Ds

1000

Figure 13.1 Smoothed, weighted deviation plot for luminescence decay of uranyl-Na mordenite regressed onto model for single exponential decay using program in Box 10.1. (Reprinted with permission from [2], copyright © 1985, Marcel Dekker, Inc.)

done separately on each half-set of the data to obtain the initial guesses of y, and r,. Nonlinear regression analysis onto the p = 2 model is then begun using these initial parameters. After convergence, the threshold testing of the new smoothed deviation plot is done as described. If no new Ds exceeds the threshold, the p = 2 hypothesis is accepted. If one or more Ds exceed the threshold, a third regression analysis with p = 3 is initiated.

Initial parameter guesses for this final nonlinear regression onto the p = 3 model are chosen by three semilogarithmic linear regressions of the data divided into three equal parts. After the regression analysis, the new smoothed deviation plot is analyzed as previously to test the p = 3 hypothesis.

The automatic classification program was tested with data simulated from the model in Table 13.1, with 1% Poisson distributed random noise added. Attention was paid to the success of classifications having relatively small ratios of lifetimes; that is, r,+i/r,. Results for simulated data with t2/t, = 4.3 and t3/t2 = 4.3, T2/r, = 2.3 all gave successful classification. Because their values were relatively close, errors for r2 and t, were in excess of 20%. Errors deceased significantly as the ratio t2Itx in the simulated data was increased [2],

Table 13.2 also lists the initial estimates of the parameters found by the program using linear regression. For the single exponential (p = 1), these values were quite similar to the final values obtained by nonlinear regression. However, for the data representing two and three exponentials, the final parameters from nonlinear regression had significantly smaller errors.

Table 13.2 Automated Classification and Analysis of Simulated Data"

True Initial Initial % Final Final %

p Parameter value value* error value error

Table 13.2 Automated Classification and Analysis of Simulated Data"

True Initial Initial % Final Final %

p Parameter value value* error value error

1

y 1

1878

1888

0.5

1888

0.5

T]

698

695

0.4

694

0.5

2

y\

9080

6738

26

8762

3.5

Tl

162

319

97

170

5.0

yi

1878

2441

30

1723

8.2

ti

698

609

12

732

4.9

3

y\

20771

10219

51

15932

23

ti

69

245

255

94

36

yi

9080

3800

58

5265

42

tl

162

457

180

199

23

1878

2354

25

1638

13

n

698

621

11

747

7

" Using program listed in Box 13.1. b Initial parameters from linear regression.

" Using program listed in Box 13.1. b Initial parameters from linear regression.

Figure 13.2 Smoothed, weighted deviation plot for luminescence decay of uranyl-Na mordenite regressed onto model for triple exponential decay using program in Box 10.1. (Reprinted with permission from [2], copyright © 1985, Marcel Dekker, Inc.)

The automatic classification method was applied to luminescence decay data for different types of zeolites (molecular sieves) containing uranyl ions [2]. Multiple exponential decays are observed corresponding to different environments for uranyl ions in the zeolite [5], When data for uranyl in a sodium mordenite zeolite were analyzed by the program in Box 13.1, the smoothed deviation plot from the p = 1 model was clearly nonrandom (Figure 13.1), and it was classified as such by the threshold analysis employed by the program. The p = 2 hypothesis was also rejected by the program. The subsequent p = 3 fit gave a smoothed deviation plot (Figure 13.2) with no points exceeding the threshold, and the p = 3 hypothesis was accepted [6]. The same conclusion was reached independently by the inorganic chemists who collected the data after examining the results of individual nonlinear regressions onto one, two, and three exponential models [5], Parameter values were in good agreement with those reported previously.

A problem in classification was noted if the data contained high frequency instrumental noise. As in most nonlinear regression applications, the method requires the best S/N possible and elimination of systematic noise. The program has the capability of analyzing only exponentials, so the data should contain no contributions from the source or from events with nonexponential responses.

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