6,487 research outputs found
Multivariate Analysis of Orthogonal Range Searching and Graph Distances
We show that the eccentricities, diameter, radius, and Wiener index of an undirected n-vertex graph with nonnegative edge lengths can be computed in time O(n * binom{k+ceil[log n]}{k} * 2^k k^2 log n), where k is the treewidth of the graph. For every epsilon>0, this bound is n^{1+epsilon}exp O(k), which matches a hardness result of Abboud, Vassilevska Williams, and Wang (SODA 2015) and closes an open problem in the multivariate analysis of polynomial-time computation. To this end, we show that the analysis of an algorithm of Cabello and Knauer (Comp. Geom., 2009) in the regime of non-constant treewidth can be improved by revisiting the analysis of orthogonal range searching, improving bounds of the form log^d n to binom{d+ceil[log n]}{d}, as originally observed by Monier (J. Alg. 1980).
We also investigate the parameterization by vertex cover number
Multivariate Analysis of Orthogonal Range Searching and Graph Distances Parameterized by Treewidth
We show that the eccentricities, diameter, radius, and Wiener index of an
undirected -vertex graph with nonnegative edge lengths can be computed in
time , where
is the treewidth of the graph. For every , this bound is
, which matches a hardness result of Abboud,
Vassilevska Williams, and Wang (SODA 2015) and closes an open problem in the
multivariate analysis of polynomial-time computation. To this end, we show that
the analysis of an algorithm of Cabello and Knauer (Comp. Geom., 2009) in the
regime of non-constant treewidth can be improved by revisiting the analysis of
orthogonal range searching, improving bounds of the form to
, as originally observed by Monier (J. Alg.
1980).
We also investigate the parameterization by vertex cover number
Analysis of approximate nearest neighbor searching with clustered point sets
We present an empirical analysis of data structures for approximate nearest
neighbor searching. We compare the well-known optimized kd-tree splitting
method against two alternative splitting methods. The first, called the
sliding-midpoint method, which attempts to balance the goals of producing
subdivision cells of bounded aspect ratio, while not producing any empty cells.
The second, called the minimum-ambiguity method is a query-based approach. In
addition to the data points, it is also given a training set of query points
for preprocessing. It employs a simple greedy algorithm to select the splitting
plane that minimizes the average amount of ambiguity in the choice of the
nearest neighbor for the training points. We provide an empirical analysis
comparing these two methods against the optimized kd-tree construction for a
number of synthetically generated data and query sets. We demonstrate that for
clustered data and query sets, these algorithms can provide significant
improvements over the standard kd-tree construction for approximate nearest
neighbor searching.Comment: 20 pages, 8 figures. Presented at ALENEX '99, Baltimore, MD, Jan
15-16, 199
On the Procrustean analogue of individual differences scaling (INDSCAL)
In this paper, individual differences scaling (INDSCAL) is revisited, considering
INDSCAL as being embedded within a hierarchy of individual difference scaling
models. We explore the members of this family, distinguishing (i) models, (ii) the
role of identification and substantive constraints, (iii) criteria for fitting models and (iv) algorithms to optimise the criteria. Model formulations may be based either on data that are in the form of proximities or on configurational matrices. In its configurational version, individual difference scaling may be formulated as a form of generalized Procrustes analysis. Algorithms are introduced for fitting the new
models. An application from sensory evaluation illustrates the performance of the
methods and their solutions
Revisiting Guerry's data: Introducing spatial constraints in multivariate analysis
Standard multivariate analysis methods aim to identify and summarize the main
structures in large data sets containing the description of a number of
observations by several variables. In many cases, spatial information is also
available for each observation, so that a map can be associated to the
multivariate data set. Two main objectives are relevant in the analysis of
spatial multivariate data: summarizing covariation structures and identifying
spatial patterns. In practice, achieving both goals simultaneously is a
statistical challenge, and a range of methods have been developed that offer
trade-offs between these two objectives. In an applied context, this
methodological question has been and remains a major issue in community
ecology, where species assemblages (i.e., covariation between species
abundances) are often driven by spatial processes (and thus exhibit spatial
patterns). In this paper we review a variety of methods developed in community
ecology to investigate multivariate spatial patterns. We present different ways
of incorporating spatial constraints in multivariate analysis and illustrate
these different approaches using the famous data set on moral statistics in
France published by Andr\'{e}-Michel Guerry in 1833. We discuss and compare the
properties of these different approaches both from a practical and theoretical
viewpoint.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS356 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Descriptive methods of data analysis for marketing data – theoretical and practical considerations
Marketing has as main objective the guidance of a firm’s activities according to current and future needs – of consumers’. This necessarily assumes the existence of a suitable information system, and also the knowledge of some modern analysis, processing and interpretation of the so complex information in the field of marketing. The descriptive methods of data analysis represent multidimensional analysis tools that are strong and effective, tools based on which important information can be obtained for market research. The paper comparatively presents some of these methods, respectively: factor analysis, main component analysis, correspondence analysis and canonical analysis.factor analysis, marketing, descriptive methods.
Multivariate Approaches to Classification in Extragalactic Astronomy
Clustering objects into synthetic groups is a natural activity of any
science. Astrophysics is not an exception and is now facing a deluge of data.
For galaxies, the one-century old Hubble classification and the Hubble tuning
fork are still largely in use, together with numerous mono-or bivariate
classifications most often made by eye. However, a classification must be
driven by the data, and sophisticated multivariate statistical tools are used
more and more often. In this paper we review these different approaches in
order to situate them in the general context of unsupervised and supervised
learning. We insist on the astrophysical outcomes of these studies to show that
multivariate analyses provide an obvious path toward a renewal of our
classification of galaxies and are invaluable tools to investigate the physics
and evolution of galaxies.Comment: Open Access paper.
http://www.frontiersin.org/milky\_way\_and\_galaxies/10.3389/fspas.2015.00003/abstract\>.
\<10.3389/fspas.2015.00003 \&g
P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches
While the P vs NP problem is mainly approached form the point of view of
discrete mathematics, this paper proposes reformulations into the field of
abstract algebra, geometry, fourier analysis and of continuous global
optimization - which advanced tools might bring new perspectives and approaches
for this question. The first one is equivalence of satisfaction of 3-SAT
problem with the question of reaching zero of a nonnegative degree 4
multivariate polynomial (sum of squares), what could be tested from the
perspective of algebra by using discriminant. It could be also approached as a
continuous global optimization problem inside , for example in
physical realizations like adiabatic quantum computers. However, the number of
local minima usually grows exponentially. Reducing to degree 2 polynomial plus
constraints of being in , we get geometric formulations as the
question if plane or sphere intersects with . There will be also
presented some non-standard perspectives for the Subset-Sum, like through
convergence of a series, or zeroing of fourier-type integral for some natural . The last discussed
approach is using anti-commuting Grassmann numbers , making nonzero only if has a Hamilton cycle. Hence,
the PNP assumption implies exponential growth of matrix representation of
Grassmann numbers. There will be also discussed a looking promising
algebraic/geometric approach to the graph isomorphism problem -- tested to
successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure
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