Abstract
In this paper, we introduce the Divide and Conquer (D&C) algorithm, a computationally attractive algorithm for determining classification rules which minimize the training sample misclassification cost in two-group classification. This classification rule can be derived using mixed integer programming (MIP) techniques. However, it is well-documented that the complexity of MIP-based classification problems grows exponentially as a function of the size of the training sample and the number of attributes describing the observations, requiring special-purpose algorithms to solve even small size problems within a reasonable computational time. The D&C algorithm derives its name from the fact that it relies, a.o., on partitioning the problem in smaller, more easily handled sub-problems, rendering it substantially faster than previously proposed algorithms. The D&C algorithm solves the problem to the exact optimal solution (i.e., it is not a heuristic that approximates the solution), and allows for the analysis of much larger training samples than previous methods. For instance, our computational experiments indicate that, on average, the D&C algorithm solves problems with 2 attributes and 500 observations more than 3 times faster, and problems with 5 attributes and 100 observations over 50 times faster than Soltysik and Yarnold's software, which may be the fastest existing algorithm. We believe that the D&C algorithm contributes significantly to the field of classification analysis, because it substantially widens the array of data sets that can be analyzed meaningfully using methods which require MIP techniques, in particular methods which seek to minimize the misclassification cost in the training sample. The programs implementing the D&C algorithm are available from the authors upon request.
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References
P.L. Abad and W.J. Banks, New LP based heuristics for the classification problem, European Journal of Operational Research 27(1993)88–100.
J.A. Anderson, Separate sample logistic discrimination, Biometrika 59(1972)19–35.
T.W. Anderson, An Introduction to Multivariate Statistical Analysis, 2nd ed., Wiley, New York, 1984.
O.K. Asparoukhov, Microprocessor system for investigation of thromboembolic complications, Unpublished Ph.D. Dissertation, Technical University of Sofia, Bulgaria, 1985 (in Bulgarian).
S.M. Bajgier and A. Hill, An experimental comparison of statistical and linear programming approaches to the discriminant problem, Decision Sciences 13(1982)604–618.
W.J. Banks and P.L. Abad, An efficient optimal solution algorithm for the classification problem, Decision Sciences 22(1991)1008–1023.
W.J. Banks and P.L. Abad, On the performance of linear programming heuristics applied on a quadratic transformation in the classification problem, European Journal of Operational Research 74(1994)23–28.
T.S. Campbell and J.K. Dietrich, The determinants of default on insured conventional residential mortgage loans, Journal of Finance 38(1983)1569–1581.
N. Capon, Credit scoring systems: A critical analysis, Journal of Marketing 46(1982)82–91.
C. Chen, Hybrid misclassification minimisation, Paper presented at the National INFORMS Meeting, Washington, DC, May 1996.
A.P. Duarte Silva, Minimizing misclassification costs in two-group classification analysis, Unpublished Ph.D. Dissertation, The University of Georgia, 1995.
A.P. Duarte Silva and A. Stam, Second order mathematical programming formulations for discriminant analysis, European Journal of Operational Research 74(1994)4–22.
R.A. Eisenbeis, Pitfalls in the application of discriminant analysis, Journal of Finance 32(1977) 875–900.
L.P. Fatti, D.M. Hawkins and E.L. Raath, Discriminant analysis, in: Topics in Applied Multivariate Analysis, D.M. Hawkins, ed., Cambridge University Press, Cambridge, England, 1982, pp. 1–77.
R.A. Fisher, The use of multiple measurements in taxonomy problems, Annals of Eugenics 7(1936) 179–188.
N. Freed and F. Glover, A linear programming approach to the discriminant problem, Decision Sciences 12(1981)68–74.
N. Freed and F. Glover, Simple but powerful goal programming formulations for the discriminant problem, European Journal of Operational Research 7(1981)44–60.
N. Freed and F. Glover, Evaluating alternative linear programming models to solve the two-group discriminant problem, Decision Sciences 17(1986)151–162.
W.V. Gehrlein, General mathematical programming formulations for the statistical classification problem, Operations Research Letters 5(1986)299–304.
L.W. Glorfeld and M.W. Kattan, A comparison of the performance of three classification procedures when applied to contaminated data, in: Proceedings of the 21st Annual Meeting of the Decision Sciences Institute, 1989, pp. 1153–1155.
F. Glover, Improved linear programming models for discriminant analysis, Decision Sciences 21 (1990)771–785.
F. Glover, S. Keene and B. Duea, A new class of models for the discriminant problem, Decision Sciences 19(1988)269–280.
W. Gochet, A. Stam, V. Srinivasan and S. Chen, Multigroup discriminant analysis using linear programming, Operations Research 45(1997)213–225.
D.J. Hand, Discrimination and Classification, Wiley, New York, 1981.
F.S. Hillier and G.J. Lieberman, Introduction to Operations Research, 5th ed., McGraw-Hill, New York, 1990.
C.J. Huberty, Issues in the use and interpretation of discriminant analysis, Psychological Bulletin 95(1984)156–171.
T. Ibaraki and S. Muroga, Adaptive linear classifier by linear programming, IEEE Transactions on Systems, Science and Cybernetics SSC-6(1970)53–62.
International Business Machines, IBM Mathematical Programming Systems Extended/370 (MPSX/370), Mixed Integer Programming/370 (MIP/370), White Plains, New York, 1975.
E.A. Joachimsthaler and A. Stam, Four approaches to the classification problem in discriminant analysis: An experimental study, Decision Sciences 19(1988)322–333.
E.A. Joachimsthaler and A. Stam, Mathematical programming approaches for the classification problem in two-group discriminant analysis, Multivariate Behavioral Research 25(1990)427–454.
L. Kim and Y. Kim, Innovation in a newly industrializing country: A multiple discriminant analysis, Management Science 31(1985)312–322.
G.J. Koehler, Discriminant functions determined by genetic search, ORSA Journal on Computing 3(1991)345–357.
G.J. Koehler and S.S. Erenguc, Minimizing misclassifications in linear discriminant analysis, Decision Sciences 21(1990)63–85.
J.S. Koford and G.F. Groner, The use of an adaptive threshold element to design a linear optimal pattern classifier, IEEE Transactions on Information Theory IT-12(1966)42–50.
P.A. Lachenbruch, C. Sneeringer and L.T. Revo, Robustness of the linear and quadratic discriminant function to certain types of non-normality, Communications in Statistics 1(1973)39–57.
J.M. Liitschwager and C. Wang, Integer programming solution of a classification problem, Management Science 24(1978)1515–1525.
M.A. Mahmood and E.C. Lawrence, A performance analysis of parametric and nonparametric discriminant approaches to business decision making, Decision Sciences 18(1987)308–326.
O.L. Mangasarian, Linear and nonlinear separation of patterns by linear programming, Operations Research 13(1965)444–452.
C.A. Markowski and E.P. Markowski, Some difficulties and improvements in applying linear programming formulations to the discriminant problem, Decision Sciences 16(1985)237–247.
C.A. Markowski and E.P. Markowski, An experimental comparison of several approaches to the discriminant problem with both qualitative and quantitative variables, European Journal of Operational Research 28(1987)74–78.
G.J. McLachlan, Discriminant Analysis and Statistical Pattern Recognition, Wiley, New York, 1992.
D.F. Morrison, Multivariate Statistical Methods, 3rd ed., McGraw-Hill, New York, 1990.
G.E. Pinches and K.A. Mingo, A multivariate analysis of industrial bond ratings, Journal of Finance 28(1973)1–18.
C.T. Ragsdale and A. Stam, An efficient heuristic method for minimizing the number of misclassified observations in discriminant analysis, Working Paper, Terry College of Business, The University of Georgia, 1991.
V. Ramanujan, N. Venkatraman and J.C. Camillus, Multi-objective assessment of effectiveness of strategic planning: A discriminant analysis approach, Academy of Management Journal 29(1986) 347–372.
P.A. Rubin, Heuristic solution procedures for a mixed-integer programming discriminant model, Managerial and Decision Economics 11(1990)255–266.
P.A. Rubin, A comment regarding polynomial discriminant analysis, European Journal of Operational Research 72(1994)29–31.
L. Schrage, LINDO: User's Manual, Release 5.0, The Scientific Press, South San Francisco, CA, 1991.
C.A.B. Smith, Some examples of discrimination, Annals of Eugenics 13(1947)272–282.
F.W. Smith, Pattern classifier design by linear programming, IEEE Transactions on Computers C-17(1968)367–372.
R.C. Soltysik and P.R. Yarnold, Fast solutions to optimal discriminant analysis problems, presented at the TIMS/ORSA National Meeting (Invited), Orlando, FL, April 1992.
R.C. Soltysik and P.R. Yarnold, ODA 1.0: Optimal Discriminant Analysis for DOS, Optimal Data Analysis, Chicago, IL, 1993.
R.C. Soltysik and P.R. Yarnold, The Warmack-Gonzalez algorithm for linear two-category multivariate optimal discriminant analysis, Computers and Operations Research 21(1994)735–745.
D.J. Spiegelhalter and R.P. Knill-Jones, Statistical and knowledge-based approaches to clinical decision-support systems, with an application to gastroenterology, Journal of the Royal Statistical Society, Series A 147, Part 1 (1984)35–77.
V. Srinivasan and Y.H. Kim, Credit granting: A comparative analysis of classification procedures, Journal of Finance 42(1987)665–683.
A. Stam and E.A. Joachimsthaler, Solving the classification problem in discriminant analysis via linear and nonlinear programming methods, Decision Sciences 20(1989)285–293.
A. Stam and E.A. Joachimsthaler, A comparison of a robust mixed-integer approach to existing methods for establishing classification rules for the discriminant problem, European Journal of Operational Research 46(1990)113–120.
A. Stam and D.G. Jones, Classification performance of mathematical programming techniques in discriminant analysis: Results for small and medium sample sizes, Managerial and Decision Economics 11(1990)243–253.
A. Stam and C.T. Ragsdale, On the classification gap in MP-based approaches to the discriminant problem, Naval Research Logistics 39(1992)545–559.
R.E. Warmack and R.C. Gonzalez, An algorithm for the optimal solution of linear inequalities and its application to pattern recognition, IEEE Transactions on Computers C-22(1973)1065–1075.
P.R. Yarnold, R.C. Soltysik and G.J. Martin, Heart rate variability and susceptibility for sudden cardiac death: An example of multivariable optimal discriminant analysis, Statistics in Medicine 13(1994)1015–1021.
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Duarte Silva, A.P., Stam, A. A mixed integer programming algorithm for minimizing the training sample misclassification cost in two-group classification. Annals of Operations Research 74, 129–157 (1997). https://doi.org/10.1023/A:1018962102794
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DOI: https://doi.org/10.1023/A:1018962102794