## Data Depth: Robust Multivariate Analysis, Computational Geometry, and ApplicationsThe book is a collection of some of the research presented at the workshop of the same name held in May 2003 at Rutgers University. The workshop brought together researchers from two different communities: statisticians and specialists in computational geometry. The main idea unifying these two research areas turned out to be the notion of data depth, which is an important notion both in statistics and in the study of efficiency of algorithms used in computational geometry. Many ofthe articles in the book lay down the foundations for further collaboration and interdisciplinary research. Information for our distributors: Co-published with the Center for Discrete Mathematics and Theoretical Computer Science beginning with Volume 8. Volumes 1-7 were co-published with theAssociation for Computer Machinery (ACM). |

### From inside the book

Results 1-5 of 6

Page 2

Therefore, among the alternatives to reliance on normal models, the

semiparametric and nonparametric approaches are even more significant in the

multivariate case than in the

such ...

Therefore, among the alternatives to reliance on normal models, the

semiparametric and nonparametric approaches are even more significant in the

multivariate case than in the

**univariate**case. For the multivariate case, however,such ...

Page 8

In the

Thus, for a real weight function w(-), define wt = w(Odn(Xi,X)), and take This

approach has been developed by Mosteller and Tukey (1977) in the

case ...

In the

**univariate**case, with for example the outlyingness function 0ln(9,X) = n ...Thus, for a real weight function w(-), define wt = w(Odn(Xi,X)), and take This

approach has been developed by Mosteller and Tukey (1977) in the

**univariate**case ...

Page 11

Formulation for the

central region" may be defined by the closed interval which has probability

weight p. (This particular choice equalizes tail probabilities.) For p = 1/2, this

gives an ...

Formulation for the

**univariate**case. The median M is given by F-1(1/2). The "pthcentral region" may be defined by the closed interval which has probability

weight p. (This particular choice equalizes tail probabilities.) For p = 1/2, this

gives an ...

Page 12

MF, also as in the

with those of D(x,F). For (as typical) D(-,F) affine invariant, QF(~) is affine

equivariant: for 6 C Kd and nonsingular d x d A, (4.1) QAX+b Special case: the

spatial ...

MF, also as in the

**univariate**case. The contours of the depth ||1 - Qp1(x)\\ agreewith those of D(x,F). For (as typical) D(-,F) affine invariant, QF(~) is affine

equivariant: for 6 C Kd and nonsingular d x d A, (4.1) QAX+b Special case: the

spatial ...

Page 17

The procedure for carrying out this depth rank test is exactly the same as that for

the Wilcoxon rank sum test in testing

generalize the depth rank test to a Kruskal-Wallis type test for scale homogeneity

of ...

The procedure for carrying out this depth rank test is exactly the same as that for

the Wilcoxon rank sum test in testing

**univariate**location shifts. Moreover, wegeneralize the depth rank test to a Kruskal-Wallis type test for scale homogeneity

of ...

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### Contents

xi | |

1 | |

17 | |

On scale curves for nonparametric description of dispersion | 37 |

Data analysis and classification with the zonoid depth | 49 |

On some parametric nonparametric and semiparametric discrimination rules | 61 |

Regression depth and support vector machine | 71 |

Spherical data depth and a multivariate median | 87 |

Impartial trimmed means for functional data | 121 |

Geometric measures of data depth | 147 |

Computation of halfspace depth using simulated annealing | 159 |

Primaldual algorithms for data depth | 171 |

An improved definition analysis and efficiency for the finite sample case | 195 |

Fast algorithms for frames and point depth | 211 |

Statistical data depth and the graphics hardware | 223 |

Depthbased classification for functional data | 103 |

### Common terms and phrases

affine affine transformations algorithm analysis applications approach arrangement assume bounds buffer cell central classification complexity compute consider containing contours convergence convex hull corresponding curves data depth data points data set defined definition denote depth function described dimension discussed distance distribution dual Editors error estimate example exists Figure follows formed frame geometry given halfspace depth hyperplane implementation intersection introduced Journal linear Mathematics matrix mean measure median methods multivariate nonparametric Note notion observations obtained optimal outlyingness parameter performance pixel plane position present probability problem procedure PROOF properties proposed quantile random rank regions regression respect robust robust estimate Rousseeuw rule sample scale curve Science simplicial depth space spherical Statistics Table Theorem univariate vector volume zonoid

### Popular passages

Page 17 - The discussion on aviation safety in this paper reflects the views of the authors, who are solely responsible for the accuracy of the analysis results presented herein, and does not necessarily reflect the official view or policy of the FAA.

Page 189 - K. Miller, S. Ramaswami, P. Rousseeuw, T. Sellares, D. Souvaine, I. Streinu and A. Struyf, Fast implementation of depth contours using topological sweep, Proceedings of the Twelfth ACM-SIAM Symposium on Discrete Algorithms, Washington, DC (2001), 690-699.

Page 157 - S. Jadhav and A. Mukhopadhyay. Computing a centerpoint of a finite planar set of points in linear time.

Page 34 - R. Liu, J. Parelius, and K. Singh, Multivariate analysis by data depth: descriptive statistics, graphics and inference (with discussions), Annals of Statistics 27 (1999), 783-858.

Page 188 - A. Marzetta, K. Fukuda and J. Nievergelt, The parallel search bench ZRAM and its applications, Annals of Operations Research (1999), 45-63.

Page 34 - Structural properties and convergence results for contours of sample statistical depth functions.

Page 168 - I. Ruts, and PJ Rousseeuw, Computing depth contours of bivariate point clouds, Computational Statistics and Data Analysis 23 (1996), 153-168.

Page 118 - Computing depth contours of bivariate point clouds. Computational Statistics and Data Analysis, 23, pp. 153-168. Struyf, A. and Rousseeuw, PJ (2000). High-dimensional computation of the deepest location. Computational Statistics and Data Analysis, to appear. Tukey, JW (1975), Mathematics and the picturing of data.

Page 85 - Steinwart. On the influence of the kernel on the consistency of support vector machines. Journal of Machine Learning Research, 2:67-93, 2002.