Smoothing splines

R. L. Eubank 1999, Nonparametric Regression and Spline Smoothing, Marcel Dekker, New York. 

Example 4.2.2 Detection of end points in potentiometric titration by spline smoothing... 

The kinetic spectra were calculated using a cubic spline smoothing calculation (12). Because the domain of the kinetic spectrum is the natural logarithm of time, small errors in peak position correspond to large errors in rate constants. The kinetic spectrum analysis was used only to obtain preliminary estimates of the rate constants and the percent dissociation by each path. These were then used as the initial values in a simplex non-linear regression (12) fit of the original data. The results of this latter treatment are the reported values. 

Fig. 2.6 Changes in Pb concentrations in ice deposited at Summit, central Greenland, from 1773 to 1992 compiled from Candelone et al. (1995) (solid circles) and Boutron et al. (1991) (solid triangles). The general time trend is shown with a spline-smoothed curve.

Woltring, H.J. (1986). A FORTRAN package for generalized, cross-validatoiy spline smoothing and differentiation, Atfvancer in Engineering Software, 8(2) 104-113. 

The best and easiest way to smooth the data and avoid misuse of the polynomial curve fitting is by employing smooth cubic splines. IMSL provides two routines for this purpose CSSCV and CSSMH. The latter is more versatile as it gives the option to the user to apply different levels of smoothing by controlling a single parameter. Furthermore, IMSL routines CSVAL and CSDER can be used once the coefficients of the cubic spines have been computed by CSSMH to calculate the smoothed values of the state variables and their derivatives respectively. 

Finally, the user should always be aware of the danger in getting numerical estimates of the derivatives from the data. Different smoothing cubic splines or polynomials can result in similar values for the state variables and at the same time have widely different estimates of the derivatives. This problem can be controlled... 

As we mentioned, the first and probably most crucial step is the computation of the time derivatives of the state variables from smoothed data. The best and easiest way to smooth the data is using smooth cubic splines using the IMSL routines CSSMH, CSVAL CSDER. The latter two are used once the cubic splines coefficients and break points have been computed by CSSMH to generate the values of the smoothed measurements and their derivatives (rj, and t] )-... 

Figure 7.1 Smoothed data for variables Xi and, x2 using a smooth cubic spline approximation (s/N O.Ol, 0.1 and I).

Figure 7.2 Computed time derivatives of xt and x using smooth cubic splines for three different values of the smoothing parameter (s N=0 01. 0.1 and I).

In addition, the program used for data smoothing with cubic splines for the shortcut methods are given for the example ... 

Roosen, C. B., and Hastie, T. J., Automatic smoothing spline projection pursuit. J. Comput. Graph. Stat., 3, 235 (1994). 

A convenient and systematic way to represent fj (rtj) (r is the distance between particles i and j) as a linear function of unknowns is to employ cubic splines [48], as shown in Figure 8-3. The advantage of using cubic splines is that the function is continuous not only across the mesh points, but also in the first and second derivatives. This ensures a smooth curvature across the mesh points. The distance is divided into 1-dimensional mesh points, thus, fj rij) in the Mi mesh (r < rq < r +i) is described by Eqs. (8-4), (8-5) and (8-6) [48],... 

In the present study we have extracted the EXAFS from the experimentally recorded X-ray absorption spectra following the method described in detail in Ref. (l , 20). In this procedure, a value for the energy threshold of the absorption edge is chosen to convert the energy scale into k-space. Then a smooth background described by a set of cubic splines is subtracted from the EXAFS in order to separate the non-osciHatory part in ln(l /i) and, finally, the EXAFS is multiplied by a factor k and divided by a function characteristic of the atomic absorption cross section (20). 

In a particle implementation of transported PDF methods (see Chapter 7), it will be necessary to estimate go(f) using, for example, smoothing splines, gi (f) will then be found by differentiating the splines. Note that this implies that estimates for the conditional moments (i.e., go) are found only in regions of composition space where the mixture fraction occurs with non-negligible probability. 

The calculation can be made for an arbitrary number of points provided their abscissa lie inside the range of x values. Figure 3.7 shows the characteristic features of spline interpolation, a very smooth aspect although with some overshooting problems, i.e., extrema located between the data points. Alternative interpolation schemes are discussed by Wiggins (1976). o... 

Craven, P. and Wahba, G., Smoothing noisy data with spline functions, Num. Math., 31, 377 03, 1979. 

Calibration graphs defined by data with non-negligible error have to be constructed by some kind of smoothing operation. In cases, in which the form of the underlying curve is known a priori, the latter can be approximated by minimizing the squares of deviations. Otherwise a spline function can be used (JJ[, 1 ). The spline function S(x) is constructed to minimize a measure of smoothness defined by... 

In spite of the great success of the spline functions"Tor radio-immunoassay standard curves caveats are voiced primarily concerning the conscientious choice of the smoothing parameters (25) and the overfitting (26). Both aspects deserve attention in other applications as weTT. ... 

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