Smooths cubic splines

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).

M65 Determination of smoothing cubic spline method of C.H. Reinsch 6500 6662... 

The approach used by Fisher et al. (1995) is a smoothed cubic spline that approximates the forward ciuve. The number of nodes to use is recommended as approximately one-third of the number of bonds used in the sample, spaced apart so that there is an equal number of bonds maturing between adjacent nodes. This is different to the theoretical approach, which is to have node points at every interval where there is a bond cash flow however, in practice using the smaller number of nodes as proposed by Fisher et al. produces essentially an identical forward rate curve, but with fewer calculations required. The resulting forward rate curve is the cubic spline that minimises the function (5.17) ... 

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

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 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. 

A method for interpolation of calculated vapor compositions obtained from U-T-x data is described. Barkers method and the Wilson equation, which requires a fit of raw T-x data, are used. This fit is achieved by dividing the T-x data into three groups by means of the miscibility gap. After the mean of the middle group has been determined, the other two groups are subjected to a modified cubic spline procedure. Input is the estimated errors in temperature and a smoothing parameter. The procedure is tested on two ethanol- and five 1-propanol-water systems saturated with salt and found to be satisfactory for six systems. A comparison of the use of raw and smoothed data revealed no significant difference in calculated vapor composition. 

Figure 5. Contours of several diffraction maxima the (2,1,3), (1,0,4), and (3,2,2) before (a,c,e) and after (b,d,f) corrections and background removal. Smoothing by cubic splines is performed during contouring. Note the nonuniformity of fiber tilt distribution appears in the corrected maxima.

Spline Fitting in One Variable. By definition, a cubic spline function in one variable consists of a set of polynomial arcs of degree three or less joined smoothly end to end. The smoothness consists of continuity in the function itself and in its first and second derivatives. 

The answer to this difficulty lies in the use of piecewise approximants, such as cubic splines, which are in general use in the mathematics literature (11). Carey and Finlayson (12) have introduced a finite-element collocation method along these lines, which uses polynomial approximants on sub-intervals of the domain, and apply continuity conditions at the break-points to smooth the solution. It would seem more straight-forward, however, to use piecewise polynomials which do not require explicit continuity... 

Fig. 4. (A) Parametrization of a shaped pulse envelope with the help of a cubic spline interpolation between a small number of anchor points (circles). (B) Approximation of the smooth pulse envelope by rectangular pulses with piecewise constant rf amplitude. (Adapted from Ewing et al., 1990, p. 123, with kind permission from Elsevier Science—NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)...

Figure 6-15 Effect of Temperature on SLR —AG (At) of a 1% High-Methoxyl Pectin/60% Sucrose Gel during Heating from 20 to 90°C, Right after a Cooling Scan from 90 to 20°C. denote AG jAt calculated after cubic spline fit the line is for AG jAt calculated after fitting a polynomial equation to the data denote the smoothed AG jAt values calculated after cubic spline fit.

Figure 6-16 Effect of Temperature on SLR of a Cured Gel (20°C, 48 hr) during Heating from 20 to 90°C. Values of AG At values calculated by derivation of a polynomial equation fitted to the data (continuous line), or after cubic spline fit and smoothed using a moving average procedure ( ) ( ) denote G values.

The applicability of the Kalman filter requires an accurate knowledge of the response of each component and an efficient procedure for background removal. Background subtraction has recently been treated with cubic splines polynomials(5,6] as smoothing interpolators between peak valleys and this has proved to be efficient for baseline resolution particularly for very low signal-to-noise ratios [7]. 

The unpredictable shifts of the pure component peaks respect to the mixtures, mainly due to the dififerent surface charging of the samples, made it necessary to find the optimal alignment of the peak doublets, before determining the percentage composition of the components in the mixtures through the Kalman filter. The smoothed Pb 4f peaks of each pure component were used in the filter model after background ranoval from the peak doublets by using the cubic splines interpolation. Also the unresolved Pb 4f peaks relative to the mixtures... 

A common method of extracting f K) from Eq. 3.82 is to assume a form of the distribution function by differentiation of a smooth fimction describing the data. The function obtained by this method is called the affinity spectrum (AS) and the method, the AS method [71]. The most general approach uses a cubic spline to approximate the data. However, a simpler procedure uses a Langmuir-Freundlich (LF) isotherm model and the AS distribution is derived from the best parameters of a fit of the experimental isotherm data to the LF model [71]. This approach yields a unimodal distribution of binding affinity with a central peak, if the range... 

Make a cubic B-spline pass through all the data points [x(z),))(/), i = 1,..., zm], A cubic spline is a cubic function of position, defined on small regions between data points. It is constructed so the function and its first and second derivatives are continuous from one region to another. It usually makes a nice smooth curve through the points. The following commands create Figure B.3. 

Another structure for expressing a nonlinear relationship between X and Y is splines [333] or smoothing functions [75]. Splines are piecewise polynomials joined at knots (denoted by Zj) with continuity constraints on the function and all its derivatives except the highest. Splines have good approximation power, high flexibility and smooth appearance as a result of continuity constraints. For example, if cubic splines are used for representing the inner relation ... 

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