Generalized Inverse Procedure

The generalized inverse procedure seeks to minimize the weighted composite objective function as represented by Eq. (28). In this procedure, the unknown model structure (parameter structure) and its corresponding parameter values are determined not only from prior information and observations but also by the accuracy requirement in model applications. Sun et al. (1997, 1998) presented a stepwise regression procedure for a simultaneous estimation of parameter structure and its corresponding parameter values. The procedure starts from a homogeneous parameter structure and gradually increases the complexity of the parameter structure. For a given set of data and a specified model reliability requirement, the method, at each level of complexity, calculates both the least-squares error as well as the parameterization error of using a simpler parameter structure to replace a more complex parameter structure. The method is most general as it considers errors in the parameter, observation, as well as prediction/management space. The established procedure allows one to determine whether a more complex parameter structure is needed or to conclude that data are insufficient to meet the specified model reliability requirement; and hence, additional data are needed.

In this procedure, we form a series of parameter structures of increasing complexity:

where G] represents a homogeneous structure, G2 a two-zone structure, and so forth; G2 is generated from Gl by dividing it into two zones and, generally, Gm+l is generated from Gm by dividing one of the zones of Gm into two zones. At each level, we calculate the residual error (RE) and the parameter structure error (SE). Specifically, the following steps are involved:

Step 1. Let G] be a homogeneous parameter structure, we solve the generalized inverse problem to find pf and the corresponding residual error RE[. In general, RE can be found by minimizing a linear combination of the norms in the parameter and observation space. Details can be found in Sun et al. (1998).

Step 2. Divide 6', into two zones to generate G2. The method suggested by Sun and Yeh (1995) can be used to optimize simultaneously the zonation pattern and its corresponding parameter values. In this step, we find a model M2(G2, pf) and its residual error RE2. RE2 must be smaller than RE! because a homogeneous parameter structure is being replaced by a two-zone structure to fit the same set of observations.

Step 3. Calculate the parameter structure error SE, by using 6', to replace G2. Details with regard to how to calculate SE, are presented in Sun et al. (1998).

Step 4. If both SE, and RE2 are large, we continue to increase the parameter structure complexity by finding the optimum three-zone parameter structure M3 (G3, p3) and RE3.

Step 5. Calculate the parameter structure error SE2 of using G2 to replace G3. If both SE2 and RE2 are large, we repeat steps 4 and 5 to obtain M4(G4, p*), RE4 and SE3, and so forth. Assume that through this procedure we have found Mm+1(Gm+1, Pm+i). REm+1, and SEm.

Step 6. Then consider the following four cases:

1. If both REm+1 and SEm are large compared to the observation error and accuracy requirement of the prediction/management problem, respectively, increase m by one and repeat step 5.

2. If both REm+1 and SEm are small, stop and use Mm+l as the identified model.

3. If REm+1 is large but SEm is small, either stop or continue to increase the complexity until REm+1 becomes small;

4. If REm+1 is small but SEm is large, additional data are required.

In case l, the identified model cannot satisfy the accuracy requirement of the given model application but, at the same time, the existing data still have the potential to provide more information. Therefore, we increase the parameter structure complexity. In cases 2 and 3, when the complexity of the parameter structure is increased, the prediction/management solution is not significantly improved; thus, we can either accept the identified model or if existing data still contain more information, we can continue to increase the parameter structure complexity. In case 4, the information contained in the existing data is insufficient to identify a reliable model and, thus, additional data are required to be collected.

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