Smoothing parameters, spline

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. 

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).

In spite of the great success of the spline functions"Tor radio-immunoassay standard curves caveats are voiced primarily concerning the conscientious choice of the smoothing parameters (25) and the overfitting (26). Both aspects deserve attention in other applications as weTT. ... 

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. 

A new type of covariate screening method is to use partially linear mixed effects models (Bonate, 2005). Briefly, the time component in a structural model is modeled using a penalized spline basis function with knots at usually equally spaced time intervals. Under this approach, the knots are treated as random effects and linear mixed effects models can be used to find the optimal smoothing parameter. Further, covariates can be introduced into the model to improve the goodness of fit. The LRT between a full and reduced model with and without the covariate of interest can be used to test for the inclusion of a covariate in a model. The advantage of this method is that the exact structural model (i.e., a 1-compartment or 2-compartment model with absorption) does not have to be determined and it is fast and efficient at covariate identification. 

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... 

The FA method gives isotherm data. To be useful in preparative chromatography, these data must be fitted to an isotherm model. There are presently no numerical procedures available to smooth the data from multidimensional plots, similar to the 2-D splines or French curves and obtain purely empirical isotherms. Therefore, the major difficulty is the selection of adequate models. The Langmuir isotherm is too simplistic in most cases, and the LeVan-Vermeulen isotherm is complicated and difficult to use as a fitting fimction. Several methods have been described to extract the "best" set of Langmuir parameters which could accormt for a set of competitive adsorption data [108]. These methods have been compared. The most suitable method seems to depend on the aim of the determination and on the deviation of the system from true Langmuir behavior [108]. 

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). ... 

The heat capacities of LUCI3 measured experimentally and smoothed by a cubic spline function were reported by Tolmach et al. (1987c). We calculated 6r>, 6 1, 0e2, 0e3, and a on the basis of all these experimental data, except three values in the temperature range 254.24-262.41 K. The characteristic parameters listed in Table 22 were obtained at a comparatively low = 0.05497 value. Comparison with the corresponding data on hexagonal lanthanide trichlorides shows that an increase in the molar volume causes an insignificant decrease in a, noticeable increases in 0 2 and 0E3, and substantial decreases in 0 and 0ei- Such modifications of these parameters change the temperature dependence of heat capacity in a quite definite way. Namely, because of the low 0 and 6ei values, heat... 

Thus, theoretically, given data for u(t) and 1(f) and the various parameters, k(f) can be found. However, the data for u(t) and I(t) must be suitable for differentiation and integration, which will usually require the fitting of splines through the data. Care must be taken to make sure the approximation spline accurately represents the data, while still allowing relatively smooth and continuous derivatives. The KBM can be used by nonexperts who are looking for a simple LAB model for storage applications, where the fluctuations of rate are less of a concern. However, this kind of model is obviously too simple to offer sufficient accuracy under a dynamic situation. 

---