Testing for nonlinearity in irregular fluctuations with long-term trends

Testing for nonlinearity in nonstationary time series: A network-based surrogate data test

The classical surrogate data tests, which are used to differentiate linear noise processes from nonlinear processes, are not suitable for nonstationary time series. In this paper, we propose a surrogate data test that can be applied on both stationary time series as well as nonstationary time series with short-term fluctuations. The method is based on the idea of constructing a network from the time series, employing a generalized symbolic dynamics method introduced in this work, and using any one of the several easily computable network parameters as discriminating statistics. The construction of the network is designed to remove the long-term trends in the data automatically. The network-based test statistics pick up only the short-term variations, unlike the discriminating statistics of the traditional methods, which are influenced by nonstationary trends in the data. The method is tested on several systems generated by linear or nonlinear processes and with deterministic or stochastic trends, and in all cases it is found to be able to differentiate between linear stochastic processes and nonlinear processes quite accurately, especially in cases where the common methods would lead to false rejections of the null hypothesis due to nonstationarity being interpreted as nonlinearity. The method is also found to be robust to the presence of experimental or dynamical noise of a moderate level in an otherwise nonlinear system.

Interpreting null models of resting-state functional MRI dynamics: not throwing the model out with the hypothesis

Null models are useful for assessing whether a dataset exhibits a non-trivial property of interest. These models have recently gained interest in the neuroimaging community as means to explore dynamic properties of functional Magnetic Resonance Imaging (fMRI) time series. Interpretation of null-model testing in this context may not be straightforward because (i) null hypotheses associated to different null models are sometimes unclear and (ii) fMRI metrics might be 'trivial', i.e. preserved under the null hypothesis, and still be useful in neuroimaging applications. In this commentary, we review several commonly used null models of fMRI time series and discuss the interpretation of the corresponding tests. We argue that, while null-model testing allows for a better characterization of the statistical properties of fMRI time series and associated metrics, it should not be considered as a mandatory validation step to assess their relevance in representing brain functional dynamics.

Dyconnmap: Dynamic connectome mapping-A neuroimaging python module

Despite recent progress in the analysis of neuroimaging data sets, our comprehension of the main mechanisms and principles which govern human brain cognition and function remains incomplete. Network neuroscience makes substantial efforts to manipulate these challenges and provide real answers. For the last decade, researchers have been modelling brain structure and function via a graph or network that comprises brain regions that are either anatomically connected via tracts or functionally via a more extensive repertoire of functional associations. Network neuroscience is a relatively new multidisciplinary scientific avenue of the study of complex systems by pursuing novel ways to analyze, map, store and model the essential elements and their interactions in complex neurobiological systems, particularly the human brain, the most complex system in nature. Due to a rapid expansion of neuroimaging data sets' size and complexity, it is essential to propose and adopt new empirical tools to track dynamic patterns between neurons and brain areas and create comprehensive maps. In recent years, there is a rapid growth of scientific interest in moving functional neuroimaging analysis beyond simplified group or time‐averaged approaches and sophisticated algorithms that can capture the time‐varying properties of functional connectivity. We describe algorithms and network metrics that can capture the dynamic evolution of functional connectivity under this perspective. We adopt the word ‘chronnectome’ (integration of the Greek word ‘Chronos’, which means time, and connectome) to describe this specific branch of network neuroscience that explores how mutually informed brain activity correlates across time and brain space in a functional way. We also describe how good temporal mining of temporally evolved dynamic functional networks could give rise to the detection of specific brain states over which our brain evolved. This characteristic supports our complex human mind. The temporal evolution of these brain states and well‐known network metrics could give rise to new analytic trends. Functional brain networks could also increase the multi‐faced nature of the dynamic networks revealing complementary information. Finally, we describe a python module (https://github.com/makism/dyconnmap) which accompanies this article and contains a collection of dynamic complex network analytics and measures and demonstrates its great promise for the study of a healthy subject's repeated fMRI scans.

Short-Time Estimation of Fractionation in Atrial Fibrillation with Coarse-Grained Correlation Dimension for Mapping the Atrial Substrate

Atrial fibrillation (AF) is currently the most common cardiac arrhythmia, with catheter ablation (CA) of the pulmonary veins (PV) being its first line therapy. Ablation of complex fractionated atrial electrograms (CFAEs) outside the PVs has demonstrated improved long-term results, but their identification requires a reliable electrogram (EGM) fractionation estimator. This study proposes a technique aimed to assist CA procedures under real-time settings. The method has been tested on three groups of recordings: Group 1 consisted of 24 highly representative EGMs, eight of each belonging to a different AF Type. Group 2 contained the entire dataset of 119 EGMs, whereas Group 3 contained 20 pseudo-real EGMs of the special Type IV AF. Coarse-grained correlation dimension (CGCD) was computed at epochs of 1 s duration, obtaining a classification accuracy of 100% in Group 1 and 84.0–85.7% in Group 2, using 10-fold cross-validation. The receiver operating characteristics (ROC) analysis for highly fractionated EGMs, showed 100% specificity and sensitivity in Group 1 and 87.5% specificity and 93.6% sensitivity in Group 2. In addition, 100% of the pseudo-real EGMs were correctly identified as Type IV AF. This method can consistently express the fractionation level of AF EGMs and provides better performance than previous works. Its ability to compute fractionation in short-time can agilely detect sudden changes of AF Types and could be used for mapping the atrial substrate, thus assisting CA procedures under real-time settings for atrial substrate modification.

Inferring correlations associated to causal interactions in brain signals using autoregressive models

The specific connectivity of a neuronal network is reflected in the dynamics of the signals recorded on its nodes. The analysis of how the activity in one node predicts the behaviour of another gives the directionality in their relationship. However, each node is composed of many different elements which define the properties of the links. For instance, excitatory and inhibitory neuronal subtypes determine the functionality of the connection. Classic indexes such as the Granger causality (GC) quantifies these interactions, but they do not infer into the mechanism behind them. Here, we introduce an extension of the well-known GC that analyses the correlation associated to the specific influence that a transmitter node has over the receiver. This way, the G-causal link has a positive or negative effect if the predicted activity follows directly or inversely, respectively, the dynamics of the sender. The method is validated in a neuronal population model, testing the paradigm that excitatory and inhibitory neurons have a differential effect in the connectivity. Our approach correctly infers the positive or negative coupling produced by different types of neurons. Our results suggest that the proposed approach provides additional information on the characterization of G-causal connections, which is potentially relevant when it comes to understanding interactions in the brain circuits.

Detecting nonlinear stochastic systems using two independent hypothesis tests

Various systems in the real world can be nonlinear and stochastic, but because nonlinear time series analysis has been developed to distinguish nonlinear deterministic systems from linear stochastic systems, there have been no appropriate methods developed so far for testing the nonlinear stochasticity for a given system. Thus, here we propose a set of two hypothesis tests, one for the nonlinearity and one for the stochasticity, independent of each other. The test for the linearity is based on Fourier-transform-based surrogate data with a nonlinear test statistic, while the test for determinism depends on the theory of ordinal patterns or permutations recently developed intensively. We demonstrate the proposed set of tests with time series generated from toy models. In addition, we show that both a foreign exchange market and a temperature series in Tokyo could be nonlinear and stochastic, as well as sometimes with determinism beyond pseudoperiodicity.

Surrogate Data Preserving All the Properties of Ordinal Patterns up to a Certain Length

We propose a method for generating surrogate data that preserves all the properties of ordinal patterns up to a certain length, such as the numbers of allowed/forbidden ordinal patterns and transition likelihoods from ordinal patterns into others. The null hypothesis is that the details of the underlying dynamics do not matter beyond the refinements of ordinal patterns finer than a predefined length. The proposed surrogate data help construct a test of determinism that is free from the common linearity assumption for a null-hypothesis.

The meaning of "coherent" and its quantification in coherent hemodynamics spectroscopy

We have recently introduced a new technique, coherent hemodynamics spectroscopy (CHS), which aims at characterizing a specific kind of tissue hemodynamics that feature a high level of covariation with a given physiological quantity. In this study, we carry out a detailed analysis of the significance of coherence and phase synchronization between oscillations of arterial blood pressure (ABP) and total hemoglobin concentration ([Hbt]), measured with near-infrared spectroscopy (NIRS) during a typical protocol for CHS, based on a cyclic thigh cuff occlusion and release. Even though CHS is based on a linear time invariant model between ABP (input) and NIRS measurands (outputs), for practical reasons in a typical CHS protocol, we induce finite "groups" of ABP oscillations, in which each group is characterized by a different frequency. For this reason, ABP (input) and NIRS measurands (output) are not stationary processes, and we have used wavelet coherence and phase synchronization index (PSI), as a metric of coherence and phase synchronization, respectively. PSI was calculated by using both the wavelet cross spectrum and the Hilbert transform. We have also used linear coherence (which requires stationary process) for comparison with wavelet coherence. The method of surrogate data is used to find critical values for the significance of covariation between ABP and [Hbt]. Because we have found similar critical values for wavelet coherence and PSI by using five of the most used methods of surrogate data, we propose to use the data-independent Gaussian random numbers (GRNs), for CHS. By using wavelet coherence and wavelet cross spectrum, and GRNs as surrogate data, we have found the same results for the significance of coherence and phase synchronization between ABP and [Hbt]: on a total set of 20 periods of cuff oscillations, we have found 17 coherent oscillations and 17 phase synchronous oscillations. Phase synchronization assessed with Hilbert transform yielded similar results with 14 phase synchronous oscillations. Linear coherence and wavelet coherence overall yielded similar number of significant values. We discuss possible reasons for this result. Despite the similarity of linear and wavelet coherence, we argue that wavelet coherence is preferable, especially if one wants to use baseline spontaneous oscillations, in which phase locking and coherence between signals might be only temporary.

A comprehensive study of the delay vector variance method for quantification of nonlinearity in dynamical systems

Although vibration monitoring is a popular method to monitor and assess dynamic structures, quantification of linearity or nonlinearity of the dynamic responses remains a challenging problem. We investigate the delay vector variance (DVV) method in this regard in a comprehensive manner to establish the degree to which a change in signal nonlinearity can be related to system nonlinearity and how a change in system parameters affects the nonlinearity in the dynamic response of the system. A wide range of theoretical situations are considered in this regard using a single degree of freedom (SDOF) system to obtain numerical benchmarks. A number of experiments are then carried out using a physical SDOF model in the laboratory. Finally, a composite wind turbine blade is tested for different excitations and the dynamic responses are measured at a number of points to extend the investigation to continuum structures. The dynamic responses were measured using accelerometers, strain gauges and a Laser Doppler vibrometer. This comprehensive study creates a numerical and experimental benchmark for structurally dynamical systems where output-only information is typically available, especially in the context of DVV. The study also allows for comparative analysis between different systems driven by the similar input.

Dynamics of intracranial electroencephalographic recordings from epilepsy patients using univariate and bivariate recurrence networks

Recently Andrezejak et al. combined the randomness and nonlinear independence test with iterative amplitude adjusted Fourier transform (iAAFT) surrogates to distinguish between the dynamics of seizure-free intracranial electroencephalographic (EEG) signals recorded from epileptogenic (focal) and nonepileptogenic (nonfocal) brain areas of epileptic patients. However, stationarity is a part of the null hypothesis for iAAFT surrogates and thus nonstationarity can violate the null hypothesis. In this work we first propose the application of the randomness and nonlinear independence test based on recurrence network measures to distinguish between the dynamics of focal and nonfocal EEG signals. Furthermore, we combine these tests with both iAAFT and truncated Fourier transform (TFT) surrogate methods, which also preserves the nonstationarity of the original data in the surrogates along with its linear structure. Our results indicate that focal EEG signals exhibit an increased degree of structural complexity and interdependency compared to nonfocal EEG signals. In general, we find higher rejections for randomness and nonlinear independence tests for focal EEG signals compared to nonfocal EEG signals. In particular, the univariate recurrence network measures, the average clustering coefficient C and assortativity R, and the bivariate recurrence network measure, the average cross-clustering coefficient C(cross), can successfully distinguish between the focal and nonfocal EEG signals, even when the analysis is restricted to nonstationary signals, irrespective of the type of surrogates used. On the other hand, we find that the univariate recurrence network measures, the average path length L, and the average betweenness centrality BC fail to distinguish between the focal and nonfocal EEG signals when iAAFT surrogates are used. However, these two measures can distinguish between focal and nonfocal EEG signals when TFT surrogates are used for nonstationary signals. We also report an improvement in the performance of nonlinear prediction error N and nonlinear interdependence measure L used by Andrezejak et al., when TFT surrogates are used for nonstationary EEG signals. We also find that the outcome of the nonlinear independence test based on the average cross-clustering coefficient C(cross) is independent of the outcome of the randomness test based on the average clustering coefficient C. Thus, the univariate and bivariate recurrence network measures provide independent information regarding the dynamics of the focal and nonfocal EEG signals. In conclusion, recurrence network analysis combined with nonstationary surrogates can be applied to derive reliable biomarkers to distinguish between epileptogenic and nonepileptogenic brain areas using EEG signals.

A complex systems analysis of stick-slip dynamics of a laboratory fault

We study the stick-slip behavior of a granular bed of photoelastic disks sheared by a rough slider pulled along the surface. Time series of a proxy for granular friction are examined using complex systems methods to characterize the observed stick-slip dynamics of this laboratory fault. Nonlinear surrogate time series methods show that the stick-slip behavior appears more complex than a periodic dynamics description. Phase space embedding methods show that the dynamics can be locally captured within a four to six dimensional subspace. These slider time series also provide an experimental test for recent complex network methods. Phase space networks, constructed by connecting nearby phase space points, proved useful in capturing the key features of the dynamics. In particular, network communities could be associated to slip events and the ranking of small network subgraphs exhibited a heretofore unreported ordering.

Improvements to surrogate data methods for nonstationary time series

The method of surrogate data has been extensively applied to hypothesis testing of system linearity, when only one realization of the system, a time series, is known. Normally, surrogate data should preserve the linear stochastic structure and the amplitude distribution of the original series. Classical surrogate data methods (such as random permutation, amplitude adjusted Fourier transform, or iterative amplitude adjusted Fourier transform) are successful at preserving one or both of these features in stationary cases. However, they always produce stationary surrogates, hence existing nonstationarity could be interpreted as dynamic nonlinearity. Certain modifications have been proposed that additionally preserve some nonstationarity, at the expense of reproducing a great deal of nonlinearity. However, even those methods generally fail to preserve the trend (i.e., global nonstationarity in the mean) of the original series. This is the case of time series with unit roots in their autoregressive structure. Additionally, those methods, based on Fourier transform, either need first and last values in the original series to match, or they need to select a piece of the original series with matching ends. These conditions are often inapplicable and the resulting surrogates are adversely affected by the well-known artefact problem. In this study, we propose a simple technique that, applied within existing Fourier-transform-based methods, generates surrogate data that jointly preserve the aforementioned characteristics of the original series, including (even strong) trends. Moreover, our technique avoids the negative effects of end mismatch. Several artificial and real, stationary and nonstationary, linear and nonlinear time series are examined, in order to demonstrate the advantages of the methods. Corresponding surrogate data are produced with the classical and with the proposed methods, and the results are compared.

Testing for nonlinearity in non-stationary physiological time series

Testing for nonlinearity is one of the most important preprocessing steps in nonlinear time series analysis. Typically, this is done by means of the linear surrogate data methods. But it is a known fact that the validity of the results heavily depends on the stationarity of the time series. Since most physiological signals are non-stationary, it is easy to falsely detect nonlinearity using the linear surrogate data methods. In this document, we propose a methodology to extend the procedure for generating constrained surrogate time series in order to assess nonlinearity in non-stationary data. The method is based on the band-phase-randomized surrogates, which consists (contrary to the linear surrogate data methods) in randomizing only a portion of the Fourier phases in the high frequency domain. Analysis of simulated time series showed that in comparison to the linear surrogate data method, our method is able to discriminate between linear stationarity, linear non-stationary and nonlinear time series. Applying our methodology to heart rate variability (HRV) records of five healthy patients, we encountered that nonlinear correlations are present in this non-stationary physiological signals.

On detecting determinism and nonlinearity in microelectrode recording signals: approach based on non-stationary surrogate data methods

Two new surrogate methods, the Small Shuffle Surrogate (SSS) and the Truncated Fourier Transform Surrogate (TFTS), have been proposed to study whether there are some kind of dynamics in irregular fluctuations and if so whether these dynamics are linear or not, even if this fluctuations are modulated by long term trends. This situation is theoretically incompatible with the assumption underlying previously proposed surrogate methods. We apply the SSS and TFTS methods to microelectrode recording (MER) signals from different brain areas, in order to acquire a deeper understanding of them. Through our methodology we conclude that the irregular fluctuations in MER signals possess some determinism.