Time-series Identification

Dynamic process models are usefulfor detailed cfynamic process analysis and in process control applications. They are also increasingly usedfor predicting process variables, such as quality. Especially when the quality variable is difficult to measure or can be measured infrequently, dynamic models can provide an inferential measurement, based on easily and frequently measurable variables. 

Depending on the process being investigated, one could develop first-principle models (lin-ear/non-linear (partial) differential equations) or simple linear empirical differential or difference equations builtfrom process data only. The first type ofmodel is always preferred however, developing first-principle models is time consuming and consequently expensive. In this chapter, the focus will therefore be on time series models, derivedfrom process measurements. It is assumed that outliers have already been removed from the process data set therefore a model fit parameter as defined in chapter 19 is used to calculate how well the model fits the data. 

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Time series identification for EC Box-Pierce Statistic = 40.8498 Degrees of freedom = 10 Significance level =. 0000... 

Finally, the value of extensive time-series records extends beyond the identification of a specific problem. Long-term time-series permits verification that decisions are effective (or not) solutions are, indeed, working (or not) and the ongoing costs and benefits of the given control program are assessed accurately. With proper design of what to measure, it can also assist in understanding the why or why not. 

U. Parlitz. Identification of true and spurious Lyapunov exponents from time series. Int. J. Bifur. and Chaos, 2(1) 155-165, 1992. 

The system identification step in the core-box modeling framework has two major sub-steps parameter estimation and model quality analysis. The parameter estimation step is usually solved as an optimization problem that minimizes a cost function that depends on the model s parameters. One choice of cost function is the sum of squares of the residuals, Si(t p) = yi(t) — yl(t p). However, one usually needs to put different weights, up (t), on the different samples, and additional information that is not part of the time-series is often added as extra terms k(p). These extra terms are large if the extra information is violated by the model, and small otherwise. A general least-squares cost function, Vp(p), is thus of the form... 

The rank modification of von Neumann testing for data independence as described in Madansky (1988) and Bartels (1982) is applied. Although steady-state identification is not the original goal of this technique, it indicates if a time series has no time correlation and can thus be used to infer that there is only random noise added to a stationary behavior. In this test a ratio v is calculated from the time series, whose distribution is expected to be normal with known mean and standard deviation, in order to confirm the stationarity of a specific set of points. 

The multifractal behavior of time series such as SRV, HRV, and BRV can be modeled using a number of different formalisms. For example, a random walk in which a multiplicative coefficient in the random walk is itself made random becomes a multifractal process [59,60], This approach was developed long before the identification of fractals and multifractals and may be found in Feller s book [61] under the heading of subordination processes. The multifractal random walks have been used to model various physiological phenomena. A third method, one that involves an integral kernel with a random parameter, was used to model turbulent fluid flow [62], Here we adopt a version of the integral kernel, but one adapted to time rather than space series. The latter procedure is developed in Section IV after the introduction and discussion of fractional derivatives and integrals. 

Model identification is an iterative process. There are several software packages with modules that automate time series model development. When a model is developed to describe data that have stochastic variations, one has to be cautious about the degree of fit. By increasing model complexity (adding extra terms) a better fit can be obtained. But, the model may describe part of the stochastic variation in that particular data which will not occur identically in other data sets. Consequently, although the fit to the training data may be improved, the prediction errors may get worse. 

It is desirable, from a practical standpoint, to model time series in as simple a form as possible. Such are linear models with a minimal number of parameters. There are two primary steps to the modeling process identification of the model form and estimation of the model s parameters. These steps are appropriately followed by testing of the model s ability to fit or predict new data. 

Regression analysis models and estimation theory models are very useful for the identification of mathematical relations and parameter values in these relations from sets of data or measurements. Regression and estimation methods are used frequently in conjunction with mathematical modehng, in particular with trend extrapolation and time series forecasting, and with econometrics. These methods are often also used to validate models. Often these approaches are called system identifi-... 

Since this work deals with the aggregated simulation and planning of chemical production processes, the focus is laid upon methods to determine estimations of the process models. For process control this task is the crucial one as the estimations accuracy determines the accuracy of the whole control process. The task to find an accurate process model is often called process identification. To describe the input-output behaviour of (continuously operated) chemical production plants finite impulse response (FIR) models are widely used. These models can be seen as regression models where the historical records of input/control measures determine the output measure. The term "finite" indicates that a finite number of historical records is used to predict the process outputs. Often, chemical processes show a significant time-dynamic behaviour which is typically reflected in auto-correlated and cross-correlated process measures. However, classic regression models do not incorporate auto-correlation explicitly which in turn leads to a loss in estimation efficiency or, even worse, biased estimates. Therefore, time series methods can be applied to incorporate auto-correlation effects. According to the classification shown in Table 2.1 four basic types of FIR models can be distinguished. 

Stochastic identification techniques, in principle, provide a more reliable method of determining the process transfer function. Most workers have used the Box and Jenkins [59] time-series analysis techniques to develop dynamic models. An introduction to these methods is given by Davies [60]. In stochastic identification, a low amplitude sequence (usually a pseudorandom binary sequence, PRBS) is used to perturb the setting of the manipulated variable. The sequence generally has an implementation period smaller than the process response time. By evaiuating the auto- and cross-correlations of the input series and the corresponding output data, a quantitative model can be constructed. The parameters of the model can be determined by using a least squares analysis on the input and output sequences. Because this identification technique can handle many more parameters than simple first-order plus dead-time models, the process and its related noise can be modeled more accurately. 

Inspired from the principle of identification dynamic time-series model by determining the model order, we present in this paper an Information Criteria (IC) approach to select the time window length in the dynamic PCA. Then a new dynamic monitoring algorithm with the proposed IC approach is presented. Application of TE process proved that the proposed IC approach is effective. The dynamic PCA with appropriate time window length and windows moving width is able to detect and identify faults and abnormal events better than conventional PCA approach. 

The System Identification Toolbox in MATLAB is a very useful toolbox when fitting models for system identification using the prediction error model. It provides a convenient and concise way of storing, accessing, and manipulating different data sets and their associated models. Although most time series analyses can be performed using the System Identification Toolbox, at times it is easier to use the econometric toolbox described below. In order to fully appreciate and use the System Identification Toolbox, it is first useful to examine in detail the special data objects that store and hold the information the Iddata and the idpoly objects. 

After identification of a cosmic spike, another approach to remove the spike is to use a median filter on several related spectra. The spectra can be related in time, for time-series data as with data taken from a process on-line analyzer, or related in space, in the case of a Raman image dataset in which spectra are taken from spatially segregated locations. 

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