Exponential model advantage

In both cases 6 = 1. The most sophisticated fitting law of this kind, known as ECS-EP [244], uses the advantages of both exponential and polynomial modelling and has five fitting parameters. As was shown in [245] it is no better than three-parameter SPEG within experimental accuracy. 

The advantage of this method is its ability to check whether the donor fluorescence decay in the absence and presence of acceptor is a single exponential or not. If this decay is not a single exponential in the absence of acceptor, this is likely to be due to some heterogeneity of the microenvironment of the donor. It can then be empirically modeled as a sum of exponentials ... 

The cluster site approximation, CSA, has the great advantage that the required number of equations to handle the necessary variables is given by the product Cn (Oates and Wentl 1996) as compared to the exponential value C" aassociated with a full CVM treatment (Inden and Pitsch 1991). The CSA is a variant of the quasichemical model proposed by Li as early as 1949, where the number of clusters that are considered to contribute to the entropy are reduced by excluding all clusters that share edges or bonds. Kikuchi (1977) has deduced the consequential changes in entropy for an f.c.c. structure (Eqs 7.32 and 7.33), which places CSA intermediate... 

ML is the approach most commonly used to fit a distribution of a given type (Madgett 1998 Vose 2000). An advantage of ML estimation is that it is part of a broad statistical framework of likelihood-based statistical methodology, which provides statistical hypothesis tests (likelihood-ratio tests) and confidence intervals (Wald and profile likelihood intervals) as well as point estimates (Meeker and Escobar 1995). MLEs are invariant under parameter transformations (the MLE for some 1-to-l function of a parameter is obtained by applying the function to the untransformed parameter). In most situations of interest to risk assessors, MLEs are consistent and sufficient (a distribution for which sufficient statistics fewer than n do not exist, MLEs or otherwise, is the Weibull distribution, which is not an exponential family). When MLEs are biased, the bias ordinarily disappears asymptotically (as data accumulate). ML may or may not require numerical optimization skills (for optimization of the likelihood function), depending on the distributional model. 

In order to take advantage of the full information contained in the AMOC data we use a two-dimensional fitting procedure A two-dimensional model function representing the number of counts as a function of positron age and energy of the annihilation quanta is fitted to the raw AMOC relief without prior data reduction. On the age axis, each positron state is represented by an exponential decay function convoluted with the time resolution function of... 

What is the advantage of using sums of exponentials to describe pharmacokinetic data in the situation of the single accessible pool model following a bolus injection or constant infusion The reason is that the integrals required to estimate the pharmacokinetic parameters are very easy to calculate ... 

The Power-Law Formalism possesses a number of advantages that recommend it for the analysis of integrated biochemical systems. As discussed above, we saw that estimation of the kinetic parameters that characterize the molecular elements of a system in this representation reduces to the straightforward task of linear regression. Furthermore, the experimental data necessary for this estimation increase only as the number of interactions, not as an exponential function of the number of interactions, as is the case in other formalisms. The mathematical tractability of the local S-system representation is evident in the characterization of the intact system and in the ease with which the systemic behavior can be related to the underlying molecular determinants of the system (see above). Indeed, the mathematical tractability of this representation is the very feature that allowed proof of its consistency with experimentally observed growth laws and allometric relationships. It also allowed the diagnoses of deficiencies in the current model of the TCA cycle in Dictostelium and the prediction of modifications that led to an improved model (see above). 

Principal Advantages. There are a number of difficulties associated with the modeling and analysis of complex nonlinear systems, (a) The functional form of the nonllnearltles Is often unknown, as are the numbers of Interactions and parameters that must be specified, (b) Once one has assumed a functional form there Is still difficulty In extracting statistical estimates of the parameter values from experimental data, (c) The amount of experimental data Itself that Is required to characterize many nonlinear mechanisms Increases exponentially with the dimensions of the problem, (d) General methods for analyzing the resulting system of nonlinear equations are not available. 

In the present section we discuss a quite different strategy with the important advantage of simple implementation. We show, using solvable decay models, that by increasing the distance of the observation point from the source, the transition from exponential to post-exponential decay occurs at higher probability densities and thus becomes more easily observable. Finally, note that when the distance is too large the exponential decay regime... 

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