric models for the dynamics, we can impose general spatio-temporal constraints onto the solution. For the purpose of estimating the parameters and evaluating the model, we employ the Akaike Bayesian Information Criterion (ABIC), which is based on the type II likelihood. As a new approach for estimating the current distribution we introduce a method which we call "Dynamic LORETA". A recursive penalized least squares (RPLS) step forms the main element of our implementation. Whereas LORETA exploits exclusively spatial information, Dynamic LORETA exploits both spatial and temporal information, such that it becomes possible to obtain improved inverse solutions. The performance of the new method is evaluated by application to simulated EEG data, and a considerable improvement over LORETA is found. We also show results for the application to clinical EEG data.
Abstract:
We present a new approach for estimating solutions of the dynamical inverse problem of EEG generation. In contrast to previous approaches, we reinterpret this problem as a filtering problem in a state space framework; for the purpose of its solution we propose a new extension of Kalman filtering to the case of spatiotemporal dynamics. The temporal evolution of the distributed generators of the EEG can be reconstructed at each voxel of a discretisation of the gray matter of brain. By fitting linear autoregressive models with neighbourhood interactions to EEG time series new classes of inverse solutions with improved resolution and localisation ability can be explored. For the purposes of model comparison and parameter estimation from given data we employ a likelihood maximisation approach. Both for instantaneous and dynamical inverse solutions we derive estimators of the time-dependent estimation error at each voxel. The performance of the algorithm is demonstrated by application to simulated and clinical EEG recordings. It is shown that by choosing appropriate dynamical models it becomes possible to obtain inverse solutions of considerably improved quality, as compared to the usual instantaneous inverse solutions.
Abstract:
The purpose of this study is to propose and investigate a new approach for discriminating between focal and non-focal hemispheres in intractable temporal lobe epilepsy, based on applying multivariate time series analysis to the discharge-free background brain activity observed in nocturnal electrocorticogram (ECoG) time series. Five unilateral focal patients and one bilateral focal patient were studied. In order to detect the location of epileptic foci, linear multiva riate autoregressive (MAR) models were fitted to the ECoG data; as a new approach for the purpose of summarizing these models in a single relevant parameter, the behaviour of the corresponding impulse response functions was studied and described by an attenuation coefficient. In the majority of unilateral focal patients the averaged attenuation coefficient was found to be almost always significantly larger in the focal hemisphere, as compared to the non-focal hemisphere. Also the amplitude of the fluctuations of the attenuation coefficient was significantly larger in the focal hemisphere. Moreover, in one patient showing a typical regular sleep cycle, the attenuation coefficient in the focal hemisphere tended to be larger during REM sleep and smaller during Non-REM sleep. In the bilateral focal patient no statistically significant distinction between the hemispheres was found. This study provides encouraging results for new investigations of brain dynamics by multivariate parametric modeling. It opens up the possibility of relating diseases like epilepsy to the properties of inconspicuous background brain dynamics, without the need to record and analyze epileptic seizures or other evidently pathological waveforms.
Abstract:
The problem of estimating unobserved states of spatially extended dynamical systems poses an inverse problem, which can be solved approximately by a recently developed variant of Kalman Filtering; in order to provide the model of the dynamics with more flexibility with respect to space and time, we suggest to combine the concept of GARCH modelling of covariance, well-known in econometrics, with Kalman Filtering. We formulate this algorithm for spatiotemporal systems governed by stochastic diffusion equations and demonstrate its feasibility by presenting a numerical simulation designed to imitate the situation of the generation of electroencephalographic recordings by the human cortex.
Abstract:
We consider the problem of detecting and quantifying nonstationary structure in time series from high-dimensional dynamical systems. This problem is relevant in particular for EEG monitoring, e.g. for the prediction of epileptic seizures, but also for practical data analysis in many other fields. Three groups of measures of nonstationarity are discussed: Correlation dimension, measures based on autoregressive modelling and cross-prediction, and measures based on entropies defined in the spectral or wavelet domains. Results both for simulated and clinical time series are shown.
Abstract:
We propose a method based on the equal-time correlation matrix as a sensitive detector for phase-shape correlations in multivariate data sets. The key point of the method is that changes of the degree of synchronisation between time series provoke level repulsions between eigenstates at both edges of the spectrum of the correlation matrix. Consequently, detailed information about the correlation structure of the multivariate data set is imprinted into the dynamics of the eigenvalues and into the structure of the corresponding eigenvectors. The performance of the technique is demonstrated by application to N_f-tori, autoregressive models, and coupled chaotic systems. The high sensitivity, the comparatively small computational effort, and the excellent time resolution of the method recommend it for application to the analysis of complex, spatially extended, nonstationary systems.
Abstract:
We present a new approach to modelling non-stationarity in EEG time series by a generalized state space approach. A given time series can be decomposed into a set of noise-driven processes, each corresponding to a different frequency band. Non-stationarity is modelled by allowing the variances of the driving noises to change with time, depending on the state prediction error within the state space model. The method is illustrated by an application to EEG data recorded during the onset of anaesthesia.
Abstract:
We address the issue of inferring the connectivity structure of spatially extended dynamical systems by estimation of mutual information between pairs of sites. The well-known problems resulting from correlations within and between the time series are addressed by explicit temporal and spatial modelling steps which aim at approximately removing all spatial and temporal correlations, i.e. at whitening the data, such that it is replaced by spatiotemporal innovations; this approach provides a link to the maximum-likelihood method and, for appropriately chosen models, removes the problem of estimating probability distributions of unknown, possibly complicated shape. A parsimonious multivariate autoregressive model based on nearest-neighbour interactions is employed. Mutual information can be reinterpreted in the framework of dynamical model comparison (i.e. likelihood ratio testing), since it is shown to be equivalent to the difference of the log-likelihoods of coupled and uncoupled models for a pair of sites, and a parametric estimator of mutual information can be derived. We also discuss, within the framework of model comparison, the relationship between the coefficient of linear correlation and mutual information. The practical application of this methodology is demonstrated for simulated multivariate time series generated by a stochastic coupled-map lattice. The parsimonious modelling approach is compared to general multivariate autoregressive modelling and to Independent Component Analysis (ICA).
Abstract:
We suggest a procedure to identify those parts of the spectrum of the equal-time correlation matrix C where relevant information about correlations of a multivariate time series is induced. Using an ensemble average over each of the distances between eigenvalues, all nearest-neighbor distributions can be calculated individually. We present numerical examples, where (a) information about cross correlations is found in the so-called "bulk" of eigenvalues (which generally is thought to contain only random correlations) and where (b) the information extracted from the lower edge of the spectrum of C is statistically more significant than that extracted from the upper edge. We apply the analysis to electroencephalographic recordings with epileptic events.
Abstract:
We discuss a model for the dynamics of the primary current density vector field within the grey matter of human brain. The model is based on a linear damped wave equation, driven by a stochastic term. By employing a realistically shaped average brain model and an estimate of the matrix which maps the primary currents distributed over grey matter to the electric potentials at the surface of the head, the model can be put into relation with recordings of the electroencephalogram (EEG). Through this step it becomes possible to employ EEG recordings for the purpose of estimating the primary current density vector field, i.e. finding a solution of the inverse problem of EEG generation. As a technique for inferring the unobserved high-dimensional primary current density field from EEG data of much lower dimension, a linear state space modelling approach is suggested, based on a generalisation of Kalman filtering, in combination with maximum-likelihood parameter estimation. The resulting algorithm for estimating dynamical solution of the EEG inverse problem is applied to the task of localising the source of an epileptic spike from a clinical EEG data set; for comparison, we apply to the same task also a non-dynamical standard algorithm.
Abstract:
Electroencephalographic (EEG) source localization is an important tool for noninvasive study of brain dynamics, due to its ability to probe neural activity more directly, with better temporal resolution than other imaging modalities. One promising technique for solving the EEG inverse problem is Kalman filtering, because it provides a natural framework for incorporating dynamic EEG generation models in source localization. Here, a recently developed inverse solution is introduced, which uses spatiotemporal Kalman filtering tuned through likelihood maximization. Standard diagnostic tests for objectively evaluating Kalman filter performance are then described and applied to inverse solutions for simulated and clinical EEG data. These tests, employed for the first time in Kalman-filter-based source localization, check the statistical properties of the innovation and validate the use of likelihood maximization for filter tuning. However, this analysis also reveals that the filter's existing space- and time-invariant process model, which contains a single fixed-frequency resonance, is unable to completely model the complex spatiotemporal dynamics of EEG data. This finding indicates that the algorithm could be improved by allowing the process model parameters to vary in space.