Smooths splines

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

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. 

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 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. 

Using inverse linear interpolation the two titration equivalence points are obtained as the zero-crossing points of the second derivative at V = 3.78 ml and V = 7.14 ml. On Fig. 4.4 the second derivative curve of the interpolating spline (SD = ) and that of the smoothing spline (SD = 8.25) are shown. The false zero-crossing of the second derivative present at interpolation is eliminated by smoothing. 

We note that another type of smoothing spline can be fitted by the traditional least squares method. In that case, however, the q subintervals on which the individual cubics are defined should be selected prior to the fit,... 

Use smoothing spline to obtain the initial estimates for peak location (the location of the maximum), peak height (function value at the maximum point)... 

Gu and Wahba (1993) used a smoothing-spline approach with some similarities to the method described in this chapter, albeit in a context where random error is present. They approximated main effects and some specified two-variable interaction effects by spline functions. Their example had only three explanatory variables, so screening was not an issue. Nonetheless, their approach parallels the methodology we describe in this chapter, with a decomposition of a function into effects due to small numbers of variables, visualization of the effects, and an analysis of variance (ANOVA) decomposition of the total function variability. 

Gu, C. and Wahba, G. (1993). Smoothing spline ANOVA with componentwise Bayesian confidence intervals . Journal of Computational and Graphical Statistics, 2, 97-117. 

Fig. 17 Energy as a function of a T-T distance and b T-T-T angle used in the simulation procedure (calculated as smoothing spline fits to Boltzmann equilibrium interpretations of the histogrammed data taken from 32 representative zeolite crystal structures). Only the central portions are shown, c The contribution to the energy sum for the merging of two symmetry-related atoms merging is only permitted when the two atoms are at less than a defined minimum distance [84], Reproduced with the kind permission of the Nature Publishing Group (http //www.nature.com/)...

The dula analysis was performed uccortling io relerence 109. To determine Ihe smooth pari of ihe background subiracictl spectrum, a modified smoothing spline algorithm was used. The EXAFS Xlk) functions were multiplied with k1 and Fourier filtered in the ranges 1.40- .50 A for the bromine F.XAI S spectra and 1.00- .40 A for the magnesium EXAFS spectra. 

Data analysis was performed with a program package, specially designed for the evaluation of liquid or amorphous systems [109]. The background removal was done by use of a modified smoothing spline algorithm, and subsequent normalization with the determined spline. The k ranees of the weiehted EXAFS functions of... 

Figures 7.5 and 7.6 show two cases of fitting the above B-splines to the data shown in Figures 7.1, 7.2, 7.3 (i.e. on the interval [1,2], exp(x) + noise). In Figure 7.5 the coefficients in the linear combination are chosen to give a reasonable balance between fitting the position and getting a smooth spline. As can be seen, the derivative estimates are remarkably good using this criterion.

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). ... 

For a smoothed spline, the level of oscillation is cmitrolled by setting a roughness penalty in the fimction, and not by reducing the number of node points. The yield curve / is chosen that minimises the objective function (5.15) ... 

Fisher, M., Nychka, D.,Zervos, D., 1995. Fitting the Term Stracture of Interest Rates with Smoothing Splines. Federal Reserve Board, Finance and Economic Discussion Series 95-1. 

In XAS analyses, it is typical to define the EXAFS, x(k), as the fractional modulation in the X-ray absorption coefficient as in Equation (2), where p, is the observed absorption coefficient and po is the absorption that would be observed in the absence of EXAFS effects. Since po cannot be directly measured, it is approximated, typically by fitting a smooth spline function through the data. Division by po normalizes the EXAFS oscillations per atom, and thus the EXAFS represents the average structure around the absorbing atoms ... 

Where P is the reaction force at the loading pin location, 0 is rotation at the loading pin obtained through DIG calculation, and b is the width of the specimen. Then, a smoothing spline fit is obtained to take the first derivative of J-integral with respect to COD (6) at each data point with following equation ... 

Nederlof, M. M., W. H. van Riemsdijk, and L. K. Koopal. 1994. Heterogeneity analysis for binding data using an adapted smoothing spline technique. Environmental Science Technology 28, no. 6 1037-1047. doi 10.1021/es00055a012. 

To approximate experimental adsorption isotherms and to calculate appropriate derivatives, we apply a procedure of smoothing splines, described by Reinsch [10]. In this approach, the experimental data are approximated by a cubic spline function g(x), which minimizes the following functional... 

The spatial derivative is also calculated by the smoothing spline approach. Since transport coefficients have a physical interpretation which results in certain restrictions (e.g. positivity), those models violating any restriction could be discarded already at this stage. 

Use data interpolation techniques, such as smoothing splines (Friedman et al., 1983) or time series modeling (Box et al., 1994). In case of time series modeling, a time series model for the variable in question is developed and the missing data during the time frame that no measurements are available are estimated with the help of the model. This is a complicated approach, since it is difficult to develop a reasonable time series model for the variable in Ae data set for which the measurements are missing. 

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