Explicit Estimation

The choice of the objective function is very important, as it dictates not only the values of the parameters but also their statistical properties. We may encounter two broad estimation cases. Explicit estimation refers to situations where the output vector is expressed as an explicit function of the input vector and the parameters. Implicit estimation refers to algebraic models in which output and input vector are related through an implicit function. 

Given N measurements of the output vector, the parameters can be obtained by minimizing the Least Squares (LS) objective function which is given below as the weighted sum ofsquares of the residuals, namely, 

For systems described by ODEs the LS objective function becomes, 

As we mentioned earlier a further complication of systems described by ODEs is that instead of a single run, a series of runs may have been conducted. If we wish to estimate the parameters using all the data simultaneously we must consider the following objective function 

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Implicit estimation offers the opportunity to avoid the computationally demanding state estimation by formulating a suitable optimality criterion. The penalty one pays is that additional distributional assumptions must be made. Implicit formulation is based on residuals that are implicit functions of the state variables as opposed to the explicit estimation where the residuals are the errors in the state variables. The assumptions that are made are the following ... 

Because the calculation of these residuals does not require any iterative calculations, the overall computational requirements are significantly less than for the explicit estimation method using Equation 14.15 and the explicit LS estimation method using Equations 14.16a and b (Englezos et al. 1990a). 

An alternative scheme, proposed by Garside et al. (16,17), uses the dynamic desupersaturation data from a batch crystallization experiment. After formulating a solute mass balance, where mass deposition due to nucleation was negligible, expressions are derived to calculate g and kg in Equation 3 explicitly. Estimates of the first and second derivatives of the transient desupersaturation curve at time zero are required. The disadvantages of this scheme are that numerical differentiation of experimental data is quite inaccurate due to measurement noise, the nucleation parameters are not estimated, and the analysis is invalid if nucleation rates are significant. Other drawbacks of both methods are that they are limited to specific model formulations, i.e., growth and nucleation rate forms and crystallizer configurations. 

For kinetic systems with well-separated constants the left and right eigenvectors can be explicitly estimated. Their coordinates are close to +1 or 0. We analyzed these estimates first for linear chains and cycles (5) and then for general acyclic auxiliary dynamical systems (34), (36) (35), (37). The distribution of zeros and +1 in the eigenvectors components depends on the rate constant ordering and may be rather surprising. Perhaps, the simplest example gives the asymptotic equivalence (for of the reaction network A,+2 with... 

Of course, similar results for perturbations of zero eigenvalue are valid for more general ergodic chemical reaction network with positive steady state, and not only for simple cycles, but for cycles we get simple explicit estimates, and this is enough for our goals. 

Provided that mixing of the chemicals in the river is fast, the initial concentration C0 can be calculated from C0 = 7 / 0. Note In Chapter 24.2 the mixing hypothesis will be substantiated by giving explicit estimates for the relevant mixing parameters. 

Pomper (1976, p. 150) explicitly estimated terminal values and integrated them into his model. He calculated the additional cash flows generated by each alternative network configuration in the final period of the planning horizon as compared to the "do-nothing" option. To determine terminal values incremental cash flows were capitalized over a period of 10 years. However, this approach was only possible because all alternative final-period network configurations were known up front due to the backtracking solution procedure employed. 

In this chapter, Bayesian and likelihood-based approaches have been described for parameter estimation from multiresponse data with unknown covariance matrix S. The Bayesian approaches permit objective estimates of 6 and E by use of the noninformative prior of Jeffreys (1961). Explicit estimation of unknown covariance elements is optional for problems of Types 1 and 2 but mandatory for Types 3 and 4. 

Explicit estimation of S requires enough data to include each independent element aij as a parameter. Replicate experiments are particularly effective for this purpose, provided that they are reported precisely enough to give error information [unlike the repeated values produced by rounding in the data of Fuguitt and Hawkins (1945, 1947)]. 

Equation (24) also allows for an explicit estimate of the metadynamics error as a function of 6s, w/tg and the system parameters. Using (10) and (11) we have... 

It should be recognized that, no matter what risk assessment scheme is used, there will finally emerge an exposure level which will be said to be acceptable. There will probably always be a tendency to view such "official levels as the dividing lines between "safe" and "unsafe" exposures. We suggest, however, that the use of a scheme that provides explicit estimates of risk, and from which policy-makers decide on the risk that is tolerable in specific circumstances, is less likely to be misinterpreted as providing such sharp distinctions. 

Although generic firms may build and own their own factories, the price they charge for the product must reflect the amount they must pay their investors for the use of the facility, This rental rate is implicit in the competitive price of the product and does not have to be explicitly estimated. 

Herring, C. Explicit estimation of ground-state kinetic energies from electron-densities. Phys. Rev. A 1986, 34, 2614-2631. 

OHTelation terms, which require explicit estimation in other treatments, are automatically encompassed in e experimental E1/2 values. 

As for the point-source analysis, on-site approaches can be used to predict finite source effects, such as directivity and azimuthal changes on the ground motion directly analyzing the early motion at the single sites, without explicitly estimating any source parameters. The idea is to use the threshold-based approach to compute threshold levels at single stations and to interpolate the... 

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