Gaussian Time Series Research Articles

Fast simulation of self-similar and correlated processes

Simulations with long-range dependent or self-similar input processes are hindered both by the slowness of convergence displayed by the output data and by the high computational complexity of the on-line methods for generating the input process. In this paper, we present optimized algorithms for simulating efficiently the occupancy process of a M/G/ ∞ system, which can be used as a sequential pseudo-random number generator of a broad class of self-similar and correlated sample-paths. We advocate the use of this approach in the simulation toolbox, as a simple method to overcome the drawbacks of other synthetic generators of Gaussian self-similar time series. Our approach to fast simulation of the M/G/ ∞ model is the decomposition of the service time distribution as a linear combination of deterministic and memoryless random variables, plus a residual term. Then, the original M/G/ ∞ system is replaced by a number of parallel, independent, virtual and easier to simulate M/G/ ∞ subsystems, the dynamics of which can be replicated sequentially or in parallel too. We report the results of several experiments demonstrating the substantial improvements attainable with this decomposition.

Bayesian non-parametric signal extraction for Gaussian time series

We consider the problem of unobserved components in time series from a Bayesian non-parametric perspective. The identification conditions are treated as unknown and analyzed in a probabilistic framework. In particular, informative prior distributions force the spectral decomposition to be in an identifiable region. Then, the likelihood function adapts the prior decompositions to the data. A full Bayesian analysis of unobserved components will be presented for financial high frequency data. Particularly, a three component model (long-term, intra-daily and short-term) will be analyzed to emphasize the importance and the potential of this work when dealing with the Value-at-Risk analysis. A second astronomical application will show how to deal with multiple periodicities.

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.

Inducing normality from non‐Gaussian long memory time series and its application to stock return data

AbstractMotivated by Lee and Ko (Appl. Stochastic Models. Bus. Ind.2007;23:493–502) but not limited to the study, this paper proposes a wavelet‐based Bayesian power transformation procedure through the well‐known Box–Cox transformation to induce normality from non‐Gaussian long memory processes. We consider power transformations of non‐Gaussian long memory time series under the assumption of anunknowntransformation parameter, a situation that arises commonly in practice, while most research has been devoted to non‐linear transformations of Gaussian long memory time series withknowntransformation parameter. Specially, this study is mainly focused on the simultaneous estimation of the transformation parameter and long memory parameter. To this end, posterior estimations via Markov chain Monte Carlo methods are performed in the wavelet domain. Performances are assessed on a simulation study and a German stock return data set. Copyright © 2009 John Wiley & Sons, Ltd.

A Bayesian approach to estimating the long memory parameter

We develop a Bayesian procedure for analyzing stationary long-range dependent processes. Specifically, we consider the fractional exponential model (FEXP) to estimate the memory parameter of a stationary long-memory Gaussian time series. In particular, we propose a hierarchical Bayesian model and make it fully adaptive by imposing a prior distribution on the model order. Further, we describe a reversible jump Markov chain Monte Carlo algorithm for variable dimension estimation and show that, in our context, the algorithm provides a reasonable method of model selection (within each repetition of the chain). Therefore, through an application of Bayesian model averaging, we incorporate all possible models from the FEXP class (up to a given finite order). As a result we reduce the underestimation of uncertainty at the model-selection stage as well as achieve better estimates of the long memory parameter. Additionally, we establish Bayesian consistency of the memory parameter under mild conditions on the data process. Finally, through simulation and the analysis of two data sets, we demonstrate the effectiveness of our approach.

Multiresolution analysis of fluctuations in non-stationary time series through discrete wavelets

We illustrate the efficacy of a discrete wavelet based approach to characterize fluctuations in non-stationary time series. The present approach complements the multifractal detrended fluctuation analysis (MF-DFA) method and is quite accurate for small size data sets. As compared to polynomial fits in the MF-DFA, a single Daubechies wavelet is used here for detrending purposes. The natural, built-in variable window size in wavelet transforms makes this procedure well suited for non-stationary data. We illustrate the working of this method through the analysis of binomial multifractal model. For this model, our results compare well with those calculated analytically and obtained numerically through MF-DFA. To show the efficacy of this approach for finite data sets, we also do the above comparison for Gaussian white noise time series of different sizes. In addition, we analyze time series of three experimental data sets of tokamak plasma and also spin density fluctuations in 2D Ising model.

Gaussian Process Time Series Model for Life Prognosis of Metallic Structures

Al 2024-T351 fatigue specimens have been modeled using a kernel-based multi-variate Gaussian Process approach. The Gaussian Process model projects fatigue affecting input variables to output crack growth by probabilistically inferring the underlying nonlinear relationship between input and output. The Gaussian Process approach not only explicitly models the uncertainty due to scatter in material microstructure parameter but it also implicitly models the loading sequence effect due to variable loading. The loading sequence effect is modeled through the Gaussian Process optimal hyperparameters by using the crack length data observed over the entire domain of spectrum loading. The performance in the crack growth prediction is evaluated for two covariance functions, a radial basis-based, anisotropic, covariance function and a neural network-based isotropic covariance function. Furthermore, the performance of different types of scaling, used to scale the input—output data space, is tested. It is found that for the radial basis-based anisotropic covariance function with normalized scaling, the prediction error is consistently lower compared to other combinations. In addition, the Gaussian Process model allows determination of the collapse load condition, which is a desirable feature for the online health monitoring and prognosis.

Optimising prediction error among completely monotone covariance sequences

We provide a characterisation of Gaussian time series which optimise the one-step prediction error subject to the covariance sequence being completely monotone with the first m covariances specified.

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.

CENTRAL LIMIT THEOREM FOR THE LOG-REGRESSION WAVELET ESTIMATION OF THE MEMORY PARAMETER IN THE GAUSSIAN SEMI-PARAMETRIC CONTEXT

We consider a Gaussian time series, stationary or not, with long memory exponent d ∈ ℝ. The generalized spectral density function of the time series is characterized by d and by a function f*(λ) which specifies the short-range dependence structure. Our setting is semi-parametric in that both d and f* are unknown, and only the smoothness of f* around λ = 0 matters. The parameter d is the one of interest. It is estimated by regression using the wavelet coefficients of the time series, which are dependent when d ≠ 0. We establish a Central Limit Theorem (CLT) for the resulting estimator [Formula: see text]. We show that the deviation [Formula: see text], adequately normalized, is asymptotically normal and specify the asymptotic variance.

Generation of Gaussian Self-similar series via Wavelets for Use in Traffic Simulations

Measurements have shown that network traffic has fractal properties such as self-similarity and long memory or long-range dependence. Long memory is characterized by the existence of a pole at the origin of the power spectrum density function (1/f shape). It was also noticed that traffic may present short-range dependence at some time scales. The use of a “realistic” aggregated network traffic generator, one that synthesizes fractal time series, is fundamental to the validation of traffic control algorithms. In this article, the synthesis of approximate realizations of a kind of self-similar random process named fractional Gaussian noise is done via wavelet transform. The proposed method is also capable of synthesizing Gaussian time series with more generic spectra than 1/f, that is, time series that also have short-range dependence. The generation is done in two stages. The first one generates an approximate realization of fractional Gaussian noise via discrete Wavelet transform. The second one introduces short-range dependence through IIR (Infinite Impulse Response) filtering at the output of the first stage. A detailed characterization of the resulting series was done, using statistical moments of first, second, third and fourth orders, as well as specific statistical tests for self-similar series. It was verified that the Whittle estimator of the Hurst parameter is more robust than the periodogram method for series that simultaneously present short-range and long-range dependence.

Convergence rates of posterior distributions for noniid observations

We consider the asymptotic behavior of posterior distributions and Bayes estimators based on observations which are required to be neither independent nor identically distributed. We give general results on the rate of convergence of the posterior measure relative to distances derived from a testing criterion. We then specialize our results to independent, nonidentically distributed observations, Markov processes, stationary Gaussian time series and the white noise model. We apply our general results to several examples of infinite-dimensional statistical models including nonparametric regression with normal errors, binary regression, Poisson regression, an interval censoring model, Whittle estimation of the spectral density of a time series and a nonlinear autoregressive model.

Haar–Fisz Estimation of Evolutionary Wavelet Spectra

SummaryWe propose a new ‘Haar–Fisz’ technique for estimating the time-varying, piecewise constant local variance of a locally stationary Gaussian time series. We apply our technique to the estimation of the spectral structure in the locally stationary wavelet model. Our method combines Haar wavelets and the variance stabilizing Fisz transform. The resulting estimator is mean square consistent, rapidly computable and easy to implement, and performs well in practice. We also introduce the ‘Haar–Fisz transform’, a device for stabilizing the variance of scaled χ2-data and bringing their distribution close to Gaussianity.

Estimating seasonal long-memory processes: a Monte Carlo study

This paper discusses extensions of the popular methods proposed by Geweke and Porter-Hudak [Geweke, J. and Porter-Hudak, S., 1983, The estimation and application of long memory times series models. Journal of Time Series Analysis, 4(4), 221–238.] and Fox and Taqqu [Fox, R. and Taqqu, M.S., 1986, Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Annals of Statistics, 14, 517–532.] for estimating the long-memory parameter of autoregressive fractionally integrated moving average models to the estimation of long-range dependent models with seasonal components. The proposed estimates are obtained from a selection of harmonic frequencies chosen between the seasonal frequencies. The maximum likelihood method given in Beran [Beran, J., 1994, Statistic for Long-Memory Processes (New York: Chapman & Hall).] and the semi-parametric approaches introduced by Arteche and Robinson [Arteche, J. and Robinson, P.M., 2000, Semiparametric inference in seasonal and cyclical long memory processes. Journal of Time Series Analysis, 21(1), 1–25.] are also considered in the study. Our finite sample Monte Carlo investigations indicate that the proposed methods perform well and can be used as alternative estimating procedures when the data display both long-memory and cyclical behavior.

Exact simulation of Gaussian Time Series from Nonparametric Spectral Estimates with Application to Bootstrapping

The circulant embedding method for generating statistically exact simulations of time series from certain Gaussian distributed stationary processes is attractive because of its advantage in computational speed over a competitive method based upon the modified Cholesky decomposition. We demonstrate that the circulant embedding method can be used to generate simulations from stationary processes whose spectral density functions are dictated by a number of popular nonparametric estimators, including all direct spectral estimators (a special case being the periodogram), certain lag window spectral estimators, all forms of Welch's overlapped segment averaging spectral estimator and all basic multitaper spectral estimators. One application for this technique is to generate time series for bootstrapping various statistics. When used with bootstrapping, our proposed technique avoids some --- but not all --- of the pitfalls of previously proposed frequency domain methods for simulating time series.

Mixture Gaussian Time Series Modeling of Long-Term Market Returns

Stochastic modeling of investment returns is an important topic for actuaries who deal with variable annuity and segregated fund investment guarantees. The traditional lognormal stock return model is simple, but it is generally less appealing for longer-term problems. In recent years, the use of regime-switching lognormal (RSLN) processes for modeling maturity guarantees has been gaining popularity. In this paper we introduce the class of mixture Gaussian time series processes for modeling long-term stock market returns. It offers an alternative class of models to actuaries who may be experimenting with the RSLN process. We use monthly data from the Toronto Stock Exchange 300 and the Standard and Poor-s 500 indices to illustrate the mixture time series modeling procedures, and we compare the fits of the mixture models to the lognormal and RSLN models. Finally, we give a numerical example comparing risk measures for a simple segregated fund contract under different stochastic return models.

VALIDITY OF THE SAMPLING WINDOW METHOD FOR LONG-RANGE DEPENDENT LINEAR PROCESSES

The sampling window method of Hall, Jing, and Lahiri (1998, Statistica Sinica 8, 1189–1204) is known to consistently estimate the distribution of the sample mean for a class of long-range dependent processes, generated by transformations of Gaussian time series. This paper shows that the same nonparametric subsampling method is also valid for an entirely different category of long-range dependent series that are linear with possibly non-Gaussian innovations. For these strongly dependent time processes, subsampling confidence intervals allow inference on the process mean without knowledge of the underlying innovation distribution or the long-memory parameter. The finite-sample coverage accuracy of the subsampling method is examined through a numerical study.The authors thank two referees for comments and suggestions that greatly improved an earlier draft of the paper. This research was partially supported by U.S. National Science Foundation grants DMS 00-72571 and DMS 03-06574 and by the Deutsche Forschungsgemeinschaft (SFB 475).

Higher-order improvements of the parametric bootstrap for long-memory Gaussian processes

Abstract This paper determines coverage probability errors of both delta method and parametric bootstrap confidence intervals (CIs) for the covariance parameters of stationary long-memory Gaussian time series. CIs for the long-memory parameter d 0 are included. The results establish that the bootstrap provides higher-order improvements over the delta method. Analogous results are given for tests. The CIs and tests are based on one or other of two approximate maximum likelihood estimators. The first estimator solves the first-order conditions with respect to the covariance parameters of a “plug-in” log-likelihood function that has the unknown mean replaced by the sample mean. The second estimator does likewise for a plug-in Whittle log-likelihood. The magnitudes of the coverage probability errors for one-sided bootstrap CIs for covariance parameters for long-memory time series are shown to be essentially the same as they are with iid data. This occurs even though the mean of the time series cannot be estimated at the usual n 1 / 2 rate.

Volatility of linear and nonlinear time series

Previous studies indicated that nonlinear properties of Gaussian distributed time series with long-range correlations, u(i), can be detected and quantified by studying the correlations in the magnitude series |u(i)|, the "volatility." However, the origin for this empirical observation still remains unclear and the exact relation between the correlations in u(i) and the correlations in |u(i)| is still unknown. Here we develop analytical relations between the scaling exponent of linear series u(i) and its magnitude series |u(i)|. Moreover, we find that nonlinear time series exhibit stronger (or the same) correlations in the magnitude time series compared with linear time series with the same two-point correlations. Based on these results we propose a simple model that generates multifractal time series by explicitly inserting long range correlations in the magnitude series; the nonlinear multifractal time series is generated by multiplying a long-range correlated time series (that represents the magnitude series) with uncorrelated time series [that represents the sign series sgn (u(i))]. We apply our techniques on daily deep ocean temperature records from the equatorial Pacific, the region of the El-Ninõ phenomenon, and find: (i) long-range correlations from several days to several years with 1/f power spectrum, (ii) significant nonlinear behavior as expressed by long-range correlations of the volatility series, and (iii) broad multifractal spectrum.

Exact maximum likelihood estimation of structured or unit root multivariate time series models

The exact likelihood function of a Gaussian vector autoregressive-moving average (VARMA) model is evaluated in two nonstandard cases: (a) a parsimonious structured form, such as obtained in the echelon form structure or the scalar component model (SCM) structure; (b) a partially nonstationary (integrated of order 1) model in error-correction form. The starting point is any algorithm for computing the exact likelihood of a Gaussian VARMA time series. Our algorithm also provides the parameter estimates and their standard errors. The small sample properties of our algorithm were studied by Monte Carlo methods. Examples with real data are provided.