Auto-regressive process

Even under the relatively simple type of model presented above, the process At does not follow an ARMA(1,1) evolution as under the case in which the demand evolves according to the elementary auto-regressive process of Example 1. In fact, under (10.19), the orders At maintain the following recursive scheme ... 

Given a set of experimental data, we look for the time profile of A (t) and b(t) parameters in (C.l). To perform this key operation in the procedure, it is necessary to estimate the model on-line at the same time as the input-output data are received [600]. Identification techniques that comply with this context are called recursive identification methods, since the measured input-output data are processed recursively (sequentially) as they become available. Other commonly used terms for such techniques are on-line or real-time identification, or sequential parameter estimation [352]. Using these techniques, it may be possible to investigate time variations in the process in a real-time context. However, tools for recursive estimation are available for discrete-time models. If the input r (t) is piecewise constant over time intervals (this condition is fulfilled in our context), then the conversion of (C.l) to a discrete-time model is possible without any approximation or additional hypothesis. Most common discrete-time models are difference equation descriptions, such as the Auto-.Regression with eXtra inputs (ARX) model. The basic relationship is the linear difference equation ... 

Another purpose of model updating is to obtain a mathematical model to represent the underlying system for future prediction. Even though there are also parameters to be identified as in the previous case, these parameters may not necessarily be physical, e.g., coefficients of auto-regressive models. In this situation, the identified parameters are not necessarily as important as the previous case provided that the identified model provides an accurate prediction for the system output. It will be shown in the following chapters that there is no direct relationship between satisfactory model predictions and small posterior uncertainty of the parameters. This point will be further elaborated in Chapter 6. Nevertheless, no matter for which purpose, quantification of the parametric uncertainty is useful for further processing. For example, it can be utilized for comparison of the identified parameter values at different stages or for uncertainty analysis of the output of the identified model. Furthermore, it will be demonstrated in Chapter 6 that quantification of the posterior uncertainty allows for the selection of a suitable class of models for parametric identification. 

To include information about process dynamics, lagged variables can be included in X. The (auto)correlograms of all x variables should be developed to determine first how many lagged values are relevant for each variable. Then the data matrix should be augmented accordingly and used to determine the principal components that will be used in the regression step. 

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. 

Unfortunately, neither PACF nor ACF lead to directly interpretable results for ARMA processes. The extended ACF tries to overcome this drawback by jointly providing information about the order of both components. For each AR order tested, the EACF first determines estimates of the AR coefficients by a sequence of regression models. Afterwards, the residuals ACF is calculated. The results are presented in a table indicating significant or non-significant auto-correlations (typically denoted by an x and o, respectively). In such a table, the rows represent the AR order p whereas columns represent... 

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