Taylor expansions statistical methods

Response surface methodology (RSM) is a method of optimization using statistical techniques based upon the special factorial designs of Box and Behenkini and Box and Wilson.I It is a scientific approach to determining optimum conditions which combines special experimental designs with Taylor first and second order equations. The RSM process determines the surface of the Taylor expansion curve which describes the response (yield, impurity level, etc.) The Taylor equation, which is the heart of the RSM method, has the form ... 

For more complex models or for input distributions for which exact analytical methods are not applicable, approximate methods might be appropriate. Many approximation methods are based on Taylor series expansion solutions, in which the series is truncated depending on the desired amount of solution accuracy and whether one wishes to consider covariance among the input distributions (Hahn Shapiro, 1967). These methods often go by names such as generation of system moments , statistical error propagation , delta method and first-order methods , as discussed by Cullen Frey (1999). 

Approximation methods can be useful, but as the degree of complexity of the input distributions or the model increases, in terms of more complex distribution shapes (as reflected by skewness and kurtosis) and non-linear model forms, one typically needs to carry more terms in the Taylor series expansion in order to produce an accurate estimate of percentiles of the distribution of the model output. Thus, such methods are often most widely used simply to quantify the mean and variance of the model output, although even for these statistics, substantial errors can accrue in some situations. Thus, the use of such methods requires careful consideration, as described elsewhere (e.g. Cullen Frey, 1999). 

Conditional First-Order (NLME) Method. Proposed by Lindstrom and Bates,this uses a first-order Taylor series expansion about conditional estimates of interindividual random effects. This estimation method is available in S-plus statistical software as the function NLME. ... 

Thus, by using a judicious combination of the LQA method and approximate evaluation of higher order terms in the Taylor-series expansion of the path, the potential energy surface information that is already available for performing statistical or dynamical calculations of the chemistry can be used to more accurately follow the path. 

Probabilistic response analysis consists of computing the probabilistic characterization of the response of a specific structure, given as input the probabilistic characterization of material, geometric and loading parameters. An approximate method of probabilistic response analysis is the mean-centred First-Order Second-Moment (FOSM) method, in which mean values (first-order statistical moments), variances and covariances (second-order statistical moments) of the response quantities of interest are estimated by using a mean-centred, first-order Taylor series expansion of the response quantities in terms of the random/uncertain model parameters. Thus, this method requires only the knowledge of the first- and second-order statistical moments of the random parameters. It is noteworthy that often statistical information about the random parameters is limited to first and second moments and therefore probabilistic response analysis methods more advanced than FOSM analysis cannot be fully exploited. 

The perturbation method starts with a Taylor series expansion of the solution, the external loading, and the stochastic stiffness matrix in terms of the random variables introduced by the discretization of the random parameter field. The unknown coefficients in the expansion of the solution are obtained by equating terms of equal order in the expansion. From this, approximations of the first two statistical moments can be obtained. The perturbation method is computationally more efficient than direct Monte Carlo simulation. However, higher-order approximations will increase the computational effort dramatically, and therefore accurate results are obtained for small coefficients of variation only. 

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