Multivariate time series models

The methodology for multivariate time series models is similar to the univariate cases except for the fact that all notation is changed into vectors and matrices such that most 

Suppose a multivariate time series of dimension L collected at T points of time, say Y The basic types of dependency, auto-regression and moving average, are used to 

Model (2.58) can easily be extended to a VARMAX model by incorporating exogenous variables. Let X be the matrix of explanatory variables, then (2.58) is reformulated 

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Frequently, concentration variations in environmental matrices not only concern themselves with one-dimensional cases, e.g. the time series of one parameter as discussed before, but also with many parameters which change simultaneously. In environmental analysis in particular, such time or local changes of environmental contaminants are very relevant [GEISS and EINAX, 1992], Multivariate time series models are available,... 

Recknagel, F., Bobbin, J., Whigham, P., and Wilson, H., 2002. Comparative application of artificial neural networks and genetic algorithms for multivariate time-series modeling of algal blooms in freshwater lakes. Journal of Hydroinformatics 4(2), p.l25-134. 

The SIC is deduced from Bayesian arguments. It consistently estimates the true order of ARMA(p, q) processes and is probably the most widely used information criterion in univariate time series analysis. The HQIC is the most recent IC and especially designed for multivariate time series models. In practice, multiple ICs are simultaneously calculated which allows the analyst to cross-check the recommendations of the various ICs. Strongly deviating recommendations may indicate an inappropriate model structure. 

In-Kwon Yeo received the PhD degree in Statistics from University of Wisconsin-Madison in 1997. He joined the Department of Control and Instrumentation Engineering, Kangwon National University as a visiting professor in 2000 and the Division of Mathematics and Statistical Informatics, Chonbuk National University as an assistant professor in Korea. He is currently an associate professor at the Department of Statistics, Sookmyung Women s University. His current research interests include data transformations, multivariate time series analysis and generalized additive models. 

Various multivariate regression techniques are outlined in Section 4.1. Section 4.2 introduces PCA-based regression and its extension to capture d3mamic variations in data. PLS regression is discussed in Section 4.3. Input-output modeling of d3mamic processes with time series models is introduced in Section 4.4 and state-space modeling techniques are presented in Section 4.5. 

Another limitation of existing SPC methods is that they require the measurements to be uncorrelated, or white, whereas, in practice, autocorrelated measurements are extremely common. A common approach for decorrelat-ing autocorrelated measurements is to approximate the measurements by a time series model, and monitor the residual error. Unfortunately, this approach is not practical, particularly for multivariate processes with hundreds... 

While the estimates of the autocorrelation coefficients for the Cg time series (lower rows in 1 to ordy change slightly, the estimates the autocorrelation coefficients for the Benzene time series (upper rows in to 3) are clearly affected since three parameters are dropped from the model. The remaining coefficients are affected, too. In particular, the lagged cross-correlations to the Cg time series change from 1.67 to 2.51 and from -2.91 to -2.67 (right upper entries in 1 and This confirms the serious effect of even unobtrusive outliers in multivariate times series analysis. By incorporating the outliers effects, the model s AIC decreases from -4.22 to -4.72. Similarly, SIC decreases from -4.05 to -4.17. The analyses of residuals. show a similar pattern as for the initial model and reveal no serious hints for cross- or auto-correlation. i Now, the multivariate Jarque-Bera test does not reject the hypothesis of multivariate normally distributed variables (at a 5% level). The residuals empirical covariance matrix is finally estimated as... 

A homogeneity index or significance coefficienf has been proposed to describe area or spatial homogeneity characteristics of solids based on data evaluation using chemometrical tools, such as analysis of variance, regression models, statistics of stochastic processes (time series analysis) and multivariate data analysis (Singer and... 

Such analysis demands many cases in the series similar to multivariate model computations. Now a multiple regression analysis with the independent variables jan, feb, mar,. .., summ and the number variable num, and the nitrate time series as the dependent variable is started. 

Fitting a smooth, multivariate model to a time series of noisy data is an effective way to produce a more precise estimate of the measured quantity at each sample time. Typically, the logarithm of intrinsic viscosity is... 

The application of embedding time-series analysis to multivariate observables [75,84,86] is desired in order to extract a good projection, revealing the dynamical structure from a limited set of observables. It is known [89,90] that an application of the embedding analysis to time series involving intermittency, like those of the Go-like model, is not straightforward and involves many problems that need to be overcome. 

Results for elements in aerosol samples which are obtained by multielement techniques from data sets from which information about the sources of the components can be extracted (Gordon 1980). Such methods which make use of data obtained at receptor points are called receptor models. The most important receptor models are chemical mass balances (CMB), enrichment factors, time series correlation, multivariate models and spatial models (Cooper and Watson 1980 Gordon 1988). Dispersion modeling has also been used to explain the... 

Furthermore, recent statistical advances have expanded the repertoire of tools with which to analyze data tfom these designs. For example, hierarchical linear models (J. E. Schwartz, Warren, Pickering, 1994), random regression models (Jacob et al., in press), or pooled cross-sectional time series (Dielman, 1983) allow for the partitioning of inter-individual and intra-individual variability from a number of different sources. Complemented by iet a a/vric techniques that allow for the examination of multiple dependent variables (Cohen, 1982), these methods offer many data analytic strategies for multivariate, replicated, repeated-measures, singlesubject designs. Several of these techniques are illustrated in the next section. 

This work required a large amount of subsidiary R D in (1) hydrodynamic sediment-plant mesocosm design, replication, and monitoring, (2) synthetic and analytical chemistry, including the synthesis of commercially unavailable standards and development analytical approaches to detect minor differences in organic chemicals between time points and treatments and (3) sensor design, time series data acquisition and wavelet analysis of non-stationary series [6], and covariance structure modeling of mesocosm and ecosystem data [1]. Basic questions (e.g., what constitutes a true spatiotemporal replicate in a multivariate, multiply colinear system What is the minimum number of indicator variables needed to characterize the states of such a system and how often do they need to be sampled in space and time ) arose and had to... 

The tools of chemometrics encompass not only the familiar (univariant) methods of statistics, but especially the various multivariant methods, together with a package of pattern-recognition methods for time-series analyses and all the known models for signal detection and signal processing. Chemometric methods of evaluation have now become an essential part of environmental analysis, medicine, process analysis, criminology, and a host of other fields. 

Given the pre-eminence of the transfer function and frequency-domain-based approaches in process and chemical engineering, these two approaches will be discussed in greatest detail in this chapter. Nevertheless, information about the state-space-based approach will also be considered. This chapter will present the basic, univariate approach to time series analysis, which will be extended in Chap. 6 to consider the multivariate case containing both stochastic and deterministic components in order to model complex processes for process control, economic analysis, and simulation development. 

Climate dataset generation The climate dataset for simulation comprises of a synthetic wind speed, significant wave height and wave period time series. These are generated using a Multivariate Auto-Regressive (MAR) model, shown in Equation 1, normalised by the mean of the data p where is the simulated wind speed at time-step t, n is the number of variables, is a variable state vector, is a matrix of the MAR model coefficients and is a noise vector with mean zero and covariance matrix of the data, order p (Box and Jenkins, 1970). 

The solvent-mediated transformation of o -L-glutamic acid to the S-form was quantitatively monitored over time at a series of temperatures [248]. The calibration model was built using dry physical mixtures of the forms, but still successfully predicted composition in suspension samples. Cornel et al. monitored the solute concentration and the solvent-mediated solid-state transformation of L-glutamic acid simultaneously [249]. However, the authors note that multivariate analysis was required to achieve this. Additionally, they caution that it was necessary to experimentally evaluate the effect of solid composition, suspension density, solute concentration, particle size and distribution, particle shape, and temperature on the Raman spectra during calibration in order to have confidence in the quantitative results. This can be a substantial experi-... 

A linear model predictive control law is retained in both cases because of its attracting characteristics such as its multivariable aspects and the possibility of taking into account hard constraints on inputs and inputs variations as well as soft constraints on outputs (constraint violation is authorized during a short period of time). To practise model predictive control, first a linear model of the process must be obtained off-line before applying the optimization strategy to calculate on-line the manipulated inputs. The model of the SMB is described in [8] with its parameters. It is based on the partial differential equation for the mass balance and a mass transfer equation between the liquid and the solid phase, plus an equilibrium law. The PDE equation is discretized as an equivalent system of mixers in series. A typical SMB is divided in four zones, each zone includes two columns and each column is composed of twenty mixers. A nonlinear Langmuir isotherm describes the binary equilibrium for each component between the adsorbent and the liquid phase. 

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