Fitting to data

It would seem better to transform chemisorption isotherms into corresponding site energy distributions in the manner reviewed in Section XVII-14 than to make choices of analytical convenience regarding the f(Q) function. The second procedure tends to give equations whose fit to data is empirical and deductions from which can be spurious. 

Curve fitting to data is most successhil when the form of the equation used is based on a known theoretical relationship between the variables associated with the data points, eg, use of the Clausius-Clapeyron equation for vapor pressure. In the absence of known theoretical relationships, polynomials are one of the most usehil forms to describe a curve. Polynomials are easy to evaluate the coefficients are linear and the degree, ie, the highest power appearing in the equation, is a convenient measure of smoothness. Lower orders yield smoother fits. 

Equation for is dimensional. Fit to data for effective area quite good for distillation. Good for absorption at low values of Nr.ai Red) correlation is too high at higher values of (Nca,L X A/fle.c)-... 

Table 6.2 summarizes the low pressure intercept of observed shock-velocity versus particle-velocity relations for a number of powder samples as a function of initial relative density. The characteristic response of an unusually low wavespeed is universally observed, and is in agreement with considerations of Herrmann s P-a model [69H02] for compression of porous solids. Fits to data of porous iron are shown in Fig. 6.4. The first order features of wave-speed are controlled by density, not material. This material-independent, density-dependent behavior is an extremely important feature of highly porous materials. 

For any distribution, the cumulative hazard function and the cumulative distribution junction are connected by a simple relationship. The probability scale for the cumulative distribution function appears on the horizontal axis at the top of hazard paper and is determined from that relationship. Thus, the line fitted to data on hazard paper... 

Figure 3. (a) Plots of rjjjj versus T for NHj decomposition on clean Rh. (b) versus T for the NO+CO reaction on clean Pt. Solid curves are fit to data using LH rate models Indicated In the text. 

It should be noted that often the model does not have to give an exact fit to data as sometimes it may be sufficient to simply have a qualitative agreement with the process. 

Best fit to data found for 90% 218Po as neutral and 10% as carrying a unit charge ... 

The solutions in Table 10.1 were fitted to data by Yokoi [3] from small fire sources with an estimation of XT, the radiation fraction, for his fuel sources. Fits by Zukoski [8], without accounting for radiation losses (Xr = 1), give values of < , 0.11 and (3 = 0.91... 

The MQH correlation for the layer temperature rise has found the empirical fit to data ... 

Fig. 1. Formation of MeS(0)Tol, showing fitted (solid line) versus experimental data during the reaction of 33.9 mM MeSTol and 83 mM tert-butyl hydroperoxide in the presence of 0.31 mM MeReO(mtp)PPh3, 1. Reactions were conducted in benzene at 298 K. The experimental progress curve was modeled by a kinetics simulation routine that gave the optimum fit to data from six such experiments.

When the function to be fitted to data does not depend linearly on the parameters, recursive methods must be used. A slightly modified version of the Newton-Raphson method (Chapter 3) will be used (Hamilton, 1964). Let jc be the vector of the n unknowns Xj and y — f(x) the m-vector of observable functions y, = /(jc). The analytical form of the functions f(x) may be the same or not. Let the vector / represent the m observations / of these functions. A vector jc is sought which minimizes the scalar c2 such that... 

If an approximate Km value for the enzyme-substrate combination of interest is known, a full-scale kinetic assay may be done immediately. However, often an approximate value is not known and it is necessary first to do a range finding or suck and see preliminary assay. For such an assay, a concentrated substrate solution is prepared and tenfold serial dilutions of the substrate are made so that a range of substrate concentrations is available within which the experimenter is confident the Km value lies. Initial velocities are determined at each substrate concentration, and data may he plotted either hyperholically (as V versus [S]) or with [S] values expressed as logio values. In the latter case, a sigmoidal curve is fitted to data with a three parameter logistic equation (O Eq. 4) ... 

Thereafter, and V ax values for substrate turnover are determined in the absence (controls) and presence of several concentrations of the inhibitor of interest. It is recommended that substrate turnover in the presence of at least four concentrations of inhibitor are examined, at concentrations between 1/3 x IC50 and 4 x IC50. Velocity data are then plotted versus substrate concentration, yielding a control plot and plots at each of the concentrations of inhibitor assessed. Hyperbolic curves are then fitted to data with the Michaelis-Menten equation, or with whichever variation of the Michaelis-Menten equation was found to describe control enzyme behavior most appropriately (see Section 4.1.4 etseq.). In this way, a pattern of changes in Km and Vmax> or both, should become apparent with changing inhibitor concentration. 

A mean square residuals is equal to 1.395. If the model contained po and five additional parameters, and if the model was fit to data from twelve experiments, what is the variance of residuals The sum of squares of residuals ... 

Other models can be fit to data from two-level factorial designs. For example, fitting the model expressed by Equation 12.8 to the data used in this section will produce the fitted model given by Equation 12.10. Some models cannot be fit to data from two-level factorial experiments for example, the model... 

Other models can be fit to data from star designs. For example, the model... 

Efficiency of full second-order polynomial models fit to data from central composite designs without replication. 

The model usually fitted to data from a full 2 factorial design is... 

In the BMD approach, a curve is fitted to discrete responses (binary, dichotomous/quantal data, i.e., yes/no) or to continuous mean effect values (a response such as weight that can assume any value in a range). The curve is usually fitted to data using the maximum likelihood approach. 

---