Time‐series clustering via quasi U‐statistics

The problem of time‐series discrimination and classification is discussed. We propose a novel clustering algorithm based on a class of quasi U‐statistics and subgroup decomposition tests. The decomposition may be applied to any concave time‐series distance. The resulting test statistics are proven to be asymptotically normal for either i.i.d. or non‐identically distributed groups of time‐series under mild conditions. We illustrate its empirical performance on a simulation study and a real data analysis. The simulation setup includes stationary vs. stationary and stationary vs. non‐stationary cases. The performance of the proposed method is favourably compared with some of the most common clustering measures available.     

A Nonparametric Prewhitened Covariance Estimator

This paper proposes a new nonparametric spectral density estimator for time series models with general autocorrelation. The conventional nonparametric estimator that uses a positive kernel has mean squared error no better than n^?4/5. We show that the best implementation of our estimator has mean squared error of order n^?8/9, provided there is sufficient smoothness present in the spectral density. This is, of course, achieved by bias reduction; however, unlike most other bias reduction methods, like the kernel method with higher‐order kernels, our procedure ensures a positive definite estimate. Our method is a generalization of the well‐known prewhitening method of spectral estimation; we argue that this can best be interpreted as multiplicative bias reduction. Higher‐order expansions for the proposed estimator are derived, providing an improved bandwidth choice that minimizes the mean squared error to the second order. A simulation study shows that the recommended prewhitened kernel estimator reduces bias and mean squared error in spectral density estimation.     

On limiting spectral distribution of large sample covariance matrices by VARMA(p,q)

We studied the limiting spectral distribution of large‐dimensional sample covariance matrices of a stationary and invertible VARMA(p,q) model. Relationship of the power spectral density and limiting spectral distribution of large population dimensional covariance matrices of ARMA(p,q) is established. The equation about Stieltjes transform of large‐dimensional sample covariance matrices is also derived. As applications, the classical M‐P law, VAR(1) and VMA(1) can be regarded as special examples.     

Stationary bootstrapping for non‐parametric estimator of nonlinear autoregressive model

We consider stationary bootstrap approximation of the non‐parametric kernel estimator in a general kth‐order nonlinear autoregressive model under the conditions ensuring that the nonlinear autoregressive process is a geometrically Harris ergodic stationary Markov process. We show that the stationary bootstrap procedure properly estimates the distribution of the non‐parametric kernel estimator. A simulation study is provided to illustrate the theory and to construct confidence intervals, which compares the proposed method favorably with some other bootstrap methods.     

The Periodogram of fractional processes1

Abstract. We analyse asymptotic properties of the discrete Fourier transform and the periodogram of time series obtained through (truncated) linear filtering of stationary processes. The class of filters contains the fractional differencing operator and its coefficients decay at an algebraic rate, implying long‐range‐dependent properties for the filtered processes when the degree of integration α is positive. These include fractional time series which are nonstationary for any value of the memory parameter (α ≠ 0) and possibly nonstationary trending (α ≥ 0.5). We consider both fractional differencing or integration of weakly dependent and long‐memory stationary time series. The results obtained for the moments of the Fourier transform and the periodogram at Fourier frequencies in a degenerating band around the origin are weaker compared with the stationary nontruncated case for α > 0, but sufficient for the analysis of parametric and semiparametric memory estimates. They are applied to the study of the properties of the log‐periodogram regression estimate of the memory parameter α for Gaussian processes, for which asymptotic normality could not be showed using previous results. However, only consistency can be showed for the trending cases, 0.5 ≤ α < 1. Several detrending and initialization mechanisms are studied and only local conditions on spectral densities of stationary input series and transfer functions of filters are assumed.     

A numerical method for factorizing the rational spectral density matrix

Improving Rozanov (1967, Stationary Random Processes. San Francisco: Holden‐day.)’s algebraic‐analytic solution to the canonical factorization problem of the rational spectral density matrix, this article presents a feasible computational procedure for the spectral factorization. We provide numerical comparisons of our procedure with the Bhansali's (1974, Journal of the Statistical Society, B36 , 61.) and Wilson's (1972 SIAM Journal on Applied Mathematics, 23 , 420) methods and illustrate its application in estimation of invertible MA representation. The proposed procedure is usefully applied to linear predictor construction, causality analysis and other problems where a canonical transfer function specification of a stationary process in question is required.     

Tests for the Equality of Two Processes' Spectral Densities with Unequal Lengths Using Wavelet Methods

Testing procedures for assessing whether two stationary and independent linear processes with unequal lengths have the same spectral densities or same auto‐covariance functions are investigated. New test statistics are proposed based on the difference of the two wavelet‐based estimates of the two spectral densities. The asymptotic normal distributions of the empirical wavelet coefficients are derived based on Bartlett type approximation of a quadratic form with dependent variables by the corresponding quadratic form with independent and identically distributed (i.i.d.) random variables. The limit distributions of the proposed test statistics are derived from those asymptotic results, and they asymptotically follow known chi‐square distributions. The advantage of those new procedures is that those test statistics are constructed very simply and can be used for two time series with arbitrary lengths. The performance of those new tests is compared with some recent test statistics, with respect to their exact levels and powers. Simulation studies show that our proposed tests are very comparable to the current tests.     

Wavelet‐Based Tests for Comparing Two Time Series with Unequal Lengths

Test procedures for assessing whether two stationary and independent time series with unequal lengths have the same spectral density (or same auto‐covariance function) are investigated. A new test statistic is proposed based on the wavelet transform. It relies on empirical wavelet coefficients of the logarithm of two spectral densities' ratio. Under the null hypothesis that two spectral densities are the same, the asymptotic normal distribution of the empirical wavelet coeffcients is derived. Furthermore, these empirical wavelet coefficients are asymptotically uncorrelated. A test statistic is proposed based on these results. The performance of the new test statistic is compared to several recent test statistics, with respect to their exact levels and powers. Simulation studies show that our proposed test is very comparable to the current test statistics in most cases. The main advantage of our proposed test statistic is that it is constructed very simply and is easy to implement.     

A new frequency domain approach of testing for covariance stationarity and for periodic stationarity in multivariate linear processes

In modelling seasonal time series data, periodically (non‐)stationary processes have become quite popular over the last years and it is well known that these models may be represented as higher‐dimensional stationary models. In this article, it is shown that the spectral density matrix of this higher‐dimensional process exhibits a certain structure if and only if the observed process is covariance stationary. By exploiting this relationship, a new L₂‐type test statistic is proposed for testing whether a multivariate periodically stationary linear process is even covariance stationary. Moreover, it is shown that this test may also be used to test for periodic stationarity. The asymptotic normal distribution of the test statistic under the null is derived and the test is shown to have an omnibus property. The article concludes with a simulation study, where the small sample performance of the test procedure is improved by using a suitable bootstrap scheme.     

Unambiguous Assignment of Short‐ and Long‐Range Structural Restraints by Solid‐State NMR Spectroscopy with Segmental Isotope Labeling

We present an efficient method for the reduction of spectral complexity in the solid‐state NMR spectra of insoluble protein assemblies, without loss of signal intensity. The approach is based on segmental isotope labeling by using the split intein DnaE from Nostoc punctiforme. We show that the segmentally ¹³C,¹⁵N‐labeled prion domain of HET‐s exhibits significantly reduced spectral overlap while retaining the wild‐type structure and spectral quality. A large number of unambiguous distance restraints were thus collected from a single two‐dimensional ¹³C,¹³C cross‐correlation spectrum. The observed resonances could be unambiguously identified as intramolecular without the need for preparing a dilute, less sensitive sample.     

DETERMINING THE NUMBER OF REGIMES IN MARKOV SWITCHING VAR AND VMA MODELS

We give stable finite‐order vector autoregressive moving average (p ^* ,q ^* ) representations for M‐state Markov switching second‐order stationary time series whose autocovariances satisfy a certain matrix relation. The upper bounds for p ^* and q ^* are elementary functions of the dimension K of the process, the number M of regimes, the autoregressive and moving‐average orders of the initial model. If there is no cancellation, the bounds become equalities, and this solves the identification problem. Our classes of time series include every M‐state Markov switching multi‐variate moving‐average models and autoregressive models in which the regime variable is uncorrelated with the observable. Our results include, as particular cases, those obtained by Krolzig (1997) and improve the bounds given by Zhang and Stine (2001) and Francq and Zakoïan (2001) for our classes of dynamic models. A Monte Carlo experiment and an application on foreign exchange rates complete the article. Copyright © 2013 John Wiley & Sons, Ltd.     

Statistical tests for a single change in mean against long‐range dependence

Statistical tests are introduced for distinguishing between short‐range dependent time series with a single change in mean, and long‐range dependent time series, with the former making the null hypothesis. The tests are based on estimation of the self‐similarity parameter after removing the change in mean from the series. The focus is on the GPH (Geweke and Porter‐Hudak, 1983) and local Whittle estimation methods in the spectral domain. Theoretical properties of the resulting estimators are established when testing for a single change in mean, and small sample properties of the tests are examined in simulations. The introduced tests improve on the BHKS ( Berkes et al., 2006 ) test which is the only other available test for the considered problem. It is argued that the BHKS test has a low power against long‐range dependence alternatives and that this happens because the BHKS test statistic involves estimation of the long‐run variance. The BHKS test could be improved readily by considering its R/S‐like regression version which estimates the self‐similarity parameter and which does not involve the long‐run variance. Yet better alternatives are to use more powerful estimation methods (such as GPH or local Whittle) and lead to the tests introduced here.     

Bi‐Spectral IRCM‐Flare Formulations based on Tetrazole‐Derivatives

The spectral performance, sensitiveness to ignition stimuli and burning rate of bi‐spectral flare formulations based on tetrazole containing fuels, 5‐phenyl‐1H‐tetrazole, 5,5′‐(1,4‐phenylene)bis(1H‐tetrazole) and 5‐(4‐nitro‐phenyl)‐1H‐tetrazole, utilizing potassium perchlorate as the oxidizer, are reported. The formulation based on 5‐(4‐nitro‐phenyl)‐1H‐tetrazole yielded the highest spectral efficiency in the β‐band (44.1 J g⁻¹ sr⁻¹). The formulation based on 5‐phenyl‐1H‐tetrazole gave the highest color ratio (θ_β/α=6.8) and was the least sensitive.     

Mechanical properties of a particle‐strengthened polyurethane foam

Quasi‐static compression tests have been performed on polyurethane foam specimens. The modulus of the foam exhibited a power‐law dependence with respect to density of the form: E* ∝ (ρ*)ⁿ, where n = 1.7. The modulus data are described well by a simple geometric model (based on the work of Gibson and Ashby) for a closed‐cell foam in which the stiffness of the foam is governed by the flexure of the cell struts and cell walls. The compressive strength of the foam is also found to follow a power‐law behavior with respect to foam density. In this instance, Euler buckling is used to explain the density dependence. The modulus of the foam was modified by addition of gas‐atomized, spherical, aluminum powder. Additions of 30 and 50 wt % Al measurably increased the foam modulus, but without a change in the density dependence. However, there was no observable increase in modulus with 5 and 10 wt % additions of the metal powder. Strength was also increased at high loading fractions of powder. The increase in modulus and strength could be predicted by combining the Gibson–Ashby model, referred to above, with a well‐known model describing the effect on modulus of a rigid dispersoid in a compliant matrix. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 74: 2724–2736, 1999     

On the Spectral Density of the Wavelet Coefficients of Long‐Memory Time Series with Application to the Log‐Regression Estimation of the Memory Parameter

Abstract. In recent years, methods to estimate the memory parameter using wavelet analysis have gained popularity in many areas of science. Despite its widespread use, a rigorous semi‐parametric asymptotic theory, comparable with the one developed for Fourier methods, is still lacking. In this article, we adapt to the wavelet setting, the classical semi‐parametric framework introduced by Robinson and his co‐authors for estimating the memory parameter of a (possibly) non‐stationary process. Our results apply to a class of wavelets with bounded supports, which include but are not limited to Daubechies wavelets. We derive an explicit expression of the spectral density of the wavelet coefficients and show that it can be approximated, at large scales, by the spectral density of the continuous‐time wavelet coefficients of fractional Brownian motion. We derive an explicit bound for the difference between the spectral densities. As an application, we obtain minimax upper bounds for the log‐scale regression estimator of the memory parameter for a Gaussian process and we derive an explicit expression of its asymptotic variance.     

A Simple Test for White Noise in Functional Time Series

We propose a new procedure for white noise testing of a functional time series. Our approach is based on an explicit representation of the L²‐distance between the spectral density operator and its best (L²‐)approximation by a spectral density operator corresponding to a white noise process. The estimation of this distance can be easily accomplished by sums of periodogram kernels, and it is shown that an appropriately standardized version of the estimator is asymptotically normal distributed under the null hypothesis (of functional white noise) and under the alternative. As a consequence, we obtain a very simple test (using the quantiles of the normal distribution) for the hypothesis of a white noise functional process. In particular, the test does not require either the estimation of a long‐run variance (including a fourth order cumulant) or resampling procedures to calculate critical values. Moreover, in contrast to all other methods proposed in the literature, our approach also allows testing for ‘relevant’ deviations from white noise and constructing confidence intervals for a measure that measures the discrepancy of the underlying process from a functional white noise process.     

Density estimation with locally identically distributed data and with locally stationary data

Abstract. We consider multivariate density estimation when the assumptions of identically distributed data or stationary data are relaxed to the assumptions of locally identically distributed data or locally stationary data. We assume that the distribution of the data is changing continuously as function of time. To estimate densities non‐parametrically with these local regularity conditions, we need time localization in addition to the usual space localization. We define a time‐localized kernel estimator that estimates the density non‐parametrically at any given point of time. The consistency of the time‐localized kernel estimator is proved and the rates of convergence of the estimator are derived under conditions on the β‐and α‐mixing coefficients. Both the time‐series setting and spatial setting are covered.     

Robust estimation for the covariance matrix of multi‐variate time series

In this article, we study the robust estimation for the covariance matrix of stationary multi‐variate time series. As a robust estimator, we propose to use a minimum density power divergence estimator (MDPDE) proposed by Basu et al. (1998) . Particularly, the MDPDE is designed to perform properly when the time series is Gaussian. As a special case, we consider the robust estimator for the autocovariance function of univariate stationary time series. It is shown that the MDPDE is strongly consistent and asymptotically normal under regularity conditions. Simulation results are provided for illustration.     

Impact of the Sampling Rate on the Estimation of the Parameters of Fractional Brownian Motion

Abstract. Fractional Brownian motion is a mean‐zero self‐similar Gaussian process with stationary increments. Its covariance depends on two parameters, the self‐similar parameter H and the variance C. Suppose that one wants to estimate optimally these parameters by using n equally spaced observations. How should these observations be distributed? We show that the spacing of the observations does not affect the estimation of H (this is due to the self‐similarity of the process), but the spacing does affect the estimation of the variance C. For example, if the observations are equally spaced on [0, n] (unit‐spacing), the rate of convergence of the maximum likelihood estimator (MLE) of the variance C is . However, if the observations are equally spaced on [0, 1] (1/n‐spacing), or on [0, n²] (n‐spacing), the rate is slower, . We also determine the optimal choice of the spacing Δ when it is constant, independent of the sample size n. While the rate of convergence of the MLE of C is in this case, irrespective of the value of Δ, the value of the optimal spacing depends on H. It is 1 (unit‐spacing) if H = 1/2 but is very large if H is close to 1.     

High‐frequency sampling and kernel estimation for continuous‐time moving average processes

Interest in continuous‐time processes has increased rapidly in recent years, largely because of high‐frequency data available in many applications. We develop a method for estimating the kernel function g of a second‐order stationary Lévy‐driven continuous‐time moving average (CMA) process Y based on observations of the discrete‐time process Y^Δ obtained by sampling Y at Δ, 2Δ, …, nΔ for small Δ. We approximate g by g^Δ based on the Wold representation and prove its pointwise convergence to g as Δ → 0 for continuous‐time autoregressive moving average (CARMA) processes. Two non‐parametric estimators of g^Δ, on the basis of the innovations algorithm and the Durbin–Levinson algorithm, are proposed to estimate g. For a Gaussian CARMA process, we give conditions on the sample size n and the grid spacing Δ(n) under which the innovations estimator is consistent and asymptotically normal as n → ∞. The estimators can be calculated from sampled observations of any CMA process, and simulations suggest that they perform well even outside the class of CARMA processes. We illustrate their performance for simulated data and apply them to the Brookhaven turbulent wind speed data. Finally, we extend results of Brockwell et al. (2012) for sampled CARMA processes to a much wider class of CMA processes.