Numerical differentiation smoothing techniques

One therefore needs a smooth density estimation techniques that is more reliable than the histogram estimates. The automatic estimation poses additional problems in that the traditional statistical techniques for estimating densities usually require the interactive selection of some smoothing parameter (such as the bin size). Some publicly available density estimators are available, but these tended to oversmooth the densities. So we tried a number of ideas based on numerical differentiation of the empirical cdf to devise a better density estimator. 

In actual computations, the numerical differentiation can introduce a large error and should be avoided. A simple solution to this would be to fit spline functions, or a piecewise polynomial and overall smooth function of E, to the numerically calculated eigenphase sum 5(E), and then to differentiate the spline functions analytically [52, 53]. In the E-matrix method [44], the analytic E dependence of the R matrix and associated matrices can be taken advantage of in the direct differentiation of these quantities. This technique was found to be useful for automatic and fast analysis of the results of E-matrix method calculations [54-56]. 

Numerical differentiation of real experimental data is a difficult procedure because of the experimental error which is typically present. This means that data smoothing is usually required before the first derivative is calculated. On the basis of the segmental technique described here, the first derivative of y with respect to x is given by... 

The mole fraction profiles were numerically smoothed and differentiated in order to determine the species molar fluxes. This process was repeated on the flux profiles in order to provide net molar rates of formation or destruction for each species. The numerical techniques have been described previously (1,2,7,10). Probable errors associated with fluxes and reaction rates so determined cannot be stated explicitly because the perturbation of the sampling probe on the one-dimensional flame assumption has not been assessed quantitatively. 

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