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 50
Page 7
... rank of x in the data set ( we elaborate on quantile and centered rank functions later ) . Using the spatial quantile function [ 9 ] , we obtain the spatial depth [ 57 ] , [ 48 ] . Parameter estimation via outlyingness : location . We ...
... rank of x in the data set ( we elaborate on quantile and centered rank functions later ) . Using the spatial quantile function [ 9 ] , we obtain the spatial depth [ 57 ] , [ 48 ] . Parameter estimation via outlyingness : location . We ...
Page 10
... Rank Functions The term " quantile " in the multivariate case has become used rather too loosely . Thus we ask : How to formulate multivariate quantile functions ? What properties are desirable ? What are the interrelations among ...
... Rank Functions The term " quantile " in the multivariate case has become used rather too loosely . Thus we ask : How to formulate multivariate quantile functions ? What properties are desirable ? What are the interrelations among ...
Page 11
... rank function . Its magnitude | QÃ1 ( x ) | 2F ( x ) 1 in a natural way measures the outlyingness of x relative to the distribution F. Since x satisfies x = QF ( Q ( x ) ) , we may think of the quantiles QF ( u ) as indexed by a ...
... rank function . Its magnitude | QÃ1 ( x ) | 2F ( x ) 1 in a natural way measures the outlyingness of x relative to the distribution F. Since x satisfies x = QF ( Q ( x ) ) , we may think of the quantiles QF ( u ) as indexed by a ...
Page 12
... rank function " . Here r || u || represents the outlyingness of QF ( u ) but not the probability weight of the central region { QF ( t ) : || t || ≤ r } demarked by { QF ( u ) , || u } = r } . Here u is not the direction from QF ( u ) ...
... rank function " . Here r || u || represents the outlyingness of QF ( u ) but not the probability weight of the central region { QF ( t ) : || t || ≤ r } demarked by { QF ( u ) , || u } = r } . Here u is not the direction from QF ( u ) ...
Page 13
... rank functions may be formlulated as mutually inverse , and the magnitude of a centered rank function defines an outlyingness function having an associated depth functions as inverse and whose contours define a quantile function . These ...
... rank functions may be formlulated as mutually inverse , and the magnitude of a centered rank function defines an outlyingness function having an associated depth functions as inverse and whose contours define a quantile function . These ...
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.