Abstract
We propose an algorithm to sample the area of the smallest convex hull containing \(n\) sample points uniformly distributed over unit square. To do it, we introduce a new coordinate system for the position of vertexes and re-write joint distribution of the number of vertexes and their locations in the new coordinate system. The proposed algorithm is much faster than existing procedure and has a computational complexity on the order of \(O(T)\), where \(T\) is the number of vertexes. Using the proposed algorithm, we numerically investigate the asymptotic behavior of functionals of the random convex hull. In addition, we apply it to finding pairs of stocks where the returns are dependent on each other on the New York Stock Exchange.
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References
Barnett V (1976) The ordering of multivariate data (with discussion). J R Stat Soc Ser A 139:318–355
Benjamini Y, Hochberg Y (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. J R Stat Soc Ser B 57:289–300
Berger VW, Permutt T (1998) Convex hull test for ordered categorical data. Biometrics 54:1541–1550
Buchta C (2005) An identity relating moments of functionals of convex hulls. Discrete Comput Geom 33:125–142
Buchta C (2006) The exact distribution of the number of vertexes of a random convex chain. Mathematika 53:247–254
Cabo AJ, Groeneboom (1994) Limit theorems for functional of convex hull. Prob Theory Relat Fields 199:31–55
Cook RD (1979) Influential observations in linear regression. J Am Stat Assoc 74:169–174
Devroye L (1982) On the computer generation of random convex hulls. Comput Math Appl 8:1–13
Devroye L (1991) On the oscillation of the expected number of points on a random convex hull. Stat Prob Lett 11:281–286
Efron B (1965) The convex hull of random set of points. Biometrika 52:331–343
Gatev E, Goetzmann WN, Rouwenhorst G (2006) Pairs trading: performance of a relative-value arbitrage rule. Rev Finan Stud 19:797–827
Graham R (1972) An efficient algorithm for determining the convex hull of a finite point set. Inf Process Lett 1:132–133
Groeneboom P (1988) Limit theorems for convex hulls. Prob Theory Relat Fields 79:327–368
Hsing T (1994) On the asymptotic distribution of the area outside a random convex hull in a disk. Ann Appl Prob 4:478–493
Huber PJ (1972) Robust statistics: a review. Ann Math Stat 43:1041–1067
Hueter I (1994) The convex hull of normal samples. J Appl Prob 26:855–875
Hueter I (1999) Limit theorems for the convex hull of random points in higher dimensions. Trans Am Math Soc 351:4337–4363
Kirkpatrick DG, Seidel R (1986) The ultimate planar convex hull algorithm. SIAM J Comput 15:287–299
Renyi A, Sulanke R (1963a) Über die knovexe Hülle von n zufällig gewählten Punkten. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 2:75–84
Renyi A, Sulanke R (1963b) Über die knovexe Hülle von n zufällig gewählten Punkten. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 3:138–147
Worton BI (1995) A convex hull-based estimator for home-range size. Biometrics 51:1206–1215
Zani S, Riani M, Corbelini A (1998) Robust bivariate boxplots and multiple outlier detection. Comput Stat Data Anal 28(3):257–270
Acknowledgments
We are grateful to the Associate Editor and the anonymous reviewers for many constructive suggestions, which greatly improve the paper. We are grateful to Prof. Youngjo Lee, Department of Statistics, Seoul National University for his general help in preparing the manuscript. Johan Lim’s research was supported by a National Research Foundation of Korea (NRF) grant funded by the Korea Government (MSIP) (No. 2011-0029104).
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Ng, C.T., Lim, J., Lee, K.E. et al. A fast algorithm to sample the number of vertexes and the area of the random convex hull on the unit square. Comput Stat 29, 1187–1205 (2014). https://doi.org/10.1007/s00180-014-0486-1
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DOI: https://doi.org/10.1007/s00180-014-0486-1