First-order estimation method

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]. 

Derivative of intensity against structure parameters and thickness can be obtained using the first order perturbation method [31]. The finite difference method can also be used to evaluate the derivatives. Estimates of errors in refined parameters can also be obtained by repeating the measurement. In case of CBED, this can also be done by using different... 

After the kinetic model for the network is defined, a simulation method needs to be chosen, given the systemic phenomenon of interest. The phenomenon might be spatial. Then it has to be decided whether in addition stochasticity plays a role or not. In the former case the kinetic model should be described with a reaction-diffusion master equation [81], whereas in the latter case partial differential equations should suffice. If the phenomenon does not involve a spatial organization, the dynamics can be simulated either using ordinary differential equations [47] or master equations [82-84]. In the latter case but not in the former, stochasticity is considered of importance. A first-order estimate of the magnitude of stochastic fluctuations can be obtained using the linear noise approximation, given only the ordinary differential equation description of the kinetic model [83-85, 87]. 

Disadvantages arise mainly from the complexity of the statistical algorithms and the fact that fitting models to data is time consuming. The first-order (EO) method used in NONMEM also results in biased estimates of parameters, especially when the distribution of inter individual variability is specified incorrectly. The first-order conditional estimation (EOCE) procedure is more accurate but is even more time consuming. The objective function and adequacy of the model are based in part on the residuals, which for NONMEM are determined based on the predicted concentrations for the mean pharmacokinetic parameters rather than on the predicted concentrations for each individual. Therefore, the residuals are confounded by intraindividual, inter individual, and linearization errors. 

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. ... 

Having estimated the change in eigenvalue by the first-order perturbation method, this matrix equation can easily be inverted to solve (to first order) for the perturbation in the density. Having this better estimate of the density, one can employ the perturbation theory to second order, and so on. 

The first order reliability method (FORM) is an approximate method for assessing the reliability of a structural system. Its basic assumption is to approximate the limit state function (g(Tj,g(x)) = 0) of the structural reliability problem by means of a hyperplane which is orthogonal to the design point vector note that this approximation is constructed in the standard normal space. Thus, the failure probability can be estimated using the Euclidean norm of X, i.e. 

These values of n can be used in Eq. (16.16), in combination with the values of true and A02 estimated via the RE kinetics, to estimate a novel pp as seen in column 6 of Table 16.1. These novel values tally impressively with the values of app estimated via the first-order kinetics method and are in column 4 of the same table. 

In the first-order reliability method, the limit-state surface in the standard normal space is replaced with the tangent plane at the point with minimum distance from the origin. The first-order estimate of the probability of failure, then, is... 

ABSTRACT Runway overrun is one of the main accident types in airline operations. Nevertheless, due to the high safety levels in the aviation industry, the probability of a runway overrun is small. This motivates the use of structural reliability concepts to estimate this probability. We apply the physically-based model for the landing process of Drees and Holzapfel (2012) in combination with a probabilistic model of the input parameters. Subset simulation is used to estimate the probability of runway overrun for different runway conditions. We also carry out a sensitivity analysis to estimate the influence of each input random variable on the probability of an overrun. Importance measures and parameter sensitivities are estimated based on the samples from subset simulation and concepts of the First-Order Reliability Method (FORM). 

Probabilistic structural reliability analysis is performed to ensure that a structure is able to withstand the required design loads. A realistic description of the environmental loads and structural response is a crucial prerequisite for structural reliability analysis of structures exposed to environmental forces. The concept of environmental contours is an efficient method of estimating extreme conditions as basis for design. See (Winterstein et d., 1993) and (Haver Winterstein 2009). It is widely used in marine structural design. See e.g., (Baarholm et al., 2010), (Fontaine et al., 2013), (Jonathan et al., 2011), (Moan 2009) and (Ditlevsen 2002). The traditional approach is to use the well-known Rosenblatt transformation introduced in (Rosenblatt 1952) to transform the environmental variables into independent standard normal variables and identify a sphere with desired radius in the transformed space. Environmental contours are then found by re-transforming the sphere back to the original space. This approach is closely related to the FORM-approximation (First Order Reliability Method), where the failure boundary in the transformed space is approximated by a hyperplane at the design point. 

The section discussed the use of first-order reliability methods in order to estimate the reliability of structures. First, the first-order second-moment (FOSM) method was presented and then extended to the advanced FOSM method. The concept of most probable point (MPP) was introduced. It was derived that the distance from the origin to the MPP, in standard normal space, is equal to the safety index or reliability index, denoted by ft. Information regarding the gradient at the MPP can be used to identify the sources of uncertainty that are significant contributors to the failure of the structure. 

Each of these methods requires two evaluations of the derivative term. The first method ean be viewed as a poor man s trapezoidal method since the two derivatives are evaluated at flie t. and t- time points. Instead of using the unknown value of the solution variable at die second time point, a first order estimation of the solution value is used based upon the pervious time point and the slope at the previous time point. In this manner an estimate of the trapezoidal rule derivatives... 

On the basis of data obtained the possibility of substrates distribution and their D-values prediction using the regressions which consider the hydrophobicity and stmcture of amines was investigated. The hydrophobicity of amines was estimated by the distribution coefficient value in the water-octanole system (Ig P). The molecular structure of aromatic amines was characterized by the first-order molecular connectivity indexes ( x)- H was shown the independent and cooperative influence of the Ig P and parameters of amines on their distribution. Evidently, this fact demonstrates the host-guest phenomenon which is inherent to the organized media. The obtained in the research data were used for optimization of the conditions of micellar-extraction preconcentrating of metal ions with amines into the NS-rich phase with the following determination by atomic-absorption method. 

Clearly the accurate measurement of the final (infinity time) instrument reading is necessary for the application of the preceding methods, as exemplified by Eq. (2-52) for the spectrophotometric determination of a first-order rate constant. It sometimes happens, however, that this final value cannot be accurately measured. Among the reasons for this inability to determine are the occurrence of a slow secondary reaction, the precipitation of a product, an unsteady instrumental baseline, or simply a reaction so slow that it is inconvenient to wait for its completion. Methods have been devised to allow the rate constant to be evaluated without a known value of in the process, of course, an estimate of A is also obtainable. 

A parameter such as a rate constant is usually obtained as a consequence of various arithmetic manipulations, and in order to estimate the uncertainly (error) in the parameter we must know how this error is related to the uncertainties in the quantities that contribute to the parameter. For example, Eq. (2-33) for a pseudo-first-order reaction defines k, which can be determined by a semilogarithmic plot according to Eq. (2-6). By a method to be described later in this section the uncertainty in itobs (expressed as its variance associated with cb. Thus, we need to know how the errors in fcobs and cb are propagated into the rate constant k. 

Although a closed-form solution can thus be obtained by this method for any system of first-order equations, the result is often too cumbersome to lead to estimates of the rate constants from concentration-time data. However, the reverse calculation is always possible that is, with numerical values of the rate constants, the concentration—time curve can be calculated. This provides the basis for a curve-... 

Wilkinson s method for the estimation of the reaction order is illustrated for first-order (left) and second-order (right) kinetic data. The first-order reaction is the decomposition of diacetone alcohol (Table 2-3 and Fig. 2-4) data to about 50 percent reaction are displayed. The slope gives an approximate order of 1.2. The second-order data (Fig. 2-2) give a precise fit to Eq. (2-59) and an order of two exactly. 

The experimental results imply that the main reaction (eq. 1) is an equilibrium reaction and first order in nitrogen monoxide and iron chelate. The equilibrium constants at various temperatures were determined by modeling the experimental NO absorption profile using the penetration theory for mass transfer. Parameter estimation using well established numerical methods (Newton-Raphson) allowed detrxmination of the equilibrium constant (Fig. 1) as well as the ratio of the diffusion coefficients of Fe"(EDTA) andNO[3]. 

Thermal methods in kinetic modelling. Methods for the estimation of thermokinetic parameters based on experiments in a reaction calorimeter will be discussed below. As mentioned in section 5.4.4.3, instantaneous heat evolved due to a single reaction is directly proportional to the reaction rate. Assume that the reaction is of first order. Then for isothermal operation ... 

Early studies of ET dynamics at externally biased interfaces were based on conventional cyclic voltammetry employing four-electrode potentiostats [62,67 70,79]. The formal pseudo-first-order electron-transfer rate constants [ket(cms )] were measured on the basis of the Nicholson method [99] and convolution potential sweep voltammetry [79,100] in the presence of an excess of one of the reactant species. The constant composition approximation allows expression of the ET rate constant with the same units as in heterogeneous reaction on solid electrodes. However, any comparison with the expression described in Section II.B requires the transformation to bimolecular units, i.e., M cms . Values of of the order of 1-2 x lO cms (0.05 to O.IM cms ) were reported for Fe(CN)g in the aqueous phase and the redox species Lu(PC)2, Sn(PC)2, TCNQ, and RuTPP(Py)2 in DCE [62,70]. Despite the fact that large potential perturbations across the interface introduce interferences in kinetic analysis [101], these early estimations allowed some preliminary comparisons to established ET models in heterogeneous media. 

---