Surrogate time series

Before we apply nonlinear techniques, for example those inspired by chaos theory, to dynamical phenomena occurring in nature, it is necessary to first ask if the use of such advanced techniques is justified "by the data". While many processes in nature seem very unlikely a priori to be linear, the possible nonlinear nature might not be evident in specific aspects of their dynamics. The method of surrogate data has become a very popular tool to address such a question. However, while it was meant to provide a statistically rigorous, foolproof framework, some limitations and caveats have shown up in its practical use. In this paper, recent efforts to understand the caveats, avoid the pitfalls, and to overcome some of the limitations, are reviewed and augmented by new material. In particular, we will discuss specific as well as more general approaches to constrained randomisation, providing a full range of examples. New algorithms will be introduced for unevenly sampled and multivariate data and for surrogate spike trains. The main limitation, which lies in the interpretability of the test results, will be illustrated through instructive case studies. We will also discuss some implementational aspects of the realisation of these methods in the TISEAN (http://www.mpipks-dresden.mpg.de/~tisean) software package.Comment: 28 pages, 23 figures, software at http://www.mpipks-dresden.mpg.de/~tisea

Constrained-Realization Monte-Carlo Method for Hypothesis Testing

We compare two theoretically distinct approaches to generating artificial (or ``surrogate'') data for testing hypotheses about a given data set. The first and more straightforward approach is to fit a single ``best'' model to the original data, and then to generate surrogate data sets that are ``typical realizations'' of that model. The second approach concentrates not on the model but directly on the original data; it attempts to constrain the surrogate data sets so that they exactly agree with the original data for a specified set of sample statistics. Examples of these two approaches are provided for two simple cases: a test for deviations from a gaussian distribution, and a test for serial dependence in a time series. Additionally, we consider tests for nonlinearity in time series based on a Fourier transform (FT) method and on more conventional autoregressive moving-average (ARMA) fits to the data. The comparative performance of hypothesis testing schemes based on these two approaches is found to depend on whether or not the discriminating statistic is pivotal. A statistic is ``pivotal'' if its distribution is the same for all processes consistent with the null hypothesis. The typical-realization method requires that the discriminating statistic satisfy this property. The constrained-realization approach, on the other hand, does not share this requirement, and can provide an accurate and powerful test without having to sacrifice flexibility in the choice of discriminating statistic.Comment: 19 pages, single spaced, all in one postscript file, figs included. Uncompressed .ps file is 425kB (sorry, it's over the 300kB recommendation). Also available on the WWW at http://nis-www.lanl.gov/~jt/Papers/ To appear in Physica

Accounting for outliers and calendar effects in surrogate simulations of stock return sequences

Surrogate Data Analysis (SDA) is a statistical hypothesis testing framework for the determination of weak chaos in time series dynamics. Existing SDA procedures do not account properly for the rich structures observed in stock return sequences, attributed to the presence of heteroscedasticity, seasonal effects and outliers. In this paper we suggest a modification of the SDA framework, based on the robust estimation of location and scale parameters of mean-stationary time series and a probabilistic framework which deals with outliers. A demonstration on the NASDAQ Composite index daily returns shows that the proposed approach produces surrogates that faithfully reproduce the structure of the original series while being manifestations of linear-random dynamics.Comment: 21 pages, 7 figure

Testing For Nonlinearity Using Redundancies: Quantitative and Qualitative Aspects

A method for testing nonlinearity in time series is described based on information-theoretic functionals -- redundancies, linear and nonlinear forms of which allow either qualitative, or, after incorporating the surrogate data technique, quantitative evaluation of dynamical properties of scrutinized data. An interplay of quantitative and qualitative testing on both the linear and nonlinear levels is analyzed and robustness of this combined approach against spurious nonlinearity detection is demonstrated. Evaluation of redundancies and redundancy-based statistics as functions of time lag and embedding dimension can further enhance insight into dynamics of a system under study.Comment: 32 pages + 1 table in separate postscript files, 12 figures in 12 encapsulated postscript files, all in uuencoded, compressed tar file. Also available by anon. ftp to santafe.edu, in directory pub/Users/mp/qq. To be published in Physica D., [email protected]

Approximate entropy as an indicator of non-linearity in self paced voluntary finger movement EEG

This study investigates the indications of non-linear dynamic structures in electroencephalogram signals. The iterative amplitude adjusted surrogate data method along with seven non-linear test statistics namely the third order autocorrelation, asymmetry due to time reversal, delay vector variance method, correlation dimension, largest Lyapunov exponent, non-linear prediction error and approximate entropy has been used for analysing the EEG data obtained during self paced voluntary finger-movement. The results have demonstrated that there are clear indications of non-linearity in the EEG signals. However the rejection of the null hypothesis of non-linearity rate varied based on different parameter settings demonstrating significance of embedding dimension and time lag parameters for capturing underlying non-linear dynamics in the signals. Across non-linear test statistics, the highest degree of non-linearity was indicated by approximate entropy (APEN) feature regardless of the parameter settings

Influence of wiring cost on the large-scale architecture of human cortical connectivity

In the past two decades some fundamental properties of cortical connectivity have been discovered: small-world structure, pronounced hierarchical and modular organisation, and strong core and rich-club structures. A common assumption when interpreting results of this kind is that the observed structural properties are present to enable the brain's function. However, the brain is also embedded into the limited space of the skull and its wiring has associated developmental and metabolic costs. These basic physical and economic aspects place separate, often conflicting, constraints on the brain's connectivity, which must be characterized in order to understand the true relationship between brain structure and function. To address this challenge, here we ask which, and to what extent, aspects of the structural organisation of the brain are conserved if we preserve specific spatial and topological properties of the brain but otherwise randomise its connectivity. We perform a comparative analysis of a connectivity map of the cortical connectome both on high- and low-resolutions utilising three different types of surrogate networks: spatially unconstrained (‘random’), connection length preserving (‘spatial’), and connection length optimised (‘reduced’) surrogates. We find that unconstrained randomisation markedly diminishes all investigated architectural properties of cortical connectivity. By contrast, spatial and reduced surrogates largely preserve most properties and, interestingly, often more so in the reduced surrogates. Specifically, our results suggest that the cortical network is less tightly integrated than its spatial constraints would allow, but more strongly segregated than its spatial constraints would necessitate. We additionally find that hierarchical organisation and rich-club structure of the cortical connectivity are largely preserved in spatial and reduced surrogates and hence may be partially attributable to cortical wiring constraints. In contrast, the high modularity and strong s-core of the high-resolution cortical network are significantly stronger than in the surrogates, underlining their potential functional relevance in the brain