Parameter Estimation

Consistency analysis parameter estimates are computed for the conditions of m= do nothing, q= Central, and t= 1990. This yields a sample of 4 observations on B_t and 6 observations on D_t. Together, these values constitute a single realization of the model's marginal joint probability of these variables under the above conditions and at these observation time points. Given the complexity of the model, this sample represents an extreme case of data sparsity relative to model complexity.

As discussed above, the sample is derived in-part from published maps and the report of several authors. In addition, several of the maps are meant to be illustrative rather than being derived from extensive field work. Hence, the sample contains some unknown amount of measurement error. The hypothesis parameter values on the other hand, are a result of literature review and the assigning of a ``reasonable'' value to each parameter. Hence, neither the sample nor the hypothesis values can be considered completely reliable. This subjective assessment of the sample's reliability relative to prior knowledge is represented by setting c_H to 0.5.

Due to computer expense constraints, only the 64 parameters that define B_t, K_t, f_t, r_t, N_t, and D_t are estimated (Table 8). The parameters of these nodes were chosen for estimation because the associated nodes are close in terms of path length to the observed nodes. Optimization is over explicit bounds, i.e., during the optimization, a parameter's value is not allowed to leave a finite interval. These intervals were chosen to be as wide as possible without containing absurd values. In Table 8, the k₀ = 1000.0, and $\sigma = 10.0$ values are the result of the algorithm setting these parameters to their lower, and upper bounds, respectively. In addition, .001 is the lower bound for $\beta_f$ and $\beta_r$ (Table 8).

For c_H = 0, g_CA = g_S = -.564. For c_H=1, g_CA = g_H = -.373. For c_H = .5, the final values of g_S and g_H corresponding to the maximized value of g_CA, are -.846 and -1.226, respectively. As expected, in order to maximize g_CA, compromises are made with respect to individually maximizing g_S and g_H.

For comparison, the solution corresponding to complete faith in the sample and none in $\mbox{\boldmath$\beta$}_H$ (c_H = 0) is also computed. For each value of c_H, the constrained optimization's starting point is $\mbox{\boldmath$\beta$}_H$. Differences between c_H=.5 and c_H=0 are minimal.

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