Theoretical equations, statistical fitting

Even with the Kelen Tudos refinement there are statistical limitations inherent in the linearization method. The independent variable in any form of the linear equation is not really independent, while the dependent variable does not have a constant variance [O Driscoll and Reilly, 1987]. The most statistically sound method of analyzing composition data is the nonlinear method, which involves plotting the instantaneous copolymer composition versus comonomer feed composition for various feeds and then determining which theoretical plot best fits the data by trial-and-error selection of r and values. The pros and cons of the two methods have been discussed in detail, along with approaches for the best choice of feed compositions to maximize the accuracy of the r and r% values [Bataille and Bourassa, 1989 Habibi et al., 2003 Hautus et al., 1984 Kelen and Tudos, 1990 Leicht and Fuhrmann, 1983 Monett et al., 2002 Tudos and Kelen, 1981]. 

Tabulated data for experimental adsorption isotherms are fitted with analytical equations for the calculation of thermodynamic properties by integration or differentiation. These thermodynaunic properties expressed as a function of temperature, pressure, and composition are input to process simulators of atdsorption columns. In addition, anaJytical equations for isotherms are useful for interpolation and cautious extrapolation. Obviously, it is desirable that the Isotherm equations agree with experiment within the estimated experimental error. The same points apply to theoretical isotherms obtained by molecular simulation, with the requirement that the analytical equations should fit the isotherms within the estimated statistical error of the molecular simulation. 

The second direction has been statistical-mechanical. This usually follows the kind of reasoning described above for virial equations of state (equation 15.34). The interactions of all particles are calculated by summing interactions of pairs, triples, quadruples, and so on. In fact, what results is another regression equation with an appropriate shape and adjustable parameters with possible physical significance. Such equations are fit to real data at the present time it is not feasible to calculate the parameters theoretically and then predict the behavior of a specific gas (except for highly idealized systems). With the advent of very fast computers it has been possible... 

It should be readily apparent that experimental conditions and ecological factors can influence the empirical constants a and b. The above exponents for the heart weight (Mh), however, do not differ significantly from the theoretical exponent of 1.0 (M °). The deviations arise from statistical fits of regressions to experimental data. It is also readily apparent that if a variable scales as M °, then the allometric equation can be made invariant by taking a ratio with M in denominator, that is, normalizing with body mass. 

The conduct of proton inventory consists of determining the Idnetic parameters of interest in a number of isotopic water mixtures of deuterium atom fraction n, so that the data set comprises values of (n). The data are then fit, by an appropriate statistical procedure, to a corresponding theoretical equation, and contributing effects are calculated from this. 

Thus far the discussion in this chapter has concentrated on the statistical treatment of random errors in calibration data and calibration equations, with only passing mention of the implications of the practicalities involved in the acquisition of these data. For example, no mention has been made of what kind of calibration data are (or can be) acquired in common analytical practice, under which circumstances one approach is used rather than another and (importantly) what are the theoretical equations to which the experimental calibration data should be fitted by least-squares regression for different circumstances. Moreover, it is important to address the question of analytical accuracy to complement the discussion of precision that we have been mainly concerned with thus far the meanings of accuracy and precision in the present context are discussed in Section 8.1. The present section represents an attempt to express in algebra the calibration functions that apply in different circumstances, while exposing the potential sources of systematic uncertainty in each case. 

We are fortunate in this example to have represented our data by a form of equation which happened to match exactly that obtained from the theoretical mechanism. Often a number of different equation types will fit a set of experimental data equally well, especially for somewhat scattered data. Hence, to avoid rejecting the correct mechanism, it is advisable to test the fit of the various theoretically derived equations to the raw data using statistical criteria whenever possible, rather than just matching equation forms. 

There are many commercial programs available for fitting data to theoretical curves. They are extremely powerful but very dangerous. Used properly, they fit data directly to nonlinear curves without the need for transformation of data to linear equations that can distort the statistics, and they allow data to be fitted to complex equations that could not be solved by hand. But a computer just gives the best fit to the particular equation, y = f(x), that you choose, and your data might not follow that equation. 

NMR spectroscopy allows testing whether in a particular polymerization the propagation follows the Bernoulli, Markov, or enantiomorphic statistical form. Attempts are usually made to fit data for dyads, triads, tetrads, and higher sequence fractions to the equation for the different models. Spectral intensities can be associated with theoretical expression involving reaction probability parameters. Theoretical intensities are compared with the observed ones. This is optimized to obtain the best-fit values of reaction probability parameters and fully characterize the structure of the macromolecule. The fitting of data can be carried out with the aid of computers. 

The Langmuir isotherm equation is the first theoretically developed adsorption isotherm. Many of the equations proposed later and which fit the experimental results over a wide range are either based on this equation, or these equations have been developed using the Langmuir concept. Thus, the Langmuir equation still retains an important position in physisorption as well as chemisorption theories. The equation has also been derived using thermodynamic and statistical approaches but we shall discuss the commonly used kinetic approach for its derivation. 

The coefficients, were treated as adjustable parameters to give the best fit to the experimental data. Other equations were evaluated and a number were found that predicted tf with reasonable accuracy. Jones et al. also used the regression method to determine values for the terms C, a, and b in Eq. (11) good agreement was found between the regression values and those obtained graphically. The statistical approach is purely empirical, with no theoretical basis. However, it does provide useful predictive equations. 

In fact, there is no theoretical justification for any of these alternative equations. The quadratic has a particular failing in that the statistical scatter of points can lead to an FWHM curve that curls upwards rather than downwards. This clearly flies in the face of physical reality. It would be useful if, whatever other FWHM fit options were... 

The theories developed for calculating the oscillatory force are based on modeling by means of the integral equations of statistical mechanics [449-453] or numerical simulations [454-457]. As a rule, these approaches are related to complicated theoretical expressions or numerical procedures, in contrast with the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, one of its main advantages being its simplicity [36], To overcome this difficulty, some relatively simple semiempirical expressions have been proposed [458,459] on the basis of fits of theoretical results for hard-sphere fluids. 

Equation 4.249 was found to fit well experimental data for the disjoining pressure of liquid films stabilized by adsorbed protein molecules bovine serum albumin (BSA) [629]. In that case, E was identified with the surface density of the loose secondary protein adsorption layer, while X turned out to be about the size of the BSA molecule. A more detailed statistical approach to the theoretical modeling of protrusion force was proposed [630]. 

From a statistical viewpoint it is difficult to state a definite answer because several problems accumulate here. First, there is not an exact solution to the comparison of two population means, for which we have to estimate simultaneously (from the limited experimental data available) their average values and their associated variances, as is the case in laboratories. Second, the equations stated above were deduced for normal distributions but the slopes derived from a least-squares fit follow approximately a Student s distribution. We must bear in mind that although the theoretical slope and intercept of the population follow a normal distribution their estimators do not because the latter (along with the variance of the regression itself) must be estimated from (usually) a very reduced number of data and the number of degrees of freedom - dof- must be taken into account. In statistical terms an intermediate pivot statistic must be introduced to obtain an approximate Student s distribution. ... 

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