Gaussian Time Series Research Articles

Generalized Gaussian time series model for increments of EEG data

Generalized Gaussian time series model for increments of EEG data

A Parametric МА-Algorithm for Gaussian Time Series Change Point Detection from Mathematical Expectation

The problem of online detection of a Gaussian time series change connected with a spontaneous abrupt change in its mathematical expectation is solved. For solving the problem, it is proposed to use a parametric controlling algorithm based on the Moving Average method or the MA- algorithm. It is noted that, despite the fact that this algorithm has been known almost since the first works in the field of statistical control, its properties, capabilities, and efficiency in comparison with other change detection algorithms are still (for a number of reasons) have been studied to a very poor extent. The purpose of this work is to thoroughly study the MA-algorithm’s characteristics with the aim to synthesize an optimal time series change detection procedure. The study was carried out using a simulation modeling method. The article presents the structure and element-wise description of the simulation experiment program, which fully replicates the MA-algorithm operation in the online mode. By using the developed program, it has been found that the conventional way of setting the control procedure deciding threshold h, the value of which should provide the preset value of the average time between false alarms, when an alarm signal about the appearance of a change is generated, although in reality the object remains in the "normal" state, is invalid. For a fixed series of the above-mentioned quantities, the correct values of the threshold h are obtained depending on the width of the moving averaging window N of the controlling MA-algorithm. Similarly, the values of the average delay time of producing an alarm signal when a change with the given fixed level  appears. Based on the data obtained, dependences of the control procedure efficiency indicator E on N for a set of different values of average time between false alarms and  have been found. It has been shown that for each such set there is a value of N at which the control procedure is most effective. This optimal procedure is compared with a similar procedure of the cumulative sum algorithm (a CUSUM algorithm), and it has been shown from the comparison results that the MA-algorithm is in general only slightly inferior to the CUSUM algorithm in efficiency, and in some cases even outperforms it.

On the asymptotic distribution of the maximum sample spectral coherence of Gaussian time series in the high dimensional regime

We investigate the asymptotic distribution of the maximum of a frequency smoothed estimate of the spectral coherence of a M-variate complex Gaussian time series with mutually independent components when the dimension M and the number of samples N both converge to infinity. If B denotes the smoothing span of the underlying smoothed periodogram estimator, a type I extreme value limiting distribution is obtained under the rate assumptions MN→0 and MB→c∈(0,+∞). This result is then exploited to build a statistic with controlled asymptotic level for testing independence between the M components of the observed time series. Numerical simulations support our results.

Spectral measure of empirical autocovariance matrices of high-dimensional Gaussian stationary processes

Consider the empirical autocovariance matrices at given non-zero time lags, based on observations from a multivariate complex Gaussian stationary time series. The spectral analysis of these autocovariance matrices can be useful in certain statistical problems, such as those related to testing for white noise. We study the behavior of their spectral measure in the asymptotic regime where the time series dimension and the observation window length both grow to infinity, and at the same rate. Following a general framework in the field of the spectral analysis of large random non-Hermitian matrices, at first the probabilistic behavior of the small singular values of a shifted version of the autocovariance matrix is obtained. This is then used to obtain the asymptotic behavior of the empirical spectral measure of the autocovariance matrices at any lag. Matrix orthogonal polynomials on the unit circle play a crucial role in our study.

Renormalization group approach to connect discrete- and continuous-time descriptions of Gaussian processes

Discretization of continuous stochastic processes is needed to numerically simulate them or to infer models from experimental time series. However, depending on the nature of the process, the same discretization scheme may perform very differently for the two tasks, if it is not accurate enough. Exact discretizations, which work equally well at any scale, are characterized by the property of invariance under coarse-graining. Motivated by this observation, we build an explicit renormalization group (RG) approach for Gaussian time series generated by autoregressive models. We show that the RG fixed points correspond to discretizations of linear SDEs, and only come in the form of first order Markov processes or non-Markovian ones. This fact provides an alternative explanation of why standard delay-vector embedding procedures fail in reconstructing partially observed noise-driven systems. We also suggest a possible effective Markovian discretization for the inference of partially observed underdamped equilibrium processes based on the exploitation of the Einstein relation.

On sparse high-dimensional graphical model learning for dependent time series

We consider the problem of inferring the conditional independence graph (CIG) of a sparse, high-dimensional stationary multivariate Gaussian time series. A sparse-group lasso-based frequency-domain formulation of the problem based on frequency-domain sufficient statistic for the observed time series is presented. We investigate an alternating direction method of multipliers (ADMM) approach for optimization of the sparse-group lasso penalized log-likelihood. We provide sufficient conditions for convergence in the Frobenius norm of the inverse PSD estimators to the true value, jointly across all frequencies, where the number of frequencies are allowed to increase with sample size. This result also yields a rate of convergence. We also empirically investigate selection of the tuning parameters based on the Bayesian information criterion, and illustrate our approach using numerical examples utilizing both synthetic and real data.

Computation of the Exact Fisher Information Matrix of a Multiple Input Single Output Time Series Models

Klein, Mélard, and Zahaf (1998) have proposed the computation of the exact Fisher information matrix of a large class of Gaussian time series models called the single-input-single-output (SISO) model, includes dynamic regression with autocorrelated errors and the transfer function model, with autoregressive moving average errors. For computing the Fisher information matrix of a SISO model, they introduced an algorithm based on a combination of two computational procedures: recursions for the covariance matrix of the derivatives of the state vector with respect to the parameters and the fast Kalman filter recursions used in the evaluation of the likelihood function. In this paper, we propose a generalization of this method for computing the Fisher information matrix of a MISO model.

An endemic–epidemic beta model for time series of infectious disease proportions

Time series of proportions of infected patients or positive specimens are frequently encountered in disease control and prevention. Since proportions are bounded and often asymmetrically distributed, conventional Gaussian time series models only apply to suitably transformed proportions. Here we borrow both from beta regression and from the well-established HHH model for infectious disease counts to propose an endemic–epidemic beta model for proportion time series. It accommodates the asymmetric shape and heteroskedasticity of proportion distributions and is consistent for complementary proportions. Coefficients can be interpreted in terms of odds ratios. A multivariate formulation with spatial power-law weights enables the joint estimation of model parameters from multiple regions. In our application to a flu activity index in the USA, we find that the endemic–epidemic beta model provides a better fit than a seasonal ARIMA model for the logit-transformed proportions. Furthermore, a multivariate approach can improve regional forecasts and reduce model complexity in comparison to univariate beta models stratified by region.

Analysis and Forecasting of Risk in Count Processes

Risk measures are commonly used to prepare for a prospective occurrence of an adverse event. If we are concerned with discrete risk phenomena such as counts of natural disasters, counts of infections by a serious disease, or counts of certain economic events, then the required risk forecasts are to be computed for an underlying count process. In practice, however, the discrete nature of count data is sometimes ignored and risk forecasts are calculated based on Gaussian time series models. But even if methods from count time series analysis are used in an adequate manner, the performance of risk forecasting is affected by estimation uncertainty as well as certain discreteness phenomena. To get a thorough overview of the aforementioned issues in risk forecasting of count processes, a comprehensive simulation study was done considering a broad variety of risk measures and count time series models. It becomes clear that Gaussian approximate risk forecasts substantially distort risk assessment and, thus, should be avoided. In order to account for the apparent estimation uncertainty in risk forecasting, we use bootstrap approaches for count time series. The relevance and the application of the proposed approaches are illustrated by real data examples about counts of storm surges and counts of financial transactions.

Small animal disease surveillance 2020/21: SARS-CoV-2, syndromic surveillance and an outbreak of acute vomiting in UK dogs.

Small animal disease surveillance 2020/21: SARS-CoV-2, syndromic surveillance and an outbreak of acute vomiting in UK dogs.

Spectral Independent Component Analysis with noise modeling for M/EEG source separation

BackgroundIndependent Component Analysis (ICA) is a widespread tool for exploration and denoising of electroencephalography (EEG) or magnetoencephalography (MEG) signals. In its most common formulation, ICA assumes that the signal matrix is a noiseless linear mixture of independent sources that are assumed non-Gaussian. A limitation is that it enforces to estimate as many sources as sensors or to rely on a detrimental PCA step. MethodsWe present the Spectral Matching ICA (SMICA) model. Signals are modelled as a linear mixing of independent sources corrupted by additive noise, where sources and the noise are stationary Gaussian time series. Thanks to the Gaussian assumption, the negative log-likelihood has a simple expression as a sum of ‘divergences’ between the empirical spectral covariance matrices of the signals and those predicted by the model. The model parameters can then be estimated by the expectation-maximization (EM) algorithm. ResultsOn phantom MEG datasets with low amplitude dipole sources (20 nAm), SMICA makes a median dipole localization error of 1.5 mm while competing methods make an error ≥7 mm. Experiments on EEG datasets show that SMICA identifies a source subspace which contains sources that have less pairwise mutual information, and are better explained by the projection of a single dipole on the scalp. With 10 sources, the number of strongly dipolar sources (dipolarity >90%) is more than 80% for SMICA while competing methods do not exceed 65%. Comparison with existing methodsWith the noisy model of SMICA, the number of sources to be recovered is controlled by choosing the size of the mixing matrix to be fitted rather than by a preprocessing step of dimension reduction which is required in traditional noise-free ICA methods. ConclusionsSMICA is a promising alternative to other noiseless ICA models based on non-Gaussian assumptions.

Fitting Time Series with Heavy Tails and Strong Time Dependence

Earlier, a model of a time series with heavy tails constructed from a Gaussian time series was developed. In the present paper, the reverse problem is considered: an estimator of the copula function is built; the copula function is a nonlinear function that maps Gaussian variables to the variables from the Frechet maximum domain of attraction. The statistical properties of this estimator are considered for a stationary time series with a low rate of covariance decay.

Properties of linear spectral statistics of frequency-smoothed estimated spectral coherence matrix of high-dimensional Gaussian time series

The asymptotic behaviour of Linear Spectral Statistics (LSS) of the smoothed periodogram estimator of the spectral coherency matrix of a complex Gaussian high-dimensional time series (yn)n∈Z with independent components is studied under the asymptotic regime where the sample size N converges towards +∞ while the dimension M of y and the smoothing span of the estimator grow to infinity at the same rate in such a way that M N→0. It is established that, at each frequency, the estimated spectral coherency matrix is close to the sample covariance matrix of an independent identically NC(0,IM) distributed sequence, and that its empirical eigenvalue distribution converges towards the Marcenko-Pastur distribution. This allows to conclude that each LSS has a deterministic behaviour that can be evaluated explicitly. Using concentration inequalities, it is shown that the order of magnitude of the supremum over the frequencies of the deviation of each LSS from its deterministic approximation is of the order of 1 M+ M N+(M N)3 where N is the sample size. Numerical simulations supports our results.

Nonlinear Spectral Analysis: A Local Gaussian Approach

The spectral distribution of a stationary time series can be used to investigate whether or not periodic structures are present in , but has some limitations due to its dependence on the autocovariances . For example, can not distinguish white iid noise from GARCH-type models (whose terms are dependent, but uncorrelated), which implies that can be an inadequate tool when contains asymmetries and nonlinear dependencies. Asymmetries between the upper and lower tails of a time series can be investigated by means of the local Gaussian autocorrelations, and these local measures of dependence can be used to construct the local Gaussian spectral density presented in this paper. A key feature of the new local spectral density is that it coincides with for Gaussian time series, which implies that it can be used to detect non-Gaussian traits in the time series under investigation. In particular, if is flat, then peaks and troughs of the new local spectral density can indicate nonlinear traits, which potentially might discover local periodic phenomena that remain undetected in an ordinary spectral analysis.

On Edgeworth models for count time series

Inspired by the Edgeworth–Portnoy model for Gaussian time series, a family of randomized moving window (RMW) and randomized moving sum (RMS) models for stationary count time series is proposed. For the RMW process, we derive Markov properties which, in turn, allow to conclude on a connection of the RMS model to the Hidden-Markov model. This connection is used to develop an efficient scheme for maximum likelihood estimation. Then, we derive marginal and serial moment properties of the RMS process. It commonly exhibits an autoregressive autocorrelation structure, but also forms of a long memory are possible. The latter holds in particular for the proposed extended RMS model, which also satisfies certain Markov properties.

The Hyvärinen scoring rule in Gaussian linear time series models

In this work we study stationary linear time-series models, and construct and analyse “score-matching” estimators based on the Hyvärinen scoring rule. We consider two scenarios: a single series of increasing length, and an increasing number of independent series of fixed length. In the latter case there are two variants, one based on the full data, and another based on a sufficient statistic.We study the empirical performance of these estimators in three special cases, autoregressive (AR), moving average (MA) and fractionally differenced white noise (ARFIMA) models, and make comparisons with full and pairwise likelihood estimators. The results are somewhat model-dependent, with the new estimators doing well for MA and ARFIMA models, but less so for AR models.

Transforming Gaussian correlations. Applications to generating long-range power-law correlated time series with arbitrary distribution.

The observable outputs of many complex dynamical systems consist of time series exhibiting autocorrelation functions of great diversity of behaviors, including long-range power-law autocorrelation functions, as a signature of interactions operating at many temporal or spatial scales. Often, numerical algorithms able to generate correlated noises reproducing the properties of real time series are used to study and characterize such systems. Typically, many of those algorithms produce a Gaussian time series. However, the real, experimentally observed time series are often non-Gaussian and may follow distributions with a diversity of behaviors concerning the support, the symmetry, or the tail properties. It is always possible to transform a correlated Gaussian time series into a time series with a different marginal distribution, but the question is how this transformation affects the behavior of the autocorrelation function. Here, we study analytically and numerically how the Pearson's correlation of two Gaussian variables changes when the variables are transformed to follow a different destination distribution. Specifically, we consider bounded and unbounded distributions, symmetric and non-symmetric distributions, and distributions with different tail properties from decays faster than exponential to heavy-tail cases including power laws, and we find how these properties affect the correlation of the final variables. We extend these results to a Gaussian time series, which are transformed to have a different marginal distribution, and show how the autocorrelation function of the final non-Gaussian time series depends on the Gaussian correlations and on the final marginal distribution. As an application of our results, we propose how to generalize standard algorithms producing a Gaussian power-law correlated time series in order to create a synthetic time series with an arbitrary distribution and controlled power-law correlations. Finally, we show a practical example of this algorithm by generating time series mimicking the marginal distribution and the power-law tail of the autocorrelation function of real time series: the absolute returns of stock prices.

A family of multivariate non‐gaussian time series models

In this article, we propose a class of multivariate non‐Gaussian time series models which include dynamic versions of many well‐known distributions and consider their Bayesian analysis. A key feature of our proposed model is its ability to account for correlations across time as well as across series (contemporary) via a common random environment. The proposed modeling approach yields analytically tractable dynamic marginal likelihoods, a property not typically found outside of linear Gaussian time series models. These dynamic marginal likelihoods can be tied back to known static multivariate distributions such as the Lomax, generalized Lomax, and the multivariate Burr distributions. The availability of the marginal likelihoods allows us to develop efficient estimation methods for various settings using Markov chain Monte Carlo as well as sequential Monte Carlo methods. Our approach can be considered to be a multivariate generalization of commonly used univariate non‐Gaussian class of state space models. To illustrate our methodology, we use simulated data examples and a real application of multivariate time series for modeling the joint dynamics of stochastic volatility in financial indexes, the VIX and VXN.

Modeling an association between malaria cases and climate variables for Keonjhar district of Odisha, India: a Bayesian approach.

Malaria, a vector-borne disease, is a significant public health problem in Keonjhar district of Odisha (the malaria capital of India). Prediction of malaria, in advance, is an urgent need for reporting rolling cases of disease throughout the year. The climate condition do play an essential role in the transmission of malaria. Hence, the current study aims to develop and assess a simple and straightforward statistical model of an association between malaria cases and climate variates. It may help in accurate predictions of malaria cases given future climate conditions. For this purpose, a Bayesian Gaussian time series regression model is adopted to fit a relationship of the square root of malaria cases with climate variables with practical lag effects. The model fitting is assessed using a Bayesian version of R2(RsqB). Whereas, the predictive ability of the model is measured using a cross-validation technique. As a result, it is found that the square root of malaria cases with lag 1, maximum temperature, and relative humidity with lag 3 and 0 (respectively), are significantly positively associated with the square root of the cases. However, the minimum and average temperatures with lag 2, respectively, are observed as negatively (significantly) related. The considered model accounts for moderate amount of variation in the square root of malaria cases as received through the results for RsqB. We also present Absolute Percentage Errors (APE) for each of the 12months (January-December) for a better understanding of the seasonal pattern of the predicted (square root of) malaria cases. Most of the APEs obtained corresponding to test data points is reasonably low.Further, the analysis shows that the considered model closely predicted the actual (square root of) malaria cases, except for some peak cases during the particular months. The output of the current research might help the district to develop and strengthen early warning prediction of malaria cases for proper mitigation, eradication, and prevention in similar settings.

Semiparametric estimation and inference on the fractal index of Gaussian and conditionally Gaussian time series data

This paper studies the properties of a particular estimator of the fractal index of a time series with a view to applications in financial econometrics and mathematical finance. We show how measurement noise (e.g., microstructure noise) in the observations will bias the estimator, potentially resulting in the econometrician erroneously finding evidence of fractal characteristics in a time series. We propose a new estimator which is robust to such noise and construct a formal hypothesis test for the presence of noise in the observations. A number of simulation exercises are carried out, providing guidance for implementation of the theory. Finally, the methods are illustrated on two empirical data sets; one of turbulent velocity flows and one of financial prices.