Anal. Chem. 1900, 5 2 , 112R-122R (139) Murphy, C. B. Anal. Chem. 1978, 50, 143R. (140) Nandi, P. N.; Deshpande, D. A.; Kher, V. G.Proc. Indian Acad. Sci., Sect. A , 1979, 88, (Pt. 1, No. 2), 113. (141) Nassau, K.; Wang, C. A.; Grasso, M. J . Am. Ceram. Soc. 1979, 62, 503. (142) Naumann. R.; Petzold, D.; Paulik, F.; Paulik, J. J . Therm. Anal. 1979, 15, 47. (143) Niume, K.; Nakamichi, K.; Takatuka, R.; Toda, F.; Uno, K.; Iwakura, Y. J . Polym. Sci., Polym. Chem. Ed. 1979, 17, 2371. (144) Pacey, R. A,; Clark, J. B. Thermochim. Acta 1979, 30, 115. (145) Padilla, R.; Sohn, H. Y. Metall. Trans. B 1979, IO, 109. (146) Palepu, R.; Moore, L. Thermochim. Acta 1979, 30, 384. (147) Paulik, F.; Paulik, J.; Buzagh-Gere, E.; Arnold, M. J . Therm. Anal. 1979, 15, 271. (148) Pawei, R. E.; Cathcart. J. V.; McKee, R. A. J , Electrochem. Soc. 1979, 126. 1105. (149) Pennings, A. J.; Zwijnenburg, A. J . Polym. Sci., Polym. Phys. Ed. 1979, 17, 1011. (150) Petranovic, N.; Susic, M. Thermochim. Acta 1979, 37, 211. (151) Phillips, D. C.; Smith, J. B. D.; Meier, J. F.; Kaczmarek, T. D. Microchem. J . 1978, 23, 165. (152) Phillips, P. J. J . Polym. Sci., Polym. Phys. Ed. 1979, 17, 409. (153) Ponge, C.; Rosso, J. C.; Carbonnel, L. J . Therm. Anal. 1979, 15, 101. (154) Prince, E. T.; Helbig, H. F.; Czanderna, A. W. J . Vac. Sci. Technol. 1979, 16. 244. (155) Prince, E. T.; Helbig, H. F.; Czanderna, A. W. Thermochim. Acta 1979, 29. 353. (156) Pusatcioglu, S. Y.; Fricke, A. L.; Hassler, J. C. J. Appl. Polym. Sci. 1979, 24, 937. (157) Rao, C. R. M.; Mehrotra, P. N. Thermochim. Acta 1979, 2 9 , 180. (158) Reichek, W.; Oppermann, H.; Wolf, E. Z. Anorg. Allg. Chem. 1979, 452, 96. 159) Reimschuessel, H. K. J . Polym. Sci., Polym. Chem. Ed. 1978, 16, 1229. 160) Reimschuessel, H. K. J . Polym. Sci., Polym. Chem. Ed. 1979, 17, 2447. 161) Reimschuessel, H. K.; Turi, E. A.; Akkapeddi, M. K. Ref. 160, p 2769. 162) Rogers, D. E.; Bibby. D. M. Thermochim. Acta 1979, 30, 303. 163) Rose, R. L. J . Phys. E 1979, 12, 13. 164) Rose, R. L.; Kelley, R. E.; Lesuer, D. R.; J , Nucl. Mater. 1979, 79, 414. 165) Rudin, A,; Samanta, M. C.; Reilly, P. M. J . Appl. Polym. Sci. 1979, 2 4 , 171. (166) Sahoo, P. K.; Bose, S. K.; Sircar, S. C. Thermochim. Acta 1979, 37, 303. (167) Sahoo, P. K.; Bose, S. K.; Sircar, S. C. Ref. 166, p 315. (168) Sandison, R. D.; Eggerding, C. L. Rev. Sci. Instrum. 1979, 5 0 , 129. (169) Sasaki, S.; Nakamura, T.; Uematsu, I.J . Polym. Sci., Po/ym. Phys. Ed. 1979, 17, 825. (170) Sawada, Y.; Uematsu, K.; Mizutani, N.; Kato, M. Thermochim. Acta 1978, 27. 45. (171) Sawada, Y.; Yamaguchi, J.; Sakurai, 0.;Uematsu, K.; Mizutani, N.; Kato, M. Thermochim. Acta 1979, 33, 127.
(172) Sefcik, M. D.; Yuen, H. K. Thermochim. Acta 1978, 2 6 , 297. (173) Seybold, K.; Meisel, T.; Cserfalvi, T. J . Therm. Anal. 1979, 15. 93. (174) Shaplygin, I . S.; Komarov, V. P.; Lazarev. V. B. Ref. 173, p 215. (175) Shibasaki, Y.; Fukuda, K. J . Polym. Sci.. Polym. Chem. Ed. 1979, 17, 2947. (176) Shimokawabe, M.; Furuichi, R.; Ishii, T. Thermochim. Acta 1979, 28, 287. (177) Silano. V.; Zahnley, J. C. Biochim. Siophys. Acta 1978, 533, 181. (178) Simonsen, K. A.; Zaharescu, M. J . Therm. Anal. 1979, 15,25. (179) Sorenson, 0. T. Thermochim. Acta 1979, 2 9 , 211. (180) Spratte, W.; Schneider. G. M. Mol. Cryst. Liq. Cryst. 1979, 51, 101. (181) Stahl, G. A. J . Polym. Sci., Polym. Chem. Ed. 1979, 17, 1883. (182) Stephens, M. A.; Tamplin, W. S. J . Chem. Eng. Data 1979, 24. 81. (183) Sutakshuto-Trivijitkasem. S.; Holm, B. J.; Oeye, H. A. Acta Chem. Scand., Ser. A 1978; 32. 969. (184) Swamianthan, V.; Madhaven. N. S. Thermochim. Acta 1979, 33, 367. (185) Theocaris, P. S.; Paipetis, S. A.; Papanicolaou, G. C. J . Appl. Polym. Sci. 1978, 22, 2245. (186) Ueda, M.; Takahashi, M.; Hishiki, S.; Imai, Y. J . Polym. Sci., Polym. Chem. Ed. 1979, 17, 2459. (187) Ueda, M.; Takahashi, M.; Imai, Y. Ref. 186, p 2477. (188) Ueda, Y.; Sayama, S.; Nishikawa, Y.; Ueda, S.; Yokoyama, S.; Makino, K. Ind. Eng. Chem., Process Des. Dev. 1979, 18, 353. (189) Valentich, J. J . Mater. Sci. 1979, 14, 371. (190) Vallebona, G. J . Therm. Anal. 1979, 16, 49. (191) Varhegyi, G.; Groma, G.; Lengyel, M. Thermochim.Acta 1979, 30, 311. (192) Varma, I.K.; Goel, R. N.; Varma, D. S. J . Polym. Sci., Polym. Chem. Ed. 1979, 17, 703. (193) Vasofsky, R.; Czanderna, A. W.; Thomas, R . W. J , Vac. Sci. Technol. 1979, 16, 711. (194) Vickers, L. P.; Donovan, J. W.; Schachman, H. K. J . Biol. Chem. 1978, 253, 8493. (195) Wahrmund, D. C.; Paul, D. R.; Barlow, J. W. J . Appl. Polym. Sci. 1978, 22,2155. (196) Ward, T. C.; Wnuk, A. J.; Henn, A. R.; Tang, S.; McGrath, J. E. Polym. Prepr., Am. Chem. Soc., Div. Polym. Chem. 1978, 1 9 ( 1 ) , 115. (197) Warne, S. St. J. J . Inst. Energy 1979, 52, 21. (198) Warne, S. St. J.; Mitchell, B. D. J . Soil Sci. 1979, 30, 111. (199) Wen, W. Y.;Lin, J. W. J . Appl. Polym. Sci. 1978, 2 2 , 2285. (200) Wendiandt, W. W. Thermochim. Acta 1978, 26, 19. (201) Wight, F. R. J . Polym. Sci., Polym. Lett. Ed. 1978. 6 1 , 121. (202) Yamaguchi, 0.; Yabuno. K.; Takeoka, K.; Shimizu, K. Chem. Lett. 1979, (4), 401. (203) Yuen, H. K.; Yosel, C. J. Thermochim. Acta 1979, 33, 281. (204) Zeldin. A. N.; Kukacka, L. E.; Fontana, J. J.; Carciello, N. R.; Reams, W. J . Appl. Polym. Sci. 1979, 2 4 , 455. (205) Zasko, J. Ref. Roum. Chim. 1970, 15,693. (206) Zsako, J. J . Therm. Anal. 1979, 15, 369. (207) Zsinka, L.; Szirtes, L.; Le Van So; Poko, S.J . Therm. Anal. 1978, 14, 245.
Chemometrics Bruce R. Kowalski Laboratory of Chemometrics, Department of Chemistry, BG- 10, University of Washington, Seattle, Washington 98 195
INTRODUCTION Many new developments in chemical analysis have followed closely on the heels of advances in electronics, physics, computer science, and various areas of engineering. For this reason there are, and will continue to be, strong interdisciplinary ties between analytical chemists and researchers in other areas of science and engineering. However, since the measurements made by analytical chemists are associated with some degree of uncertainty and an analytical result is usually derived from a mathematical formula, it is difficult to conceive of a more perfect marriage than analytical chemistry and statistics and applied mathematics. Unfortunately, a t this time the relationship between these fields can be characterized only as an engagement. This is not to say that there has been a complete absence of statistical and mathematical methods in analytical chemistry. Prior to the 1960’s, much of the statistical methodology that could be implemented without computers had been introduced in the literature by chemical statisticians working at the National Bureau of Standards and various industrial laboratories. As the computer entered the analytical 112R
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laboratory, however, significant changes occurred. These changes were noted and reported in 1976 by Shoenfeld and DeVoe, the authors of t h e Fundamental Review section entitled “Statistical and Mathematical Methods in Analytical Chemistry” ( A 5 ) . T h a t review, and its more exhaustive predecessor ( A I ) ,are absolutely required reading for serious students of chemical analysis. Besides writing a n excellent critical review of their topic from 1972 to 1976, Shoenfeld and DeVoe injected an element of optimism for the future of statistical and mathematical methods in analytical chemistry, offered a note of caution to those using multivariate methods to deduce causation, and suggested that analytical chemists pay much more attention to such fundamentals as experimental design, measurement characterization, and the testing of assumptions. They even provided the editor of ANALYTICAL CHEMISTRY with a new title for t h e present review. Chemometrics has been defined as the application of mathematical and statistical methods to chemical measurements ( A 2 ) . A more detailed definition of chemometrics and information about the Chemometrics Society can be obtained C 1980 American Chemical Society
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Bruce R. Kowalskl is orofessor of chemistrv at the University of Washington In 1969 he received his Ph D. from the University of Washington and joined the Shell Development Co. in Emeryville, Calif. After spending one year at the Lawrence Livermore Laboratory, he became an assistant professor of chemistry at Colorado State University in 1972. He then moved to Seattle where he has been since December 1973. As an analytical chemist and co-founder of the Chemometrics Society, his research interests focus on improving the analytical measurement process and obtaining more useful chemical information by using advanced mathematical and statistical methods.
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from the literature ( A 4 ) . Of special note is the emphasis placed on chemometrics during the celebration of ANALYTICAL CHEMISTRY’S 50th Anniversary. At a symposium covering the highlights of the science and the Journal over each of the past five decades, the advances of the 1970’s were covered from the viewpoint of the analytical chemist as an information scientist ( A 3 ) . This present review covers the period from January 1976 to December 1979. This review should not be used as a substitute for a thorough literature search in any area of chemometrics. By request from the editor, the review leans more toward a critical review of this emerging area of chemistry. T h e reader should not get the impression that only analytical chemists are interested in, or practicing, chemometrics as, for example, the Chemometrics Society has two subdivisions; analytical chemistry and physical organic chemistry. As with the last review, the computer played a role in the preparation of this review. Table I shows the expanded list of words used to search the literature as represented by the Chemical Abstracts Service computer tapes. T h e numbers in the tables are the numbers of references (“hits” by computer science jargon) found for each entry. T h e computer search was used to support the author’s knowledge of the literature of chemometrics in analytical chemistry. In all honesty, very few unknown references surfaced from the mountain of government reports, obscure journal references, and other literature not germane to this purpose. Nevertheless, the numbers in t h e table give a fairly representative estimate of the emphasis received by each topic over the four-year period. There are some exceptions. For example, very few of the references retrieved using the key words “experimental design”, had anything at all to do with the important and formal branch of statistics with that name. Likewise, few “calibration” references dealt with the mathematical aspects of calibration as do the papers discussed later in this review. Prior to 1976, there were few chemometricians and even fewer journals in which to present the fruits of their efforts. During t h e past four years, this situation has changed markedly. T h e number of chemometricians, especially analytical chemometricians, has grown rapidly, and the number of journals more or less specializing in the publication of chemometrics papers has also grown. Now, besides the standard journals that readily accept high quality relevant works from chemometricians (e.g., ANALYTICAL CHEMISTRY) there are a number of journals (see Table 11) trying to attract these works as well. Although there is no journal called “Chemometrics” at the time of this writing, there will no doubt be one in the future. I t is this author’s opinion that another journal would be a mistake. There are too many journals in proportion to the number of quality papers in, or related to, chemometrics. Another journal would only dilute the literature still further.
BOOKS T h e 1976 review by Shoenfeld and DeVoe recommended a number of good books on the basic mathematical and statistical methods (e.g., linear algebra) that must be mastered to understand and appreciate the chemical applications of the methods covered in their review. At that time, they could recommend only one book, relevant to their review subject, t h a t was written for chemists by chemists. During the final four years of the past decade, a number of books and chapters
Table I. Computer Search of Chemical Abstracts from Jan. 1 9 7 6 to Oct. 1 9 7 9 no. key word( s )
of ref.
no. of
key word( s )
calibration 2 3 1 2 curve fitting che mo me tri cs 8 spectral resolution sampling theory 2 deconvolution multivariate analysis 5 2 factor analysis parameter estimation 68 principal components time series analysis 20 feature selection spectral analysis 1102 Fourier transform optimal control 1 1 9 information theory systems analysis 80 signal processing evolutionary operation 1 peak fitting operations research 6 digital filtering regression 9 1 2 least squares mathematical analysis 116 nonlinear regression statistics 207 2 nonparametric statistics pattern recognition 2 3 2 simplex data reduction 70 nonlinear Cali brat ion 266 multiule regression exDerimental design
ref. 120 27 133 175
46 8 34 227 44
6 4
315 30 1 27
7 61
Table 11. Journals SDecializing in Chemometrics Related Papers Techno metrics Analytica Chimica Acta : Computer Techniques and 0 pt imizat ion Analytical Letters : Chemometrics Sect ion Journal of Chemical Information and Computer Science Computers in Chemistry
have been made available that should be quite helpful to those interested in learning more about what analytical chemists have done with chemometrics. All of the following books are recommended by the author of this review and have formed part of the reference collection for the author’s formal graduate course in chemometrics. The papers presented a t the “Chemometrics: Theory and Application” Symposium, 172nd American Chemical Society meeting have been published as an ACS Symposium Series publication (B6).Its chapters cover chemical applications of optimization, pattern recognition, factor analysis, modeling, and various tools to statistics. A slightly more introductory book that comes closest to a textbook has been edited by Massart, Dijkstra, and Kaufman with contributions from This is an excellent Wold, Vandeginste, and Michotte (€7). book that starts with fundamental aspects of analytical chemometrics: precision, accuracy, reliability, sensitivity, and specificity. T h e book then proceeds in a logical manner to cover optimization, multivariate methods, factor analysis, principal components analysis, pattern recognition, operations research, modeling, mathematical programming, and finally the systems approach t o analytical chew istry. J. R. DeVoe (B2)is the editor of an ACS Symposium Series book covering six topics ranging from systematic error in chemical analysis to the optimization of analytical methods. This is an important book as i t covers topics of interest to all analytical chemists. Ten topics on various areas of chemometrics are included in a book edited by R. F. Hirsch (B5). T h e book, titled “Statistics”, is the collection of papers from the 1977 Eastern Analytical Symposium and really covers more than the title promises. A book edited by Fujiwara and Mark, Jr., (B3) has the interesting title “Information Chemistry” and has three chapters with only the first of direct interest to chemometrics. The other two chapters cover topics of chemical information storage and retrieval and computer networks and may be of indirect interest to the readers of this review. Finally, Griffiths has edited an extremely good collection of chapters on “Transform Techniques in Chemistry” (B4). The contributors are chemists and nonchemists considered to be experts in the use of Fourier Transforms and other domain transformers for building vastly improved analytical instruments and processing spectral data. ANALYTICAL CHEMISTRY, VOL. 52, NO. 5, APRIL 1980
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During the past four years, no doubt several excellent basic books on statistics and mathematics have appeared. The scope of this review does not allow a review of all of these books. However, a new book by Box, Hunter, and Hunter ( B l ) certainly deserves the strongest recommendation for its quality as well as for its many chemical examples. All of the above books are collections of contributions from a number of chemists, most of whom are analytical chemists. This fact, and the pronounced redundancy of chapter authors and subject material in these books, is quite natural for a new area of science a t a first stage of evolution. As chemometrics matures through stages of evolution, textbooks will be written, the notation and terminology of the field will be standardized, and a useful organization of the field will be formed. T h e previous authors of this review had some difficulty organizing the material they reviewed. This difficulty was shared, but to a lesser degree, by the present author. T h e remaining sub-titles used in this review came about from a manual cluster analysis. The nine topics finally selected seem t o describe most of t h e tools or objectives of present day analytical chemometrics. In some sense, these topics represent the form of the substance of chemometrics as it is practiced today. It will be most interesting to see how this outline of chemometrics changes in the coming years. The present form is not a perfect outline of chemometrics, as the reader will find papers that may perhaps fit better in other sections and a number of papers that fit well in two or more sections of this review. At this point, the author offers his sincere apology for the possibly inaccurate or unrepresentative form and erroneous, biased, or incomplete substance of this review.
constants on kinetic methods of analysis. He concludes that although kinetic methods are usually faster than equilibrium methods based on the same chemistry, they are not as precise. The propagation of random errors in the trace analysis of organics in environmental samples has been investigated (C10). T h e stages from sampling through analysis are considered. T h e paper should be read by all environmental analytical chemists. A number of papers have been published on expressing and maximizing the precision and accuracy of analytical methods (C14, C15, C17, C20, C21) and other topics such as approximation of distributions ( C l , C2) and goodness-of-fit measure improvements (C13). As the limits of precision, detection and resolution for analytical methods are pushed further and further, the more advanced methods of statistics, such as those usually discussed a t the annual Gordon Research Conference on Statistics in Chemistry and Chemical Engineering, will begin to find application. Likewise, as society’s problems become technically more complex, it will be more difficult for scientists to ignore the use of sampling theory and experimental design in large studies involving thousands of measurements as is not often the case. Finally, Creedy ( C 6 ) has shown that statistics is applicable to chemists as well as chemistry. In a study that uses data from chemists in the United Kingdom, he uses multivariate methods to relate chemists’ personal attributes to their salaries.
STATISTICAL APPLICATIONS
Calibration Curves. There can be no doubt of the importance of calibration in analytical chemistry, or, for that matter in any rieasurement science. The standard statistical methods offer much for the estimation of the uncertainty of analytical measurements and the propagation of these estimates through the calibration method to the final analytical result. Unfortunately, all b u t the most basic of statistical concepts (e.g., standard deviation) are often ignored and many, if not the majority of, analysis results are reported without uncertainty estimates of any kind. Even more unfortunate is the sad fact that these results are considered as “truth” in making important decisions. One of the most encouraging items to report in this review is the considerable amount of attention paid by analytical chemists to calibration methods in the past four years. Much of the work would not be considered novel to a trained statistician. However, the topic is important enough that a periodic reintroduction to the statistics of calibration, combined with some novel twist of course, is welcome. Each calibration method assumes some form of a mathematical model that is used to relate an analytical response to a concentration (the estimation of other model parameters is reviewed later). At a basic level, some attention has been paid to the evaluation of linearity of clinical analytical procedures by the method of least squares ( 0 1 1 ) , the use of polynomial approximations t o statistical distributions so as to aid the calculation of confidence limits to linear least squares fits via hand held calculators ( D l , 0 3 ) , and the improved results when using a regression model forced to pass through the origin when appropriate (026). Using basic statistical methods to report confidence limits with linear calibration curves usually requires that a variance of the data points be uniform (homoscedastic) over the range of calibration. In the case of counting experiments, the variances inherently are not homoscedastic. Schwartz (024) utilizes a variable transformation method that takes advantage of the a priori knowledge of counting variances to derive confidence limits for this calibration problem. Analytical chemists no longer avoid nonlinear calibration curves or nonlinear limits of linear calibration curves. However, the reliable estimation of confidence limits and detection limits is more difficult, usually requiring more exact estimates of measurement variances. Some very excellent papers have appeared over the past four years that begin to provide analytical chemists with the tools t o work with nonlinear calibration models. Midgley ( D I 7 ) shows how nonlinearities can be detected and corrected with ion-selective electrodes. Liteanu and co-workers (D16) have made a detailed study of ion sensitive membrane electrode detection limits in the nonlinear domain. Ingle, Jr., and Wilson (014) report on the
I n some ways, this section is a catch-all for a number of topics, each of which is not represented by enough papers to warrant a separate section. This section would actually be the largest section in the review if the statistical applications in the Resolution, and State and Parameter Estimation sections were not segregated to show current emphases. Even for its size, i t is perhaps the most important section of this review for at least two reasons. First, the topics of sampling, experimental design, accuracy and precision optimization, etc. are extremely important but still, very often ignored. Second, the few papers reviewed here are really quite excellent accounts of very good work. Again the books cited in this review offer a selection of very readable introductions to most of the statistics required of analytical chemists. There have been publications warning of imwroDer uses of statistics in chemistrv _ (C18. . , IB6. . -w 192 and p 2191). In an effort to standardize the use of statistics in analvtical chemistry, the Analytical Chemistry Division of the fnternational Union of Pure and Applied Chemistry (C22) in 1975 approved certain nomenclature, symbols, and units for data interpretation. This effort should be expanded with the help of noted practicing statisticians. Scilla and Morrison ( C l 9 )provide equations which estimate the degree of heterogeneity present in in-situ microsampling of solids by ion microprobes. They also recommend procedures for establishing the number of replicate analyses required to achieve a desired precision. The proper sampling of lots of material has also been carefully investigated (C12, C16). Experimental design and the analysis of variance have been subjects of very informative tutorial articles (CS, C9) and have been applied to method comparison by least squares ((25)and laboratory comparison when unequal members of replicates result ( C l l ) . T h e topic of error propagation has surfaced in some very fine papers. Hayes and Schoeller (C7) show that in pulse counting experiments, deadtime uncertainties lead to total count uncertainties. They derive an expression for the maximum count rate for any measurement. They also show t h a t the minimum counting time varies inversely with the third power of the required precision and that, since higher precisions require lower count rates, 0.1% is the highest precision attainable with reasonable counting times. In a special report, Cooper ( C 4 ) shows how computer data acquisition and data processing with small word-size computers can lead t o serious error propagation. Carr (C3) studied the effect of random variations in rate 114R
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difficulties they encountered with nonlinearities in estimating detection limits for molecular fluorescence and chemiluminescence measurements. And Mitchell and co-workers ( 0 1 8 ) approach the nonlinear calibration problem with a piecewise linear method and show that accuracy and precision of analysis results from atomic absorption spectrophotometry are markedly improved. In three papers published in this journal, Schwartz (022, 023,025) tackles the problems associated with using nonlinear calibration curves directly. Several methods for estimating confidence limits for analyses obtained using nonlinear calibration curves of arbitrary form are proposed, tested, and extended to handle cases where the variances of the data along the curve are nonunifoim. These papers should be quite useful in the future as analytical chemists attempt to extend the operational range of analytical methods. An example of this which uses a cubicis the paper by Frigieri and Rossi (D8) spline function to extend the operational range of spectrochemical analysis using microphotometer readings of photographic emulsion from the gross fog level of the emulsion to very high absorbence values approaching saturation. Interferences. Calibration, and chemical analysis, in the presence of interferences usually lead to incorrect analysis results. Interferences must be detected and then either eliminated or compensated for in some way. The most difficult type of interference to deal with is a sample interference or matrix effect which occurs when the sample to be analyzed has different concentrations of matrix elements than the standard used for calibration; a not uncommon case. Analytical chemists working with “real life samples” spend much of their time characterizing and compensating for interferences and, it is most encouraging that, in recent years, there seems to be new research emphasis on this topic. The bulk of the literature is in the area of X-ray fluorescence analysis where interelement interferences are common. The approaches taken vary from theoretical treatments to empirical curve fitting (02, 0 4 , 0 1 0 , 0 1 2 , 0 2 0 ) . A paper by Birks and coworkers ( 0 6 ) reports on a combined approach utilizing mathematical expressions with fundamental parameters and empirical coefficients to correct for interelement effects in multicomponent X-ray fluorescence analysis. The authors obtain accuracies of 1.2% relative for major components and 1.6% relative for minor constituents. Alternatively, Claisse and Thinh ( 0 5 ) compare X-ray line intensities from a sample to the line intensities of standards of a similar type of material. T h e authors claim accuracies that approach the ultimate accuracies of X-ray fluorescence analysis. Perone and co-workers ( 0 9 )take a completely different and most imaginative approach to the interference problems associated with analysis by ion-selective electrodes in a flowing system. In one experiment they use a computer controlled flow system to mix standards in one cell to match the electrode potentiais of a n unknown solution flowing in a second cell. Their method, which allowed a sodium and potassium analysis to 1% relative, uses either a simplex or Newton-Raphson algorithm to match the electrode responses by adjusting standard concentrations. The authors experienced little difficulty with electrode potential drift but a new electrode calibration procedure that minimizes errors in the presence of drift could presumably aid the analysi. ( D l 5 ) . T h e important method of standard additions has also received attention since the previous review. Hosking and coworkers (013) have investigated the shortcomings of the standard addition method as applied to atomic absorption analysis of calcium in the presence of silicon and aluminum. Franke and co-workers (07)and Ratzlaff ( 0 1 9 )do a thorough job of investigating the conditions that should be used to improve the precision of the method. The former paper concludes by proving that, when the linear model is valid, optimum precision is obtained by a single addition of the largest amount of standard that can be made while remaining in the linear response range of the instrument. Finally, the method of standard additions has been generalized to provide a means of detecting and quantifying interferences and performing multicomponent analyses in the presence of interferences no matter how complex (021). The “Generalized Standard Addition Method” can be used for the simultaneous analysis of any number of analytes using analytical sensors that are far from fully-selective. Interferences are not wasted but are used to aid an analysis giving rise to
a conservation of analytical signal.
RESOLUTION Resolution is one of the most important elements of analytical chemistry. T h e word, resolution, ha5 different connotations in chemistry. The entire Geld of chromatography has the common goal of resolving the components of mixtures. In another sense, the spectroscopist often makes instrumental adjustments in order to increase spectral resolut,ion, usually at the expense of sensitivity. In still another sense, an atmospheric chemist might make several analytical measurements on several samples taken on a geographic sampling grid in order to resolve the sources of air pollution in a city. In this section, studies dedicated to resolution of one type or another are reviewed. I t is a common goal that has received a considerable amount of well deserved attention during the review period. T h e reader will note that the dominant resolution goal in the review period is spectral resolution and the tools most often used are factor analysis (usually always principal components analysis), multiple linear regression analysis, and the Fourier convolution theorem. Following a review of advances in factor analysis and its application to the noiispectral resolution problem, the spectral resolution research is reviewed in three parts. Factor Analysis. The last review covered the introductory papers of chemical factor analysis and only a few tutorial papers have been published since (EIO,E19, E20). Significant and useful advances in the determination of the number of factors and experimental error in a data matrix have been made by Malinowski (E13,E13) and Cochran and Horne (E5). These points are extremely important and, unfortunately, brushed aside in many of the chemical factor analysis papers. An earlier paper attempts a careful examination of the different results received with the four dispersion matrices that can be factor analyzed (E7). The paper also tests various measures proposed for estimating the “true” number of factors in a data matrix and then offers an alternate method that is most effective when estimates of the measurement uncertainties are known. In fact, the “uncertainty perturbation” method can be used to test the results of almost any multivariate data analysis method. By factor analyzing a data matrix consisting of the ion current intensities at all or selected mass-to-charge ratios for a collection of compounds, various research groups have been able to decipher fiagmentation patterns ( E l 7 , ElR), classify compounds on the basis of kinetic and mechanistic dissociation reactions of parent ions (E16),and determine the relative number and sequence of nucleosides in underivatized nucleotides ( E 3 ) . Factor analysis has also been used with success to analyze collections of Fourier transform infrared spectra of blended and cross-linked polymers ( E Z ) ,carbon-13 NMR chemical shifts of alkyl halides ( E X ) ,the relat,ionship between metal chelate stability and analytical reagent selectivity ( E 6 ) ,gas chromatography data (Ed,E12), chromatographic profiles of food products in relation to sensory data ( E l ,E15), and data from environmental studies (E8, E9,E l l , [B6, p !0]1. Mixture Resolution by Factor Analysis. This section reviews studies aimed a t resolving the components of mixtures by using factor analysis, principal components analysis, or matrix rank analysis to analyze a matrix of spectral intensities. The mathematical methods used and the experimental approaches are all roughly equivalent even though data from different instruments are analyzed. It is quite interesting to note that even thoiJgh the last fundamental review orl statistics (A5)covered the tools to solve the spectral resolution problem and Reeves and co-workers ( F 7 ) have published an excellent report on the methods available to solve the problem using matrix algebra, some of the papers in this section apply only parts of complete solution methods, propose a new modification, but do not completely resolve mixture spectra into component spectra. The resolution of an unresolved gas chromatography peak can be accomplished by factor anaiyzing a number of mass spectra measured as the mixture is eluted. If the components have equal elution times, they presumably cannot be resolved by factor analysis, although Halket ( F 2 ) claims the contrary under certain conditions, and least squares procedures (next section) must be used. However, if the elution times are not ANALYTICAL CHEMISTRY, VOL. 52, NO. 5, APRIL 1980
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exactly equal, a number of studies have shown that some degree of resolution is possible ( F l , F4, F6, F8, F9). Malinowski and McCue (F6) use the target transformation method to yield the identity of the components, i f they are contained in a spectrum library, and a quantitative analysis of the mixture. Mixture resolution by spectral analysis via matrix analysis has also been applied to NMR spectra (F5),infrared spectra (FIO), and molecular fluorescence spectra (F3, FI1). M i x t u r e Resolution by Multiple Regression Analysis. When the identity of the components of a mixture are known, reference spectra available for each component, and the component spectral intensities add linearly, multiple linear regression analysis, using minimization algorithms of one form or another ( G l ) , can be used to quantitatively analyze a mixture spectrum. Often, because of measurement error, the least squares criteria produces negative quantities for component amounts. To overcome this, Leggett (G9)suggests the use of constrained least squares and simplex optimization. However, in an excellent paper by Gayle and Bennett (G5), simulation studies are used to investigate the consequences of assuming that the component identities areknown, when they are not. It is no surprise that incorrect results are obtained. What is most interesting is that the unconstrained least squares criteria used by conventional multiple regression programs gives an indication of the incorrect assumption while the constrained calculation methods do not. Least squares regression analysis has been recently applied to mixture analysis via kinetic and spectral analysis with a vidicon spectrometer (GIO),X-ray fluorescence spectra (G11, G12), molecular fluorescence spectra (G13),mass spectra (G4), electronic, vibrational, and emission spectra ( G I , G7), potential-step voltammetry ( G 2 ) ,X-ray diffractometry ( G 8 ) , time-resolved phosphorescence spectra ( G 6 ) , and kinetic analysis (G3). In the last paper mentioned ( G 3 ) ,Connors presents a method that is similar to the generalized standard addition method developed by Saxberg and Kowalski ( 0 2 1 ) . However, instead of making standard reagent additions, the different time decay characteristics of the components are used to affect resolution. Mixture Resolution by Alternative Methods. When the components of a mixture are known, regression analysis methods can be used for mixture resolution. When the components have not all been identified, the various methods related to factor analysis can be applied if more spectra than the number of components are available. However, these are not the only approaches available to the analytical chemist for simple peak resolution or complete mixture resolution. A number of alternative approaches have appeared in the chemical literature during this review period. Simple reference spectrum subtraction has been suggested for mixture resolution using polarographs ( H 4 ) , infrared spectra ( H I 5 ) ,and mass spectra ( H 2 ) . Alternative graphical methods have been developed for kinetic ( H 7 ) ,potentiometric (HI 7 ) ,and spectrophotometric ( H a ) analysis of mixtures. These methods are simple to use and can be very effective under certain situations. Perone and co-workers (H9,H21, H22) have been successful a t applying feature selection and classification methods of pattern recognition to separate severely overlapped voltammetric waveforms. They have applied their methods on-line with computer controlled instrumentation, in contrast to several of the published mixture resolution studies which have amounted to feasibility studies. Lam, Forst, and Bank ( H 1 3 ) use the simplex method for spectral deconvolution of energy dispersive X-ray spectra and electron microprobe analysis. Fourier deconvolution and other deconvolution methods are used in electron spectroscopy (H3, H5, H6, H l l , H16, H20), X-ray analysis ( H 1 2 ) , photoluminescence spectrometry (HI&?), single photon counting fluorescence spectrometry ( H I ) ,metastable peak mass spectrometry (HIO),and low energy electron diffraction studies (H14). Finally, O’Haver and Green ( H I 9 ) propose the use of derivative methods to resolve overlapping Gaussian bands. They claim the lowest errors in most cases studied.
STATE AND P A R A M E T E R E S T I M A T I O N T h e analytical chemist is fortunate in having many theoretical and proven empirical mathematical models with which ll6R
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to work. Since the models usually are not perfect, and since measurements contain uncertainty, the models must be “fit” to the measurements using appropriate algorithms and “goodness of fit” criteria. Early analytical chemists were forced to linearize their models and use the least squares criterion for model fitting. The availability of computers has lifted the dependence on linear modeling and analytical chemists are now using sophisticated mathematical methods and powerful computer algorithms to fit multiparameter models which are often nonlinear in their parameters and contain many more than one or two parameters. When the primary interest of a study is to determine the parameters of a model from experimental data, many scientists and engineers regard the study as parameter estimation. When providing estimates of the data or the derivatives of the data is the goal, the study is often called state estimation. Parameter estimation is usually discussed in two parts, linear parameter estimation and nonlinear parameter estimation, depending upon whether the model parameters are linear or not. When in doubt, try to solve the normal equations derived from the model. The attempt will be unsuccessful for nonlinear models unless a model transformation is made. State estimation is usually concerned with smoothing, filtering, or prediction depending upon whether the data estimation is in the past, present, or future, respectively. It is unfortunate indeed that systems engineers, mathematicians, statisticians, and researchers in other disciplines all have different terminology and notations for these topics with little hope of standardization foreseen. This author prefers the simplicity of state and parameter estimation but warns the reader of the inadvertent deception in the literature. P a r a m e t e r Estimation. The September 1978 and November 1979 issues of ANALYTICAL CHEMISTRY contain Instrumentation articles on linear and nonlinear parameter estimation written by experts in the field (17, 18). These reports cover both the development of models and the use of currently available methods for estimating model parameters. Their view of these topics is primarily from the engineering viewpoint. The interested reader would be well advised to supplement the reading of these important articles with selections from the statistics literature (cf., B l ) . Pattengill and Sands (129)report on the statistical significance of the parameters derived by linear least squares analysis and Deming and Morgan (113) consider the use of linear models in clinical chemistry. DeMaine and co-workers propose automatic curve fitting (111,112). They have developed computer programs to fit curves with automatically selected models and then provide tests of reliability. Seelig and Blount use the Kalman filter for recursive estimation of concentration values from anodic stripping voltammograms (134,135) and achieve greater precision than other methods used for ASV data analysis. Although their papers might well be considered as calibration papers as concentrations are the parameters in their model, the recursive Kalman filter is one of the most important tools of estimation theory. The multiparametric curve fitting methods of Meites are based on fitting models to experimental data using nonlinear regression analysis. The methods enjoy continued utilization by Meites and co-workers (126, 128), Campbell (19), and Meloun and Cermak (124). With the exception of Campbell’s paper (19),most of the applications of multiparametric curve fitting have been to potentiometric titrations. These studies are most interesting as they demonstrate t h a t analysis is possible even when no discernible inflection in the titration curve can be seen with the naked eye; one of the best pattern recognizers in existence. Meites (123) has recently prepared an excellent review on the use of nonlinear regression analysis and has proposed some new methods of data analysis as well. Other investigators have also reported the application of nonlinear regression analysis to potentiometric and spectrophotometric data (11,13,116,132). Frazer and co-workers (115) use computer raphics to examine Gran plots and error functions in orcfer to determine the equivalence point of potentiometric titrations. The estimation of rate constants is an active area of analytical chemistry as kinetic methods are used more frequently. Mieling and Pardue (125) describe a new approach based on multiple regression to analyze data from first-order kinetic analyses. The method is insensitive to extraneous parameters
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such as p H and temperature. Methods based on pattern recognition (114),cubic-spline interpolation and simplex optimization (118),and other techniques (119,133,136)have also been used to estimate kinetic mechanisms and parameters with success. Recent studies, where models are fit to experimental data in order to estimate useful chemical parameters, are many. They include, but are not limited to, the statistical analysis of membrane electrode selectivities (122),characterization of plasma discharge lamps for atomic analysis (13I),modeling of pulse height distributions for semiconductor X-ray detectors (137), infrared (14,15, Z I O ) , Raman (161,and NMR (127)band shape analysis, the determination of polymer properties from Fourier transform infrared spectra (12),and for chromatographic retention time interpolation (117 ) and band broadening studies (130). Time series analysis is considered by some to be in the realm of a parameter estimation problem. The relationship between autocorrelation and analysis of variance in time series analysis has been studied (120) as well as the effect of missing data (121). State Estimation. T o the author‘s knowledge, there are no published chemical examples of digital prediction, in the classic sense, by the standard, state estimation methods. This is probably due to the time frame in which chemical research is done. Unlike the engineer tracking a space vehicle, the analytical chemist need not predict a spectrum before it is measured; it is simply measured when required. Digital filtering implies a real time filtering and smoothing implies a data analysis after the data have been collected. During the review period, Enke and Nieman (J5) published a special report on smoothing by polynomial least squares. It is presumed that this report reduced the subject to practice in chemistry as very few other papers have appeared on this subject. Betty and Horlick (J2) and Madden ( J 6 ) have published useful reports on the most popular Savitsky-Golay filters, a more efficient polynomial smoothing algorithm has been introduced ( J 3 ) ,and the importance of data smoothing before numerical differential has been recognized ( J 4 ) . Finally, it is most interesting to contemplate the future of smoothing spectral data in the light of the introduction of tapped analog delay lines introduced to analytical chemistry by Betty and Horlick ( J I ) . These devices can be used to store a time sequence of an analog signal. The stored signal can be multiplied by predetermined weights and summed to effect transverse filtering; effectively an analog crosscorrelation of the imput waveform with a weight function. Filtering was born in the analog domain with simple passive RC filters. It then evolved to complex digital filtering which provided very real improvements. With the new analog storage devices, sophisticated smoothing could very well become a totally analog operation.
PATTERN RECOGNITION T h e subject of pattern recognition received a large share of attention in the last review, most likely due to the number of publications appearing in chemical journals during the review period on that topic. The techniques of pattern recognition promised to relieve the analytical chemist from the overwhelming burden of extracting useful chemical information from the voluminous amount of data errupting from computerized analytical instrumentation. Although the review discussed a few chemical applications of pattern recognition, mostly structure elucidation via spectrum analysis, the majority of the papers appearing from 1972 to 1976 emphasized improvements in methodology. In this review, pattern recognition has again dominated other areas of chemometrics on the basis of the number of papers published in chemical journals during the review period. However, the reader will note the change of emphasis from method development t o chemical application which is a sign of maturity and a clear demonstration of utility. Resides the chapters on pattern recognition in the books mentioned earlier, a few more introductory articles have appeared which should prove valuable to the beginner. An enlightening article by Wold and co-authors ( K I ) proposes four levels of pattern recognition in chemistry. The first level is the classic categorization of samples into one of a number of defined classes. Almost all of the published works in chemistry are at this level. The second level is a n extension
of level one to allow outlier samples, not belonging to any of the defined classes, and/or the situation, called by asymmetric case, where one class has no systematic structure and cannot be modeled. An interesting example of level two can be found in a somewhat obscure journal (K6). Level three demands the use of classification methods such as Wold’s SIMCA method ( K 9 ) that extracts model defining parameters from defined classes which can then be related to properties external to the classification problem. At the time this review was written, there were no published examples of level three in analytical chemistry. The published examples are in medicinal chemistry (cf., K 2 ) . Level four is really a generalization of multivariate analysis. Classes of samples are modeled by latent variables or unobservable properties derived from several measurements. The classes are then related by a path relationship. The new method from Herman Wold, called Latent Variables with Partial Least Squares (PLS) Estimation ( K 8 ) is the tool for level four problems. Gerlach and Kowalski, in collaboration with H. Wold ( K 3 ) have recently applied this new method to estimate the impact of a mining operation on local water quality by analyzing water chemistry measurements. The generality of the PLS approach is supported by the fact that multiple linear regression analysis, principal components analysis, and canonical correlation analysis are all special cases in the framework of H. Wold’s method. Parsons and co-workers (K5) offer a well written report on the applications of pattern recognition published by the end of 1977. Their report is both a review and a research paper and is certainly recommended reading. One year earlier, MacDonald (K4) published a report that emphasized the time share aspects of pattern recognition and computer programs available at the time of publication. Finally, Wilkins ( K 7 ) emphasizes mass spectral data analysis in a report on interactive pattern recognition. Methods. There can be little doubt that the most published pattern recognition method in the chemical literature is the linear learning machine. This method has come under close scrutiny recently and has drawn a considerable amount of fire from unsatisfied users (L4, L10, L12, L14, L16). Useful criteria for the confidence that can he attached to classification results have been proposed ( L I I ,L15). The simplex method has been suggested as an improvement to linear discriminant analysis ( L I , L2, L5, L6) but the feature selection problems and other problems associated with the use of linear discriminant functions still prescribe cautious application. For the spectral identification problem using any method, McLafferty ( L 9 ) suggests performance measures for prediction and evaluation. Two studies (L8, L13) have been reported that attempt to evaluate the performance of classification methods commonly employed by chemical pattern recognizers. These papers should prove useful to chemists with classification problems. As an aid to displaying multivariate chemical information, Lin and Chen (L7)propose a new three-dimensional display system. They claim that their Representation Space Transformation fulfills the criterion of preserving the proximity relationship between sample points on pairs of reference points. Their claim has been disputed by Drack ( L 3 ) who states that the method is, however, still useful. Finally, the use of measurement uncertainties has been added to several pattern recognition methods (R6, p 14) by propagating the uncertainty of the measurements, through the calculations to the results. This is an important improvement and can significantly change the results of an application. The pattern recognition methods available to chemists are many and varied. In view of the decided shift in emphasis toward applications, to be discussed next, it seems that chemists are satisfied with the depth and breadth of methods available to them in readily available computer programs. It is hoped that the future will see a healthy balance between method development and application. Very few tools can tolerate a long period of use without a proper amount of maintenance and improvement. Applications. When more than three or four measurements are made on a collection of samples, pattern recognition methods and the methods of multivariate statistical analysis can he used to find useful relationships among samples or among measurements. The methods are quite general and ANALYTICAL CHEMISTRY, VOL. 52, NO. 5, APRIL 1980
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might be considered as tools for the examination of n-dimensional plots. Since all scientists rely heavily on plots, and since the covariance of measurements can be as important as the variance, it is no surprise that pattern recognition methods have enjoyed such widespread application in chemistry. Chemistry is certainly a multivariate science. In covering the chemical applications of pattern recognition, only those studies utilizing measurements made via chemical analysis were selected. T h e range of problems that can be solved by analyzing chemical measurements with these tools is so large that this review will no doubt fail to be all inclusive. I n fact, the very process of selecting an analytical method is the subject of a very interesting pattern recognition application (M37). Also, pattern recognition has been used for resolution of spectra and parameter estimation so other applications can be found in other sections of this review. The largest area of application remains to be the computer analysis of spectral data. However, the applications have expanded considerably in breadth and quality. For example, Perone and co-workers ( M 8 , M47) combined the nearest neighbor classification rule, enzymatic hydrolysis, and mass spectrometric procedures to develop a promising technique for identifying amino acid sequences in polypeptides. The technique, demonstrated to be 100% accurate for the Nterminal amino acid test and 97% accurate for the C-terminal test, has been fully automated (M47) and should prove to be quite useful for biochemical research. Other quite interesting work using pattern recognition to analyze mass spectra of steroids and other large compounds is being done in Vienna by Varmuza and co-workers (M31, M32, M39, M40). This work, coupled with the applications of pattern recognition to Infrared, Raman, and NMR spectra ( M l l , M12, M29, M35, M45, M46) show marked progress toward the most significant goal of automated structure elucidation. Other interesting spectrum and waveform applications include the classification of organic compounds from voltammetric data ( M 9 ) ,quantitative analysis by anodic stripping voltammetry ( M 4 ) , electrode process analysis ( M I 9 ) , the analysis of pyrolysis gas chromatograms ( M 2 0 ) , chemical characterization from carbon Auger spectra (M16) and the classification of atomic emission spectral lines (M27,M28). Clinical chemistry continues to rely more heavily on new methods of chemical analysis. Here is where the application of pattern recognition may have the largest impact in the future. The data that have been analyzed can be the standard chemical measurements which, individually are proven indicators of health disorders but analyzed in concert, may have even more revealing information. For example, Massart and co-workers (M13) have used discriminant analysis on blood constituent analyses to diagnose thyroid diseases. Other workers have diagnosed calcium related disorders (M22) and liver diseases ( M I 4 ) ,also from standard clinical analytical measurements. Bacteria have been classified by measured resistance to standard antibiotics ( M 6 ) and fungi have been classified using pyrolysis-gas chromatograms of the fungi (M3). T h e latter work is quite interesting, as the products from pyrolysis were represented as areas under numbered peaks in the chromatogram with no identification of the peaks attempted. This last work points to the application of pattern recognition methods to medical diagnosis using data that are not normally and routinely used for that purpose but are known to contain useful information. Zlatkis and co-workers (M48) have analyzed the volatile compounds in only 70 FL of serum of control and virus-infected patients by capillary column gas chromatography. The nearest neighbor classification method classified 85.7% of the test cases correctly and preliminary results indicate the possibility of direct assessment of virus susceptibility. In another excellent study, Novotny and coworkers (M24) use capillary column gas chromatography to first separate the volatile components of urine in diabetes mellitus patients. These data are then used for metabolic profiling with excellent success. Webb and co-workers (M41,M43, M44) analyze trace metal concentrations in animal human heart tissue to classify the exact origin of the tissue. Then, using the knowledge gained from these studies, they examine the changes in serum levels of 14 trace metals after patients have experienced acute myocardial infarctions (M42). 118R
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Using pattern recognition methods and concepts similar to those mentioned above, various investigators have reported progress in odor research ( M 7 , M25), flavor research in food products and wine ( M I ,M21), forensic chemistry (MIO,M33, M34), archaeometry ( M 5 , M23, M26), the interpretation of groundwater chemistry ( M 2 ) , and eveh space exploration (M30). Finally, an application area that has seen some activity and promises more for the future is the application of pattern recognition to the multitude of chemical data from environmental studies. The characterization of atmospheric particulate composition ( M 1 5 ) , the identification of sources of pollutants (MI@, and air pollution control (M36)are examples of pattern recognition applications to atmospheric pollution. Water pollution studies include the interpretation of the chemistry in lake sediments (MI71 and the description of water quality (M38).
OPTIMIZATION Mathematical methods to efficiently search for the optimal conditions for analytical methods, or provide a maximum response per amount of analyte from an analytical instrument, continue to be a focus of research. In the last review, i t was noted that the simplex optimization technique was beginning to find application in analytical chemistry and that a number of other methods could also be used. I t is quite natural that Deming, who introduced the technique to analytical chemists, has co-authored the first major review paper on simplex optimization in analytical chemistry ( N 3 ) . Two additional introductory papers have appeared on simplex optimization, one written primarily to chemists in food science ( N 4 ) and one to chemical educators (N13). Using these sources, the literature referenced in the last review (A5), and chapters in the chemometric books (B6, B 7 ) , analytical chemists now have a number of opportunities to learn more about this useful method. A useful improvement to the simplex optimization method has been made by Denton and co-workers ( N I I ) . Their method uses a few additional measurements per simplex and a cubic least-squares fit to accelerate the size of the step made for any one simplex. They show how this improvement leads to rapid convergence using fewer measurements (consequently less time and cost). Additionally, failure of the method to find the optimum in the presence of noise due to premature diminuation of the simplex, adherence to false ridges on the response surface, or an inability to find the global minimum is reduced by using their method. The paper is quite complete as the authors use a computer controlled atomic emission spectrometer (four parameters are controlled) to demonstrate the advantages of their simplex method. New applications of simplex optimization include the preparation and operation of microwave excited electrodeless discharge lamps for atomic analysis ( N 8 ) ,the optimization of a phosphate determination by photometric analysis using molybdenum blue (N15),some very excellent applications in chromatography ( N 6 ,NIO, N12, N16),the optimization of an automatic analysis of calcium ( N 9 ) ,and an application to energy dispersive X-ray fluorescence ( N 7 ) . Watson and Carr (N16) use simplex optimization to allow the analyst to specify realistic chromatographic performance goals (Le., analysis time and peak-to-peak resolution) and then optimize the analysis. T h e example used in their paper is a gradient separation of five PTH-amino acids and involves the optimization of five variables. The authors also use factorial analysis in the region of the optimum to learn more about the operational characteristics of their analytical method. Rubin and Bayne (N22) optimize the response of a nitrogen-phosphorus detector for a gas chromatograph in a textbook demonstration of how to use optimization methods. They begin their three-parameter optimization with a statistically designed and evaluated factorial experiment in order to approach the optimum conditions for the detector. They then use simplex designs to define the optimum set of conditions. Optimization without the simplex method is the topic of other works in analytical chemistry. Davis and Pevnick ( N l ) develop an optimization theory to optimize factors affecting enzyme activity in the kinetic analysis of various substrates. Perone and co-workers ( N 1 4 ) specify desired performance characteristics for anodic stripping voltammetric analysis; then
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they use a theoretical model and on-line computer control to achieve the desired response. Erni and Mueller (N5)optimize a complex wet chemical continuous flow determination of nitrogen and phosphorus by first establishing a mathematical model for each separate stage of the analysis. Then the nonlinear optimization problem, with constraints, is solved. Finally, Massart and co-workers (N2) use operations research techniques to select optimal functional probes for gas chromatography.
INFORMATION THEORY During the past four years, there has been a renewed interest in applying the principles of information theory, developed primarily by Shannon, to analytical chemistry. The contributions published during this review period come almost entirely from the laboratories of Dijkstra in The Netherlands and Eckschlager in East Germany. While Eckschlager is primarily concerned with the mathematical agspects of information theory (04) and how it can express the “information efficiency” of analytical methods ( 0 2 ) , Dijkstra’s interest in applying information theory is to improve analytical methods. For example, the coding of spectral data in mass spectrometry (07, OS) and infrared spectrometry (03,05) is significantly improved with the aid of information theory. Also, in the area of chromatography, the information content of TLC identification procedures can be determined ( 0 1 ) . The four operations in analytical chemistry that have been aided in the past by information theory are: chemical classification, analytical method selection, optimal combination of analytical methods, and chemical information storage and retrieval. These topics have been reassessed with respect to the entropy of the events a priori the analytical process and the entropy of the conditional events at the input of the process with respect to the events at the output of the process (06).
The preoccupation of these analytical chemists with the information content of analytical measurements is a healthy sign. I t is perhaps the recognition of analytical chemistry as a modern information science. However, certain facts should not be overlooked. Measurements do not contain only information as they are associated with uncertainty and it is the propriety of Statistics to estimate and express uncertainty. Also, it may be a long way from measurements, even very certain measurements, to useful information. Knowledge is the end product of science. Measurements, which hopefully lead to information which can, with a proper experimental design, advance knowledge, are the means to that end.
TRANSFORM DOMAINS Every chemist is aware of the advantages gained by applying Fourier transform mathematics to infrared and nuclear magnetic resonance spectrometry. The waveforms that are actually measured during the experiment are of very little use when analyzed by the chemist. However, Fourier analysis transforms the information to a domain more familiar to the chemist. Practically everything that an analytical chemist needs to know about transform techniques in chemistry is contained in the book edited by Griffiths (B4). However, some advances in Fourier analysis have been published recently. Also, the applications of Fourier analysis have broadened and a small amount of work on other types of transforms is now in print. Recent studies have reported important results concerning the use of Fourier transforms in spectrometry. Winefordner and co-workers (P3)showed that if spectral components of major interest are near components of higher intensity, the multiplexing nature of Fourier transform spectrometry can lead to lower signal-to-noise ratios for these components than conventional dispersive spectrometry, even under photon noise limited situations. The advantages and disadvantages of the commonly used practice of zero-filling prior to Fourier transformation have been addresses (P4, P7, PIO) and methods for correcting wedging errors in absorbence subtraction FT-IR have been proposed jP5). In Fourier transform mass spectrometry, Marshall (P9)has derived the theoretical expressions for signal-to-noise ratio and mass resolution. Data processing in electrochemistry has come to rely more heavily on the use of Fourier theory, principally through the
efforts of Smith. Smith (P15,P16) wrote a two-part account of how the fast Fourier transform could be used to acquire and enhance electrochemical data in the Instrumentation pages of the 1976 issues of ANALYTICAL CHEMISTRY.In addition to these introductory articles, Smith, Bond, and coworkers (P13, P14) have published accounts of the construction and application of a synchronous data generation and sampling system to assist broad band electrochemical relaxation measurements. They show that the reversible electrochemical response can be routinely obtained from a quasi-reversible faradaic admittance spectrum. Carr (P2) has also used Fourier analysis to study the transient behavior of potentiometric enzyme electrodes. Derivative and wavelength modulation spectrometry are tools usually used to resolve overlapping spectral bands and the Instrumentation article by O’Haver (P12)might be best included in the section on resolution. However, there are applications of wavelength modulation that d o not involve recording derivative spectra. These tools should be thought of as more than just spectral resolution tools. Burke and co-workers (PI 7) have significantly improved the interesting technique of correlation chromatography. By frequency modulating a steady sample flow, partition coefficients remain constant even while working in the nonlinear region of an isotherm. By this modification, the nonlinear problems caused by multiple injections are eliminated. Autocorrelation analysis continues to be an aid for time series waveform analysis with missing values (P8) and for general laboratory signal processing using discrete time analog signal processing devices ( P I ,P6). Improvements have been made to the normal-to-sequency-ordered Hadamard matrix conversion algorithm (P19),and the Abel Inversion has been used to obtain spatially resolved information from a plasma discharge (P12). Finally, Walg and Smit (P18) have developed general purpose Fourier analysis computer software. The software listing is available from the authors and should provide an opportunity for many analytical chemists to use these important tools.
IMAGE ANALYSIS This section has been included as a separate section because of its potential for the future and certainly not because of the number of papers published during the review period. The papers in this section could just have easily been included in various other sections and may perhaps be found in other sections in future reviews. They are bound together here by a common thread; dimensionality. A simple black and white photograph of an object on a background is often referred to as an image of the object. It is also a two-dimensional light intensity waveform with two spatial axes that can be calibrated in units of length and a grey scale representation of reflected or emitted light. Now, if a camera could be constructed that was sensitive only to, for example, molecules containing sulfur atoms, the image would become a chemical image with, again, two spatial dimensions of chemical information. Instruments to obtain both real and virtual chemical images are currently being used in chemical research. Since some of these instruments have the potential to collect many more measurements than chemists, and most computers, are capable of analyzing, it is fortunate indeed that many powerful image analysis methods have been developed for analyzing medical and satellite images. A recent Instrumentation article is available for a more detailed introduction to image generation and analysis in analytical chemistry (Q6). Perhaps the first application of an image analysis method, two-dimensional cross correlation via Fourier theory, was published by Morrison and co-workers ( Q 2 ) in 1977. In that article, image cross correlation is used to align different secondary ion mass spectrometric (SIMS) images produced by an ion microscope. In a later paper, Fasset and Morrison ( Q I ) use image analysis methods to observe differently oriented grains of polycrystalline iron by SIMS. Steiger and Rudenauer (Q8) demonstrate the generation of elemental concentration maps for all elements from SIMS images. Virtual images have been produced by a molecular fluorescence imaging spectrometer (Q5). The units of the two-dimensional image produced, although spatial in origin, can be calibrated as the excitation and emission wavelengths. ANALYTICAL CHEMISTRY, VOL. 52, NO. 5, APRIL 1980
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Virtual images have also been collected and analyzed by computer controlled instrumentation used to characterize and optimize complex enzyme systems (Q3,Q4). In scanning Auger microscopy, progress has been made with real image generation and processing systems (627). The development and use of high dimensional image (real and virtual) generation instrumentation and image analysis methods is certain to experience considerable growth during the next years.
CONCLUSION Analytical chemistry has long been considered an applied science. Within our science, however, many have argued for the need for basic research support. This argument is steadily gaining ground, and support for it clearly exists in this review. On the one hand, analytical chemists are using analytical measurements to solve problems in food science, forensic science, the environment, and elsewhere. The large number of chemical applications of pattern recognition, for example, provides evidence that analytical chemists are expanding their roles in analytical chemistry as an applied science. On the other hand, papers reviewed in the Information Theory, Calibration, and other sections of this review provide evidence that the very nature of the chemical measurement process is being investigated from a theoretical point of view. T h e analytical chemist doing basic research must keep abreast of the developments in electronics, physics, and now mathematics and statistics in the hope of developing a completely new analytical instrument or method or substantially improving existing ones. This is no different than theoretical chemists studying physics and mathematics in the hope of advancing our basic knowledge of chemistry. Whenever two disciplines derive substantial benefits from communication with each other, it is not long before formal or informal bridges are built between the disciplines. The bridges originally are built simply to facilitate communication but, in the long run, they tend to strengthen both disciplines. Examples are many. Chemistry's part of the bridge to biology is biochemistry. The corresponding part from biology to chemistry is molecular biology. Between chemistry and physics is physical chemistry and chemical physics. As stated in the introduction, it is hard to imagine a more natural and vital interaction than that between analytical chemistry and applied mathematics and statistics. Analytical chemists have much to be gained from a stronger interaction with these fields. Mathematicians and statisticians will in turn discover fertile areas of application of their methods and many challenging problems to solve. This review is an attempt to report the structure and scme of the substance of analytical chemometrics from 1976 through 1979. While much of the work published in the last review period dealt primarily with the introduction of methodology, this review seems to show a strong tendency toward application. Method development and application must attain a balance if a field of science is to grow with science. It is hoped that in the next four-year period this balance will be achieved and that analytical chemists will continue to improve analytical methods and extract more useful chemical information from analytical methods with the growing number of tools of chemometrics.
ACKNOWLEDGMENT T h e author expresses his appreciation for the support he received while preparing this review from Ivar Ugi of the Technical University of Munich and the Alexander von Humboldt Foundation. LITERATURE CITED INTRODUCTION
(Al) (A2) (A3) (A4) (A5)
Currie, L. A.; Filliben, J. J.; DeVoe, J. R. Anal. Chem. 1972, 44, 497R. Kowalski, B. R. J . Chem. Inf. Comput. Sci. 1975, 15, 201. Kowalski, B. R. Anal. Chem. 1978. 5 0 , 1309A. Kowalski, B. R. Chem. Ind. (London) 1978, 22, 882. Shoenfeld, P. S.; DeVoe, J. R. Anal. Chem. 1976, 48, 403R.
BOOKS
(BI) Box, G. E. P.; Hunter, W. G.; Hunter, J. S. "Statistics for Experimenters"; Wiley-Interscience: New York. 1978. (82) DeVoe, J. R., Ed. "Validation of the Measurement Process", ACS Symposium Series 63; American Chemical Society: Washington, D.C., 1977.
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(83) Fujiwara, S.; Mark, H. B., Jr., Eds. "Information Chemistry": University of Tokyo Press: Tokyo, 1975. 184) Griffiths. P. R.. Ed. "Transform Techniaues in Chemistry"; Plenum Press: New York, 1978. (B5) Hirsch, R. F., Ed. "Statistics"; The Franklin Institute Press: Philadelphia, P a . 1978. (B6) --Kowakki, B. R., Ed. "Chemometrics: Theory and Application", ACS Symposium Series 52; American Chemical Society: Washington, D.C., 1977. (87) Massart, D. L.; Dijkstra, A.; Kaufman, L., Eds. "Evaluation and Optimization of Laboratory Methods and Analytical Procedures"; Elsevier: Amsterdam, 1978. STATISTICAL APPLICATIONS
( C l ) Algie, S. H. Anal. Chem. 1977, 49, 186. (C2) Buys, T. S.; de Clerk, K. Anal. Chem. 1976, 48, 585. (C3) Carr. P. W. Anal. Chem. 1978, 50, 1603. (c4) Cooper, J. W. Anal. Chem. 1978, 50, 801A. (C5) Cornbleet. P. J.; Gochman, H. Clin. Chem. 1979, 25, 432 (C6) Creedy, J. Chem. Br. 1977, 73, 146. (C7) Hayes, J. M.; Schoeiler, D. A. Anal. Chem. 1977, 49, 306. (C8) Hendrix, C. D. ChemTech 1979, 9 , 167. (C9) Hirsch, R. F. Anal. Chem. 1977, 49, 691A. (C10) Janardan. K. G.; Schaeffer, D. J. Anal. Chem. 1979, 5 1 , 1024. (C11) Jordan, D. C.; de Alvarc, R. Anal. Chem. 1979, 57,1079. (C12) Kateman, G.; Muskens, P. J. W. M. Anal. Chlm. Acta 1978, 103, 11. (C13) Keiter. P. B.; Carr, J. D. Anal. Chem. 1979, 57, 1857. (C14) Liddell, P. R. Anal. Chem. 1978, 48, 1931 (C15) Midgley, D. Anal. Chem. 1977, 49, 510. (C16) Muskens, P. J. W. M.; Kateman, G. Anal. Chim. Acta 1978, 103, 1. IC171 PieDmeier. E. H. Anal. Chem. 1976. 48. 1296. ( C l 8 i Rowe, P. N. ChemTech 1976, 6. 266. (C19) Scilla, G. J.; Morrison, G. H. Anal. Chem. 1977, 4 9 , 1529. (C20) Tattershail, B. W. Anal. Chem. 1979, 57, 1160. (C21) Youmans, H. L.; Brown, V. H. Anal. Chem. 1976, 48, 1152. (C22) Internationai Union of Pure and Applied Chemistry, Nomenclature, Symbols. Units and Their Usage in Spectrochemical Analysis-11. Data Interpretation, Anal. Chem. 1976, 48, 2294. CALIBRATION
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Factor Analysis (El) Aishima, T. Agric. Bioi. Chem. 1979, 43, 1711. (E2) Antoon, M. K.; D'Esposito, L.; Koenig, J. L. Appl. Spectrosc. 1979, 3 3 , 351. (E3) Burgard, D. R.; Perone, S. P.; Wiebers. J. L. Anal. Chem. 1977. 49, 1444. (E4) Chastrette, M. J . Chromatogr. Scl. 1976, 14. 357. (E5) Cociiran, R. N.; Horne, F. H. Anal. Chem. 1977, 49, 846. (E61 Duewer, D. L.; Freiser, H. Anal. Chem. 1977, 49. 1940. (E71 Duewer, D. L.; Kowalski, B. R.; Fasching, J. L. Anal. Chem. 1976, 48, 2002. (E8) Gatz, D. F. J . Appl. Meteorol. 1978, 17, 600. (E9) Harper, A. M.; Kowalski. B. R. Anal. Lett 1979, 72, 693. (E10) Howery, D. G. Am. Lab. 1976, 8 , 14. ( E l l ) Joung, H. M.; Miller, W. W.; Mahannah, C. N.; Guitjens, J. C. J , fnvlron. Qual. 1979, 8 , 95. (E12) Laffort, P.; Patte, F. J . Chromatogr. 1976, 726. 625. (E13) Malinowski, E. R. Anal. Chem. 1977, 49, 606. (E14) Malinowski. E. R. Anal. Chem. 1977, 49, 612. (E151 Powers. J. J.; Godwin, D. R.; Borgmann. R. E.. ACS Symposium Series 51; American Chemical Society: Washington, D.C.. 1977; p 51. (E161 Rozett, R. W.; Petersen, E. M. Anal. Chem. 1976, 48, 817. (Eli') Rozett. R. W.; Petersen. E. M. Am. Lab. 1977, 9 , 107. (E181 Rozett, R. W.; Petersen, E. M. Adv. Mass Spectrom. 1978, 7 8 , 993.
CHEMOMETRICS fE19) Swain. C. G.; Brvndza. H. E.: Swain. M. S. J . Chem. Inf. ComDut. Sci. 1979, 19, 19. . (€20) Weiner. P. H. ChemTech 1977, 7 ,321. (€21) Wiberg, K. E.; Pratt, W. E.; Bailey, W. F. Tetrahedron Lett. 1978, 49, 4861. '
Mixture Resolution by Factor Analysis ( F l ) Halket, J. M. J . Chromatogr. 1979, 175, 229. (F2) Halket, J. M. I n "Recent Developments in Chromatography and Electrophoresis", Frigerio, A., Renoz, L., Eds.; Elsevier: Amsterdam, 1979: p 327. (F3) Ho, C.-N.; Christian, G. D.; Davidson, E. R. Anal. Chem. 1978, 50, 1108. (F4) Knorr, F. J.; Futrell, J. H. Anal. Chem. 1979, 57, 1236. (F5) Maiinowski, E. R. Anal. Chim. Acta 1978, 703, 339. (F6) Malinowski, E. R.; McCue, M. Anal. Chem. 1977, 49, 284. (F7) Metzler, D. E.; Harris, C. M.; Reeves, R. L.; Lawton, W. H.; Maggio, M. S. Anal. Chem. 1977, 49, 864A. (F8) Rasmussen, G. T.; Hohne, B. A.; Wieboldt, R. C.; Isenhour. T. L. Anal. Chim. Acta 1979, 772, 151. (F9) Ritter, G. L.; Lowry, S. R.; Isenhour, T. L.; Wiikins, C. L. Anal. Chem. 1976, 48,591. (F10) Shurvell, H. F.; Chitumbo, K.;Dunham, A.; Chawell, L. Can. J . Spec.. trosc. 1976, 27, 53. (F11) Warner, I. M.; Christian, G. D.; Davidson, E. R.; Callis, J. E. Anal. Chem. 1977, 49, 564. Mlxture Resolutlon by Multlple Regresslon Analysis ( G l ) Allen, G. C.; McMeeking, R. F. Anal. Chim. Acta 1978, 703,73. (G2) Boudreau, P. A.; Perone, S. P. Anal. Chem. 1979, 57. 81 1. (G3) Connors, K. A. Anal. Chem. 1979, 57, 1155. (G4) Fausett, D. W.; Weber, J. H. Anal. Chem. 1978, 50, 722. (G5) Gayle, J. 8.; Bennett, H. D. Anal. Chem. 1978, 5 0 , 2085. (G6) Goeringer. D. E.; Pardue, H. L. Anal. Chem. 1979, 57, 1054. (G7) Gold, H. S.: Rechsteiner, C. E.; Buck, R. P. Anal. Chem. 1976, 48, 1540. (G8) Huang, T. C.; Parrish, W. Adv. X-Ray Anal. 1978, 2 7 , 275. (G9) Leggett, D. J. Anal. Chem. 1977, 49, 276. (G10) Ridder, G. M.; Margerum, D. W. Anal. Chem. 1977, 49, 2098. (G11) Schwalbe, L. A.; Morris, R . A. Anal. Chem. 1978, 5 0 , 2155. (G12) Statham, P. J. Anal. Chem. 1977, 49, 2149. (G13) Warner, I. M.; Davidson. E. R.; Christian, G. D. Anal. Chem. 1977, 49, 2155. Mlxture Resolutlon by Alternatlve Methods (Hl) Andre, J. C.; Vincent, L. M.; O'Connor. D.; Ware, W. R. J . Phys. Chem. 1979. 83. 2285. (H2) Atwater; B. L. (Fell); Venkataraghavan, R.; McLafferty, F. W. Anal. Chem. 1979, 5 1 , 1945. (H3) Beatham, N.; Orchard, A. F. J . Electron Spectrosc. 1976, 9 , 129. (H4) Bond, A. M.; Grabaric, B. S. Anal. Chem. 1976, 4 8 , 1624. (H5) Carley, A. F.; Joyner, R. W. J . Nectron Spectrosc. 1978, 73, 411. (H6) Carley, A. F.; Joyner, R. W. J . Electron Spectrosc. 1979, 76, 1. (H7) Conners, K. A. Anal. Chem. 1978. 48, 87. (He) Connors. K. A.; Chukwuenweniwe, J. E. Anal. Chem. 1979, 5 1 , 1262. (H9) DePalma. R. A.; Perone, S. P. Anal. Chem. 1979, 57,825. (H10) Evers, E. A. I. M.; Noest. A. J.; Akkerman, 0. S. Org. Mass Spectrom. 1977, 72,419. (H11) Gurker. N. J . Nectron Spectrosc. 1978, 74, 1. (H12) Ingamells, C. 0.; Fox, J. J. X-Ray Spectrom. 1979, 8 , 79. (H13) Lam, C. F.; Forst, A,; Bank, H. Appl. Spectrosc. 1979, 33, 273. (H14) Landman, U. Faraday Discuss. Chem. Soc. 1976, 6 0 , 230. (H15) Lynch, P. F.; Brady, M. M. Anal. Chem. 1978, 50, 1518. (H16) Matthew, J. A. D.; Underhill, P. R. J. Electron Spectrosc. 1978, 74, 371. (H17) McCallum, C.; Midgley, D. Anal. Chem. 1976, 48, 1232. (H18) McKinnon, A. E.; Szabo, A. G.; Miller, D. R. J . Phys. Chem. 1977, 87, 1564. (H19) O'Haver, T. C.; Green, G. L. Anal. Chem. 1976, 48, 312. (H20) Onsgaard, J. H.; Morgen, P.; Creaser, R. P. J . Vac. Sci. Technol. 1978, 75, 44. (H21) Thomas, 0.V.; DePalma, R. A.; Perone. S. P. Anal. Chem. 1977. 49. 1376. (H22) Thomas, Q. V.; Perone, S. P. Anal. Chem. 1977, 49, 1369. PARAMETER ESTIMATION (11) Alcock, R. M.; Hartley. F. R.; Rogers, D. E. J . Chem. Soc., Dalton Trans. 1978, 2 , 115. (12) Antoon, M. K.: Koenig, J. H.; Koenig, J. L. Appl. Spectrosc. 1977, 37. 518. (13) Arena, G.; Rizzarelli, E.; Sammartano, S.;Rigano, C. Talanta 1979, 26, 1
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OPTIMIZATION
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X-Ray Diffraction D.
K. Smith* and K. L. Smith
D e p a r t m e n t of Geosciences, T h e Pennsylvania S t a t e University, University Park, Pennsylvania
This review is concerned with the field of X-ray diffraction and its applications in research and analytical characterization of compounds and materials. The emphasis of the review will be on new developments in applications arid instrumentation in both single crystal and powder diffraction, but no attempt has been made to cover all studies in which X-ray diffraction has been employed. In particular, the topic of crystal structure analysis has been ignored e x c q t in the area of new instrumentation. Likewise, studies on compound characterizatiun have been skipped where the emphasis was on the compound 122 R
0003-2700/80/0352-122RSOl .OO/O
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rather than the method. T o cover all aspects of the utilization
of X-ray diffraction would be prohibitive within limits of this
review because the method is one of the most widely used analytical tools in scientific research and in industry. This review covers the time period from mid 1977 through late 1979, trying to pick up where the last review ( 1 ) left off and to be as complete as the literature allows in December 1979. The reterences were located primarily by searching the various scieatific abstract journals Some specific journals were scanned directly as were t h e programs of specific meetings c 1980 American Cheniical Society
