Data Depth: Robust Multivariate Analysis, Computational Geometry, and ApplicationsRegina Y. Liu, Robert Joseph Serfling, Diane L. Souvaine American Mathematical Soc. - 246 pages The 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
Page 1
... parameter space , for example on the multivariate space of " regression fits " in univariate multiple regression . Section 4 examines quantile and centered rank functions as entities closely related to each other and connects them with ...
... parameter space , for example on the multivariate space of " regression fits " in univariate multiple regression . Section 4 examines quantile and centered rank functions as entities closely related to each other and connects them with ...
Page 7
... Parameter estimation via outlyingness : location . We mention two ways to use outlyingness functions for location parameter estimation . MINIMIZE OUTLYINGNESS , OR MAXIMIZE DEPTH . For location estimation , the parameter space and the ...
... Parameter estimation via outlyingness : location . We mention two ways to use outlyingness functions for location parameter estimation . MINIMIZE OUTLYINGNESS , OR MAXIMIZE DEPTH . For location estimation , the parameter space and the ...
Page 8
... parameter space . In this case , a natural location estimator is given by choosing the parameter value with minimal outlyingness ( or maximal depth ) . That is , minimize Odn ( 0 , X ) = sup On ( u'0 , u'X ) , || u || = 1 which we may ...
... parameter space . In this case , a natural location estimator is given by choosing the parameter value with minimal outlyingness ( or maximal depth ) . That is , minimize Odn ( 0 , X ) = sup On ( u'0 , u'X ) , || u || = 1 which we may ...
Page 9
... parameter spaces instead of data spaces . Here we take a quick look at the various extended notions of depth function that have appeared . DATA DEPTH ON CIRCLES AND SPHERES ( Liu and Singh , 1992 ) . REGRESSION DEPTH ( Rousseeuw and ...
... parameter spaces instead of data spaces . Here we take a quick look at the various extended notions of depth function that have appeared . DATA DEPTH ON CIRCLES AND SPHERES ( Liu and Singh , 1992 ) . REGRESSION DEPTH ( Rousseeuw and ...
Page 11
... parameter . Equivalent univariate formulation . A “ median - oriented quantile function " QF ( u ) , with u = 2p - 1 and median M = Q ( 0 ) , is defined by QF ( u ) = F - 1 1+ 2 -1 < u < 1 , with sign of u corresponding to direction ...
... parameter . Equivalent univariate formulation . A “ median - oriented quantile function " QF ( u ) , with u = 2p - 1 and median M = Q ( 0 ) , is defined by QF ( u ) = F - 1 1+ 2 -1 < u < 1 , with sign of u corresponding to direction ...
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
algorithm Annals of Statistics bivariate breakdown point buffer cell center-outward central region classification Computational Geometry Computer Science containing convergence convex hull Data Analysis data depth data points data set defined denote density depth contours depth function depth measures depth-based dimensional distribution dual error rate estimate example Figure finite functional data given halfspace depth hyperplane hyperplane arrangement integer programs iterations kernel Lemma linear location depth logistic regression lower bound Mathematics Mathematics Subject Classification matrix methods Multivariate Analysis multivariate data ncomplete O(n² optimal outlyingness parameter pixel point set problem PROOF properties quantile function random rank tests regression depth robust Rousseeuw sample scale curve Section Serfling simplicial depth simplicial median simulated spatial spherical depth spherical median stencil buffer subset support vector machine symmetric Theorem triangles trimmed mean Tukey univariate weak convergence zonoid depth
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.