Estimation Method

Classes of Estimation Methods Table 1.1.1 summarizes the property estimation methods considered in this book. Quantitative property-property relationships (QPPRs) are defined as mathematical relationships that relate the query property to one or several properties. QPPRs are derived theoretically using physicochemical principles or empirically using experimental data and statistical techniques. By contrast, quantitative structure-property relationships (QSPRs) relate the molecular structure to numerical values indicating physicochemical properties. Since the molecular structure is an inherently qualitative attribute, structural information has first to be expressed as a numerical values, termed molecular descriptors or indicators before correlations can be evaluated. Molecular descriptors are derived from the compound structure (i.e., the molecular graph), using structural information, fundamental or empirical physicochemical constants and relationships, and stereochemcial principles. The molecular mass is an example of a molecular descriptor. It is derived from the molecular structure and the atomic masses of the atoms contained in the molecule. An important chemical principle involved in property estimation is structural similarity. The fundamental notion is that the property of a compound depends on its structure and that similar chemical stuctures (similarity appropriately defined) behave similarly in similar environments. 

Quantitative property-property relationships (QPPRs) Property 

Quantitative structure-property relationships (QSPRs) Molecular descriptor 

Group contribution models (GCMs) Fragment constants 

Similarity-based models Between isomeric compounds Molecular descriptor 

The advantage of the maximum-likelihood approach is that it will make full use of the collected infonnalion this property is known as efficiency. Typically in functional genomic experiments, the number of independent samples collected by investigators is small relative to the dimensionality of the data being examined. Thus, it behooves investigators to have available analytical methods that are efficient. 

The other dominant approach for estimation is known as Bayesian estimation. We do not attempt to give a detailed exposition of the topic here much greater detail can be found in Gehnan et al. (2004) and Carhn and Louis (2000). In this approach, the parameters are treated as random variables themselves. This is in contrast to the maximum-likehhood estimation procedure, in which the parameters are treated as unknown constants that only take one possible value. In the Bayesian framework, the goal is to make inference about the parameters. This is done in the following way. The parameters themselves have a certain probabUistic model that we call a prior distribution. We then construct a posterior probability model for the parameters by the use of Bayes s mle  

L(parameters data)g(parameters) J L(parameters data)g(parameters) 

We make some comments about the terms in Eq. (8.2). The left-hand side of (8.2) is referred to as the posterior distribution of the parameters given the observed data. The function g corresponds to the probability model we wish to specify for the parameters of the model. It is known as the prior distribution. Both the likelihood and prior distributions are needed to perform Bayesian inference. As more and more data are collected, the influence of the prior distribution relative to the likelihood function diminishes. The integral in the denominator represents a sum taken over all possible values of the parameters that can go into the function. The denominator serves as a so-called normalizing constant so that the area under the curve for the posterior distribution is equal to 1. 

As a simple example, we consider the following model. Let Xi. be independent and identically distributed observations from a normal distribution with mean m and variance s.  

In this example the off-diagonal elements of il are assumed to be 0, but they do not have to be independent and are often not. s is assumed to be normally distributed with mean 0 and variance a2. In this example, is one term, but it does not have to be. If consists of more than one term, then collectively the set of residuals is the X matrix. 

Suppose Y = f(x, 0, t ) + g(z, e) where nr] — (0, il), (0, ), x is the set of subject-specific covariates x, z, O is the variance-covariance matrix for the random effects in the model (t ), and X is the residual variance matrix. NONMEM (version 5 and higher) offers two general approaches towards parameter estimation with nonlinear mixed effects models first-order approximation (FO) and first-order conditional estimation (FOCE), with FOCE being more accurate and computationally difficult than FO. First-order (FO) approximation, which was the first algorithm derived to estimate parameters in a nonlinear mixed effects models, was originally developed by Sheiner and Beal (1980 1981 1983). FO-approximation expands the nonlinear mixed effects model as a first-order Taylor series approximation about t) = 0 and then estimates the model parameters based on the linear approximation to the nonlinear model. Consider the model 

Sheiner and Beal (1980 1981 1983) proposed taking a first-order Taylor series approximation around the set of r S evaluated at r = 0 to find the variance. Recall that Taylor series approximations, which are linear polynomials, take a function and create an approximation to the model around some neighborhood. The derivatives of Eq. (7.86) to the model are 

the first-order Taylor series approximation to Eq. (7.86) evaluated at r = 0 is 

Notice that the approximation is exact, but that The reason is that the residual in Eq. (7.93) includes the truncation error for the approximation plus the residual error term, s. Equation (7.93) can be thought of as a linear model 

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The second consideration is the geometry of the molecule. The multipole estimation methods are only valid for describing interactions between distant regions of the molecule. The same is true of integral accuracy cutoffs. Because of this, it is common to find that the calculated CPU time can vary between different conformers. Linear systems can be modeled most efficiently and... 

W. J. Lyman, W. F. Reehl, D. H. Rosenblatt, Handbook of Chemical Property Estimation Methods American Chemical Society, Washington (1990). 

Due to the noncrystalline, nonequilibrium nature of polymers, a statistical mechanical description is rigorously most correct. Thus, simply hnding a minimum-energy conformation and computing properties is not generally suf-hcient. It is usually necessary to compute ensemble averages, even of molecular properties. The additional work needed on the part of both the researcher to set up the simulation and the computer to run the simulation must be considered. When possible, it is advisable to use group additivity or analytic estimation methods. 

An area that has used chemical stmctures for predictive purposes quite successfully is the estimation of thermophysical properties of compounds. There has been an extensive compilation of estimation methods (81), and prediction of physical properties has been automated using these techniques (82). More recendy, the use of group contribution techniques to design new molecules that have specified properties has been described (83). This approach to compound design is being used to develop replacement materials for chloroduorocarbons. 

Factor Methods. A more detailed product cost estimation method is to relate manufacturing cost items to a few calculated items, such as raw materials, labor, and utihties by means of simple factors (1,2). Internal accounting groups often develop factors to use with this method. This factor method is very popular. 

Data compilations, the first recourse for an engineering calculation requiring physical property or parameter data, are often incomplete or do not contain data within the appropriate range of temperature or pressure (6—9). For this reason, correlation and estimation methods play an important role in apphed thermodynamics. 

A.ssessmentofUNIFy C. UNIFAC is a method to predict the activity of binary Hquid solutions in the absence of all data except stmctural information. Because state-of-the-art real fluid estimation methods are empirical or semi-empirical, the use of more data results in improved activity estimation. 

Heat Capacity. The multiple property estimation methods for constant pressure ideal-gas heat capacities cover a broad range of organic compounds (188,216,217). Joback s method (188) is the easiest to use however, usage of all these methods has been recommended only over the range 280—1100 K (7). An accurate method for ideal-gas heat capacities (constant pressure), limited to hydrocarbons, has been presented (218) that involves a fit of seven variables, and includes steric, ring, branching, alkene, and even allene corrections. 

Hctivity Coefficients. Most activity coefficient property estimation methods are generally appHcable only to pure substances. Methods for properties of multicomponent systems are more complex and parameter fits usually rely on less experimental data. The primary group contribution methods of activity coefficient estimation are ASOG and UNIEAC. Of the two, UNIEAC has been fit to more combinations of groups and therefore can be appHed to a wider variety of compounds. Both methods are restricted to organic compounds and water. 

Enthalpy of Formation The ideal gas standard enthalpy (heat) of formation (AHJoqs) of chemical compound is the increment of enthalpy associated with the reaction of forming that compound in the ideal gas state from the constituent elements in their standard states, defined as the existing phase at a temperature of 298.15 K and one atmosphere (101.3 kPa). Sources for data are Refs. 15, 23, 24, 104, 115, and 116. The most accurate, but again complicated, estimation method is that of Benson et al. " A compromise between complexity and accuracy is based on the additive atomic group-contribution scheme of Joback his original units of kcal/mol have been converted to kj/mol by the conversion 1 kcal/mol = 4.1868 kJ/moL... 

Ideal gas absolute entropies of many compounds may be found in Daubert et al.,"" Daubert and Danner," JANAF Thermochemical Tables,TRC Thermodynamic Tables,and Stull et al. ° Otherwise, the estimation method of Benson et al. " is reasonably accurate, with average errors of 1-2 J/mol K. Elemental standard-state absolute entropies may be found in Cox et al." Values from this source for some common elements are listed in Table 2-389. ASjoqs may also be calculated from Eq. (2-52) if values for AHjoqs and AGJoqs are known. 

Multicomponent Mixtures No simple, practical estimation methods have been developed for predicting multicomponent hquid-diffusion coefficients. Several theories have been developed, but the necessity for extensive activity data, pure component and mixture volumes, mixture viscosity data, and tracer and binaiy diffusion coefficients have significantly limited the utihty of the theories (see Reid et al.). 

Accuracy of Pyrometers Most of the temperature estimation methods for pyrometers assume that the objec t is either a grey body or has known emissivity values. The emissivity of the nonblack body depends on the internal state or the surface geometry of the objects. Also, the medium through which the therm radiation passes is not always transparent. These inherent uncertainties of the emissivity values make the accurate estimation of the temperature of the target objects difficult. Proper selection of the pyrometer and accurate emissivity values can provide a high level of accuracy. 

Published data and shortcut estimating methods can be used to calculate the approximate manufacturing cost of a new product. However, most companies have extensive data on various items of cost such as overheads, property taxes, etc. These data should be used whenever possible to give the estimate that is most vahd for a particular company. 

Piping Estimation The cost of fabrication and installation of process-plant piping appears to range from 18 to 61 percent of the FOB equipment cost as indicated in Table 9-56. This would normally represent about 7 to 15 percent of the installed plant cost and is obviously a significant item. The various available piping-estimation methods are as follows ... 

The performances and estimating methods of welded PHEs match those of gasketed PHEs in most cases, but normally the Compabloc, with larger depth of corrugations, can be lower in overall coefficient. Some extensions of the design operating conditions are possible with welded PHEs, most notably is that ciyogenic applications are possible. Pressure vessel code acceptance is available on most units. 

For ordered, or structured, packings, pressure-drop estimation methods have been reviewed by Fair and Bravo [Chem. Eng. Progr, 86(1), 19 (1990)]. It is not common practice to use the packing factor approach for predicling pressure drop or flooding. For operation below the loading point, the model of Bravo et [Hydrocarbon... 

Isermann R., Process Fault Detection Based on Modeling and Estimation Methods—A Survey, Automatica, 20(4), 1984, 387 04 (Fault detection survey article)... 

Two standard estimation methods for heat of reaction and CART are Chetah 7.2 and NASA CET 89. Chetah Version 7.2 is a computer program capable of predicting both thermochemical properties and certain reactive chemical hazards of pure chemicals, mixtures or reactions. Available from ASTM, Chetah 7.2 uses Benson s method of group additivity to estimate ideal gas heat of formation and heat of decomposition. NASA CET 89 is a computer program that calculates the adiabatic decomposition temperature (maximum attainable temperature in a chemical system) and the equilibrium decomposition products formed at that temperature. It is capable of calculating CART values for any combination of materials, including reactants, products, solvents, etc. Melhem and Shanley (1997) describe the use of CART values in thermal hazard analysis. 

The frequency analysis step involves estimating the likelihood of occurrence of each of the undesired situations defined in the hazard identification step. Sometimes you can do this through direct comparison with experience or extrapolation from historical accident data. While this method may be of great assistance in determining accident frequencies, most accidents analyzed by QRA are so rare that the frequencies must be synthesized using frequency estimation methods and models. 

Regardless of the estimating method, a process contingency should be added to the total plant cost for feasibility studies. As discussed earlier, this contingency depends on the status of the project. For most factored estimates, on a first of a kind process that is fairly well defined, the process contingency should be 30%. 

Program created for DOT, EPA, and FEMA to aid emergency preparedness personnel in assessing the sequence and nature of events that may follow an accident. ARCHIE incorporates several estimation methods that may be used to assess the vapor discharge, fire, and explosion impacts associated with episodic discharges of hazardous materials. 

Cran, G. W. Graphical Estimation Methods for Weibull Distribution. Microelectronics and Reliability, Vol. 15, 1976, p. 47. 

Wagle [92] presents an estimate method for the average relative volatility of two components, related to the normal boiling points and the latent heats of vaporization of the two components, in the temperature range of their boiling points ... 

Definitive effort-hour estimating data is used to develop specific detail estimates. Composite effort-hour data is presented as a guide for quick estimating methods. All material quantities have been consolidated into an average for a given amount of material and do not include any specialty items. All effort-hour units and material quantities are based on a typical installation, and each material type should be reviewed carefully for various differences. 

It should be emphasized that the estimation methods presented previously apply to any hazard paper and, in addition, to a nonparametric fit to the data obtained by drawing a smooth curve through data on any hazard paper. 

LymanWJ. 1990. Adsorption coefficient for soils and sediment. In Handbook of chemical property estimation methods. Environmental behavior of organic compounds. Lyman WJ, Reehl WE, Rosenblatt DH, eds. Washington, DC American Chemical Society. ... 

The potentiometric titration was carried out in order to determine the functional groups present in the biomass surface. During the titration experiments, the C02-free condition was always maintained to avoid the influence of inorganic carbon on the solution pH. Detailed potentiometric titration procedure and estimation method of functional groups are available in the previous reports [4,6]. 

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