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
Results 6-10 of 81
Page 6
... data by depth value , center- outward : X [ 1 ] ,, X [ n ] · • Depth ... data set X = ( X1 , ... , Xn ) in Rd can be defined in various ways . Below we sketch ... points . As a general class of examples , for given univariate location and ...
... data by depth value , center- outward : X [ 1 ] ,, X [ n ] · • Depth ... data set X = ( X1 , ... , Xn ) in Rd can be defined in various ways . Below we sketch ... points . As a general class of examples , for given univariate location and ...
Page 7
... points x1 , ... , æ in Rd , and define n Odn ( x , X ) = k Σh ( x ; Xi Xix ) , i.e. , the average distance of a from point subsets of size k drawn from the data set . This leads to depth functions using ( 3.2 ) for unbounded h and ( 3.3 ) ...
... points x1 , ... , æ in Rd , and define n Odn ( x , X ) = k Σh ( x ; Xi Xix ) , i.e. , the average distance of a from point subsets of size k drawn from the data set . This leads to depth functions using ( 3.2 ) for unbounded h and ( 3.3 ) ...
Page 11
... point of some pth central region . In this sense , the associated p indicates the " outlyingness " of x . The pth ... points a Rd generate a quantile function QF ( u ) , u Є Bd - 1 ( 0 ) . Here the outlyingness parameter || u || also ...
... point of some pth central region . In this sense , the associated p indicates the " outlyingness " of x . The pth ... points a Rd generate a quantile function QF ( u ) , u Є Bd - 1 ( 0 ) . Here the outlyingness parameter || u || also ...
Page 19
... Data A data depth , roughly speaking , is a measure of how deep or how outlying a given point is with respect to a multivariate data cloud or its underlying distribu- tion . It gives rise to a natural center - outward ordering of the sample ...
... Data A data depth , roughly speaking , is a measure of how deep or how outlying a given point is with respect to a multivariate data cloud or its underlying distribu- tion . It gives rise to a natural center - outward ordering of the sample ...
Page 20
... point to the largest , while the former starts from the middle sample point and move outwards in all directions . Figure 1 helps illustrates this property by showing the depth ordering of a random sample of 500 points drawn from a ...
... point to the largest , while the former starts from the middle sample point and move outwards in all directions . Figure 1 helps illustrates this property by showing the depth ordering of a random sample of 500 points drawn from a ...
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.