COMPUTATION OF THE PRESS STATISTIC

Computation of the true prediction errors e (A ), k = 1,2,is a tremendous task in dynamic system identification where we typically face a large amount of data (M) and possibly high dimensionality (n) of the parameter vector 6. It will be shown here that by using the orthogonal decomposition algorithm, the computation of the PRESS residuals is simplified to an extent that its calculation can be viewed as a byproduct of the algorithm. The following theorem presents the cornerstone for computation of the PRESS statistic. 

Theorem 3.1 Let (.) denote the ith column of W and gi represent the ith estimated auxiliary parameter. Then the PRESS residuals e fc(fc), A = 1,2. M, for the original model with n parameters are given by 

Proof Prom Equation (3.13), we can write the conventional residuals in terms of the orthogonalized data matrix and the auxiliary parameter estimates 

We also note that from the definitions of (k) and in Equations (3.2) and (3.4) we have 

Prom the expression for the PRESS residuals e fc(fc) in Equation (3.21), the result in Equation (3.24) follows. 

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Chapter 3 presents the development of the PRESS statistic as a criterion for structure selection of dynamic process models which are linear-in-the-parameters. Computation of the PRESS statistic is based on the orthogonal decomposition algorithm proposed by Korenberg et al. (1988) and can be viewed as a by-product of their algorithm since very little additional computation is required. We also show how the PRESS statistic can be used as an efficient technique for noise model development directly fi-om time series data. 

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