Prediction error model

Statistical Prediction Errors (Model and Sample Diagnostic) Figure 5-28 shows the stss tical prediction errors for all four components for the samples in the validat n set. For MCB and ODCB the maximum value is —0.004 and for EB and C5IM the maximum value is —0.01. These errors are small compared to thecsncentration ranges. 

Statistical Prediction Errors (Model and Sample Diag Jostic) Uncertainties in the concentrations can be estimated because the predicted concentrations are regression coefficients from a linear regression (see Equations 5.7-5.10). These are referred to as statistical prediction errors to distinguish them from simple concentration residuals (c — c). Tlie statistical prediction errors are calculated for one prediction sample as... 

Statistical Prediction Errors (Model and Sample Diagnostic) From the S matrix it is possible to predict all four components (caustic, salt, water concentration, and temperature). However, in this application the interest is only in caustic and, therefore, only the results for this component are presented. The statistical prediction errors for the caustic concentration for the validation data vary from 0.006 to 0.028 wt.% (see Figure 5-54). The goal is to predict the caustic concentration to 0.1 wt.% (lo), and the statistical prediction errors indicate that the precision of the method is adequate. Also, there do not appear to be any sample(s) that have an unusual error when compared to the rest of the samples. 

The updated PDF p 0 S jf, A, C) in Equation (3.67) is given by Equation (3.60) with Equation (3.61) for the general case of uncertain excitation. The formulation presented here is based on the spectral density estimators obtained from the measured data V and it depends on the class of structural, excitation and prediction-error models chosen to describe the system. The updated parameter vector is obtained by minimizing the objective function J 0) = — In p 0 C)p S jf 0, A, C) with the likelihood function p S 0, A, C) given by Equation (3.61). Eurthermore, the updated PDF of the model parameter vector 0 can be... 

Irrespective of the situation, process system identification is focused on determining the values for Gp and G/ as accurately as possible. Since most of the applications assume that the controller is digital, the system identification methods considered here will focus on the discrete time implementation of system identification. For this reason, the models for each of the blocks will be assumed to be linear, rational functions of the backshift operator Such models are most often referred to as transfer functions. The most general plant model is the prediction error model, which has the following form ... 

This section will examine the principles and key results for modelling an open-loop process modelled using the general prediction error model given by Eq. (6.4). The foundation for such modelling is the prediction error method, which uses the fact that most models in system identification are used for predicting future values of the process. 

Select an appropriate (prediction error) model and determine the corresponding one-step ahead optimal predictors (Eq. (6.27)). 

Theoretically, the main concerns lie with identifiability of a process, that is, given a data set and model structure (order of polynomials), what are the conditions for there to be a unique solution to the parameter estimates. For open-loop experiments, the identifiability constraint for a prediction error model can be simply written as... 

When fitting a prediction error model to the data, it is assumed that the true model of the process is linear. Since very few chemical processes are tmly linear, it is necessary to check the original process if a linear model is sufficient over the region of consideration of the variables. Two common tests are ... 

Finally, the prediction error model assumes that the parameter values do not change with respect to time, that is, they are time invariant. A quick and simple test of the invariance of the model is to split the data into two parts and cross validate the models using the other data set. If both models perform successfully, then the parameters are probably time invariant, at least over the time interval considered. 

If it is assumed that a prediction error model is being fit, then general conditions for identifiability based on the orders of the polynomials can be obtained. A process is identifiable from routine operating data if (Shardt and Huang 2011)... 

The second method is called direct identification, where the fact that the process is running in closed loop is ignored. In this type of identification, both the process and error structures must be simultaneously estimated. Thus, either a Box-Jenkins or a general prediction error model should be fit. Since this is one of the more common approaches to closed-loop system identification, it is necessary to examine the properties of this approach. It will be assumed that the prediction error method will be used. 

All prediction error models can be fit using standard, linear regression. 

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. 

Bayesian inference can be applied at model class level to assess the plausibility of several alternative model classes based on the available observations d this is referred to as Bayesian model class selection or MCS (Beck and Yuen 2004 Yuen 2010). The set of alternative model classes commonly concern (mechanical) prediction model classes Mm-, but here the method will be used to distinguish between alternative probabi-hstic prediction error models Mjy The following, however, is elaborated for a set of general model classes Mi. 

Because the spatial correlation structure is unknown, a set of three alternative prediction error model classes is determined for E, an uncorrelated model class A, a model class B with an exponential correlation function, and a model class C with an exponentially damped cosine correlation function. Each of these model classes is parameterized by a number of prediction error parameters as follows ... 

The PDFs of the parameters characterizing this correlation model are estimated as weU as the PDF of the stiffness parameter Om- For 151 sensors (and the selected cosine prediction error model class), a posterior mean value of = 18.95 GPa is fotmd, with a standard deviation of (t(0m) = 0.80 GPa. 

This application shows that the Bayesian MCS approach can effectively be used to estimate a suitable correlation structure. Moreover, the MCS approach can also be applied to distinguish between different types of prediction error models (e.g., Gaussian vs. non-Gaussian) or between alternative descriptions for systematic prediction errors. 

This reference entry describes the use of Bayesian model class selection to determine, among a set of possible model classes, the prediction error model class most suited for Bayesian model updating, according to the available experimental data. It is demonstrated that, provided sufficient information is available, the Bayesian MCS approach is an effective tool to this end, ensuring a more realistic joint structural-probabilistic model and corresponding Bayesian model updating results. 

In addition, the chapter will provide an overview of htunan reliability quantification techniques, and the relationship between these techniques and qualitative modeling. The chapter will also describe how human reliability is integrated into chemical process quantitative risk assessment (CPQRA). Both qualitative and quantitative techniques will be integrated within a framework called SPEAR (System for Predictive Error Analysis and Reduction). 

The identification of plant models has traditionally been done in the open-loop mode. The desire to minimize the production of the off-spec product during an open-loop identification test and to avoid the unstable open-loop dynamics of certain systems has increased the need to develop methodologies suitable for the system identification. Open-loop identification techniques are not directly applicable to closed-loop data due to correlation between process input (i.e., controller output) and unmeasured disturbances. Based on Prediction Error Method (PEM), several closed-loop identification methods have been presented Direct, Indirect, Joint Input-Output, and Two-Step Methods. 

The %HIA, on a scale between 0 and 100%, for the same dataset was modeled by Deconinck et al. with multivariate adaptive regression splines (MARS) and a derived method two-step MARS (TMARS) [38]. Among other Dragon descriptors, the TMARS model included the Tig E-state topological parameter [25], and MARS included the maximal E-state negative variation. The average prediction error, which is 15.4% for MARS and 20.03% for TMARS, shows that the MARS model is more robust in modeling %H1A. 

Fig. 36.10. Prediction error (RMSPE) as a function of model complexity (number of factors) obtained from leave-one-out cross-validation using PCR (o) and PLS ( ) regression.

An optimization criterion for determining the output parameters and basis functions is to minimize the output prediction error and is common to all input-output modeling methods. The activation or basis functions used in data analysis methods may be broadly divided into the following two categories ... 

The residuals can be calculated from a given set of calibration samples in a different way. Cross validation is an important procedure to estimate a realistic prediction error like PRESS. The data for k samples are removed from the data matrix and then predicted by the model. The residual errors of prediction of cross-validation in this case are given by... 

In MPC a dynamic model is used to predict the future output over the prediction horizon based on a set of control changes. The desired output is generated as a set-point that may vary as a function of time the prediction error is the difference between the setpoint trajectory and the model prediction. A model predictive controller is based on minimizing a quadratic objective function over a specific time horizon based on the sum of the square of the prediction errors plus a penalty... 

All regression methods aim at the minimization of residuals, for instance minimization of the sum of the squared residuals. It is essential to focus on minimal prediction errors for new cases—the test set—but not (only) for the calibration set from which the model has been created. It is relatively easy to create a model— especially with many variables and eventually nonlinear features—that very well fits the calibration data however, it may be useless for new cases. This effect of overfitting is a crucial topic in model creation. Definition of appropriate criteria for the performance of regression models is not trivial. About a dozen different criteria— sometimes under different names—are used in chemometrics, and some others are waiting in the statistical literature for being detected by chemometricians a basic treatment of the criteria and the methods how to estimate them is given in Section 4.2. 

---