Auto-regressive model

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. 

In this section, the extended Kalman filter is formulated for the time-varying auto-regressive model of order p, which is abbreviated as the TVAR(p) model ... 

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 ... 

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... 

Much attention has also been devoted to modal identification without measuring the input time history. In particular, a lot of effort has been dedicated to the case of free vibration (or impulse response) and to the case of ambient vibration. In the former case, often time-domain methods based on auto-regressive moving average (ARMA) models are employed, using least squares as the core ingredient in their formulations. However, it was found that the least-squares method yields biased estimates [76], A number of methods have been developed to eliminate this bias, including the instrumental matrix with delayed observations method [76], the correlation fit method [275], the double least-squares method [114,202] and the total least-squares method [92]. A detailed comparison of these methods can be found in Cooper [61],... 

Various forms of General Auto-Regressive Conditional Heteroske-dastic (GARCH) models have been used to estimate return volatility. Such models express current volatility as a function of previous returns and forecasts. For instance, the GARCH(1,1) model takes the form ... 

Univariate Forecasts of a given variable demand are based on a model fitted only to present and past observations of a given time series. There are several different univariate models, like Extrapolation of Trend Curves, Simple Exponential Smoothing, Holt Method, Holt-Winters Method, Box-Jenkins Procedure, and Stepwise Auto-regression, which can be regarded as a subset of the Box-Jenkins Procedure. 

Supply ehains have to make two eategories of decisions—he long-term strategic and short-term operational decisions. A lot of data is required for making decisions. A model-based approach is extremely helpful to reduce the sample size. The model consists of an auto regressive (AR) and a moving average (MA) part. The ARMA... 

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 ... 

In order to be able to establish confidence intervals, it also assumed that Ck is Gaussian distributed. The ARX model (1) is characterized by 3 numbers Ua, the auto-regressive order ny, the exogeneous order and rik, the pure time delay between input and output. A regression model is an ARXOlO model (with [ua, ny, n-k] = [0,1,0]). 

ARMAX (auto-regressive moving average with exogenous variable) equation, 13-4 Arterial and venous trees, in CV modeling, 10-8-10-9 Arterial macrocirculatory hemodynamics, 56-1-56-10... 

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). 

Auto-regressive-moving-average (ARMA) models... 

Forward Dynamic Neural MIMO NARX model used in this paper is a combination between the Multi-Layer Perception Neural Networks (MLPNN) stiucturc aird the Auto-Regressive with exogenous input (ARX) model. Due to this corrrbirratiorr. Forward MIMO NARX model possesses both of powerful universal approximating feature from MLPNN stmcturc aird strong predictive feature from nonlinear ARX model. 

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