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LEARNING OPTIMIZATION FROM NATURE Genetic Algorithms and Simulated Annealing
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n science and technology, as well as in everyday life, we constantly encounter situations in which we must consider several factors and use our best judgment to balance the potential risks and benefits of some course of action. Our judgments are based on whether the outcomes of our previous decisions were acceptable. This trial-and-error learning method is similar to the way in which many numerical optimization algorithms operate. For these numerical optimization approaches however, the actions and factors underlying the decision-making proquantifiable and the goal is to find the opttmal actton The natural world provides many examples of optimization. From biological processes such as evolution to physical processes such as annealing, many events seek optimal conditions. In Darwin's theory of evolution, the genetic code of a species evolves to adapt itself more favorably to its environment. Species that survive the natural selection process pass their genetic code to their offspring, whereas species that fail to adapt become extinct. During the annealing process, as a molten metal cools, its particles arrange themselves in a low-energy state. Metals in which the particles are aligned in the lowest energy configuration (i e have been annealed) are much stronger than metals that have been cooled too quicklv in an imDroper alignment of the comoonent particles which leads to brittleness 236 A
ing using simulated annealing to track missile trajectories, develop computer war game models, reduce risks and increase profits in financial markets, and model seismic waveforms (1). Genetic algoriihms have been applied to a wide range of fields as well, including adaptive filtering, robotics, job scheduling, image analysis, and protein folding (2,3). Whereas other areas of science have been quick to embrace these powerful new methods, analytical chemistry has lagged slightly behind. However, the increase in the number of publications involving these methods in recent years indicates that analytical chemists are starting to reap the benefits of these methods. A genetic algorithm Researchers have recently targeted software package has been designed specifthese natural optimization processes as models for two numerical optimization algo- ically for chemists (4). rithms: genetic algorithms and simulated In this Report, we present a short reannealing. Unlike many traditional methods view of the underlying concepts of numerof optimization, these algorithms are espeical optimization, give an introduction to cially suited to the large-scale optimization the working principles of simulated anproblems that have abounded due to the nealing and genetic algorithms, compare proliferation of computers. Since the 1980s, these techniques with other numerical workers in computer science, physics, and optimization methods, and explore their engineering have made tremendous strides potential for application to analytical toward understanding not only how these chemistry. algorithms work but how to apply them to challenging optimization problems includNumerical optimization Defined mathematically, numerical optimization algorithms seek the set of n conditions (xj, ,2, x3 xn) that minimize eo Ronald E. Shaffer maximize a response f[x) (5) .The variable Naval Research Laboratory settings at the minimum or maximum f(x) Gary W . S m a l l Ohio University are either the global optimum (the highest
Numerical optimization techniques can be used to quantify traditional trial-and-error learning methods
Analytical Chemistry News & Features, April 1, 1997
0003-2700/97/0369-236A/$14.00/0 © 1997 American Chemical Society
or lowest function value over all possible/(r)) or a local optimum (the highest or lowest value within a neighborhood). Finding the global optimum for a complex application involving multiple variables is somewhat akin to "finding a needle in a haystack" and requires a powerful optimization algorithm. Optimization problems can be divided into two general classes: direct and indirect Problems that can be solved analytically through direct means, such as logical mathematical steps (e.g., calculus), constitute one class of problems. Direct methods require that the mathematical form of the function to be optimized be known; examples include finding the inflection point of a curve or the root(s) of a polynomial emiah'on Many optimization problems are not this straightforward, however, because the relationships between the variables and the response may be obscure. In these cases, indirect methods that operate by taking "educated guesses" about the settings for the n variables (iterative trialand-error approaches) must be used. Decisions about what guesses to make are called the search heuristics. In the indirect optimization approach at each iteration the variables modified in some logical way and presented to the objective function (fix)) to determine whether the new combination of variable settings is an improvement Simulated annealing and genetic algorithms are both considered indirect optimization
the success of any optimization method. Although the choice of response function is application dependent, it is usually based on some quantifiable property that best describes the goals of the application, such as S/N, selectivity, classification accuracv precision or similarity to a target value
To apply an indirect numerical optimization method to a specific problem, one must carefully select the variables and define a valid measure of the response of the system. A graph of the system response as a function of one or more of its independent variables, termed a response surface plot, is often used in optimization methodology to help make these decisions. The response surface plot offers a visual means of understanding the influence of the selected variables on the measurement system as shown in Figure 1 In this plot, two variables influence the response of the system, interaction among the variables is evident, and multiple local optima are observed. The response function score (f(x)) numerically encodes the aggregate performance of a combination of variable settings. The design of an appropriate response function is critical to
Many optimization problems in science require that a priori constraints be placed on the values that the variables can assume. For example, in the optimization of the conditions for a GC experiment, care must be taken to prevent variable settings that may damage the instrument (e.g., if the column temperature becomes too hot it destroy the stationary phase) This tvoe of problem is termed a constrained ODtimization (5); some optimization methods such as linear programmine signed to incorrjorate these constraints Unconstrained Optimization algorithms simulated and e-enetic algorithms ran be annlied to these proh lems by plaring a severe nenaltv on the resnnnse fi ction score h er inval'ii or dangerous variable settings occur. Optimization in analytical chemistry The field of analytical chemistry, like other areas of science and technology, contains many situations in which optimization is required. Although some of these applications are mundane—such as deter-
Analytical Chemistry News & Features, April 1, 1997 237 A
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Figure 1 . Response surface plot. The plot features a response that is a function of variables 1 and 2 and contains multiple local optima. The global optimum for this function occurs when variables 1 and 2 are equal to 0.
mining the best operating conditions to separate two compounds in an HPLC or GC experiment—analytical chemistry does provide some intriguing and novel applications of optimization. Applications of optimization in analytical chemistry can be subdivided into online and off-line optimizations. An on-line optimization involves real-time instrumental measurements and modifications of the instrumental or experimental variables, such as finding the best operating conditions for an ICP-AES instrument (6). With present technology, on-line optimization involves only a few key variables and can be performed using simple optimization algorithms or heuristic approaches. At a recent artificial-life conference in Japan, however, researchers presented some initial steps toward evolvable hardware that offer great potential for developing self-optimizing instrumentation (7). Integrated circuits, termed field programmable gate arrays (FPGAs), allow researchers to modify a circuit design almost instantaneously. By coupling the programmable circuit technology with powerful optimization algorithms, such as genetic algorithms or simulated annealing researchers hope to develop autonomous robots that can modirV their control circuits in real time Analytical instrumentation could benefit from this technology as well; it is easy to envision instruments that can self-correct for base238 A
line drift or change its operating parameters to account for fluctuating environmental conditions. Although the implementation of truly self-optimizing analytical instrumentation may be decades away, the application of optimization methods in off-line settings is commonly performed today. Off-line optimization uses data that have been downloaded from the instrument to a computer for subsequent analysis. Analytical chemists increasingly tackle problems in which multiple chemical species contribute to the instrumental response. The multidimensional data obtained from these measurements are often complicated and optimization tools C3.ii be used to refine the qualitative and quantitative analyses of the data Two applications that have recently received considerable attention are spectroscopic wavelength selection and curve fitting. Use of optimization algorithms to select subsets of wavelengths for calibration model building has been shown to greatly improve the predictive ability of the model (8-10). For complex chromatographic or spectral data, ambiguous curve fitting is a major concern. Research has shown that optimization algorithms that are robust to the initial starting conditions and less likely to be trapped in local optima are critical for this application (11 12) Other noteworthy applications of optimization in the recent literature include
Analytical Chemistry News & Features, April 7, 1997
choosing calibration samples for multivariate regression (13), selecting molecular descriptors for quantitative structureactivity relationships (14), and model-free analyzing of thin and multilayer films using X-ray fluorescence (15). Many optimization techniques used by analytical chemists are based on previous knowledge or heuristic and experimental design methods such as factorial designs (16) and Simplex optimization (17)) or they require the estimation of the derivative of the response function (gradient ascent/ descent [5,16]). For most simple optimization applications, these methods are able to find the global optimum with minimal effort. For unusually challenging optimization problems, however they tend to fail either because they are too time-consuming use narrowly defined heuristics become entrapped in local optima or because the computation of the derivative is not numerically stable or feasible frenetic algorithms annealing are most challenging optimization problems in which these other methods fail Because of the proliferation of computers for off-line optimization and our desire for on-line opumization, the need will continue to grow for powerful optimization methods such as these. Simulated annealing Simulated annealing (SA) is a numerical optimization technique based on the principles of statistical thermodynamics (1, 18,19). As mentioned previously, annealing refers to the metallurgical process in which a solid material is first melted and then allowed to cool by slowly reducing the temperature. During the cooling process, the particles of the material arrange themselves into a low-energy state. The collective energy states of the ensemble of particles may be considered the "configuration" of the material and the probability that a particle is in any given enerov state can be calculated from the Boltzmann distribution As the temperature of the material decreases the Boltzmann distribution tend 0, ,he probability oo accepting the perturbed system follows the Metropolis criterion, which is dependent on AE the Boltzmann constant k and a fixed absolute temperature T. As the temperature decreases the probability of adopting a detrimental configuration lessens Through the use of this criterion the material will eventually reach its equilibrium configuration (i e become annealed) This fundamental rnnppnt pun KP niimerical nntimizatinn prnn
lems History. SA is the term used for a family of optimization algorithms that couple Monte Carlo methods with a criterion for choosing new configurations such as those based on Boltzmann or Cauchy distributions. Although there are many versions of SA we will focus on the more standard Boltzmann annealing algorithms that rely on the Metropolis criterion for deciding when to accept or reject new variable settings. The original design of SA is attributed to Kirkpatrick and colleagues (21) who used the algorithm for large-scale combinatorial applications such as the traveling-salesman problem Later SA was adapted by Vanderbilt and Louie for applications involving continuous real-number (22) application of SA in the analytical chemistry literature was hv Kalivas and en workers (??) in 1989 Methods. For numerical optimization, the response function score (R =f(x)) replaces the energy term in the Metropolis criterion. Based on its roots in statistical thermodynamics, SA implies minimization but can be configured for maximization simply by reversing the sign or computing the inverse of the response function score (i.e., poor variable settings will result in a large response function score). Figure 2 shows the steps involved in the SA algorithm The new variable settings (x') which obtained by randomly perturbing the current configuration (x x x x , may be thought of as movements or steps on the response mirfacp T h e con-
cept of temperature is retained but is now combined with k and used as a critical control parameter. The probability P of accepting a detrimental step (i.e., AR > 0) is governed by AR and T. After the new step is taken, a random number P is drawn from a uniform random distribution on the interval [0,1]. If p < P, the eetrimental ltep is rejected and a new random step is taken from the current position. Ifp>P, ,he detrimental lsep is accepted and the new configuration replaces the old one. A new step is then taken relative to this configuration. This criterion allows the possibility of x' being accepted as the new configuration, even when its response function score is worse than that of the current configuration x Large detrimental steps (AR ») likely to be accepted than those that are
smaller This feature allows the algorithm to "walk" from local optima Additional stpn*; are taken until
some termination criterion is rearViprl fp a
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secutive steps') Analogous to the physical process, when T is reduced slowly, the probability of accepting a poor step decreases with time. The schedule by which T is reduced is called the cooling schedule and is critical to the success of SA. Another crucial parameter is the step size, which controls the magnitude of the perturbation taken at each step and is application dependent. For continuous variable optimization problems, the perturbation is caused by multiplying n Gaussian-distributed random deviates by the step size and adding the resulting values to the current variable settings; whereas for subset selection and combinatorial applications the perturbation is caused by randomly changing q
walk on the response surface, in which each new step is based on the current variable setting (configuration). Through the use of Markov chains (sequences of states in which the outcome of each state depends only on the previous one), SA can be shown to converge asymptotically to the global optimum given certain conditions (18). Genetic algorithms Genetic algorithms (GAs) are optimization techniques based on the concepts of natural selection and genetics (2,3). In a GA, the variables being optimized (xD %, % , . . . , xn) are represented as genes on a chromosome. The fitness of a chromosome is determined by computing the response function score (fitness =f(x)). Most discussions of GAs assume maximization problems (i.e., a larger fitness score corresponds to a better chromosome); however, minimization problems can be handled by simple modification of the fit(e ff -f(x) ovf(x) ) GAs feature a of candidate solutions (a DODulation of chromosomes) that evolves with time in a manner similar to the evolution of a natural populatinn of spprie^ Thmittrh
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in SA) History. The concept of using evolution to guide the optimization process originated with Box and Wiison in work that ted to the Simplex optimization technique (24). Holland was the first to use the populationbased genetic aspects of evolution by introducing a computational method, termed the recombination operator, for implementing sexual reproduction (25). In 1975, he developed the schema theorem, which provided a platform for understanding how recombination and mutation couple with natural selection to provide a powerful optimization procedure However it was not until the 1980s that the first practical applications of GAs were reported (2) Luccsius and Kateman were the first to applv GAs ,o probin analytical chemistrv (26 27) Methods. From a computational standpoint, the implementation of a GA is actually quite simple. A GA consists of four basic steps (Figures 3 and 4). First, the initial population of chromosomes is created either randomly or by randomly perturbing an input chromosome. The population size AL a user-controlled option, remains constant throughout the optimization. In the second step, evaluation the fitness of each chromosome in the population is computed. The third step is the exploitation, or natural selection, step. The chromosomes with the largest fitness scores are placed one or more times into a mattng subset in a semirandom fashion. Chromosomes not selected for the mating subset are removed from the population. A popular method for performing exploitation is binary tournament selection, in which pairs of chromosomes are randomly selected from the pop240 A
ulation. The chromosome with the better fitness score is placed in the mating subset, and both chromosomes are returned to the gene pool. This process continues until the mating subset is full. This method gives chromosomes with higher fitness scores a higher probability of being included in the mating subset than chromosomes with lower fitness scores. The fourth step, exploration, consists of applying the recombination and mutation operators. Two chromosomes (parents) from the mating subset are randomly selected to be mated. The probability Pr that these chromosomes are recombined (mated) is a user-controlled option and usually is set to a high value (e.g., 0.90)) .f the parents are allowed to mate, a recombination operator is used to exchange genes between the two parents to produce two children. If they are not allowed to mate the parents are placed into the next generation unchanged The most common recombination operator is the one-point crossover in which a crossover point is randomly selected along the chromosome and the J- t h a t n n i n t arp QwnnnpH h p t w p p n
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crossover, have proven to be more effective than one-point crossover in many applicaFollowing recombination, a mutation operator is used to increase the diversity in the population. In nature, the effects of mutation on an individual are sometimes devastating (e.g., birth defects and cancer), but in a GA, a mutation is simply a random modification of a particular gene. The probability P mat a mutation will occur is a user-controlled option and generally is set to a low value (e.g., 0.01) to prevent the destruction of good chromosomes. In many this type of mutation like an unfavorable naturally occurring mutation results in an unfit chromosome that is removed from the population during natural selection After the exploration step, the population consists of newly created chromosomes, and steps two through four are repeated. This process continues for a user-selected number of generations or until some termination criterion is reached (e.g., the majority of the chromosomes in the population are the same).
Analytical Chemistry News & &eatures, April 1, 1997
Figure 3. Genetic algorithms flow chart.
Variable representation. A unique feature of a GA is the procedure for representing the variables (xlt x2, x3, ..,, x„) on the chromosome. .n its simplest form, each chromosome consists of a vector or string of binary digits (bits), in which each gene is represented by either 0 or 1 (binary coding). In such a scheme, mutation is performed simply by changing 1 to 0 or 0 to 1. Foo rertain aaplications, however, more complicated representations, such as using a vector of integers (integer coding) or real numbers (real number coding), may be necessary to represent the variables (2,3,26,27). Each time the fitness of a chromosome is computed, the chromosome must be converted from its genetic form (vector of binary digits) to the variables used by the application. This conversion process, sometimes termed coding/decoding or mapping, is often confusing to the beginner, but is necessary for adapting the genetic operators to a wide variety of appllcations. Binary representation offers the advantages that the theory of binary-coded GAs has been well developed and that, by converting a real number value for a particular variable to a binary string, a fixed resolution is placed on the number of possible values that variable can have. A fixed resolution has the advantage of reducing the size of the search space. However, this capability also reduces the precision of the mapping procedure and can be a disadvantage for applications. A disad-
Figure 4. A worked example of a genetic algorithm.
vantage of the binary representation, in which the genes are individual bits, results from the fact that, for applications in which the variables require real numbers or integers, the importance of a gene is determined by its location on the chromosome. A change in the most significant bit of a binary representation will have a far greater impact on the variable setting than will a mutation on the least significant bit. This problem has been termed a Hamming cliff and can be partially corrected through the use of gray codings Despite these disadvantages many GA practitioners prefer a binary representation over other representations (e g integer or real) because it provides a clear distinction nptwppn the rnmnnter cr\df> to imple ment the GA and the rnHp to calrnlate the fitness score T h e GA code can be used without mortification for many annlica
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ing process and computing the fitness score. In this sense, a binary-coded GA may be operated as a black-box optimization algorithm. Theory. The search heuristics of a GA have been described by Holland's schema theorem (25). A conceptual discussion of the operation of a GA can be performed using a subset of the schema theorem, termed the building-block hypothesis, the underlying principal of which is that certain
genes or groups of genes, termed the building blocks, have a greater impact on the fitness functton than do other segments so the chromosome. In the early stages of the optimization (the first few generations), the GA attempts to determine which segments of the chromosome contain these building blocks. Once the building blocks are located, the optimal settings for these genes can be determined. By focusing the optimization on the portions of the chromosome that have the greatest impact on the fitness function the GA has reduced the complexity of the optimization problem Once these building blocks are established subsequent generations can focus on the remaining unoptimized segments of the chromosome The building-block hypothesis emphasizes the importance of the recombination operator in a GA Recombination is the only means of placing building blocks located on different segments of two parent chromosomes onto a child chromosome. Because of the numerous chromosomes in a given population and the ability to process many building blocks in a given generation, GAs are said to possess the property of implicit parallelism, thereby making them efficient optimization algorithms. A comparison of GAs and SA
Besides their common inspiration, GAs and SA share many advantages and disadvantages. Their primary advantage is their
ability to move from a local optimum and explore diverse regions of the response surface. Thus, in principle, the potential for finding the global optimum is independent of the initial variable settings (the starting point). Both SA and GAs are easy to implement and can be configured for any application. The primary disadvantages of SA are that it is difficult to select appropriate configuration parameters (e.g., Tand the step size) and that it typically requires more response function evaluations than do traditional optimization methods such as Simplex optimization and gradient methods For applications in which the buildingblock hypothesis is followed, GAs use implicit parallelism to process information quickly and tend to require fewer response function evaluations than do other iterative stochastic optimization algorithms, such as SA One of the potential pitfalls with GAs is that they are known to have trouble in locating the exact global optimum. Even for elementary optimization problems, GAs can reach the global optimum region quickly but fail to find the exact optimum The major drawback of GAs however is the configuration of the algorithm The practitioner must make many choices such as the representation population size P P recombination method and mating subset method.The lack of established guidelines for choosing the GA configurations makes the application of GAs difficult at times. It is inevitable that, when new algorithms are developed, researchers will attempt to compare and contrast the new methods with the established ones. Comparisons of GAs and SA wiih other global optimization methods are no exception. In the computer science literature, many comparative studies have been reported. Even in the analytical chemistry literature, comparisons of GAs and SA can be found (8,9,28). The consensus from these studies is that there is no "optimal" optimization method. Wolpert and Macready (29) have developed what they call the "No Free Lunch (NFL) Theorem". Their hypothesis is that, used without any knowledge about the response surface (e.g., interaction among the variables, steepness or smoothness of the response surface, pres-
Analytical Chemistry News & Features, April 1, 1997 241 A
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MOLECULAR DIVERSITY AND COMBINATORIAL CHEMISTRY: LIBRARIES AND DRUG DISCOVERY This important new volume reports progress on chemical, enzymatic, phage, and cell-derived libraries. It explores the synergy between structure-based design and combinatorial libraries and presents applications of coinDiri3toria.i libraries to drug discovery and new synthetic catalysis. Its 27 chapters are divided into six sections covering: • Strategies • Library Design: Solid-, Solution-, and Liquid-Phase Combinatorial Synthesis • Biology-Based Chemical Libraries • Automated Solid-Phase Synthesis • Analytical Methods and Screening • Applications Molecular Diversity ynd dombinatorial Chemistry also discusses automation of organic synthesis as well as new methodologies for monitoring solidphase organic synthesis. Irwin M. Chaiken, University of Pennsylvania Editor Kim D. Janda The Scripps Research Institute Editor Conference Proceedings Series 328 pages (1996) Clothbound ISBN 0-8412-3450-7 $109.95 ORDER FROM American Chemical Society 1155 Sixteenth Street, NW Washington, DC 20036 Or CALL TOLL FREE 1-800-ACS-9919 and use your credit card! FAX: 202-872-6067. ACS Publications Catalog now available on Internet: URL http://pubs.acs.org
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Report ence of discontinuities), any iteratively improving global optimization algorithm will perform, on average, the same for any given optimization problem. For an analytical chemist, the practical implication of this theorem is that some knowledge of the system being optimized is needed to select the appropriate algorithm. Intuitively, this makes sense because each algorithm exploits different aspects of the response surface (i.e. Markov chain analysis in SA and the schema theorem in GAs). One potential problem that occurs when researchers perform comparison studies involving GAs and SA iS that of the success attributed to the "winning" algorithm may be caused by having the algorithm properly configured (8); given the magnitude of configuration options for GAs and SA this is not a trivial issuea As the NFL theorem suggests, the future successes of SA and GAs are tied together. There always will be optimization problems that can be solved best by one algorithm and not the other. Thus, when comparing global optimization algorithms, the features of the response surface that were exploited by the "winning" algorithm should be determined. This information can be used to develop guidelines for determining the best optimization algorithm for specific classes of problems. By introducing the NFL theorem to the analytical chemistry community we hope to inspire future research in this area Portions of this manuscript were written while one of the authors (R.E.S.) held a National Research Council-Naval Research Laboratory Research Associateship. Thanks to Cynthia Meredith for editing previous drafts of this manuscript.
(9) Lucasius, C. B.; Beckers, M.L.M.; Kateman, G. Anal. Chim. Acta 1994,286, 135-53. (10) Bangalore, A S.; Shaffer, R E.; Small, G. W.; Arnold, M. A Anal. Chem. 19969 68,4200-12. (11) De Weijer, A. P.. Lucasius, C. B.; Kateman, G.; Heuvel, H. M.; Mannee, H. Anall Chem. .994, 66,23-31. (12) Ferry, A; Jacobsson, P. Appl. Spectrosc. 1995,49, 273-78. (13) Kalivas, J. H. Chemom. Intell. Lab. Syst. 1992,15,1-12. (14) Wessel, M. D.; Sutter, J. M.; Jurs, P. C. Anal. Chem. .996, 68, 4237-43. (15) Dane, A. D.. Timmermans, P.A.M.; van Sprang, H. A; Buydens, L.C.M. Anal. Chem. .996, 68, 2419-25. (16) Bayne, C. K.; Rubin, I. B. Practical Experimental Designs and Optimization Methods for Chemists; VCH: Deerfield Beach, FL, 1986. (17) Walter, F. H.; Parker, L. R; Morgan, S. L.; Deming, S. N. Sequential Simplex Optimization; CRC: Boca Raton, FL, 1991. (18) van Laarhoven, P.J.M.; Aarts, E.H.L. Simulated Annealing: Theory and Applications; D. Reidel: Dordrecht, The Netherlands, 1987. (19) Adaption of Simulated Annealing to Chemical Optimization Problems; Kalivas, J. H.J Ed.; Elsevier: Amsterdam, 1995. (20) Metropolis, N.; Rosenbluth, A; Rosenbluth, M.;Teller, A;Teller, E.J. Chem. Phys. 1953,21,1087-92. (21) Kirkpatrick, S.; Gelatt, C. D.; Vecchi, M. P. Science 1983,220, 671-80. (22) Vanderbilt, D.; Louie, S. G.J. Comput. Phys. .984,36, 259-71. (23) Kalivas, J. H.; Roberts N.. Sutter, J. M. Anal. Chem. 1989, 61, ,024-30. (24) Box, G.E.P.; Wilson, K B.J. R. Stat. Soc. 1951,13,1-45. (25) Holland, J. H. Adaptation in Natural and Artificial Systems; University of Michigan Press: Ann Arbor, 1975. (26) Lucasius, C. B.; Kateman, G. Chemom. Intell. Lob. Syst. 1.93,19,9-33. (27) Lucasius, C. B.; Kateman, G. Chemom. Intell. Lab. Syst. 1994,25, 99-145. (28) Shaffer, R E.; Smalll G. W. Anal. Chim. Acta 1996,331,157-75. (29) Wolpert, D. H.; Macready, W. G. IEEE Trans. Evolutionary Computation, in press.
References (1) Ingber, L./. Math. .omput. Model. 1193, Ronald E. Shaffer is an NRC Postdoctoral Research Associate at the Naval Research 18, 29-57. (2) Goldberg, D. E. Genetic Algorithms in Laboratory. His research interests include Search, Optimization, and Machine Learnthe development of chemometric methods ing; Addison-Wesley: Reading, MA, ,989. for analyzing complex chemical sensor data. (3) Mitchell, M. An Introduction to Genetic Algorithms; MIT Press: Cambridge, MA, Gary W. .mallls Professor of Chemistry and Director of the Center for Intelligent 1996. (4) Brown, S. D.Anal. Chem. .996, 68, 678- Chemical Instrumentation at Ohio Univer79. sity. His research interestt focus on the de(5) Press, W. H.. Teukolsky, S. A; Vetterling, velopment of chemical sensors based on W. T; Flannery, B. P. Numerical Recipes infrared spectroscopy. Address corresponin C; Cambridge University Press: New dence to Shaffer at the Naval Research LabYork, 1995. (6) Galley, P. J.; Horner, J. A; Hieftje, G. M. oratory Chemistry Division Code 6116 Spectrochim. Acta 1995,50B, 87-107. Washington DC 20375 or to Small at the (7) Normile, D. Science 1996,272,1872-73. Department of Chemistry Ohio University (8) Horchner, U.; Kalivas J. U.Anall.him. Athens OH 45701 Acta 1995,311,1-13.
Analytical Chemistry News & Feature:, April 1, 1997
