6,209 research outputs found
Orthogonal polynomial kernels and canonical correlations for Dirichlet measures
We consider a multivariate version of the so-called Lancaster problem of
characterizing canonical correlation coefficients of symmetric bivariate
distributions with identical marginals and orthogonal polynomial expansions.
The marginal distributions examined in this paper are the Dirichlet and the
Dirichlet multinomial distribution, respectively, on the continuous and the
N-discrete d-dimensional simplex. Their infinite-dimensional limit
distributions, respectively, the Poisson-Dirichlet distribution and Ewens's
sampling formula, are considered as well. We study, in particular, the
possibility of mapping canonical correlations on the d-dimensional continuous
simplex (i) to canonical correlation sequences on the d+1-dimensional simplex
and/or (ii) to canonical correlations on the discrete simplex, and vice versa.
Driven by this motivation, the first half of the paper is devoted to providing
a full characterization and probabilistic interpretation of n-orthogonal
polynomial kernels (i.e., sums of products of orthogonal polynomials of the
same degree n) with respect to the mentioned marginal distributions. We
establish several identities and some integral representations which are
multivariate extensions of important results known for the case d=2 since the
1970s. These results, along with a common interpretation of the mentioned
kernels in terms of dependent Polya urns, are shown to be key features leading
to several non-trivial solutions to Lancaster's problem, many of which can be
extended naturally to the limit as .Comment: Published in at http://dx.doi.org/10.3150/11-BEJ403 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
An IRT Analysis of Motive Questionnaires: The Unified Motive Scales
Multiple inventories claiming to assess the same explicit motive (achievement, power, or affiliation) show only mediocre convergent validity. In three studies (N = 1685) the structure, nomological net, and content coverage of multiple existing motive scales was investigated with exploratory factor analyses. The analyses revealed four
approach factors (achievement, power, affiliation, and intimacy) and a general avoidance factor with a facet structure. New scales (the Unified Motive Scales; UMS) were developed using IRT, reflecting these underlying dimensions. In comparison to existing questionnaires, the UMS have the highest measurement precision and provide short (6-item) and ultra-short (3-item) scales. In a fourth study (N = 96), the UMS demonstrated incremental validity over existing motive scales with respect to several outcome criteria
A semiparametric regression model for paired longitudinal outcomes with application in childhood blood pressure development
This research examines the simultaneous influences of height and weight on
longitudinally measured systolic and diastolic blood pressure in children.
Previous studies have shown that both height and weight are positively
associated with blood pressure. In children, however, the concurrent increases
of height and weight have made it all but impossible to discern the effect of
height from that of weight. To better understand these influences, we propose
to examine the joint effect of height and weight on blood pressure. Bivariate
thin plate spline surfaces are used to accommodate the potentially nonlinear
effects as well as the interaction between height and weight. Moreover, we
consider a joint model for paired blood pressure measures, that is, systolic
and diastolic blood pressure, to account for the underlying correlation between
the two measures within the same individual. The bivariate spline surfaces are
allowed to vary across different groups of interest. We have developed related
model fitting and inference procedures. The proposed method is used to analyze
data from a real clinical investigation.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS567 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Max-stable random sup-measures with comonotonic tail dependence
Several objects in the Extremes literature are special instances of
max-stable random sup-measures. This perspective opens connections to the
theory of random sets and the theory of risk measures and makes it possible to
extend corresponding notions and results from the literature with streamlined
proofs. In particular, it clarifies the role of Choquet random sup-measures and
their stochastic dominance property. Key tools are the LePage representation of
a max-stable random sup-measure and the dual representation of its tail
dependence functional. Properties such as complete randomness, continuity,
separability, coupling, continuous choice, invariance and transformations are
also analysed.Comment: 28 pages, 1 figur
The Topology ToolKit
This system paper presents the Topology ToolKit (TTK), a software platform
designed for topological data analysis in scientific visualization. TTK
provides a unified, generic, efficient, and robust implementation of key
algorithms for the topological analysis of scalar data, including: critical
points, integral lines, persistence diagrams, persistence curves, merge trees,
contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots,
Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due
to a tight integration with ParaView. It is also easily accessible to
developers through a variety of bindings (Python, VTK/C++) for fast prototyping
or through direct, dependence-free, C++, to ease integration into pre-existing
complex systems. While developing TTK, we faced several algorithmic and
software engineering challenges, which we document in this paper. In
particular, we present an algorithm for the construction of a discrete gradient
that complies to the critical points extracted in the piecewise-linear setting.
This algorithm guarantees a combinatorial consistency across the topological
abstractions supported by TTK, and importantly, a unified implementation of
topological data simplification for multi-scale exploration and analysis. We
also present a cached triangulation data structure, that supports time
efficient and generic traversals, which self-adjusts its memory usage on demand
for input simplicial meshes and which implicitly emulates a triangulation for
regular grids with no memory overhead. Finally, we describe an original
software architecture, which guarantees memory efficient and direct accesses to
TTK features, while still allowing for researchers powerful and easy bindings
and extensions. TTK is open source (BSD license) and its code, online
documentation and video tutorials are available on TTK's website
Constructing a bivariate distribution function with given marginals and correlation: application to the galaxy luminosity function
We show an analytic method to construct a bivariate distribution function
(DF) with given marginal distributions and correlation coefficient. We
introduce a convenient mathematical tool, called a copula, to connect two DFs
with any prescribed dependence structure. If the correlation of two variables
is weak (Pearson's correlation coefficient ), the
Farlie-Gumbel-Morgenstern (FGM) copula provides an intuitive and natural way
for constructing such a bivariate DF. When the linear correlation is stronger,
the FGM copula cannot work anymore. In this case, we propose to use a Gaussian
copula, which connects two given marginals and directly related to the linear
correlation coefficient between two variables. Using the copulas, we
constructed the BLFs and discuss its statistical properties. Especially, we
focused on the FUV--FIR BLF, since these two luminosities are related to the
star formation (SF) activity. Though both the FUV and FIR are related to the SF
activity, the univariate LFs have a very different functional form: former is
well described by the Schechter function whilst the latter has a much more
extended power-law like luminous end. We constructed the FUV-FIR BLFs by the
FGM and Gaussian copulas with different strength of correlation, and examined
their statistical properties. Then, we discuss some further possible
applications of the BLF: the problem of a multiband flux-limited sample
selection, the construction of the SF rate (SFR) function, and the construction
of the stellar mass of galaxies ()--specific SFR () relation. The
copulas turned out to be a very useful tool to investigate all these issues,
especially for including the complicated selection effects.Comment: 14 pages, 5 figures, accepted for publication in MNRAS
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