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Data Depth and Maximum Feasible Subsystems

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Graph Theory and Combinatorial Optimization

Abstract

Various data depth measures were introduced in nonparametric statistics as multidimensional generalizations of ranks and of the median. A related problem in optimization is to find a maximum feasible subsystem, that is a solution satisfying as many constrainsts as possible, in a given system of linear inequalities. In this paper we give a unified framework for the main data depth measures such as the halfspace depth, the regression depth and the simplicial depth, and we survey the related results from nonparametric statistics, computational geometry, discrete geometry and linear optimization.

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References

  • Agmon, S. (1954). The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6:382–392.

    MATH  MathSciNet  Google Scholar 

  • Alon, N., Bárány, I., Füredi Z., and Kleitman, D. (1992). Point selections and weak ε-nets for convex hulls. Combinatorics, Probability and Computing, 1(3):189–200.

    Article  MathSciNet  Google Scholar 

  • Aloupis, G., Cortes, C., Gomez, F., Soss, M., and Toussaint, G. (2002). Lower bounds for computing statistical depth. Computational Statistics and Data Analysis, 40(2):223–229.

    Article  MathSciNet  Google Scholar 

  • Aloupis, G., Langerman, S., Soss, M., and Toussaint, G. (2001). Algorithms for bivariate medians and a Fermat Toricelli problem for lines. Proceedings of the 13th Canadian Conference on Computational Geometry, pp. 21–24.

    Google Scholar 

  • Amaldi, E. (1991). On the complexity of training perceptrons. In: T. Kohonen, K. Mäkisara, O. Simula, and J. Kangas (eds.), Artificial Neural Networks, pp. 55–60. Elsevier, Amsterdam.

    Google Scholar 

  • Amaldi, E. (1994). From Finding Maximum Feasible Subsystems of Linear Systems to Feedforward Neural Network Design. Ph.D. thesis, Dept. of Mathematics, EPF-Lausanne, 1994.

    Google Scholar 

  • Amaldi, E. and Hauser, R. (2001). Randomized relaxation methods for the maximum feasible subsystem problem. Technical Report 2001-90, DEI, Politecnico di Milano.

    Google Scholar 

  • Amaldi, E. and Kann, V. (1995). The complexity and approximability of finding maximum feasible subsystems of linear relations. Theoretical Computer Science, 147:181–210.

    Article  MathSciNet  Google Scholar 

  • Amaldi, E. and Kann, V. (1998). On the approximability of minimizing nonzero variables or unsatisfied relations in linear systems. Theoretical Computer Science, 209(1–2):237–260.

    Article  MathSciNet  Google Scholar 

  • Amaldi E. and Mattavelli, M. (2002). The MIN PCS problem and piecewise linear model estimation. Discrete Applied Mathematics, 118:115–143.

    Article  MathSciNet  Google Scholar 

  • Amaldi, E., Pfetsch M.E., and Trotter, L.E., Jr. (2003). On the maximum feasible subsystem problem, IISs, and IIS-hypergraphs. Mathematical Programming, 95(3):533–554.

    Article  MathSciNet  Google Scholar 

  • Amenta, N., Bern, M., Eppstein D., and Teng, S.H. (2000). Regression depth and center points. Discrete and Computational Geometry, 23:305–323.

    Article  MathSciNet  Google Scholar 

  • Andrzejak A. and Fukuda, K. (1999). Optimization over k-set polytopes and efficient k-set enumeration. In: Proceedings of the 6th International Workshop on Algorithms and Data Structures (WADS'99), pp. 1–12. Lecture Notes in Computer Science, vol. 1663, Springer, Berlin.

    Google Scholar 

  • Applegate, D., Bixby, R., Chvatal, V., and Cook, W. (1998). On the solution of traveling salesman problems. Proceedings of the International Congress of Mathematicians, Vol. III. Documenta Mathematica, 1998:645–656.

    MathSciNet  Google Scholar 

  • Avis, D. (1993). The m-core properly contains the m-divisible points in space. Pattern Recognition Letters, 14(9):703–705.

    Article  MATH  Google Scholar 

  • Avis, D. and Fukuda, K. (1992). A pivoting algorithm for convex hulls and vertex enumeration of arrangements of polyhedra. Discrete and Computational Geometry, 8:295–313.

    Article  MathSciNet  Google Scholar 

  • Avis, D. and Fukuda, K. (1996). Reverse search for enumeration. Discrete Applied Mathematics, 65(1–3):21–46.

    Article  MathSciNet  Google Scholar 

  • Bárány, I. (1982). A generalization of Carathéodory's theorem. Discrete Mathematics, 40:141–152.

    Article  MATH  MathSciNet  Google Scholar 

  • Bárány, I. Personal communication.

    Google Scholar 

  • Bárány, I. and Onn, S. (1997a). Colourful linear programming and its relatives. Mathematics of Operations Research, 22:550–567.

    MathSciNet  Google Scholar 

  • Bárány, I. and Onn, S. (1997). Carathéodory's theorem, colourful and applicable. In: Intuitive Geometry (Budapest, 1995), pp. 11–21. Bolyai Society Mathematical Studies, vol. 6. Janos Bolyai Math. Soc., Budapest.

    Google Scholar 

  • Barnett, V. (1976). The ordering of multivariate data. Journal of the Royal Statistical Society. Series A, 139(3):318–354.

    MathSciNet  Google Scholar 

  • Bebbington, A.C. (1978). A method of bivariate trimming for robust estimation of the correlation coefficient. Journal of the Royal Statistical Society. Series C, 27:221–226.

    Google Scholar 

  • Bennett, K.P. and Bredensteiner, E.J. (1997). A parametric optimization method for machine learning. INFORMS Journal of Computing, 9(3):311–318.

    Google Scholar 

  • Bennett, K.P. and Mangasarian, O.L. (1992). Neural network training via linear programming. In: P.M. Pardalos (ed.), Advances in Optimization and Parallel Computing, pp. 56–67. North-Holland, Amsterdam.

    Google Scholar 

  • Boros, E. and Z. Fiüredi, Z. (1984). The number of triangles covering the center of an n-set. Geometriae Dedicata, 17:69–77.

    Article  MathSciNet  Google Scholar 

  • Burr, M.A., Rafalin E., and Souvaine, D.L. (2003). Simplicial depth: An improved definition, analysis, and efficiency for the finite sample case. DIMACS, Technicai Reports no. 2003-28.

    Google Scholar 

  • Chakravarti, N. (1994). Some results concerning post-infeasibility analysis. European Journal of Operational Research, 73:139–143.

    Article  MATH  Google Scholar 

  • Cheng, A.Y. and Ouyang, M. (2001). On algorithms for simplicial depth. In: Proceedings of the 13th Canadian Conference on Computational Geometry, pp. 53–56.

    Google Scholar 

  • Chinneck, J.W. (1996a). Computer codes for the analysis of infeasible linear programs. Operational Research Society Journal, 47(1):61–72.

    MATH  MathSciNet  Google Scholar 

  • Chinneck, J.W. (1996b). An effective polynomial-time heuristic for the minimum-cardinality IIS set-covering problem. Annals of Mathematics and Artificial Intelligence, 17(1–2):127–144.

    Article  MATH  MathSciNet  Google Scholar 

  • Chinneck, J.W. (1997). Finding a useful subset of constraints for analysis in an infeasible linear program. INFORMS Journal of Computing, 9(2):164–174.

    Article  MATH  MathSciNet  Google Scholar 

  • Chinneck, J.W. (2001). Fast heuristics for the maximum feasible subsystem problem. INFORMS Journal of Computing, 13(3):210–223.

    Article  MathSciNet  Google Scholar 

  • Chinneck, J.W. and Dravnieks, E.W. (1991). Locating minimal infeasible constraint sets in linear programs. ORSA Journal on Computing, 3(2):157–168.

    Google Scholar 

  • Clarkson, K.L., Eppstein, D., Miller, G.L., Sturtivant, C., and Teng, S.H. (1996). Approximating center points with iterative Radon points. International Journal of Computational Geometry and Applications, 6:357–377.

    Article  MathSciNet  Google Scholar 

  • Cole, R. (1987). Slowing down sorting networks to obtain faster sorting algorithms. Journal of the ACM, 34:200–208.

    Article  MathSciNet  Google Scholar 

  • Cole, R., Sharir, M., and Yap, C. (1987). On k-hulls and related problems. SIAM Journal on Computing, 16(1):61–67.

    Article  MathSciNet  Google Scholar 

  • Danzer, L., Grtinbaum, B., and Klee, V. (1963). Helly's theorem and its relatives. In: Proceedings of Symposia in Pure Mathematics, vol. 7, pp. 101–180. Amer. Math. Soc., Providence, RI.

    Google Scholar 

  • Dempster, A.P. and Gasko, M.G. (1981). New tools for residual analysis. The Annals of Statistics, 9:945–959.

    MathSciNet  Google Scholar 

  • Donoho, D.L. (1982). Breakdown properties of multivariate location estimators. Ph.D. qualifying paper, Dept. Statistics, Harvard Univ.

    Google Scholar 

  • Donoho, D.L. and Gasko, M. (1992). Breakdown properties of location estimates based on halfspace depth and projected outlyingness. The Annals of Statistics, 20(4):1803–1827.

    MathSciNet  Google Scholar 

  • Donoho, D.L. and Huber, P.J. (1982). The notion of breakdown point. In: P.J. Bickel. K.A. Doksum and J.I. Hodges, Jr. (eds.), Festschrift for Erich L. Lehmann in honor of his sixty-fifth birthday, pp. 157–184. Wadsworth, Belmont, CA.

    Google Scholar 

  • Eddy, W. (1982). Convex hull peeling. In: H. Caussinus (ed.), COMPSTAT, pp. 42–47. Physica-Verlag, Wien.

    Google Scholar 

  • Edelsbrunner, H., O'Rourke, J., and Seidel, R. (1986). Constructing arrangements of lines and hyperplanes with applications. SIAM Journal on Computing, 15:341–363.

    Article  MathSciNet  Google Scholar 

  • Eppstein, D. (2003). Computational geometry and statistics. Mathematical Sciences Research Institute, Discrete and Computational Geometry Workshop.

    Google Scholar 

  • Ferrez, J.A., Fukuda, K., and Liebling, T. (2005). Solving the fixed rank convex quadratic maximization in binary variables by a parallel zonotope construction algorithm, European Journal of Operations Research, to appear.

    Google Scholar 

  • Fukuda, K. (2002). cddlib reference manual, cddlib Version 092b. Swiss Federal Institute of Technology, Zürich.

    Google Scholar 

  • Fukuda, K. and Rosta, V. (2004). Exact parallel algorithms for the location depth and the maximum feasible subsystem problems. In: C.A. Floudas and P.M. Pardalos (eds.), Frontiers in Global Optimization, pp. 123–134. Kluwer Academic Publishers.

    Google Scholar 

  • Gil, J., Steiger, W., and Wigderson, A. (1992). Geometric medians. Discrete Mathematics, 108(1–3):37–51.

    Article  MathSciNet  Google Scholar 

  • Gleeson, J. and Ryan, J. (1990). Identifying minimally infeasible subsystems of inequalities. ORSA Journal on Computing, 2(1):61–63.

    Google Scholar 

  • Gnanadesikan, R. and Kettenring, J.R. (1972). Robust estimates, resid uals and outlier detection with multiresponse data. Biometrics, 28:81–124.

    Google Scholar 

  • Greenberg, H.J. and Murphy, F.H. (1991). Approaches to diagnosing infeasible linear programs. ORSA Journal on Computing, 3(3):253–261.

    Google Scholar 

  • Hettmansperger, T. and McKean, J. (1977). A robust alternative based on ranks to least squares in analyzing linear models. Technometrics, 19:275–284.

    Article  MathSciNet  Google Scholar 

  • Hodges, J. (1955). A bivariate sign test. The Annals of Mathematical Statistics, 26:523–527.

    MATH  MathSciNet  Google Scholar 

  • Huber, P.J. (1977). Robust covariances. In: S.S. Gupta and D.S. Moore (eds.), Statistical Decision Theory and Related Topics. II, pp. 165–191. Academic, New York.

    Google Scholar 

  • Huber, P.J. (1985). Projection pursuit (with discussion). The Annals of Statistics, 13:435–525.

    MATH  MathSciNet  Google Scholar 

  • Jadhav, S. and Mukhopadhyay, A. (1994). Computing a centerpoint of a finite planar set of points in linear time. Discrete and Computational Geometry, 12:291–312.

    MathSciNet  Google Scholar 

  • Johnson, D.S. and Preparata, F.P. (1978). The densest hemisphere problem. Theoretical Computer Science, 6:93–107.

    Article  MathSciNet  Google Scholar 

  • Kann, V. (1992). On the Approximability of NP-Complete Optimization Problems. Ph.D. Thesis, Department of Numerical Analysis and Computing Science, Royal Institute of Technology, Stockholm.

    Google Scholar 

  • van Kreveld, M., Mitchell, J.S.B., Rousseeuw, P.J., Sharir, M., Snoeyink, J., and Speckmann, B. (1999). Efficient algorithms for maximum regression depth. In: Proceedings of the 15th Symposium on Computational Geometry, pp. 31–40. ACM.

    Google Scholar 

  • Krishnan, S., Mustafa, N.H., and Venkatasubramanian, S. (2002). Hardware-assisted computation of depth contours. In: 13th ACM-SIAM Symposium on Discrete Algorithms, pp. 558–567.

    Google Scholar 

  • Langerman, S. and Steiger, W. (2000). Comnputing a maximal depth point in the plane. In: Proceedings of the Japan Conference on Discrete and Computational Geometry (JCDCG 2000).

    Google Scholar 

  • Langerman, S. and Steiger, W. (2003). The complexity of hyperplane depth in the plane. Discrete and Computational Geometry, 30(2):299–309.

    Article  MathSciNet  Google Scholar 

  • Liu, R. (1990). On a notion of data depth based on random simplices. The Annals of Statistics, 18(1):405–414.

    MATH  MathSciNet  Google Scholar 

  • Liu, R. (1992). Data depth and multivariate rank tests, L 1. In: Y. Dodge (ed.), Statistical Analysis and Related Methods, pp. 279–294. Elsevier, Amsterdam.

    Google Scholar 

  • Liu, R. (1995). Control charts for multivariate processes. Journal of the American Statistical Association, 90:1380–1388.

    Article  MATH  MathSciNet  Google Scholar 

  • Liu, R. (2003). Data depth: Center-outward ordering of multivariate data and nonparametric multivariate statistics. In: M.G. Akritas and D.N. Politis (eds.), Recent advances and Trends in Nonparametric Statistics, pp. 155–167. Elsevier.

    Google Scholar 

  • Liu, R., Parelius, J., and Singh, K. (1999). Multivariate analysis by data depth: Descriptive statistics, graphics and inference (with discussion). The Annals of Statistics, 27:783–858.

    MathSciNet  Google Scholar 

  • Liu, R. and Singh, K. (1992). Ordering directional data: concepts of data depth on circles and spheres. The Annals of Statistics, 20:1468–1484.

    MathSciNet  Google Scholar 

  • Liu, R. and Singh, K. (1993). A quality index based on data depth and multivariate rank tests. Journal of the American Statistical Association, 88:257–260.

    MathSciNet  Google Scholar 

  • Liu, R. and Singh, K. (1997). Notions of limiting P-values based on data depth and bootstrap. Journal of the American Statistical Association, 91:266–277.

    Article  MathSciNet  Google Scholar 

  • Lopuhaä, H.P. (1988). Highly efficient estimates of multivariate location with high breakdown point. Technical report 88-184, Delft Univ. of Technology.

    Google Scholar 

  • Lopuhaä, H.P. and Rousseeuw, P.J. (1991). Breakdown points of affine equivariant estimators of multivariate location and covariance inatrices. The Annals of Statistics, 19:229–248.

    MathSciNet  Google Scholar 

  • Mahalanobis, P.C. (1936). On the generalized distance in statistics. Proceedings of the National Academy India, 12:49–55.

    Google Scholar 

  • Mangasarian, O.L. (1994). Misclassification minimization. Journal of Global Optimization, 5(4):309–323.

    Article  MATH  MathSciNet  Google Scholar 

  • Mangasarian, O.L. (1999). Minimum-support solutions of polyhedral concave programs. Optimization, 45(1–4):149–162.

    MATH  MathSciNet  Google Scholar 

  • Maronna, R.A. (1976). Robust M-estimates of multivariate location and scatter. The Annals of Statistics, 4:51–67.

    MATH  MathSciNet  Google Scholar 

  • MatouÅ¡ek, J. (1992). Computing the center of a planar point set. In: J.E. Goodman, R. Pollack, and W. Steiger (eds.), Discrete and Computational Geometry, pp. 221–230. Amer. Math. Soc.

    Google Scholar 

  • MatouÅ¡ek, J. (2002). Lectures on Discrete Geometry, Graduate Texts in Mathematics, Springer-Verlag, New York.

    Google Scholar 

  • Miller, K., Ramaswami, S., Rousseeuw, P., Sellares, T., Souvaine, D., Streinu I., and Struyf, A. (2001). Fast implementation of depth contours using topological sweep. Proceedings of the Twelfth ACM-SIAM Symposium on Discrete Algorithms, Washington, DC, pp. 690–699.

    Google Scholar 

  • Motzkin, T.S. and Schoenberg, I.J. (1954). The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6:393–404.

    MathSciNet  Google Scholar 

  • Mulmuley K. and Schwarzkopf, O. (1997). Randomized algorithms. In: Handbook of Discrete and Computational Geometry, Chapter 34, pp. 633–652. CRC Press.

    Google Scholar 

  • Naor N. and Sharir, M. (1990). Computing a point in the center of a point set in three dimensions. Proceedings of the 2nd Canadian Conference on Computational Geometry, pp. 10–13.

    Google Scholar 

  • Niinimaa, A., Oja, H., and Tableman, M. (1990). On the finite-sample breakdown point of the bivariate median. Statistics and Probability Letters, 10:325–328.

    Article  MathSciNet  Google Scholar 

  • Oja, H. (1983). Descriptive statistics for multivariate distributions. Statistics and Probability Letters, 1:327–332.

    Article  MATH  MathSciNet  Google Scholar 

  • Onn, S. (2001). The Radon-split and the Helly-core of a point configuration. Journal of Geometry, 72:157–162.

    Article  MATH  MathSciNet  Google Scholar 

  • Orlik, P. and Terao, H. (1992). Arrangements of Hyperplanes. Springer.

    Google Scholar 

  • Parker, M. (1995). A Set Covering Approach to Infeasibility Analysis of Linear Programming Problems and Related Issues. Ph.D. thesis, Department of Mathematics, University of Colorado at Denver.

    Google Scholar 

  • Parker M. and Ryan, J. (1996). Finding the minimum weight IIS cover of an infeasible system of linear inequalities. Annals of Mathematics and Artificial Intelligence, 17(1–2):107–126.

    Article  MathSciNet  Google Scholar 

  • Pfetsch, M. (2002). The Maximum Feasible Subsystem Problem and Vertex-Facet Incidences of Polyhedra. Ph.D. thesis, Technischen Universitat Berlin.

    Google Scholar 

  • Rafalin, E. and Souvaine, D.L. (2004). Computational geometry and statistical depth mreasures, theory and applications of recent robust methods. In: M. Hubert, G. Pison, A. Struyf and S. Van Aelst (eds.). Theory and applications of recent robust methods, pp. 283–295. Statistics for Industry and Technology, Birkhäuser, Basel.

    Google Scholar 

  • Reay, J.R. (1982). Open problems around Radon's theorem. In: D.C. Kay and M. Breen (eds.), Convexity and Related Combinatorial Geometry, pp. 151–172. Marcel Dekker, Basel.

    Google Scholar 

  • Reiss, S. and Dobkin, D. (1976). The complexity of linear programming. Technical Report 69, Department of Computer Science, Yale University, New Haven, CT.

    Google Scholar 

  • Rossi, F., Sassano, A., and Smriglio, S. (2001). Models and algorithms for terrestrial digital broadcasting. Annals of Operations Research, 107(3):267–283.

    Article  MathSciNet  Google Scholar 

  • Rousseeuw, P.J. (1985). Multivariate estimation with high breakdown point. In: W. Grossman, G. Pflug, I. Vincze and W. Wertz (eds.), Mathematical Statistics and Applications, Vol. B, pp. 283–297. Reidel, Dordrecht.

    Google Scholar 

  • Rousseeuw, P.J. and Hubert, M. (1999a). Regression depth. Journal of American Statistical Association, 94:388–402.

    Article  MathSciNet  Google Scholar 

  • Rousseeuw, P.J. and Hubert, M. (1999b). Depth in an arrangement of hyperplanes. Discrete and Computational Geometry, 22:167–176.

    Article  MathSciNet  Google Scholar 

  • Rousseeuw, P.J. and Leroy, A.M. (1987). Robust Regression and Outlier Detection. Wiley, New York.

    Google Scholar 

  • Rousseeuw, P.J. and Ruts, I. (1996). Computing depth contours of bivariate clouds. Computational Statistics and Data Analysis, 23:153–168.

    Article  Google Scholar 

  • Rousseeuw, P.J., Ruts, I., and Tukey, J.W. (1999). The bagplot: A bivariate boxplot. The American Statistician, 53(4):382–387.

    Article  Google Scholar 

  • Rousseeuw, P.J. and Struyf, A. (1998). Computing location depth and regression depth in higher dimensions. Statistics and Computing, 8:193–203.

    Article  Google Scholar 

  • Ryan, J. (1991). Transversals of IIS-hypergraphs. Congressus Numerantium, 81:17–22.

    MATH  MathSciNet  Google Scholar 

  • Ryan, J. (1996). IIS-hypergraphs. SIAM Journal on Discrete Mathematics, 9(4):643–653.

    Article  MATH  MathSciNet  Google Scholar 

  • Sankaran, J.K. (1993). A note on resolving infeasibility in linear programs by constraint relaxation. Operations Research Letters, 13:19–20.

    Article  MATH  MathSciNet  Google Scholar 

  • Sierksma, G. (1982). Generalizations of Helly's theorem: Open problems. In: D.C. Kay and M. Breen (eds.), Convexity and related Combinatorial Geometry, pp. 173–192. Marcel Dekker, Basel.

    Google Scholar 

  • Singh, K. (1993). On the majority data depth. Technical Report, Rutgers University.

    Google Scholar 

  • Struyf, A. and Rousseeuw, P.J. (2000). High-dimensional computation of the deepest location. Computational Statistics and Data Analysis, 34:415–426.

    Article  Google Scholar 

  • Teng, S.H. (1991). Points, Spheres and Separators: A Unified Geometric Approach to Graph Partitioning. Ph.D. Thesis, Carnegie-Mellon Univ. School of Computer Science.

    Google Scholar 

  • Titterington, D.M. (1978). Estimation of correlation coefficients by ellipsoidal trimming. Journal of the Royal Statistical Society. Series C, 27:227–234.

    MATH  Google Scholar 

  • Tukey, J.W. (1974a). Order statistics. In mimeographed notes for Statistics 411, Princeton Univ.

    Google Scholar 

  • Tukey, J.W. (1974b). Address to International Congress of Mathematics, Vancouver.

    Google Scholar 

  • Tukey, J.W. (1975). Mathematics and the picturing of data. In: Proceedings of the International Congress of Mathematicians, Vol. 2, pp. 523–531.

    MATH  MathSciNet  Google Scholar 

  • Tukey, J.W. (1977). Exploratory Data Analysis. Addison-Wesley, Reading, MA.

    Google Scholar 

  • Tverberg, H. (1966). A generalization of Radon's theorem, Journal of the London Mathematical Society, 41:123–128.

    MATH  MathSciNet  Google Scholar 

  • Wagner, M., Meller, J., and Elber, R. (2002). Large-scale linear programing techniques for the design of protein folding potentials. Technical Report TR-2002-02, Old Dominion University.

    Google Scholar 

  • Wagner, U. (2003). On k-Sets and Applications. Ph.D. Thesis, Theoretical Computer Science, ETH Zurich.

    Google Scholar 

  • Wagner, U. and Welzl, E. (2001). A continuous analogue of the upper bound theorem. Discrete and Computational Geometry, 26(3):205–219.

    MathSciNet  Google Scholar 

  • Yeh, A. and Singh, K. (1997). Balanced confidence sets based on the Tukey depth. Journal of the Royal Statistical Society. Series B, 3:639–652.

    MathSciNet  Google Scholar 

  • Zuo, Y. and Serfling, R. (2000). General notions of statistical depth functions. Annals of Statististics, 28(2):461–482.

    Article  MathSciNet  Google Scholar 

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Authors
  1. Komei Fukuda
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  2. Vera Rosta
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Editors and Affiliations

  1. McGill University & GERAD, Montréal, Canada

    David Avis

  2. École Polytechnique de Montréal & GERAD, Montréal, Canada

    Alain Hertz

  3. Université du Québec a Montréal and GERAD, Montréal, Canada

    Odile Marcotte

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Fukuda, K., Rosta, V. (2005). Data Depth and Maximum Feasible Subsystems. In: Avis, D., Hertz, A., Marcotte, O. (eds) Graph Theory and Combinatorial Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-387-25592-3_3

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