Mathematics and Statistics

Molecular modeling has evolved as a synthesis of techniques from a number of disciplines—organic chemistry, medicinal chemistry, physical chemistry, chemical physics, computer science, mathematics, and statistics. With the development of quantum mechanics (1,2) ia the early 1900s, the laws of physics necessary to relate molecular electronic stmcture to observable properties were defined. In a confluence of related developments, engineering and the national defense both played roles ia the development of computing machinery itself ia the United States (3). This evolution had a direct impact on computing ia chemistry, as the newly developed devices could be appHed to problems ia chemistry, permitting solutions to problems previously considered intractable. 

Evidence of the appHcation of computers and expert systems to instmmental data interpretation is found in the new discipline of chemometrics (qv) where the relationship between data and information sought is explored as a problem of mathematics and statistics (7—10). One of the most useful insights provided by chemometrics is the realization that a cluster of measurements of quantities only remotely related to the actual information sought can be used in combination to determine the information desired by inference. Thus, for example, a combination of viscosity, boiling point, and specific gravity data can be used to a characterize the chemical composition of a mixture of solvents (11). The complexity of such a procedure is accommodated by performing a multivariate data analysis. 

R. G. Brereton, Chemometrics Applications of Mathematics and Statistics to Taboratory Systems, E. Horwood, New York, 1990. 

Throughout the 1970s, appHcations of pattern recognition were found in the chemical sciences. Other methods of multivariate mathematics and statistics were borrowed or invented, and a new discipline called chemometrics arose. In 1974, the Chemometrics Society was formed, and the first Chemometrics newsletter came out in 1976 (12). 

B. R. Kowalski, ed., Chemometrics, Mathematics, and Statistics in Chemisty, NATO ASI Series C, Mathematical and Physical Sciences, Vol. 138, D. Reidel Publishing Company, Dordrecht, The Netherlands, 1984. 

C J Brookes, I G Betteley and S M Loxston Mathematics and Statistics for Chemists, John Wiley, New York, 1966, p. 304... 

P.J. Lewi, Multidimensional data representation in medicinal chemistry. In Chemometrics. Mathematics and Statistics in Chemistry (B.R. Kowalski, Ed.), Reidel, Dordrecht, 1984, pp. 351-376. 

A. Rescigno, Mathematical foundations of linear kinetics. In Pharmacokinetics Mathematical and Statistical Approaches to Metabolism and Distribution of Chemicals and Drugs. (J. Eisenfeld and M. Witten, Eds.), North-Holland, Amsterdam, 1988. 

D. Rodbard, Mathematics and statistics of hgand assays an illustrated guide, in Ligand Assay Analysis of International Developments on Isotopic and Nonisotopic Immunoassay, ed. J. Langan and J.J. Clapp, Masson, New York, pp. 45-99 (1981). 

Mathematical Chemistry Research Unit, Department of Chemistry and Department of Mathematics and Statistics, University of Saskatchewan, 110 Science Place, Saskatoon, Canada, S7N 5C9... 

Department of Mathematics and Statistics, School of Mathematical Sciences, Monash University, Clayton, Vic. 3800, Australia Lisa.Elhott sci.monash.edu.au... 

In-Kwon Yeo received the PhD degree in Statistics from University of Wisconsin-Madison in 1997. He joined the Department of Control and Instrumentation Engineering, Kangwon National University as a visiting professor in 2000 and the Division of Mathematics and Statistical Informatics, Chonbuk National University as an assistant professor in Korea. He is currently an associate professor at the Department of Statistics, Sookmyung Women s University. His current research interests include data transformations, multivariate time series analysis and generalized additive models. 

Advanced mathematical and statistical techniques used in analytical chemistry are often referred to under the umbrella term of chemometrics. This is a loose definition, and chemometrics are not readily distinguished from the more rudimentary techniques discussed in the earlier parts of this chapter, except in terms of sophistication. The techniques are applied to the development and assessment of analytical methods as well as to the assessment and interpretation of results. Once the province of the mathematician, the computational powers of the personal computer now make such techniques routinely accessible to analysts. Hence, although it would be inappropriate to consider the detail of the methods in a book at this level, it is nevertheless important to introduce some of the salient features to give an indication of their value. Two important applications in analytical chemistry are in method optimization and pattern recognition of results. 

The early structure-activity studies [69, 70] were limited in the number of compounds studied but showed that reasonable correlations could be drawn between the structure of compounds and their biological activity without a complete understanding of the underlying mechanisms involved. Chou and Jurs [71] expanded the approach to structure-activity relationships by applying computer-assisted mathematical and statistical methods to a large set of N-nitroso compounds. These methods... 

Time profiles in vitro and in vivo represent distribution functions in a mathematical and statistical sense. For example, a release profile Fj)(t) in vitro expresses the distribution of drug released at time t the corresponding probability distribution function (PDF) profile fo(t) characterizes the rate of release. Similarly, a plasma concentration profile fp(t) represents the distribution of drug in the plasma at any time t, i.e., absorbed but not yet eliminated its cumulative distribution function (CDF) equivalent FP(t) represents the drug absorbed and already eliminated. 

Chemometrics has been defined as A chemical discipline that uses statistical and mathematical methods, to design or select optimum procedures and experiments, and to provide maximum chemical information by analyzing chemical data. In shorter words it is focused as Chemometrics concerns the extraction of relevant information from chemical data by mathematical and statistical tools. Chemometrics can be considered as a part of the wider field chemoinformatics which has been defined as The application of informatics methods to solve chemical problems (Gasteiger and Engel 2003) including the application of mathematics and statistics. 

The book is at an introductory level, and only basic mathematical and statistical knowledge is assumed. However, we do not present chemometrics without equations —the book is intended for mathematically interested readers. Whenever possible, the formulae are in matrix notation, and for a clearer understanding many of them are visualized schematically. Appendix 2 might be helpful to refresh matrix algebra. 

In the earlier time of chemometrics until about 1990, a number of books have been published that may be rather of historical interest. Chemometrics—Applications of Mathematics and Statistics to Laboratory Systems (Brereton 1990), Chemical Applications of Pattern Recognition (Jurs and Isenhour 1975), Factor Analysis in... 

Relevant collections of papers—mostly conference proceedings—have been published Chemometrics—Exploring and Exploiting Chemical Information (Buydens and Meissen 1994), Chemometrics Tutorials (Jonker 1990), Chemometrics Theory and Application (Kowalski 1977), Chemometrics—Mathematics and Statistics in Chemistry (Kowalski 1983), and Progress in Chemometrics Research (Pomerantsev 2005). 

The computer, with its enormous power in data processing and its possibilities in automation and control, has added a new dimension both to the instrumental analytical method and the application of mathematics and statistics in analytical chemistry. The introduction of the computer was one of the main factors initiating a new analytical subdiscipline, chemometries, which has a strong mathematical character. 

Chemometrics is the chemical discipline that uses mathematical and statistical methods select optimal measurement procedures and experiments, and (b) to provide maximum chemical information by analyzing chemical data."(I)... 

However, society likes to have decisions made in a black and white manner and to know whether something is there or not. This situation suggests that the analytical error should drop to zero. While this result is the goal of all analytical work, it is simply not realistic. Our basic need, then, is to simplify error determinations and explanations and to educate the public both for the reasons and for the interpretations of error. The goal of this volume is to further the use of mathematical and statistical tools—the field of chemometrics—for chemical and, specifically, trace chemical analyses of pesticides and environmental contaminants. 

B.R. Kowalski, Chemometrics Mathematics and Statistics in Chemistry, D. Reidel, Dordrecht, Netherlands, 1980. 

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