Disturbance term

When experimental data are collected over time or distance there is always a chance of having autocorrelated residuals. Box et al. (1994) provide an extensive treatment of correlated disturbances in discrete time models. The structure of the disturbance term is often moving average or autoregressive models. Detection of autocorrelation in the residuals can be established either from a time series plot of the residuals versus time (or experiment number) or from a lag plot. If we can see a pattern in the residuals over time, it probably means that there is correlation between the disturbances. 

In this case we assume the disturbance term, e , is not white noise, rather it is related to through the following transfer function (noise filter)... 

The ARMA method offers the possibility of combining both autoregressive and moving average processes. The disturbance term from autoregressive processes may be described by a moving average. 

To perform a linearized stability analysis of the fluid dynamical system, we express all flow quantities q, into a steady mean (Q) and an unsteady disturbance term eq ) that is considered one order of magnitude smaller than the mean quantities, so that... 

This model is of the same form as the one used for RLS and hence the updating equations for 0 and P are the same as in the previous section. However, the disturbance terms, i, -r, and hence the updating equations cannot be... 

An electromagnetic disturbance termed the evanescent wave penetrates the rarer medium to a finite depth. It has a wavelength A. and is continuous with the sinusoidal field of the standing wave, but the electric field amplitude E decreases exponentially with distance from the surface z as... 

A number of issues arise in connection with this sort of model. An extremely important one concerns the assumptions regarding the distribution of log(ey). It might usually be implicitly assumed that log(ey) has a Normal distribution. More important, however, would be the assumptions made about the correlation structure of these disturbance terms and also the variances. Discussion of correlations is deferred until considering random-effect models. However the variance has some relevance to a planning issue below. 

To further cope with process nonlinearities, the disturbance term d can be expressed by combining two terms which is the disturbance due to plant/model mismatch, and d, the disturbance due to nonlinearities, d, can be deter-mined at every sampling time by minimizing the output prediction error between the linear and nonlinear models [5]. 

The models described in Equations (7.70) and (7.71) are non-linear. It is possible to transform some non-linear models into linear models these types of models are referred to as intrinsically linear. It is important to note that the disturbance term is also transformed when transforming a model. We present two examples of models that can be transformed and are thereby intrinsically linear. For catalytic reactions, the reaction rate may contain an inhibition term in the denominator. The reaction rate at constant temperature can, for some cases, be expressed as... 

If a disturbance variable is known or can be measured, it can be included in the step-response model. Let d denote a measured disturbance and Sf its step-response coefficients. Then the standard step-response model in Eq. 20-6 can be modified by adding a disturbance term, N-l... 

X = nxk) matrix of observations of the independent variables p = ( X 1) vector of parameters u = nx 1) vector of disturbance terms n = number of observations. 

The disturbance terms, represented by the vector m, are random variables with zero expectation, that is. 

There are several components in an intervention model a deterministic component describing the intervention(s), the associated response of the system to the intervention, and a stochastic disturbance term. The overall modeling strategy is to obtain reasonable initial representations for these components and to iterate to a final model based on intermediate estimates, diagnostic checks, and model interpretations (Liu and Hudak 2004). 

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