Polynomial fitting

Lane improved on these tables with accurate polynomial fits to numerical solutions of Eq. 11-17 [16]. Two equations result the first is applicable when rja 2... 

At present, the data base used for the fit was not specially selected to avoid homologous proteins. Thus, a further improvement can be expected from using data for one of the specially prepared lists of PDB files (cf. Hobohm et al. [9]). We also expect further improvements from replacing the polynomial fits in the potential estimation procedure by piecewise cubic fits though at the moment it is not clear how to select the number of nodes needed to get a good but not overfitting approximation to the density. Finally, we are considering... 

The larger the value of n, the more uniform is the size distribution. Other types of distribution functions can be found in Reference 1. Distribution functions based on two parameters sometimes do not accurately match the actual distributions. In these cases a high order polynomial fit, using multiple parameters, must be considered to obtain a better representation of the raw data. 

The measured relationships between piezoelectric polarization and strain for x-cut quartz and z-cut lithium niobate are found to be well fit by a quadratic relation as shown in Fig. 4.4. In both materials a significant nonlinear piezoelectric effect is indicated. The effect in lithium niobate is particularly notable because the measurements are limited to much smaller strains than those to which quartz can be subjected. The quadratic polynomial fits are used to determine the second- and third-order piezoelectric constants and are summarized in Table 4.1. Elastic constants determined in these investigations were shown in Chap. 2. 

Other strength criteria are described by Sendeckyj [2-28]. Tennyson, MacDonald, and Nanyaro addressed the next logical step in a curve-fitting procedure, namely a third-order polynomial fit to failure data [2-29], However, the added complexity of their criterion has limited its use even though they identified some loading conditions under which their criterion is necessary to properly describe the actual failure behavior. 

SEC measurements were made using a Waters Alliance 2690 separation module with a 410 differential refractometer. Typical chromatographic conditions were 30°C, a 0.5-ml/min flow rate, and a detector sensitivity at 4 with a sample injection volume of 80 fil, respectively, for a sample concentration of 0.075%. All or a combination of PEO standards at 0.05% concentration each were used to generate a linear first-order polynomial fit for each run throughout this work. Polymer Laboratories Caliber GPC/SEC software version 6.0 was used for all SEC collection, analysis, and molecular weight distribution overlays. 

The spheres represent (roughly) the 2p atomic orbitals on C and N, and half an electron resides in each sphere. The mutual potential energy of this charge distribution can be easily calculated from elementary electrostatics. For small distances, a polynomial fit was used instead. 

Figure 3.3 Cumulative ( ) and instantaneous ( ) initiator efficiency (/) of AIBN as initiator in S polymerization (50% v/v toluene, 70 °C) as a function of monomer conversion (lines are a polynomial fit to the datapoints).1,32...

Fig. 11-26 Decade-averaged data of Northern hemisphere tree ring records from 1750-1979 and 7th-degree polynomial fit of the data. The vertical extension of blocks represents 95% confidence limits of the mean. The open circles give the change of —0.65% in atmospheric CO2 observed from 1956 to 1978 by Keeling et al. (1979). (Adapted from Peng et al, 1983.)...

A sum-over-states expression for the coefficient A for the expansion of the diagonal components faaaa was derived by Bishop and De Kee [20] and calculations were reported for the atoms H and He. However, the usual approach to calculate dispersion coefficients for many-electron systems by means of ab initio response methods is still to extract these coefficients from a polynomial fit to pointwise calculated frequency-dependent hyperpolarizabiiities. Despite the inefficiency and the numerical difficulties of such an approach [16,21], no ab initio implementation has yet been reported for analytic dispersion coefficients for frequency-dependent second hyperpolarizabiiities which is applicable to many-electron systems. 

The needed thermochemistry for many thousands of molecules is available from standard sources such as the JANAF tables. " Polynomial fits of this data in the form required by our kinetics software are also available. However, experimental thermochemical data is often lacking for many of the intermediate species that should be included in a detailed kinetics mechanism. Standard methods have been developed for estimating these properties, discussed in detail by Benson. ... 

The Dunham polynomial fit expansion of the theoretical curves involves polynomials of 9th and 10th degree leading to the data in columns A and B respectively. 

A seventh degree polynomial fit to the theoretical curves has been used (columns A and B). In column A use is made of the variational MR SD-Cl energies while in column B estimated full-CI energies 111] are utilised... 

FIGURE 35.3 Free-energy functions for reactant (AE) and product Ag (AE) of an electron transfer reaction as calculated using umbrella sampling within a simple dipolar diatomic solvent. AG° is the reaction free energy. Solid lines are polynomial fittings to the simulated points. Dashed lines are parabolic extrapolations from the minimum of the curves. (From King and Warshel, 1990, with permission from the American Institute of Physics.)... 

It is not possible to extract in a direct way theoretical information on the model. It is to be considered as an empirical model builder such as a spline or polynomial fitting procedure. 

However, an important question that needs to be answered is "what constitutes a satisfactory polynomial fit " An answer can come from the following simple reasoning. The purpose of the polynomial fit is to smooth the data, namely, to remove only the measurement error (noise) from the data. If the mathematical (ODE) model under consideration is indeed the true model (or simply an adequate one) then the calculated values of the output vector based on the ODE model should correspond to the error-free measurements. Obviously, these model-calculated values should ideally be the same as the smoothed data assuming that the correct amount of data-filtering has taken place. 

The above integrals can be calculated since we have measurements of all the state variables as a function of time. In particular, to obtain a good estimate of these integrals it is strongly recommended to smooth the data first using a polynomial fit of a suitable order. Therefore, the integrals P r(tj) should be calculated as... 

Based on the above expressions, it is obvious that by this approach one needs to estimate numerically the time derivatives of Xv, S and P as a function of time. This can be accomplished by following the same steps as described in Section 7.1.1. Namely, we must first smooth the data by a polynomial fitting and then estimate the derivatives. However, cell culture data are often very noisy and hence, the numerical estimation of time derivatives may be subject to large estimation errors. 

Similarly we can estimate the specific secretion rate. It is obvious from the previous analysis that an accurate estimation of the average specific rates can only be done if the integral Jxvdt is estimated accurately. If measurements of biomass or cell concentrations have been taken very frequently, simple use of the trapezoid rule for the computation of Jxvdt may suffice. If however the measurements are very noisy or they have been infrequently collected, the data must be first smoothed through polynomial fitting and then the integrals can be obtained analytically using the fitted polynomial. 

This assumption can be relaxed when the experimental error in the independent variable is much smaller compared to the error present in the measurements of the dependent variable, In our case the assumption of simple linear least squares implies that Xvdt is known precisely. Although we do know that there are errors in the measurement of Xv, the polynomial fitting and the subsequent integration provides a certain amount of data filtering which could allows us to assume that experimental error in Jxvdt is negligible compared to that present in S(t,) or P(t,). 

Fig. 4.2.4 Velocity profile for a 0.6% aqueous carboxymethylcellulose solution obtained by MRI. The data points are the MRI data. The line connecting these is the result of an eighth-order even polynomial fit.

To make a third-order polynomial fit of y = y(x), all we need is to enter... 

Figure 8-7. Coarse-graining of CCI4. (a) Black COM-COM RDF from EFP-MD, (b) red RDF from CG-MD, (c) orange circles the effective COM pair-force, (d) green polynomial fit of the force matching data, (e) blue the effective COM pair potential...

Figure 3b. A plot of the Mean Cnrvatnre data versns the 40% Compression Deflection valnes for the urethane foams made with constant tin catalyst concentration calculation by RS/1 for a polynomial fit.

The figure shows a polynomial fit to a dataset calculated according to a standard least squares algorithm (solid line) this is compared with a series of attempts to find a fit to the same data using a genetic algorithm. 

Convert those formulas into a set of coefficients that can be used to multiply the data spectrum by, to produce the value of the derivative according the specified polynomial fit, at the point of the center of the set of data. As we shall see, however, their paper ignores some key points. 

However, this collection of convolution coefficients appears nowhere in the S-G tables. The nine-point S-G second derivative with a Quadratic or Cubic polynomial fit has the coefficients ... 

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