Smoothing techniques

A common smoothing technique is linear interpolation. This approach fills in gaps in the market-observed yield curve caused by associated gaps in the set of observed bond prices by interpolating missing yields from actual yields. 

Linear interpolation is simple but not accurate enough to be recommended. Market analysts use multiple regression or spline-based methods instead. One technique is to assume that the discount factors represent a functional form—that is, a higher-order function that takes 

Linear interpolation is simple but not accurate enough to be recommended. Market analysts use multiple regression or spline-based 

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Lee, R. L., Gresho, P. M. and Sani, R. L., 1979. Smoothing techniques for certain primitive variable solutions of the Navier-Stokes equations. Int. J. Numer. Methods Eng. 14, 1785-1804. 

Craven, P. Wahba, 6. tlumer. Math. 1979, 31, 377-403. Utreras, F. in "Smoothing Techniques for Curve Estimation" Gasser, Th. Rosenblatt, M., eds. Lecture Notes in Mathematics, No. 757, Springer Berlin, 1979, p. 196-232. Utreras, F. "A Package for Smoothing Noisy Data With Splines" Techn. Report, Nr. Ma-80-B-209, Departamento de Matematicas, Universidad de Chile, Santiago, 1980, 45 pp. Wahba, G. personal communication. 

In Section II, the deconvolution examples used noise-free simulated spectra. Any real spectrum will be corrupted by noise. The noise can be reduced by smoothing, but smoothing generally attenuates high spectral frequencies in data. There is an operational conflict, however, because it is these same high spectral frequencies that we wish to enhance by deconvolution. In this section, the effects of noise on deconvolution are demonstrated and several smoothing techniques are evaluated. 

In testing a smoothing technique, two criteria should be considered. Obviously, a smoothing technique should reduce the magnitude of the noise and the impact of noise on the deconvolved spectrum. Second, a smoothing technique should not seriously affect the deconvolution process or the deconvolved spectrum. That is, if the smoothing is too severe, it will further... 

A. Economou, P. R. Fielden, A. J. Packham, E. J. Woollaston and R. D. Snook, Comparison of data smoothing techniques as applied to electrothermal atomic absorption spectrometry. Analyst, 122(11), 1997, 1331-1334. 

This attenuation can be made by smoothing methods that allow data extrapolation, if the number of experimental points is reduced. The smoothing methods should also allow rate calculation from these points. The most commonly used smoothing techniques of experimental data are presented below. 

Perform PCA over each composition 1 region separately (so if there are three compounds, perform three PCA calculations) and then take die loadings of the first PC as an estimate of the pure spectrum. PCA is used as a smoothing technique, the idea being diat die noise is banished to later PCs. 

J. Pillardy, L. Piela, Smoothing techniques of global optimization distance scaling method in searches for most stable Lennard-Jones atomic clusters, J. Comp. Chem. 19 (1998)... 

A wide variety of mathematical manipulation schemes are available to smooth spectral data, and in this section we shall concentrate on smoothing techniques that serve to average a section of the data. They are all simple to implement on personal computers. This ease of use has led to their widespread application, but their selection and tuning is somewhat empirical and depends on the application in-hand. 

Whatever smoothing technique is employed, the aim is to reduce the effects of random variations superimposed on the analytically useful signal. This transform can be simply expressed as... 

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