Laplacian estimation

The FOCE method uses a first-order Taylor series expansion around the conditional estimates of the t] values. This means that for each iteration step where population estimates are obtained the respective individual parameter estimates are obtained by the FOCE estimation method. Thus, this method involves minimizations within each minimization step. The interaction option available in FOCE considers the dependency of the residual variability on the interindividual variability. The Laplacian estimation method is similar to the FOCE estimation method but uses a second-order Taylor series expansion around the conditional estimates of the 77 values. This method is especially useful when a high degree of nonlinearity occurs in the model [10]. 

The Laplacian estimation method as implemented in the NONMEM program (a program the authors use for nonlinear mixed effects modeling) is used to provide... 

Since the likelihood integral is intractable for the nonlinear case, most software packages approximate it with the assumption being that the approximation is adequate. History has shown that these approximations perform well under most circumstances in pharmacokinetics and any bias introduced by the approximation is of acceptable magnitude. One test is to fit the final model with a more accurate estimation routine, i.e., fit the final model using Laplacian estimation if the model was developed with FOCE, and compare the two model fits. If the approximation is adequate, the parameters obtained from the two models will not be substantially different. Note that it is inappropriate to compare objective function values under the two estimation models. 

Note All models with using Laplacian estimation. The symbol R denotes R-matrix singularity. —denote that the parameter was not included in the model. 

The FO method was the first algorithm available in NONMEM and has been evaluated by simulation and used for PK and PD analysis [9]. Overall, the FO method showed a good performance in sparse data situations. However, there are situations where the FO method does not yield adequate results, especially in data rich situations. For these situations improved approximation methods such as the first-order conditional estimation (FOCE) and the Laplacian method became available in NONMEM. The difference between both methods and the FO method lies in the way the linearization is done. 

The NONMEM program implements two alternative estimation methods, the first-order conditional estimation and the Laplacian methods. The first-order conditional estimation (FOCE) method uses a first-order expansion about conditional estimates (empirical Bayes estimates) of interindividual random effects, rather than about zero. In this respect, it is like the conditional first-order method of Lindstrom and Bates.f Unlike the latter, which is iterative, a single objective function is minimized, achieving a similar effect as with iteration. The Laplacian method uses second-order expansions about the conditional estimates of the random effects. ... 

Jonsson (25) showed from a simulation study that the use of the standard mixed effects modeling approach may produce biased parameter estimates when ordered categorical data with a skewed distribution are analyzed using the Laplacian method. Increasing interindividual variability and skewness in the distribution of the data increase the bias associated with the estimation of those parameters. The conse-... 

Whether one is able to fit mixture models with distinct random effects parameters for each subpopulation is dependent on the nature of the underlying mixture. Are the subpopulations close together in mean, how much data is available (per subject and total), and which type of estimation is being used (first order, hybrid, Laplacian) Now to complete the attempt at applying a two subpopulation mixture model to this data, the probability model and number of subpopulations must be communicated to NONMEM via the mix block. Within the mix abbreviated code the number of subpopulations are communicated with the variable nspop and the probabilities associated with the subpopulations with the variable p (i) (or its alias Mixp(i)), where i indexes the subpopulation. Thus, the code would be... 

One may use the cutoff function technique (see Section 4.3) and change t/r fr) near the points r = R to regularize it. The modified function belongs to the domain of definition of the Laplacian operator (an alternative method is mollification, see Sect. 2.17, 2.18 in [22]) and has approximately the same mean energy. Then simple estimates (see e.g. Section V.5.3 in [25]) and relations of Equation (2.4) demonstrate that all of the values R(r), (r) are uniformly bounded, for some constant Co and large enough R, and the following simple estimates hold for any r ... 

The NLME function in S-Plus offers three different estimation algorithms a FOCE algorithm similar to NONMEM, adaptive Gaussian quadrature, and Laplacian approximation. The FOCE algorithm in S-Plus, similar to the one in NONMEM, was developed by Lindstrom and Bates (1990). The algorithm is predicated on normally distributed random effects and normally distributed random errors and makes a first-order Taylor series approximation of the nonlinear mixed effects model around both the current parameter estimates 0 and the random effects t). The adaptive Gaussian quadrature and Laplacian options are similar to the options offered by SAS. 

The model estimates were obtained using FOCE-I. Of interest was how these values would compare using a different estimation algorithm. Hence, the model was estimated using FO-approximation, FOCE, Laplacian, and generalized least-squares (GLS). The results are shown in Table 9.18. FOCE and Laplacian produced essentially the same results. Similarly, FOCE-I and GLS produced essentially the same results as well. Conditional methods tend to produce different results from... 

A typical VMC computation to estimate the energy or other expectation values for a given 4/x(R) might involve the calculation of the wavefunction value, gradient, and Laplacian at several millions points distributed in configuration space. Computationally this is the most expensive part. So a desirable feature of TVR), from the point of view of Monte Carlo, is its compactness. It would be highly impractical to use a trial wavefunction represented, for example, as a Cl expansion of thousands (or more) of Slater determinants. 

FORTRAN source code in which the maximum likelihood is evaluated with one of two different first-order expansions (FO or FOCE) and a second-order expansion about the conditional estimates of the random effects (Laplacian) S-PLUS algorithm utilizing a generalized least-squares (GLS) procedure and Taylor series expansion about the conditional estimates of the interindividual random effects... 

Vol. 1862 B. Helffer, F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians (2005)... 

Consider a wave with wavelength A, frequency ct>, and a wave amplitude of (. Also, let us consider a characteristic velocity of U for the liquid. The characteristic fluid velocity due to the motion of the interface can be estimated based on the amplitude of the disturbance and its characteristic time or u Cm. Therefore, the second derivative of velocity with respect to space, or the Laplacian of the velocity is estimated according to V u f//A, and, the time derivative of the velocity according to du/dt U(b. The order of the magnitude of the convective term can... 

Estimate the Laplacian of the gray level at each pixel P. Since the Laplacian is a second-difference operator, it is positive on one side of an edge and negative on the other side thus its zero-crossings define the locations of edges. (First differences should also be computed to estimate the steepnesses of these edges.)... 

To estimate (ys), one may note that the velocity profile in the liquid is influenced by the soUd zones only in a region of their size, a, in all directions (see Fig. 1) this behavior actually reflects the Laplacian character of the Stokes equation obeyed by the fluid velocity. One, therefore, expects (ys) U/a, where U is the slip velocity of the fluid on the shear-free zones, so that we eventually obtain Ff = A(f>sr ilJ/a. Now if one recalls the definition of the effective slip length, as given by the Navier BC, Ff also reads Ff = ArnV/bee, with F [/ the averaged slip velocity over the superhydrophobic surface. Combining the two independent estimates, one deduces ... 

---