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** Best estimates of parameter values in nonlinear **

Each average value of velocity can be used to best describe some particular property of the ensemble of molecular velocities. For example, in a gas all molecules have the same average kinetic energy. Hence, the root-mean-square velocity is the best estimate of velocity to use for computing parameters that are a function of kinetic energy [Pg.32]

Statistics can effectively be used to provide a best estimate of the value of a repeatedly measured variable, establish the reliability of such an estimate (confidence interval), estimate parameter values of a model from experimental data, help to discriminate between rival models on the basis of goodness of fit, and guard against acceptance of a model whose superior fit may well be due to chance. It can also help to design experimental data gathering to be most efficient [48], On the other hand, statistics alone cannot be relied upon to identify or verify reaction pathways or mechanisms. [Pg.65]

Sensitivity Analysis When solving differential equations, it is frequently necessary to know the solution as well as the sensitivity of the solution to the value of a parameter. Such information is useful when doing parameter estimation (to find the best set of parameters for a model) and for deciding if a parameter needs to be measured accurately. See Ref. 105. [Pg.475]

A line can be defined uniquely, by stating, in addition to fl, any one point on the line, conventionally on the line where X = 0. The value of Y in the population at this point is the parameter a, which is called the Y intercept. The best estimate of a is [Pg.17]

Maximum likelihood estimation Criterion under which the best estimate is the one which maximizes the likelihood of the observed event. Maximum likelihood estimation is the classical statistical criterion for estimating unknown parameter values from observed data (Sielken, Ch. 8). [Pg.398]

Hugoniot data have been fitted by the equation = Cq + su + qu, where Uj is the shock velocity and the associated particle velocity. Griineisen parameters have been obtained from best estimates of zero pressure thermodynamic parameters, which are sometimes of dubious value. The pressures and velocities describing the valid range of the fits do not necessarily indicate the onset or completion of a transition. [Pg.382]

The models in chemical kinetics usually contain a number of unknown parameters, whose values should be determined from experimental data. Regression analysis is a powerful and objective tool in the estimation of parameter values. The task in regression analysis can be stated as follows the value of the dependent variable (y) is predicted by the model a function (/), contains independent variables (x) and parameters (/ ). The independent variable is measured experimentally, at different conditions, i.e. at different values of the independent variables (x). The goal is to find such numerical values of the parameters (/ ) that the model gives the best possible agreement with the experimental data. Typical independent variables are reaction times, concentrations, pressures and temperatures, while molar amounts, concentrations, molar flows [Pg.431]

Eq. (3) is a one-parameter correlation (k being the unknown parameter) that permits estimation of gas-liquid interfacial areas under high pressure provided the high pressure liquid hold-up pL is known the expression between brackets can be seen as an enhancement factor of the interfacial area due to pressure effect. Values of pUi and a0 at atmospheric pressure are also required. Using available interfacial area data of Wammes et al. [5] and the data of this work, the best estimate of K is 4.9 104. Predictions of present results and of those reported by [Pg.496]

Identification. In most cases, the mathematical models of interest in industry contain a few parameters whose values, essentially unknown a priori, must be computed on the basis of the available experimental data. In the case considered here, chemical kinetics is the main field in which this problem is of concern. Identification provides methods for obtaining the best estimates of those parameters and for choosing (i.e., identifying) the best mathematical model among different alternatives. [Pg.1]

The methane-methanol binary is another system where the EoS is also capable of matching the experimental data very well and hence, use of ML estimation to obtain the statistically best estimates of the parameters is justified. Data for this system are available from Hong et al. (1987). Using these data, the binary interaction parameters were estimated and together with their standard deviations are shown in Table 14.1. The values of the parameters not shown in the table (i.e., ka, kb, kc) are zero. [Pg.246]

If this criterion is based on the maximum-likelihood principle, it leads to those parameter values that make the experimental observations appear most likely when taken as a whole. The likelihood function is defined as the joint probability of the observed values of the variables for any set of true values of the variables, model parameters, and error variances. The best estimates of the model parameters and of the true values of the measured variables are those which maximize this likelihood function with a normal distribution assumed for the experimental errors. [Pg.98]

Ri. The value of R, will then be compared to the experimentally determined value of Rt for the 57Fe resonance. The result of this comparison taken together with other information will lead to better estimates of R, which will be called R . Finally, the values of R, (our best estimate of Rf) for a particular isotope will be used to decide if that isotope may be of interest in the studies of catalytic and chemical phenomena. It should be noted here that the analysis of the Mossbauer isotopes in terms of the ratio R, provides a simple physical feeling" for the associated nuclear parameters. The treatment in this section is based on nuclear parameters available from a variety of sources (/- 7, 30,85). However, these parameters are not available in a form readily usable for chemists. [Pg.154]

The choice of appropriate reaction conditions is crucial for optimized performance in alkylation. The most important parameters are the reaction temperature, the feed alkane/alkene ratio, the alkene space velocity, the alkene feed composition, and the reactor design. Changing these parameters will induce similar effects for any alkylation catalyst, but the sensitivity to changes varies from catalyst to catalyst. Table II is a summary of the most important parameters employed in industrial operations for different acids. The values given for zeolites represent best estimates of data available from laboratory and pilot-scale experiments. [Pg.293]

** Best estimates of parameter values in nonlinear **

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