Data smoothing

Smoothed data presented at rounded temperatures, such as are available in Tables 6.2 and 6.4, plus the C° values at 298 K listed in Table 6.1 and 6.3, are especially suitable for substitution in the foregoing parabolic equations. The use of such a parabolic fit is appropriate for interpolation, but data extrapolated outside the original temperature range should not be sought. 

In Fig. 9 the relaxation time shows a very smooth variation with e (analogously smooth data has been obtained for Dj e)). In particular, no evidence for a critical anomaly as e — e, is seen. The straight line in Fig. 9 represents a law = T2 ec)Qxp[ ec — e)/k T], which indicates a simple thermally activated behavior. 

The exponential shape of the filter follows directly by elaborating eq. (40.14) for a few consecutive data points (see Table 40.4). From this table we can see that a smoothed data point at time i is the average of all data points measured before, weighted with an exponentially decaying weight X< ) with d the distance of that data point from the measurement to be smoothed. Such shapes are also found for electronic filters with a given time constant. The effect of exponential smoothing is visualized in the plot of the a , and values (Fig. 40.25) listed in... 

Fig. 40.25. Effect of exponential smoothing on the data points listed in Table 40.3 (solid line original data dotted line smoothed data).

The time-derivatives can be estimated analytically from the smoothed data... 

However, an important question that needs to be answered is "what constitutes a satisfactory polynomial fit " An answer can come from the following simple reasoning. The purpose of the polynomial fit is to smooth the data, namely, to remove only the measurement error (noise) from the data. If the mathematical (ODE) model under consideration is indeed the true model (or simply an adequate one) then the calculated values of the output vector based on the ODE model should correspond to the error-free measurements. Obviously, these model-calculated values should ideally be the same as the smoothed data assuming that the correct amount of data-filtering has taken place. 

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).

In this equation, u(T) represents the uncertainty of the observed data in the vicinity of T and is approximated by fitting a polynomial of order 1-3 to the estimated uncertainties as a function of temperature (other symbols appear in the glossary). Uncertainties in the smoothed data for the high temperature range are calculated using ... 

Figure 7 The effect of chamber pressure on the rate of primary drying, (a) 0.18 M methylprednisolone sodium succinate 2 mL in molded vials (2.54 cm2), shelf temperature +45°C. (Smoothed data from Ref. 6.) (b) Dobutamine hydrochloride and mannitol (4% w/w in water), 12 mL in tubing vials (5.7 cm2) and shelf surface temperature +10°C. (MJ Pikal. Unpublished data.) (Modified from Ref. 1.)... 

As the smoothed data in Table A.4 are given at closely successive, equally spaced temperatures, we can use these values to form the temperature intervals. For Cpm between any two temperatures, we can take the arithmetic mean between the listed experimental values. The values of AT and Cpm then are tabulated as an example in columns 3 and 4 of Table A.5. Column 5 lists the area for the given interval. Finally, the sums of the areas of the intervals from 10.00 K are tabulated in column 6. The areas between any two of the temperatures listed in column 1 can be obtained by subtraction. 

Hahn, G.J. (1979c), What Do I Gain from Smoothing Data , CHEMTECH, 9, 492-493. 

Figure 3.8. Illustration of peak distortion when using a smoother with a too-iarge window width (49 point). (Solid—raw data dashed—smoothed data.)...

Fig. 1 Raw and smoothed data of conversion (x) versus clock time (t) for different runs showing varying degrees of catalytic activity.

Fig. 16. Graphically smoothed data for calculated ternary diffusion coefficients in a system with a uniform concentration of dextran T500 (Hw 500,000) with an imposed 5 kg m-3 concentration gradient of PVP 360 the dextran concentration is varied...

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. 

There are two basic approaches to the calculation of vapor compositions from boiling point-liquid composition data or vapor pressure-liquid composition data (a) the coexistence equation (i) which requires the smoothing of experimental T-x or H-x data first, or (b) a correlating equation which relates the excess free energy with liquid composition. Various equations have been proposed, but Barker (2), who pioneered this method, employed Scatchard s equation (3). Raw or smoothed data are used, but the smoothing process may introduce unwarranted errors. 

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. 

Some smoothed data of expansion ratio appear in Figure 6.10(c) as a function of particle size and ratio of flow rates at minimum bubbling and fluidization. The rather arbitrarily drawn dashed line appears to be a conservative estimate for particles in the range of 100 pm. 

Malian- " and co-workers correlated liquid thermal conductivity data for benzene between JlfC and I26 C, OgiwaraIM iuid co-workers present smoothed data from 20 C to 7OX. Data for benzene arc spotty hut urc available from 0°C to 250 C.-H,-m 3I J, fcthyl benzene data arc available at SOX and HO C. 5 and for various temperatures between - 80 and I40 C.n Data for propylbcnzcnc71 and cumene- TVi,t arc available from ITC to 140 0. The method of Robbins and Kmgrea was used to extend the data to 200 C. 

FIGURE 10.6 Smoothed data after inverse Fourier transformation. The rectangular smoother shown in Figure 10.5 was used. 

FIGURE 10.9 Smoothed data from application of the trapezoidal smoother shown in Figure 10.8. The true, noise-free signal is shown as a dotted line. 

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