Taylor series multivariable

The coefficients of the multivariable Taylor series expansion of G J) about the point where the Schwinger probes vanish are elements of the ROMs. Thus G J) is known as the generating functional for ROMs. Mathematically, the RDMs of the functional G J) are known as the moments. The moment-generating functional G(y) may be used to define another functional W J), known as the cumulant-generating functional, by the relation... 

In most models developed for pharmacokinetic and pharmacodynamic data it is not possible to obtain a closed form solution of E(yi) and var(y ). The simplest algorithm available in NONMEM, the first-order estimation method (FO), overcomes this by providing an approximate solution through a first-order Taylor series expansion with respect to the random variables r i,Kiq, and Sij, where it is assumed that these random effect parameters are independently multivariately normally distributed with mean zero. During an iterative process the best estimates for the fixed and random effects are estimated. The individual parameters (conditional estimates) are calculated a posteriori based on the fixed effects, the random effects, and the individual observations using the maximum a posteriori Bayesian estimation method implemented as the post hoc option in NONMEM [10]. 

If g(0), be it univariate or multivariate, is a nonlinear function then an approach repeatedly seen throughout this book will be used—the function will first be linearized using a first-order Taylor series and then the expected value and variance will be found using Eqs. (3.55) and (3.56), respectively. This is the so-called delta method. If g(0) is a univariate, nonlinear function then to a first-order Taylor series approximation about 0 would be... 

If g(0) is a nonlinear function of two or more model parameters then the multivariate delta method can be used. For a function of two variables a first-order Taylor series approximation around 0 and 0j can be written as... 

For multivariate functions the Taylor series approximation to a function evaluated at (xq, yo) is given by... 

PLS and other projection methods have a theoretical foundation based on perturbation theory of a multivariable system, This derivation shows that projection models can approximate any data table as long as there is a certain degree of similarity between the objects (observations, matrix rows) and the greater the similarity between the objects and the greater the number of model components, the better the approximation, This is very similar to the derivation of polynomials as Taylor series that can approximate any continuous function in a limited interval. 

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