Functional estimation problem generalization

Numerically, the LSA approach may be implemented by calcnlations that involve estimation by (general) linear regression, e.g., nse of cnbic spline functions." The intrinsic problems of nonlinear estimation common in more structured methods can thereby be avoided or significantly reduced. 

Goal A desired end-point (i.e., result) typically characterized by multiple parameters, at least one of which is specified. Examples include specified task goals (e.g., move an object of specified mass from point A to point B in 3 sec) or estimated task performance (maximum mass, range, speed of movement obtainable given a specified elemental performance resource availability profile), depending on whether a reverse or forward analysis problem is undertaken. Whereas function describes the general process of a task, the goal directly relates to performance and is quantitative. 

Since Pfi is a function of only one standard normal random variable ui, its mean value can be point-estimated from Eq. 35. For a problem with n variables, if the probability moments of is estimated using w-point estimate, only mn function calls of P/x) are required for estimating the general probability of failure. 

For the study of flow stability in a heated capillary tube it is expedient to present the parameters P and q as a function of the Peclet number defined as Pe = (uLd) /ocl. We notice that the Peclet number in capillary flow, which results from liquid evaporation, is an unknown parameter, and is determined by solving the stationary problem (Yarin et al. 2002). Employing the Peclet number as a generalized parameter of the problem allows one to estimate the effect of physical properties of phases, micro-channel geometry, as well as wall heat flux, on the characteristics of the flow, in particular, its stability. 

When the model equations are linear functions of the parameters the problem is called linear estimation. Nonlinear estimation refers to the more general and most frequently encountered situation where the model equations are nonlinear functions of the parameters. 

As seen in Chapter 2 a suitable measure of the discrepancy between a model and a set of data is the objective function, S(k), and hence, the parameter values are obtained by minimizing this function. Therefore, the estimation of the parameters can be viewed as an optimization problem whereby any of the available general purpose optimization methods can be utilized. In particular, it was found that the Gauss-Newton method is the most efficient method for estimating parameters in nonlinear models (Bard. 1970). As we strongly believe that this is indeed the best method to use for nonlinear regression problems, the Gauss-Newton method is presented in detail in this chapter. It is assumed that the parameters are free to take any values. 

Statistical theory provides for the construction of a function incorporating differences between the measured and corresponding calculated values, with the best estimates being the properties that minimize that function. This procedure is shown in detail in the following section. Generally, the implementation is significantly impacted by the functionality of the properties. Three classes of problems are identified ... 

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