Smoothing parameter

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 Eq. (12), SE is the standard error, c is the number of selected variables, p is the total number of variables (which can differ from c), and d is a smoothing parameter to be set by the user. As was mentioned above, there is a certain threshold beyond which an increase in the number of variables results in some decrease in the quality of modeling. In fact, the smoothing parameter reflects the user s guess of how much detail is to be modeled in the training set. 

Steinhauer and Gasteiger [30] developed a new 3D descriptor based on the idea of radial distribution functions (RDFs), which is well known in physics and physico-chemistry in general and in X-ray diffraction in particular [31], The radial distribution function code (RDF code) is closely related to the 3D-MoRSE code. The RDF code is calculated by Eq. (25), where/is a scaling factor, N is the number of atoms in the molecule, p/ and pj are properties of the atoms i and/ B is a smoothing parameter, and Tij is the distance between the atoms i and j g(r) is usually calculated at a number of discrete points within defined intervals [32, 33]. 

By including characteristic atomic properties, A. of atoms i andj, the RDF code can be used in different tasks to fit the requirements of the information to be represented. The exponential term contains the distance r j between the atoms i andj and the smoothing parameter fl, which defines the probability distribution of the individual distances. The function g(r) was calculated at a number of discrete points with defined intervals. 

One of the disadvantages of the method is that one must determine the smoothing parameter by optimisation. When the smoothing parameter is too small (Fig. 33.16a) many potential functions of a learning class do not overlap with each other, so that the continuous surface of Fig. 33.15 is not obtained. A new object u may then have a low membership value for a class (here class K) although it clearly belongs to that class. An excessive smoothing parameter leads to a too flat surface (Fig. 33.16b), so that discrimination becomes less clear. The major task of the... 

Fig. 33.16. Influence of the smoothing parameters on the potential surfaces of classes which are (a) too small and (b) too large.

Table 7.1 Estimated parameter values with short cut methods for different values of the smoothing parameter (s/N) in IMSL routine CSSMH...

Once we have the smoothed values of the state variables, we can proceed and compute 22. All these computed quantities (rji, r 2, linear least squares regression. In Figure 7.1 the original data and their smoothed values are shown for 3 different values of the smoothing parameter "s" required by CSSMH. An one percent (1%) standard... 

Subsequent use of linear regression yields the two unknown parameters ki and k2. The results are shown in Table 7.1 for the three different values of the smoothing parameter. 

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).

Fig. 8.9. Gaussian functions with different smoothing parameters Oi and o2 (a,c) and the corresponding potential functions (b,d)...

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. ... 

InEq. (3), X is a smoothing parameter, the value ofwhich must be defined or optimized. The fingerprint representing a database molecule, j, is matched against the fingerprints for each of the active and inactive molecules in the training-set and its score is then computed as... 

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. 

The choice of error in temperature measurement and smoothing parameter was made as follows. The error in temperature was assumed to be 0.2°C for all liquid mixtures, and 0.05°C for single liquids. The rationale for the different treatment lies in lumping all the error in the ordinate. Hence, in the case of single... 

Table IA. Effect of Smoothing Parameters for the Excess Water Region of the 1-Propanol—Water—Ammonium Chloride System...

Tables IA and IB show how the smoothing parameters are selected for the 1-propanol-water-ammonium chloride system (9). For the water-rich region the smaller of the two parameters gives an unwarranted inflection point which...

A comparison of columns 4 and 8 reveals no clear pattern, which is perhaps of greater significance. The use of raw data yields smaller values of the vapor composition sample deviations in four out of six cases, but the effects are small and could be masked by errors in the vapor compositions themselves. It seems likely that the greatest source of error lies in determination of vapor composition. Thus there is very little difference in using raw or smoothed data. A typical example of the fit is shown in Figure 2. The optimum smoothing parameters used in run 1 were found to be the same as required for run 2, and are listed in columns 11 and 12 of Table II. 

Qi, Q2 = smoothing parameter in water-rich and alcohol-rich region... 

Figure 6 shows the results obtained for N(E) for several values of A. We do not obtain satisfactory results for A = 0, but for a wide range of A > 0 we obtain quite stable results that are relatively insensitive to the particular value of this smoothing parameter. This is precisely the behavior one wishes to see. It is also significant that the results in Fig. 6 are accurate for some ways into the classically forbidden tunneling regime, in this case for energies as much as 0.1 eV or so below the barrier, down to a transmission probability of 10"3. 

Umax — number of cases which join the series a - smoothing parameter... 

Exponential smoothing is intended for calculation of one step ahead forecasts. All further forecasts x t+2), a(/+3),. .. relate to the recent forecasted value x(t+ ), x t+2),. .. and also, in dependence on the value of the smoothing parameter, to more recent, real values ... 

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