Smoothing and Interpolation

Spectral smoothing increases the signal-to-noise ratio of the spectrum at the expense of resolution. Thus, while smoothing produces smoother looking bands, some information content may be lost. Furthermore, it may cause changes in the measured peak position and band shape. Interpolation represents the inverse of smoothing. That is, the resolution is artificially enhanced to provide a better definition of band shape. This may be achieved by polynomial fitting of the spectral data points. 

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Stress-strain-time data are usually presented as creep curves of strain versus log time. Sets of such curves, seen in Fig. 2-27, can be produced by smoothing and interpolating data on a computer. These data may also be presented in other ways, to facilitate the selection of information to meet specific design requirements. Sections may be taken t... 

The parameters of this expansion, as well as the number N of Lorentzian functions, are determined (from the experimental data) by a non-linear least squares fit along with statistical tests. It can be noticed that this expansion has no physical meaning but is merely a numerical device allowing for smoothing and interpolation of the experimental data. Nevertheless, this procedure proves to be statistically more significant than the Cole-Cole equation and thus to account much better for the representation of experimental data. The two physically meaningful parameters, i.e., C(0) and (Xo), can then be easily deduced from the quantities involved in (71)... 

The preceeding methods may be characterized by two words smoothing and interpolating. That most theoretical curves are smooth may be... 

We have attempted to depart from the fruitless procedure of using more least-squares coefficients to obtain a best possible fit of a mass of PpT data, which do not have the accuracy needed to define an equation of state consistent with the known behavior of specific heats. The present type of equation ot state is more highly constrained than any previously known, and therefore serves as a reliable smoothing and interpolation function by which means the inaccuracies and inconsistencies of various experimental data may be observed and intercompared. We prefer to replace the question How accurate is the equation of state by the question How accurate and consistent are the experimental data used here In present work, the answer is that, generally, densities are within a few tenths of one percent over the entire fluid domain (except for the critical region where they are much larger, but are within experimental uncertainties) (see Tables II and III). 

This is the way the tabulated values of AfH°stre actually constructed, but of course before this is done, the experimental values of (Hp — Hp ) for the substance and its elements must be smoothed and interpolated to give values at even temperature intervals. To do this, they are fitted statistically to a function, which is commonly... 

In the following sections we describe techniques for deciding how well two units join, and limit ourselves to doing this solely on the properties of the units. In addition to this, we can also use the techniques mentioned in Section 14.7 where we explored further techniques for smoothing and interpolation of the units on either side of the join. 

Figure 12.20 Ways of displaying FO plots against time (a) raw output from a PDA (b) raw output, but plotted with symbols rather than lines and (c) smoothed and interpolated output from a PDA.

Fig. 6 Energy diagram for Cs2NaLnCl6 relative to the top of the valence band. The band gaps are smoothed and interpolated in curve (d). Curve (a) represents the relative position of the Ln 4f ground state (b) the relative position of the lowest Ln " 4f 5d state and (c) the relative position of the Ln 4f vibrationally excited ground state...

If the distance between two impedance spectra measured on different frequency vectors is to be calculated, the impedance spectra need to be preprocessed. Therefore the impedance spectra are to be functional approximated, e.g. by utilizing smoothing and interpolation methods, e.g. roughness penalty approach. After that, the impedance spectra can be resolved on equal grids and the functional distance measure can be calculated over an mutual bandwidth. 

The pressure sensor may have changed its offset correction value compared with the laboratory calibration. The negative of the pressure sensor s pre-cast deck value replaces the laboratory offset correction. Calibration may be performed for all sensors at this stage (see Section 3.6.3). However, to save computing time it may be delayed until after the data have been reduced, but before smoothing and interpolation starts. 

Algebraic methods - in these techniques calculation of grid coordinates is based on the use of interpolation formulas. The algebraic methods are fast and relatively simple but can only be used in domains with smooth and regular boundaries. 

Minimization of this quantity gives a set of new coefficients and the improved instanton trajecotry. The second and third terms in the above equation require the gradient and Hessian of the potential function V(q)- For a given approximate instanton path, we choose Nr values of the parameter zn =i 2 and determine the corresponding set of Nr reference configurations qo(2n) -The values of the potential, first and second derivatives of the potential at any intermediate z, can be obtained easily by piecewise smooth cubic interpolation procedure. 

The ab initio and interpolated potential functions are coupled using a smooth switching function, written in terms of the energy difference between the electronic states ... 

A remarkable fact is that Equation 2.19 is the same as Equation 2.14, and Equation 2.20 is the same as Equation 2.17 if one ignores the Dirac delta function, although Equations 2.14 and 2.17 result from the ensemble approach, while Equations 2.19 and 2.20 result from the smooth quadratic interpolation. Thus, the expressions given by Equations 2.19 and 2.20 are fundamental to evaluate the chemical potential (electronegativity) and the chemical hardness. 

Now, as in the case of the energy, up to this point, we have worked with the nonsmooth expression for the electronic density. However, in order to incorporate the second-order effects associated with the charge transfer processes, one can make use of a smooth quadratic interpolation. That is, with the two definitions given in Equations 2.23 and 2.24, the electronic density change Ap(r) due to the electron transfer AN, when the external potential v(r) is kept fixed, may be approximated through a second-order Taylor series expansion of the electronic density as a function of the number of electrons,... 

The values of the two derivatives, /°(r) and A/(r), at the reference point N0, may be approximated through this smooth quadratic interpolation between the points pWn (r), pNo(r), and Pnm(r), when combined with the two conditions,... 

Polynomials do not play an important role in real chemical applications. Very few chemical data behave like polynomials. However, as a general data treatment tool, they are invaluable. Polynomials are used for empirical approximations of complex relationships, smoothing, differentiation and interpolation of data. Most of these applications have been introduced into chemistry by Savitzky and Golay and are known as Savitzky-Golay filters. Polynomial fitting is a linear, fast and explicit calculation, which, of course, explains the popularity. 

Each signal processing method discussed here involves same function which is either interpolating or smoothing, and is either local or global approximation of the data. This results in the two-way classification of the methods shown in Figure 4.2, where the quadrants of each card list methods of the same family for the particular application. 

Figure 4.10 Graphical representation of (a) baseline correction, (b) smoothing and (c) simplified view of a Savitzky Golay filter (moving window, five points per window, linear interpolation).

As described in [1] the aqueous feed and the air feed are preheated separately and are mixed in the mixing chamber with the organic feed. The total mixed mass flow Mfeed(t) is assumed to constitute one single homogeneous phase even below the critical temperature. Density p and specific heat cpof feed and fluid within the reactor are calculated from tabulated values of the pure phases by smooth cubic interpolation [3]. Non-ideal mixing effects on density and specific heat are neglected. 

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