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and NBS Phosphate Rock 120b, yielded results that differ from the reported values by 6.8% and 0.5%, respectively. Table I lists the F- results that were obtained for 15 geological standard reference materials representing a variety of matrices. The hydropyrolytic-ion chromatographic results are generally in agreement with those reported even though the reported results often represent noncertified, literature, average, or magnitude values. Comparison with an Ion-Selective Electrode. The Fcontent of the 15 hydropyrolyzed reference materials was redetermined by an ion-selective electrode (ISE) to further verify the F- results obtained by the hydropyrolytic-ion chromatographic method. Fluoride concentrations listed in Table I are in general agreement (&lo%)with those determined by ion chromatography. In three instances, NBS Fly Ash 1633, NBS Portland Cement 633, and NBS River Sediment 1645, the F- concentrations determined by ISE are in better agreement with the hydropyrolytic-ion chromatographic results than with previously reported data. Good agreeement between the two methods is also obtained for BCR Fly Ash 38 and NBS Estuarine Sediment 1646 for which no reported F- concentrations were found. Poor agreement is obtained for samples containing less than 100 ppm P such as NBS Coals 163213 and 1635 and NBS Fly Ash 1633a. This disparity can be partially attributed to the poor precision of the ion-selective electrode (9.8% RSD) compared to the precision of the ion chromatograph (0.7% RSD) at 0.5 ppm F-, the hydropyrolyzed concentration of F- from a 100 ppm sample. Imprecise determination of the F- concentration of the sample and standard addition spikes can severely alter the slope of the standard additions regression, thereby affecting the accuracy of F- determinations below 100 ppm.
CONCLUSION The data presented in this report show that a hydropyrolytic procedure can be used to quantitatively extract Ffrom geological samples. The F- content of the relatively interference-free extract can be determined either by ion chromatography or by an ion-selective electrode. However, ion chromatography is preferred when maximum selectivity and/or sensitivity is required.
ACKNOWLEDGMENT The authors thank F. G. Walthall of the United States Geological Survey and R. K. Leininger of the Indiana State Geological Survey for providing several of the rock standards
and G. A. Witt for performing many of the analyses. Registry No. F-,16984-48-8;Moo3, 1313-27-5.
LITERATURE CITED 8iological Effects of Atmosphere Pollutants-Fluorides ; National Academy of Sciences, National Research Council: Washington, DC, 1971; 295 pp. Gluskoter, H. J.; Pierand, L. H.; Pfefferkorn, H. W. J. Sediment. Petrol. 1970, 40, 1363-1366. Crossley, H. E. J. SOC. Chem. Ind., London 1944, 6 3 , 289-292. Thomas, J., Jr.; Giuskoter, H. J. Anal. Chem. 1974, 4 6 , 1321-1323. Annu. Book ASTM Stand. Part 26, D 3761-79. Troii, G.; Farzeneh, A,; Cammann, K. Chem. Geol. 1977, 2 0 , 295-305. Edmond, C. R. Anal. Chem. 1989, 41, 1327-1328. Gimeno Adeiantado, J. V.; Peris Martinez, V.; Checa Moreno, A.; Bosch Reig, F. Talanta 1985, 3 2 , 224-226. Leon-Gonzaiez, M. E.; Santos-Delgado, M. J.; Polo-Diez, L. M. Anal. Chim. Acta 1985. 778, 331-335. Butler, J. N. Ion-Selective Electrodes, NBS Spec. Publ.; Durst, R. A,. Ed.; National Bureau of Standards: Washington, DC, 1969; Chapter 5. Nicholson, K.; Duff, E. J. Anal. Lett. 1981, 74, 887-912. Ganiiang, G.; Bing, Y.; Yang, L. fuel 1984, 63, 1552-1555. Coerdt. W.; Mainka, E. Fresenius’ Z . Anal. Chem. 1985, 320, 503-506. Evans, K. L.; Tarter, J. G.; Moore, C. B. Anal. Chem. 1981, 5 3 , 925-928. Chakraborti, D.; Hiilman, D. C. J.; Zinagro. R. A.; Irgoiic, K. J. fresen/us’ Z . Anal. Chem. 1984, 319, 556-559. Hili, R. A. HRC CC, J. Hlgh Resolut. Chromatogr. Chromatogr. Commun. 1983, 6 , 275-276. Warf, J. C.; Cline, W. D.; Tevebaugh, R. D. Anal. Chem. 1954, 2 6 , 342-346. Whitehead. D.; Thomas, J. E. Anal. Chem. 1985, 5 7 , 2421-2423. Farzaneh, A.; Troll, G. Geochem. J . 1977, 1 7 , 177-181, Bock, R. A Handbook of Decomposition Methods in Analytical Chemistry; International Textbook Co., 1979. Cabeklu, N. C.; Leng, B.; Moss, J. H. J. fluorine Chem. 1975, 6 , 357-366. Giadney, E. S.;Burns, C. E.; Perrin, D. R.; Roelandts, I.; Gills, T. E. 1982 Compliatlon of Elemental Concentration Data for NBS Biological, Geological, and Environmental Standard Reference Materials ; NBS Publication 260-68, 231 pp. Dickinson Laboratorles Inc., El Paso, TX, private communicatlon. Gcdbeer, W. C.; Swaine, D. J. fuel 1987, 66, 794-798. Gonska. H.; Griepink, B.; Colombo, A.; Muntau, H. The Certification of the Contents of Arsenic, Cadmium, Chromium, Cobalt, Fluorine , Manganese, Mercury, Nickel, Lead, and Zinc in a Coal; Commission of the European Communities Report EUR 9473 EN, Brussels, Luxembourg, 1984. Clayton, E.; Dale, L. S. Anal. Lett. 1985, 18, 1533-1538. Bird, J. R.; Clayton. E. Nucl. Instrum. Meth. Phys Res. 1983, 218, 505-528. Bettineili, M. Analyst (London) 1983, 708, 404-407. National Bureau of Standards Certlficate of Analysis, SRM 633, Portland Cement, Washington, DC. 1983. National Bureau of Standards Certificate of Analysis, SRM 1645, River Sediment, Washington, DC, 1982.
RECEIVED for review May 4,1987. Accepted October 1,1987.
CORRESPONDENCE Procedure for Increasing the Accuracy of the Initial Data Point Slope Estimation by Least-Squares Polynomial Filters Sir: The least-squares polynomial filter for estimating the smoothed value and derivative of an initial point in the data set has potential applications in initial rate estimation in reaction kinetics (1,2),thermal lens measurements (3), and decay measurements (4)and has been applied by Harris et al. (5,6)to increase the precision in determining initial light intensities in thermal lens absorption measurements. Even though the Savitzky-Golay procedure (7), or its modified version (B), is excellent in smoothing and differentiation of
most of the data points, they do not allow initial-point smoothing and slope calculation since these procedures utilize a polynomial fit to a segment of data points to estimate the midpoint of the segment. However, a procedure has been described by Harris et al. (9,lO)in which a polynomial fit to a segment of data to estimate the initial point and its derivative has been utilized. In this paper accuracy of the initial point slope calculation by polynomial filters has been examined and a procedure for
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improving its accuracy has been described.
THEORY AND DISCUSSIONS The new procedure for initial point and its slope calculation is developed by combining the midpoint smoothing and differentiation procedure and the initial-point smoothing and differentiation procedure. The theory behind the midpoint smoothing and slope calculation appears in several publications (e.g. ref 7 and 8), whereas the theory behind the initial-point smoothing and slope calculation appears in ref 9. It should be pointed out that the basic difference between the initial-point and the midpoint smoothing and differentiation procedures is that in the initial-point procedure the very first point of the data segment used in polynomial fitting is smoothed and differentiated whereas in the midpoint procedure the midpoint is smoothed and differentiated. By sliding polynomial filters and calculating smoothed and slope values, one can obtain derivative curves. Using data from a known function, one can compare the calculated derivatives with the acutal ones to determine errors involved and hence accuracy of the procedure used in slope calculation. Such errors were defined as systematic errors since no random noise was superimposed on the data set. In the present work all the coefficients for slope estimation were calculated by minimizing a residual function as defined in ref 8 followed by matrix inversion and multiplication. Test data sets were generated by using functions of the form Y = B exp(-AX2) with different values of X, A , and B and also functions of the form Y = a + a,X + a2X2+ a3X3with different values of coefficients and X. For illustration purposes, results from a set of 40 data points (20 on each side of the Gaussian peak) generated by the first function with A = 1, B = 1, and X values ranging from 0.06 to 1.32 has been presented. The initial-point procedure was used with filter lengths of 5, 7, 9, 11 etc. and polynomial of order 2 for the initial point slope estimation at X = 0.06. These slope values were then compared with the true value in order to obtain the accuracy of this procedure. For the above-mentioned data set the systematic error was increased from 2% for the filter length of 5 to 18% for a filter length of 11. Such an increase in systematic error due to increased filter length is consistent with the findings of Harris et al. (9). To minimize such errors, Harris et al. (9) suggested the use of a small filter length and a higher order polynomial. Even though the use of a small filter length (3 or 5 ) and a higher polynomial order may reduce systematic errors, overall erros (due to random errors) in slope calculation are increased (8) when the data points are superimposed with a significant amount of random noise. Presumably, any gain of accuracy achieved by the lowering of systematic errors is more than offset by the loss of precision due to high noise level on the data points which are not well smoothed by small filter lengths or high-order polynomials. It should be mentioned that the noise is reduced approximately as the square root of the filter length used in smoothing (7). Besides, even for relatively smaller filter lengths of 7 or 9, systematic errors for initial slope values may be substantial, 5% and lo%,respectively, for the above-mentioned data set. Figure 1 represents three derivative curves obtained from the above-mentioned data set with their negative slope values plotted against X values. The upper curve is obtained by sliding the least-sqaures fit and calculating the slopes at first points of the data segments (the FPT procedure). The middle curve is the true derivative curve. The lower curve is the derivative curve obtained by sliding the least-squares fit and calculating the slope values of midpoints (the MPT procedure). The upper and the lower curves are derived by using filter length of 11 and polynomial of order 2. A polynomial of order 2 is used in such tests as it is known to provide the best smoothing of the data set (8) when superimposed with
0
0.4
0.8 X
1.2
Figwe 1. Three derivative curves. The upper one is obtained by the first-point smoothing and slopealculation procedure (FPT), the middle curve is the true derivative curve, and the lower curve is obtained by the midpoint smoothing and slope-calculation procedure (MPT).
Table I. Comparison of Systematic Errors at Different Filter Lengths for Initial Point Slope Estimation by the FPT Method ( 9 ) and the Present Method % error in initial point slope
filter length 5 7 9
11
estimation FPT method present work
2 5 10 18
0
1 3 7
random noise. It is interesting to notice that the MPT procedure provides more accurate slope values than the FPT procedure. The MPT cullre, at least for the first few available slope values, runs almost parallel with the FPT curve, indicating nearly a fixed percent error for all the slope values at the beginning as calculated by the FPT smoothing and the slope calculation ,procedure. Hence by comparing the first available MPT slope value with the corresponding slope value given by the FPT procedure, one can compute percent deviation and can correct the calculated slope of the initial point (as well as those that cannot be calculated by the midpoint smoothing procedure). Percent deviation is defined as % deviation = [(MPT(Xi) - FPT
(X,)) X 1001/FPT(Xi)
Here MPT(Xi) stands for the first available slope value obtained by the midpoint smoothing and the differentiation procedure and FPT(Xi) stands for the corresponding slope value obtained by the fiit-point smoothing and differentiation procedure. Since the lower curve of Figure 1 is obtained by using the midpoint procedure for the filter length of 11, the 6th-point slope value is the first available slope to be compared with the 6th-point slope of the upper curve for proper corrections. The application of this correction factor significantly improves the accuracy of the initial point slope estimation. Table I shows comparison between results obtained by this new procedure with those obtained by the first-point procedure as described by Harris et al. (9). A data set of 40 points for this comparison was generated by using the function of the form Y = B exp(-AX2) with A = 1, B = 1, and X values ranging from 0.06 to 1.32. It should be mentioned that the calculated systematic error, in the initial point slope estimation, not only depends upon the filter length but also depends upon the functional form from which the test data points were obtained and the density of data points defined for a unit interval of abscissa values. Additional sets of test data points were generated by using
ANALYTICAL CHEMISTRY, VOL. 60, NO. 4, FEBRUARY 15, 1988 1
Table 11. Effect of Number Density and the Successive Slope Change (Average) on the Accuracy of the Initial Point Slope Estimation by the FPT and the Present Method
% av change
FPT
mesent
method
work
4
47
6 8
34
82 40 24 16 8 6 4
16
20
11
14 8 12 16 NUMBER DENSITY
20
Flgure 2. Initial point slope value, as calculated by the least-squares procedures, changes as the density of data points belonging to the same cubic function is increased. The straight line represents the true initial point slope value. The curve that almost converges to the line is obtained by the present procedure, and the inner curve is obtained by the FPT procedure.
a cubic function of the form mentioned above with each coefficient equal to unity and with X values ranging from 0.06 to 10, but with different densities of data points for a given range of X values. Figure 2 shows how the initial-point slope value, calculated by the FPT method (9) as well as the present method, gradually changes and finally levels off as the density of data is increased from 4 to 20. The filter length used in these tests was equal to 7. The upper straight line represents the true initial-point slope value (1.1308), the curve next to it was obtained by the present recommended correction procedure, and the lower curve was obtained by the FPT procedure. While the FPT-slope value levels off with a systematic error of 4%, the initial-slope value obtained by the correction procedure levels off with almost no error. Table I1 shows relationship among the number density, percent change in the slope value between successive data points, and percent systematic errors by the FPT method as well as the present method. Since in least-squares procedures several data points are used at a time for polynomial fitting, an averge change in slope for the first few data points is presented instead of change in slope between the first and the second data points. It should be pointed out that the percent error, the number density, and the slope change values, as shown in Table 11, change when data from a different function is used even for the same filter length of 7. For example, for a data set from the Gaussian function the number density of 4 gives an average change in successive slope value of 26% and the percent error of 1% (present procedure) in the initial point slope calculation. For data from the cubic function the number density of 8 corresponds to a 26% change in successive slope and an error of 11TO. Even though the numerical values change, a general trend is followed in every test result. As the number density is increased, the successive slope change is decreased and the percent error is decreased. Thus, the accuracy of the initial point slope calculation is increased. Even though in every test case the new procedure provides a more accurate estimate of the initial point slope value, it
in sloDe
26 21 15 14
10
4
% error in t h e i n i t i a l p o i n t slope estimate
(from pt i t o 7)
no. density
GO
371
75 26
11 5 1 0.6 0.1
is recommended that a fairly good data density or a relatively small change in successive slope values is necessary for accurate results. Besides, if the data points are superimposed with significant random noise, one first needs to obtain a well-smoothed data set before applying this correction procedure. This has been tested by superimposing 100 data points from the cubic function with 3% and 5% random noise. A filter length of 7 provided a well-smoothed data set and hence an accurate estimate of the initial point slope value (about 3% error). However, when data points from the Gaussian function were superimposed with a 3% random noise, a poor smoothing was achieved by a filter length of 7 and hence the initial-point slope calculation procedure could not be applied for an accurate slope estimation. This new procedure for initial point slope estimation by the combination of the midpoint slope calculation procedure with the initial-point slope calculation procedure not only provides an accurate estimate of the initial point slope value but also provides accurate estimates of the edge point values, as was seen in each of the test cases. ACKNOWLEDGMENT
The author acknowledges helpful discussions with C. A. Hollingsworth,P. E. Siska, and D. W. Martin of the University of Pittsburgh. LITERATURE CITED (1) Laidler, K. J. Chemlcal Kinetics; Harper: New York, 1987. (2) Mark, H. B., Jr.; Rechnitz, 0. A. Kinetics h Analytical Chemistry; Elving, P. J.; Kolthoff, I.M., Eds.; Wiley-Interscience: New York, 1968; Vol. 24. (3) Brannon, J. H.; Madge, D. J. Phys. Chem. 1978, 82, 705-709. (4) Blrks, J. 6. photophysics of Aromatic Molecules : Wiley-Interscience: New York, 1970. (5) Carter, C. A.; Harris, J. M. Anal. Chem. 1983, 5 5 , 1256-1261. (6) Leach, R. A.; Harris, J. M. Anal. Chem. 1984, 56, 1481-1487. (7) Savitzky, A.; Golay, M. J. E. Anal. Chem. 1984, 36, 1627-1839. (8) Khan, A. Anal. Chem. 1987. 59, 854-657. (9) Leach, R. A,; Carter, C. A,; Harris J. M. Anal. Chem. 1984, 5 6 , 2304-2307. (IO) Baedecker, P. A. Anal. Chem. 1985, 5 7 , 1477-1479
Arshad Khan
Chemistry Department The Pennsylvania State University Dubois, Pennsylvania 15801
RECEIVED for review August 14,1987. Accepted October 29, 1987.
