Gaussian Time Series Models Research Articles

Generalized Gaussian time series model for increments of EEG data

Generalized Gaussian time series model for increments of EEG data

Generalized Gaussian time series model for increments of EEG data

Generalized Gaussian time series model for increments of EEG 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.

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.

Goodness‐of‐fit tests forβARMA hydrological time series modeling

AbstractWe address the issue of performing portmanteau testing inference using time series data that assume values in the standard unit interval. The motivation involves modeling the time series dynamics of the proportion of stocked hydroelectric energy in the South of Brazil. Our focus lies in the class of beta autoregressive moving average (βARMA) models. In particular, we wish to test the goodness‐of‐fit of such models. We consider several testing criteria that have been proposed for Gaussian time series models and introduce two new tests. We derive the asymptotic null distribution of the two proposed test statistics in two different scenarios, namely, when the tests are applied to an observed time series and when they are applied to the residuals from a fittedβARMA model. It is worth noticing that our results imply the asymptotic validity of standard portmanteau tests in the class ofβARMA models that are, under the null hypothesis, asymptotically equivalent to our test statistics. We use Monte Carlo simulation to assess the relative merits of the different portmanteau tests when used with fittedβARMA models. The simulation results we present show that the new tests are typically more powerful than a well‐known test whose test statistic is also based on residual partial autocorrelations. Overall, the tests we propose perform quite well. Finally, we model the dynamics of the proportion of stocked hydroelectric energy in Brazil. The results show that theβARMA model outperforms three alternative models and an exponential smoothing algorithm.

Rachel S. McCrea and Byron J. T. Morgan, Analysis of Capture–Recapture Data

Analysis of Capture–Recapture Data by McCrea and Morgan is an excellent, easy to read monograph about capture–recapture models. In this book, the authors provide a concise overview of traditional closed population capture–recapture models (Models M0, Mb, Mh, etc.), individual covariate models, and open population models such as the Cormack–Jolly– Seber, Jolly–Seber models, multi-state models, andmore recent developments such as occupancy models, state-space models, and integrated population models. The authors write “In this book we aim to cover the many modern developments in the area of capture–recapture and related models, and to set them in historical context of relevant research over the past 100 years.” The book does a good job of achieving this objective. And it is a very easy to read because it is well organized and the writing is clear and concise. I would recommend this book as a reference for the quantitative ecologist or statistician interested in knowing what’s out there. And I’m glad to have it on my bookshelf. The main topical chapters focus on specific classes of capture–recapture models and contain a mix of classical concepts and methods as well as more recent innovations. For example, the chapter on closed population models contains material on N-mixture models and also spatial capture–recapture models1. There is a strong emphasis on mark-recovery models which are important when band recoveries are available. I think chapters 11 and 12 are the strongest. Chapter 11, on state-space models (covering Gaussian time-series models and Kalman filtering), is not specifically a capture–recapture topic but sometimes used as a component of integrated population models and the concepts are of general interest in population modeling. This chapter also covers state-space formulations of CJS and

Computational Aspects of Bayesian Spectral Density Estimation

Gaussian time-series models are often specified through their spectral density. Such models present several computational challenges, in particular because of the nonsparse nature of the covariance matrix. We derive a fast approximation of the likelihood for such models. We propose to sample from the approximate posterior (i.e., the prior times the approximate likelihood), and then to recover the exact posterior through importance sampling. We show that the variance of the importance sampling weights vanishes as the sample size goes to infinity. We explain why the approximate posterior may typically be multimodal, and we derive a Sequential Monte Carlo sampler based on an annealing sequence to sample from that target distribution. Performance of the overall approach is evaluated on simulated and real datasets. In addition, for one real-world dataset, we provide some numerical evidence that a Bayesian approach to semiparametric estimation of spectral density may provide more reasonable results than its frequentist counterparts. The article comes with supplementary materials, available online, that contain an Appendix with a proof of our main Theorem, a Python package that implements the proposed procedure, and the Ethernet dataset.

The effect of Operation 24 Hours on reducing collision in the City of Edmonton

In the City of Edmonton, in order to reduce the prevalence of collisions, the Operation 24 Hours program (OPS24) was developed by using existing police and transportation services resources. The program uses traditional manned police speed enforcement method, which are supplemented by traffic safety messages displayed on permanent and mobile dynamic messaging signs (DMS). In this paper, collision data analysis was performed by looking at the daily number of collisions from 2008 to 2011 that covers 28 Operation 24 Hours (OPS24) events. The objective of the collision data analysis is to analyze if there is a reduction in collision frequencies after OPS24 was held and examined how long the collision reduction effect last. Weather factors such as temperature, thickness of snow, and wind gust have been considered by many as a great influence on collision occurrences, especially in a city with long and cold winter such as Edmonton. Therefore, collision modeling was performed by considering these external weather factors. To analyze the linear and periodic trend of different collision types (injury, fatal, and property damage only (PDO)) and examine the influence of weather factors on collisions, negative binomial time series model that accounts for seasonality and weather factors was used to model daily collision data. The modeling also considered collision proportion to account for missing traffic volume data; the Gaussian time series model that accounts for seasonality and weather factors was used to model collision proportion. To estimate the collision trend and test for changes in collision levels before/after OPS24, interrupted time series model with segmented regression was used. While for estimating how long the effect of the OPS24 last, change point method was applied.

Improved prediction limits for a general class of Gaussian models

In this article we consider the problem of prediction for a general class of Gaussian models, which includes, among others, autoregressive moving average time-series models, linear Gaussian state space models and Gaussian Markov random fields. Using an idea presented in Sjostedt-De Luna and Young (2003), in the context of spatial statistics, we discuss a method for obtaining prediction limits for a future random variable of interest, taking into account the uncertainty introduced by estimating the unknown parameters. The proposed prediction limits can be viewed as a modification of the estimative prediction limit, with unconditional, and eventually conditional, coverage error of smaller asymptotic order. The modifying term has a quite simple form and it involves the bias and the mean square error of the plug-in estimators for the conditional expectation and the conditional variance of the future observation. Applications of the results to Gaussian time-series models are presented.

Generating random AR( [formula omitted]) and MA( [formula omitted]) Toeplitz correlation matrices

Methods are proposed for generating random ( p + 1 ) × ( p + 1 ) Toeplitz correlation matrices that are consistent with a causal AR( p ) Gaussian time series model. The main idea is to first specify distributions for the partial autocorrelations that are algebraically independent and take values in ( − 1 , 1 ) , and then map to the Toeplitz matrix. Similarly, starting with pseudo-partial autocorrelations, methods are proposed for generating ( q + 1 ) × ( q + 1 ) Toeplitz correlation matrices that are consistent with an invertible MA( q ) Gaussian time series model. The density can be uniform or non-uniform over the space of autocorrelations up to lag p or q , or over the space of autoregressive or moving average coefficients, by making appropriate choices for the densities of the (pseudo)-partial autocorrelations. Important intermediate steps are the derivations of the Jacobians of the mappings between the (pseudo)-partial autocorrelations, autocorrelations and autoregressive/moving average coefficients. The random generating methods are useful for models with a structured Toeplitz matrix as a parameter.

Bayesian Variable Selection in Markov Mixture Models

Bayesian methods for variable selection and model choice have become increasingly popular in recent years, due to advances in Markov Chain Monte Carlo (MCMC) computational algorithms. Several methods have been proposed in literature in the case of linear and generalized linear models. In this article, we adapt some of the most popular algorithms to a class of nonlinear and non Gaussian time series models, i.e., the Markov Mixture Models (MMM). We also propose the “Metropolization” of the algorithm of Kuo and Mallick (1998), in order to tackle variable selection efficiently, both when the complexity of the model is high, as in MMM, and when the exogenous variables are strongly correlated. Numerical comparisons among the competing MCMC algorithms are also presented via simulation examples.

AN ANALYSIS OF DAILY PEAK STREAM DISCHARGE USING A NON-GAUSSIAN TIME SERIES MODEL

Daily peak stream discharge data, collected over time, are typically characterized by a few large peaks separated by runs of small values, where peaks correspond to the occurrence of storms. Furthermore, the peak discharge on the first day of a storm has little or no relationship to the previous day's discharge. These characteristics are not present in standard Gaussian time series models in which a zig-zag behavior not conducive to runs of small values is observed and the present value always depends on the previous value. However, they can be successfully captured with non-Gaussian time series models.

Construction of the exact Fisher information matrix of Gaussian time series models by means of matrix differential rules

The Fisher information matrix is of fundamental importance for the analysis of parameter estimation of time series models. In this paper the exact information matrix of a multivariate Gaussian time series model expressed in state space form is derived. A computationally efficient procedure is used by applying matrix differential rules for the derivatives of a matrix function J= J( θ) with respect to its vector argument. An algorithm is given. It is sketched for the general state space structure without specifying a parametrization. It is then detailed for the vector autoregressive moving average (VARMA) model, with a given parametrization, where explicit recurrent relations are developed.

Computation of the exact information matrix of Gaussian dynamic regression time series models

In this paper, the computation of the exact Fisher information matrix of a large class of Gaussian time series models is considered. This class, which is often called the single-input–single-output (SISO) model, includes dynamic regression with autocorrelated errors and the transfer function model, with autoregressive moving average errors. The method is 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. It is much faster than existing procedures. An expression for the asymptotic information matrix is also given.

Developments in Time Series Analysis: In Honor of Maurice B. Priestly.

List of Contributors. Foreword. Preface. Books and Papers by M. B. Priestley. About the Volume and Summary of Papers. Linear Time Series Models. Estimation and Asymptotics for Times Series Models. Spectral Analysis of Stationary Time Series. Nonparametric Statistical Inference in Time Series. Nonlinear and Non Gaussian Time Series Models. Index.

Developments in Time Series Analysis: In Honour of Maurice B. Priestley.

List of Contributors. Foreword. Preface. Books and Papers by M. B. Priestley. About the Volume and Summary of Papers. Linear Time Series Models. Estimation and Asymptotics for Times Series Models. Spectral Analysis of Stationary Time Series. Nonparametric Statistical Inference in Time Series. Nonlinear and Non Gaussian Time Series Models. Index.