Frequency domain generalized empirical likelihood method

This paper is concerned with a version of empirical likelihood method for spectral restrictions, which handles stationary time series data via the frequency domain approach. The asymptotic properties of frequency domain generalized empirical likelihood are studied for either strictly stationary processes with vanishing cumulant spectral density function of order 4 or linear processes generated by iid innovations with possibly non‐zero fourth order cumulant. Several statistics for testing parametric restrictions, over‐identified spectral restrictions, and additional spectral restrictions are shown to have the limiting chi‐squared distributions. Some numerical results are presented to investigate the finite sample performance of the proposed procedures. Copyright © 2013 John Wiley & Sons, Ltd.     

Inference for the Fourth‐Order Innovation Cumulant in Linear Time Series

The rescaled fourth‐order cumulant of the unobserved innovations of linear time series is an important parameter in statistical inference. This article deals with the problem of estimating this parameter. An existing nonparametric estimator is first discussed, and its asymptotic properties are derived. It is shown how the autocorrelation structure of the underlying process affects the behaviour of the estimator. Based on our findings and on an important invariance property of the parameter of interest with respect to linear filtering, a pre‐whitening‐based nonparametric estimator of the same parameter is proposed. The estimator is obtained using the filtered time series only; that is, an inversion of the pre‐whitening procedure is not required. The asymptotic properties of the new estimator are investigated, and its superiority is established for large classes of stochastic processes. It is shown that for the particular estimation problem considered, pre‐whitening can reduce the variance and the bias of the estimator. The finite sample performance of both estimators is investigated by means of simulations. The new estimator allows for a simple modification of the multiplicative frequency domain bootstrap, which extends its considerable range of validity. Furthermore, the problem of testing hypotheses about the rescaled fourth‐order cumulant of the unobserved innovations is also considered. In this context, a simple test for Gaussianity is proposed. Some real‐life data applications are presented.     

CONSISTENT ESTIMATION OF THE FOURTH-ORDER CUMULANT SPECTRAL DENSITY

Abstract. In this paper we consider the estimation of the fourth-order cumulant spectral density. Indeed this is the first case where the cumulant depends on lower-order product moments for a mean-zero stationary process. The proposed estimator of the fourth-order cumulant spectral density is constructed by replacing product moments with appropriately weighted estimates of product moments according to the definition of the fourth-order cumulant spectral density. Asymptotic unbiasedness and consistency are shown to hold for these estimators under stationarity and absolute summability of cumulants up to various orders with no restrictions on the frequencies. An expression for the asymptotic variance is also obtained.     

DETECTING SINUSOIDS IN NON-GAUSSIAN NOISE

Abstract. Spectral analysis is a well-established procedure for detecting harmonic signals in a noisy environment. Much research has been done on methods that use second-order statistics (i.e. the autocovariance function and power spectrum) such as Whittle's test, Bartlett's test, Hannan's test and the Priestley P ( Λ ) test. When the noise is non-Gaussian, statistics of order greater than two can provide more information to detect the periodicities in noisy data. We direct our main attention to the third (fourth) order cumulant and bispectral (trispectral) methods. New test statistics are derived and are shown to be more powerful than other methods based on second-order statistics under a mixed spectrum condition. The asymptotic power functions of the new test statistics and other tests are studied. Some Monte Carlo simulations are used to evaluate the performance of the new methods with moderate sample sizes.     

THE RECURSIVE PROPERTY OF THE INVERSE OF THE COVARIANCE MATRIX OF A MOVING-AVERAGE PROCESS OF GENERAL ORDER

Abstract. The inverse of the covariance matrix of a moving-average process of general order is considered. A recursive relationship between the inverses for any two consecutive orders has been established. Illustrative situations are derived and exact expressions are discussed.     

The occurrence of trace elements in coal

The standard comparison of relationships of trace element concentrations in coal is based on a comparison of mean values of elemental concentrations. This method is extended by including some functions of covariances among trace element concentrations. In the proposed method, the comparison is based on: the pattern of the mass spectrum of mean values of concentrations; the spectrum of the covariance matrix; the eigenvectors of the covariance matrix; and the order of eigenvectors. The method has been applied to the coal of the Istrian coal basin and the same genesis of the elements calcium, sulphur, iron, titanium and vanadium throughout the considered coal seam has been established.     

An Extended Portmanteau Test for VARMA Models With Mixing Nonlinear Constraints

Abstract. The portmanteau test is a widely used diagnostic tool for univariate and multivariate time‐series models. Its asymptotic distribution is known for the unconstrained vector autoregressive moving‐average (VARMA) case and for VAR models with constraints on the autoregressive coefficients. In this article, we give conditions under which the test can be applied to constrained VARMA models. Unfortunately, it cannot generally be applied to models with constraints that simultaneously affect the ARMA polynomial coefficients and the covariance matrix of the innovations (mixing constraints). This happens in latent‐variable models such as dynamic factor models (DFM). In addition, when there are constraints on the covariance matrix it seems convenient to check the goodness of fit using the zero‐lag residual covariances. We propose an extended portmanteau test that not only checks the autocorrelations of the residuals but also whether their covariance matrix is consistent with the constraints. We prove that the statistic is asymptotically distributed as a chi‐square for ARMA models under the assumption that the innovations have Gaussian‐like fourth‐order moments. We also show that the test is appropriate for the DFM, Peña–Box model and factor‐structural vector autoregression (FSVAR).     

A NEW DIAGNOSTIC TEST OF MODEL INADEQUACY WHICH USES THE MARTINGALE DIFFERENCE CRITERION

Abstract. Let {x(t)} denote a discrete-time random process. Given a sample of increments e (t) = x(t) - x(t - 1) from the time series, we wish to test formally whether the sample is consistent with the assumption that {e(t)} is a martingale difference. It is shown that the martingale criterion is more general than the white noise criterion in analyzing fitted residuals for signs of model inadequacy. In this paper we present such a test which approximately achieves a given type 1 error probability for samples. We assume that (1) the process is strictly stationary, (2) all its k th-order cumulant functions exist and (3) the k th-order cumulants are absolutely summable and satisfy a mixing condition. The martingale assumption implies that most third-order cumulants of the increment process are zero, and thus the third-order cumulant sequence is sparse. This result is used to derive test statistics based on a modified sample bispectrum. The test can be regarded as a two-dimensional portmanteau test of serial dependence. The large-sample results are demonstrated through the use of artificial data. Finally, the test is applied to a daily financial series.     

两股无预混流体湍流混合与反应过程的数值模拟

旋转填充床作为新型的高效反应传质设备,广泛应用于快速反应过程,如制备纳米粉体材料.但是旋转床内的混合机理以及混合与反应的相互关系尚不清楚.作为旋转填充床内流动、混合与反应过程一种初步的简化,针对两股无预混反应物料在通过丝网之后的湍流混合、扩散及反应行为进行了数值模拟研究,以求为实际应用中涉及多层丝网的旋转床内的混合反应机理的研究提供基础.对两股无预混反应物料的湍流流动与组分混合过程,应用双方程k-ε湍流模型进行了模拟计算.对于所涉及的化学反应过程,建立了瞬时封闭模型进行模拟.模拟的混合层厚度、平均浓度及浓度协方差等与文献公认结果进行了对比,二者吻合良好,平均浓度的偏差不超过10%.     

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.     

The performance of the emulsion polymerization of styrene in a continuous loop tubular reactor

The emulsion polymerization of styrene in a continuous loop tubular reactor (CLTR) has been used to study the process kinetics under eigth levels of sodium lauryl sulfate (SLS) concentration, eight levels of potassium persulfate (KPS) concentration, and three levels of styrene monomer (St) concentration. A critical surfactant concentration for the overshoot phenomenon in the monomer conversion exists between 8.02 and 12.40 g-SLS/1-water. Overshoot was not eliminated by changing concentrations of the initiator and monomer. The overshoot in the number concentration of polymer particles was observed in all the polymerizations studied. The dependences of the rate of polymerization and the number concentration of particles in the steady state on the SLS and KPS concentrations are presented. The growth of the average cumulant diameter of particles shows a transient period followed by a steady state. The dependences of the volumetric growth rate of polymer particles on diameter are of the order of 1 for the lower SLS concentration and 0.5 for the higher SLS concentration, respectively. The steady state cumulant diameter depends on an order of ?0.24 for the SLS concentration.     

ON THE ESTIMATION OF THE INVERSE CORRELATION FUNCTION

Two new methods for estimating the inverse covariance and inverse correlation functions of a time series are proposed. One of them is based on an orthogonality property, the other is suggested by interpolation considerations. The two methods are shown to be asymptotically equivalent, and their asymptotic distribution is derived. The asymptotic distribution turns out to be the same as that of the autoregressive estimates of the inverse correlations. The problem of choosing an estimation method in practice is discussed.     

CLASSIFICATION OF TEXTURES USING SECOND-ORDER SPECTRA

Abstract. We consider the classification of textures as realizations of stationary random fields using non-parametric estimates of their second-order spectra. The random fields in each class are assumed to be stationary with the same spectrum, which we estimate from a finite sample by smoothing its periodogram. The classification rule can be interpreted as maximizing a mean square convergent approximation to the averaged log-likelihood if the random fields are Gaussian, and in general as minimizing the discrepancy of the periodogram from the spectrum of the class. The limiting behaviour of the probability of misclassification as the sample size tends to infinity is studied under certain cumulant conditions. The classification rule is illustrated with real texture data.     

Asymptotically pointwise optimal rules of sequential estimation of mean vector when an information matrix has some structure in a multivariate normal population

The exact formulas of Bayes stopping times are often difficult to derive. Bickel and Yahav (1965) had provided the large sample approximation known as the "Asymptotically pointwise optimal" (A.P.O.) rule. The A.P.O. rule for the problem of the mean of a multivariate normal distribution, for a completely unknown covariance matrix , has been developed by the present author. This paper gives the A.P.O. rule of the mean of a multivariate normal ditribution for a covariance matrix with some structure. Also the result is shown to be asymptotically" non-deficient" in the sence of Woodroofe.     

Kinetic parameter estimation based on spectroscopic data with unknown absorbing species

In many studies, kinetic parameter estimation from spectroscopic data is performed with the absorbing species known beforehand, as this provides a straightforward link between the reaction models to the spectroscopic data. In practice, however, the absorbing species are generally unknown and they are only estimated based on professional experience and prior knowledge of the kinetic reaction. In this work, we propose an optimization strategy with both continuous and discrete decision variables in order to estimate kinetic parameters from spectroscopic data with unknown absorbing species. Also included in our approach is an estimability analysis for kinetic parameters based on the Gram‐Schmidt orthogonalization procedure, along with covariance estimation. Four case studies were considered, which demonstrate the effectiveness of our approach. The first and second have simulated data and illustrate our approach with known solutions. The third and fourth are based on actual experiments from spectroscopy data sets. © 2018 American Institute of Chemical Engineers AIChE J, 64: 3595–3613, 2018     

REDUCTION OF THE ASYMPTOTIC BIAS OF AUTOREGRESSIVE AND SPECTRAL ESTIMATORS BY TAPERING

Abstract. The asymptotic bias to terms of order T ^-1, where T is the observed series length, is studied for estimators of the coefficients and disturbance variance in an AR( p ) model. Reduction of the asymptotic bias by tapering is established and, if the tapering function is defined appropriately to depend on T , not only is the asymptotic bias reduced, but the asymptotic distribution of the estimators is not altered. In addition, the asymptotic biases of other time series parameter estimators constructed from the sample covariance function, such as several types of spectral estimators, can also be reduced by tapering.     

TESTING EQUALITY OF MEANS WHEN THE OBSERVATIONS ARE FROM FUNCTIONAL TIME SERIES

There are numerous examples of functional data in areas ranging from earth science to finance where the problem of interest is to compare several functional populations. In many instances, the observations are obtained consecutively in time, and thus, the classical assumption of independence within each population may not be valid. In this article, we derive a new, asymptotically justified method to test the hypothesis that the mean curves of multiple functional populations are the same. The test statistic is constructed from the coefficient vectors obtained by projecting the functional observations into a finite dimensional space. Asymptotics are established when the observations are considered to be from stationary functional time series. Although the limit results hold for projections into arbitrary finite dimensional spaces, we show that higher power is achieved by projecting onto the principle components of empirical covariance operators that diverge under the alternative. Our method is further illustrated by a simulation study as well as an application to electricity demand data.     

酯型阳离子易染共聚酯纤维的研究──Ⅱ．共聚酯及其纤维热稳定性的研究

采用ＴＧＡ等手段对以聚酯二醇作为第四组份的酯型ＥＣＤＰ热稳定性进行了研究，并与以聚醚二醇作为第四组份的醚型ＥＣＤＰ进行了比较．结果表明：酯型ＥＣＤＰ具有优良的热稳定性，经１８０℃热处理３０ｍｉｎ后，酯型ＥＣＤＰ纤维强度损伤低于６％，克服了醚型ＥＣＤＰ纤维耐热性差的缺点。     

Asymptotically pointwise optimal rules for estimating the mean in general exponential dtstributions for squared loss

Since exact formulae for Bayes stopping tirncs in sequential analysis are often difficult to derive.Bickle and Yahav considered an attractive large sample approximation known as " Asymptotically pointwise optimal" (A.P.O.) rulp. The (A.P.O.) rule. for the problem of cst the mean of a multivariate normal distribution has been treated by some authors but only for the case where the covariance matrix is scalar times a known matrix.In this paper, A.P.O. rules are given for a general exponeritial fitmily fror which the results for milltic,aria.tr norrnal mean vector with c-ompli.tely unknow covariance matrix are obtained. We also consider a multinomial distribution case.     

Error propagation analysis in color measurement and imaging

We apply multivariate error-propagation analysis to color-signal transformations. Results are given that indicate how linear, matrix, and nonlinear transformations influence the mean, variance, and covariance of color-measurements and color-images. Since many signal processing paths include these steps, the analysis is applicable to color-measurement and imaging systems. Expressions are given that allow image noise or error propagation for a spectrophotometer, colorimeter, or digital camera. In a computed example, error statistics are propagated from tristimulus values to CIELAB coordinates. The resulting signal covariance is interpreted in terms of CIELAB error ellipsoids and the mean value of color-difference measures, and . The application of this analysis to system design is also illustrated by relating a tolerance to equivalent tristimulus-value error statistics. © 1997 John Wiley & Sons, Inc. Col Res Appl, 22, 280–289, 1997