Spline function problems

Use of Cubic Spline Functions in Solving Calibration Problems... 

R = 0 leads back to the problem of interpolation by spline functions. It should be noted at this point that the condition stated by eq. (4) is not sufficient for the construction of calibration curves and additional considerations have to take effect. A reformulation of the problem stated in Equations (2) and (3) gives us with 8y. = 1 for all i calibration points another look at the problem that clarifies the role of the integral in Equation (2) as balanced against a value of R. Find S (x) to... 

WEGSCHEIDER Cubic Spline Functions and Calibration Problems 173... 

The problem of Example 4.1.3 is revisited here. We determine the smoothing spline function and its derivatives assuming identical standard errors d = 0.25 in the measured pH. 

A typical curve-fitting problem is illustrated in Figure 8. A spline function S(x) is required which interpolates a set of observed values (represented by f, f etc.) in an interval a x b. If we choose a set of knots kj, k, ..., in the interval, then Curry and Schoenberg (12) show that S(xf has a unique representation in a s x b of the form... 

Figure 8. Example of a typical curvefitting problem in one variable illustrating the use of spline functions ((9) knots (- -) observed data)...

Numerically, the LSA approach may be implemented by calcnlations that involve estimation by (general) linear regression, e.g., nse of cnbic spline functions." The intrinsic problems of nonlinear estimation common in more structured methods can thereby be avoided or significantly reduced. 

Problems, however, arise if the intervals between the knots are not narrow enough and the spline begins to oscillate (cf. Figure 3.13). Also, in comparison to polynomial filters, many more coefficients are to be estimated and stored, since in each interval, different coefficients apply. An additional disadvantage is valid for smoothing splines, where the parameter estimates are not expectation-true. The statistical properties of spline functions are, therefore, more difficult to describe than in the case of linear regression (cf. Section 6.1). ... 

Figure 6 An illustration of the problems that can arise when parts of parameter space are nonphysical. Assume that one is interested in sampling the potential for energies less than the maximum of the spline fit function. Classical trajectories coming from the right cannot surmount the barrier and will behave almost tihe same on the spline fit function as on the true potential. However, the GA can sample anywhere and has the possibility of accessing the nonphysical region to the left of the maximum in the spline function. One solution is to add a penalty function whose value is zero except for regions of small distance. The sum of die spline fit function and the penalty function will be at least as large as the true potential in this classically forbidden. region and will thereby push solutions to larger distances.

Generally, these expressions require numerical integrations. In the case of spherical symmetries, other approaches can be used such as hypergeometric [84] and spline functions [86] that reduce the problem to analytical functions. However, this might increase the number of fit parameters so extra care must be taken to ensure that the density profile is physically meaningful. 

Errors in advection may completely overshadow diffusion. The amplification of random errors with each succeeding step causes numerical instability (or distortion). Higher-order differencing techniques are used to avoid this instability, but they may result in sharp gradients, which may cause negative concentrations to appear in the computations. Many of the numerical instability (distortion) problems can be overcome with a second-moment scheme (9) which advects the moments of the distributions instead of the pollutants alone. Six numerical techniques were investigated (10), including the second-moment scheme three were found that limited numerical distortion the second-moment, the cubic spline, and the chapeau function. 

A common problem for both methods lies in the use of potentials that do not possess the correct net attractiveness. This can have the consequence that continuum feamres appear shifted in energy. In particular, there is evidence that the LB94 exchange-correlation potential currently used for the B-spline calculations, although possessing the correct asymptotic behavior for ion plus electron, is too attractive, and near threshold features can then disappear below the ionization threshold. An empirical correction can be made, offsetting the energy scale, but this can mean that dynamics within a few electronvolts of threshold get an inadequate description or are lost. There is limited scope to tune the Xa potential, principally by adjustment of the assumed a parameter, but for the B-spline method a preferable alternative for the future may well be use of the SAOP functional that also has correct asymptotic behavior, but appears to be better calibrated for such problems [79]. 

Here the B-spline Bim(zf, Xj) is the ith B-spline basis function on the extended partition Xj (which contains locations of the knots in the Zj direction), and is a coefficient. We use cubic splines and sufficient numbers of uniformly spaced knots so that the estimation problem is not affected by the partition. The estimation problem now involves determining the set of B-spline coefficients that minimizes Eq. (4.1.26), subject to the state equations [Eqs. (4.1.24 and 4.1.25)], for a suitable value of the regularization parameter. At this point, the minimization problem corresponds to a nonlinear programming problem. 

In addition, we are interested in functions that are at least twice continuously differentiable. One can draw several such curves satisfying (4.27), and the "smoothest" of them is the one minimizing the integral (4.19). It can be shown that the solution of this constrained minimization problem is a natural cubic spline (ref. 12). We call it smoothing spline. 

The additional problem we face is determining the optimal value for p. It is important to note that the squared distance F2(p) increases with the value of p. Therefore, the algorithm can be viewed as starting with an interpolating spline obtained at p = 0, and then "streching" this function by gradually increasing the value of p until (4.27) holds. To find this particular p we solve the nonlinear equation /2... 

Spline Fitting in Two Variables. The methods described in the previous section may be extended to functions of two variables (9). The problem now is to find a surface S(x,y) which interpolates data values in a rectangular region a xsb, c ysd. By analogy with the previous section, we specify h "interior" knots in the x direction and l in the y direction. These knots, k, k2 ", kh and nl n2 ni divi< e... 

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