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.     

活性污泥法处理染色废水过程模型辨识与参数估计

本文用Ｂｏｘ－Ｊｅｎｋｉｎｓ时间序列分析方法建立了活性污泥法处理染色废水过程中，进、出水高锰酸盐指数（ＯＣｉ，ＯＣｅ）单序列随机模型以及ＯＣｅ对ＯＣｉ序列含量有噪声项的传递函数模型，建立的模型较好地拟合了实际过程，传递函数比噪声项显得重要，模型对ＯＣｉ序列预测平均误差１０．２６％。     

SOME DOUBLY STOCHASTIC TIME SERIES MODELS

Abstract. We consider time series models obtained by replacing the parameters of autoregressive models by stochastic processes. Special attention is given to the problem of finding conditions for stationarity and to the problem of forecasting. For the first problem we are only able to obtain solutions in special cases, and the emphasis is on techniques rather than obtaining the most general results in each case. For the second problem more complete results are obtained by exploiting similarities with discrete time (nonlinear) filtering theory. The methods introduced are illustrated on two standard examples, one of state space type and one where the parameter process is a Markov chain.     

THE ADAPTIVE AUTOREGRESSIVE MODELS FOR THE SYSTEM DYNAMICS AND PREDICTION OF BLAST FURNACE

The blast furnace can be viewed as a time-varying stochastic system. The Adaptive Autoregressive (AAR) models are proposed to characterize such systems. AAR identification is a method of successive parameter estimation by using recursive formulas with variable forgetting factors to closely track time-varying parameters. A simple example is presented to illustrate the parameter tracking capability of the AAR models. Based on the prediction errors, the AAR models of blast furnace are compared with the conventional time series models. Through this comparison, the AAR models prove to be superior to the other time series models, since the latter are suitable only for time-invariant systems. It is concluded that during smooth operation, just the AAR scalar model is required for forecasting as operational guide. When the operation is uneven, the AAR vector model provides the better results. To control the performance of this process the data should be sampled under uneven operating condition, where the AAR vector model is the best among all the models considered and can properly express the dynamic relationship between the input and output variables.     

A maximum-likelihood method for estimating parameters,stochastic disturbance intensities and measurement noise variances in nonlinear dynamic models with process disturbances

An improved approximate maximum likelihood algorithm is developed for estimating measurement noise variances along with model parameters and disturbance intensities in nonlinear stochastic differential equation (SDE) models. This algorithm uses a Laplace approximation and B-spline basis functions for approximating the likelihood function of the parameters given the measurements. The resulting Laplace approximation maximum likelihood estimation (LAMLE) algorithm is tested using a nonlinear continuous stirred tank reactor (CSTR) model. Estimation results for four model parameters, two process disturbance intensities and two measurement noise variances are obtained using LAMLE and are compared with results from two other maximum-likelihood-based methods, the continuous-time stochastic method (CTSM) of Kristensen and Madsen (2003) and the Fully Laplace Approximation Estimation Method (FLAEM) (Karimi and McAuley, 2014). Parameter estimations using 100 simulated data sets reveal that the LAMLE estimation results tend to be more precise and less biased than corresponding estimates obtained using CTSM and FLAEM.     

Wiener–Kolmogorov Filtering and Smoothing for Multivariate Series With State–Space Structure

Abstract. Wiener–Kolmogorov filtering and smoothing usually deal with projection problems for stochastic processes that are observed over semi‐infinite and doubly infinite intervals. For multivariate stationary series, there exist closed formulae based on covariance generating functions that were first given independently by N. Wiener and A.N. Kolmogorov around 1940. In this article, we consider multivariate series with a state–space structure and, using a new purely algebraic approach to the problem, we prove the equivalence between Wiener–Kolmogorov filtering and Kalman filtering. Up to now, this equivalence has only been partially shown. In addition, we get some new recursions for smoothing and some new recursions to compute the filter weights and the covariance generating functions of the errors. The results are extended to nonstationary series.     

Nonlinear modelling of periodic threshold autoregressions using Tsmars

We present new methods for modelling nonlinear threshold-type autoregressive behaviour in periodically correlated time series. The methods are illustrated using a series of average monthly flows of the Fraser River in British Columbia. Commonly used nonlinearity tests of the river flow data in each month indicate nonlinear behaviour in certain months. The periodic nonlinear correlation structure is modelled nonparametrically using TSMARS, a time series version of Friedman's extended multivariate adaptive regression splines (MARS) algorithm, which allows for categorical predictor variables. We discuss two methods of using the computational algorithm in TSMARS for modelling and fitting periodically correlated data. The first method applies the algorithm to data from each period separately. The second method models data from all periods simultaneously by incorporating an additional predictor variable to distinguish different behaviour in different periods, and allows for coalescing of data from periods with similar behaviour. The models obtained using TSMARS provide better short-term forecasts for the Fraser River data than a corresponding linear periodic AR model.     

NONPARAMETRIC ESTIMATORS FOR TIME SERIES

Abstract. Kernel multivariate probability density and regression estimators are applied to a univariate strictly stationary time series X _r We consider estimators of the joint probability density of X _t at different t -values, of conditional probability densities, and of the conditional expectation of functionals of X _v given past behaviour. The methods seem of particular relevance in light of recent interest in non-Gaussian time series models. Under a strong mixing condition multivariate central limit theorems for estimators at distinct points are established, the asymptotic distributions being of the same nature as those which would derive from independent multivariate observations.     

On the stochastic simulation of particulate systems

The stochastic chemical kinetics approach provides one method of formulating the stochastic crystallization population balance equation (PBE). In this formulation, crystal nucleation and growth are modelled as sequential additions of solubilized ions or molecules (units) to either other units or an assembly of any number of units. Monte Carlo methods provide one means of solving this problem. In this paper, we assess the limitations of such methods by both (1) simulating models for isothermal and nonisothermal size-independent nucleation, growth and agglomeration; and (2) performing parameter estimation using these models. We also derive the macroscopic (deterministic) PBE from the stochastic formulation, and compare the numerical solutions of the stochastic and deterministic PBEs. The results demonstrate that even as we approach the thermodynamic limit, in which the deterministic model becomes valid, stochastic simulation provides a general, flexible solution technique for examining many possible mechanisms. Thus the stochastic simulation permits the user to focus more on modelling issues as opposed to solution techniques.     

Efficient Bayesian PARCOR approaches for dynamic modeling of multivariate time series

A Bayesian lattice filtering and smoothing approach is proposed for fast and accurate modeling and inference in multivariate non-stationary time series. This approach offers computational feasibility and interpretable time-frequency analysis in the multivariate context. The proposed framework allows us to obtain posterior estimates of the time-varying spectral densities of individual time series components, as well as posterior measurements of the time-frequency relationships across multiple components, such as time-varying coherence and partial coherence. The proposed formulation considers multivariate dynamic linear models (MDLMs) on the forward and backward time-varying partial autocorrelation coefficients (TV-VPARCOR). Computationally expensive schemes for posterior inference on the multivariate dynamic PARCOR model are avoided using approximations in the MDLM context. Approximate inference on the corresponding time-varying vector autoregressive (TV-VAR) coefficients is obtained via Whittle's algorithm. A key aspect of the proposed TV-VPARCOR representations is that they are of lower dimension, and therefore more efficient, than TV-VAR representations. The performance of the TV-VPARCOR models is illustrated in simulation studies and in the analysis of multivariate non-stationary temporal data arising in neuroscience and environmental applications. Model performance is evaluated using goodness-of-fit measurements in the time-frequency domain and also by assessing the quality of short-term forecasting.     

A flexible class of parametric transition regression models based on copulas: application to poliomyelitis incidence

This paper presents an extension of a general parametric class of transitional models of order p. In these models, the conditional distribution of the current observation, given the present and past history, is a mixture of conditional distributions, each of them corresponding to the current observation, given each one of the p-lagged observations. Such conditional distributions are constructed using bivariate copula models which allow for a rich range of dependence suitable to model non-Gaussian time series. Fixed and time varying covariates can be included in the models. These models have the advantage of straightforward construction and estimation for the analysis of time series and more general longitudinal data. A poliomyelitis incidence data set is used to illustrate the proposed methods, contrary to other researches' conclusions whose methods are mainly based on linear models, we find significant evidence of a decreasing trend in polio infection after accounting for seasonality.     

Multivariate modelling of the autoregressive random variance process

The autoregressive random variance (ARV) model proposed by Taylor (Financial returns modelled by the product of two stochastic processes, a study of daily sugar prices 1961–79. In Time Series Analysis: Theory and Practice 1 (ed. O. D. Anderson). Amsterdam: North-Holland, 1982, pp. 203–26) is useful in modelling stochastic changes in the variance structure of a time series. In this paper we focus on a general multivariate ARV model. A traditional EM algorithm is derived as the estimation method. The proposed EM approach is simple to program, computationally efficient and numerically well behaved. The asymptotic variance--covariance matrix can be easily computed as a by-product using a well-kno wn asymptotic result for extremum estimators. A result that is of interest in itself is that the dimension of the augmented state space form used in computing the variance–covariance matrix can be shown to be greatly reduced, resulting in greater computational efficiency . The multivariate ARV model considered here is useful in studying the lead–lag (causality) relationship of the variance structure across different time series. As an example, the leading effect of Thailand on Malaysia in terms of vari ance changes in the stock indices is demonstrated.     

浮选数模及计算机模拟在选矿试验中的应用

可浮性k值的分布密度函数f(x)是常数k_m和k_p所唯一确定的г分布。概率微元f(k)dk的实验室闭路浮选可以用马尔科夫随机过程来描述,所导出的中矿浮选动力学模型是一组无穷递减等差数列。模型参数用回归法确定。只要进行一些较简单的试验,即可用按本模型编的程序在计算机上进行多种模拟试验。经实际验证模拟结果与试验结果吻合较好。     

Computationally efficient methods for two multivariate fractionally integrated models

Abstract.  We discuss two distinct multivariate time-series models that extend the univariate ARFIMA (autoregressive fractionally integrated moving average) model. We discuss the different implications of the two models and describe an extension to fractional cointegration. We describe algorithms for computing the covariances of each model, for computing the quadratic form and approximating the determinant for maximum likelihood estimation and for simulating from each model. We compare the speed and accuracy of each algorithm with existing methods individually. Then, we measure the performance of the maximum likelihood estimator and of existing methods in a Monte Carlo. These algorithms are much more computationally efficient than the existing algorithms and are equally accurate, making it feasible to model multivariate long memory time series and to simulate from these models. We use maximum likelihood to fit models to data on goods and services inflation in the United States.     

Optimal discrete stochastic control theory for process application

The design of discrete feedback controllers which minimize some linear function of the variances of the output deviations from target subject to possible constraints on the variances of the inputs, for linear systems subject to stochastic disturbances, is treated from two points of view: (1) using transfer function models to characterizing the process dynamics and autoregres-sive-moving-average models to characterize the stochastic disturbances, and then solving the optimal control problem using an approach due to Box and Jenkins and a discrete version of the Wiener-Newton theory; and (2) using state variable models to characterize both the dynamic and stochastic parts of the system, and then solving the optimal control problem using the results of dynamic programming and Kalman filtering. Practical considerations such as model forms, their identification and estimation, and the development of variance relationships that are necessary for the application of these two approaches in the process industries are discussed. The relationship between and a comparison of these two approaches is made.     

Discriminant Analysis of Time Series in the Presence of Within‐Group Spectral Variability

Many studies record replicated time series epochs from different groups with the goal of using frequency domain properties to discriminate between the groups. In many applications, there exists variation in cyclical patterns from time series in the same group. Although a number of frequency domain methods for the discriminant analysis of time series have been explored, there is a dearth of models and methods that account for within‐group spectral variability. This article proposes a model for groups of time series in which transfer functions are modelled as stochastic variables that can account for both between‐group and within‐group differences in spectra that are identified from individual replicates. An ensuing discriminant analysis of stochastic cepstra under this model is developed to obtain parsimonious measures of relative power that optimally separate groups in the presence of within‐group spectral variability. The approach possesses favourable properties in classifying new observations and can be consistently estimated through a simple discriminant analysis of a finite number of estimated cepstral coefficients. Benefits in accounting for within‐group spectral variability are empirically illustrated in a simulation study and through an analysis of gait variability.