Multivariate time series

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. 

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,... 

WR DeVries and SM Wu. Evaluation of process control effectiveness and diagnosis of variation in paper basis weight via multivariate time series analysis. IEEE Trans, on Automatic Control, 23 702-708, 1978. 

GC Reinsel. Elements of Multivariate Time Series Analysis. Springer-Verlag, New York, NY, 2nd edition, 1997. 

A Singhal and DE Seborg. Pattern matching in multivariate time series databases using a moving window approach. Ind. Engg. Chem. Research, 41 3822-3838, 2002. 

This decomposition can be interpreted as a PCA on the bandpass-filtered process. An inverse Fourier transform using the first few eigenvectors provides a reduced representation of the original multivariate time series [25]. When anarrowband frequency-domain structure is present in the dynamics, it can be shown that Equation 5.20 provides stronger optimal decomposition of the multivariate time series than does conventional PCA [25]. 

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 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... 

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. 

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... 

COM 07a] CoMMANDEUR J., Bjleveld F., Bergel R., a multivariate time series analysis applied to SafetyNet data, SafetyNet Deliverable D7.7 of the EU FP6 project SafetyNet, 2007. 

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... 

In the present time with almost unlimited computer facilities in the analytical laboratory, analytical chemists should be able to obtain substantial benefits from the application of time series, information theory, multivariate statistics, a.o. factor analysis and pattern recognition, operations research, numerical analysis, linear algebra, computer science, artificial intelligence, etc. This is in fact what chemo-metricians have been doing for the past decades. 

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. 

Sequential signals are surprisingly widespread in chemistry, and require a large number of methods for analysis. Most data are obtained via computerised instruments such as those for NIR, HPLC or NMR, and raw information such as peak integrals, peak shifts and positions is often dependent on how the information from the computer is first processed. An appreciation of this step is essential prior to applying further multivariate methods such as pattern recognition or classification. Spectra and chromatograms are examples of series that are sequential in time or frequency. However, time series also occur very widely in other areas of chemistry, for example in the area of industrial process control and natural processes. 

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. 

What can a scalar time series tell us about the multivariate state (or phase) space buried in the observations The so-called embedology attributed to Whitney [75] and Takens [76] provides us with an essential clue to the answer of such a question. A detailed description of the mathematical proof is beyond the scope of this review, and here we focus on describing the brief concept and methodology. 

Many analytical measures cannot be represented as a time-series in the form of a spectrum, but are comprised of discrete measurements, e.g. compositional or trace analysis. Data reduction can still play an important role in such cases. The interpretation of many multivariate problems can be simplified by considering not only the original variables but also linear combinations of them. That is, a new set of variables can be constructed each of which contains a sum of the original variables each suitably weighted. These linear combinations can be derived on an ad hoc basis or more formally using established mathematical techniques. Whatever the method used, however, the aim is to reduce the number of variables considered in subsequent analysis and obtain an improved representation of the original data. The number of variables measured is not reduced. 

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. 

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... 

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... 

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