6,635 research outputs found
Correlation, hierarchies, and networks in financial markets
We discuss some methods to quantitatively investigate the properties of
correlation matrices. Correlation matrices play an important role in portfolio
optimization and in several other quantitative descriptions of asset price
dynamics in financial markets. Specifically, we discuss how to define and
obtain hierarchical trees, correlation based trees and networks from a
correlation matrix. The hierarchical clustering and other procedures performed
on the correlation matrix to detect statistically reliable aspects of the
correlation matrix are seen as filtering procedures of the correlation matrix.
We also discuss a method to associate a hierarchically nested factor model to a
hierarchical tree obtained from a correlation matrix. The information retained
in filtering procedures and its stability with respect to statistical
fluctuations is quantified by using the Kullback-Leibler distance.Comment: 37 pages, 9 figures, 3 table
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
Hypothesis Testing For Network Data in Functional Neuroimaging
In recent years, it has become common practice in neuroscience to use
networks to summarize relational information in a set of measurements,
typically assumed to be reflective of either functional or structural
relationships between regions of interest in the brain. One of the most basic
tasks of interest in the analysis of such data is the testing of hypotheses, in
answer to questions such as "Is there a difference between the networks of
these two groups of subjects?" In the classical setting, where the unit of
interest is a scalar or a vector, such questions are answered through the use
of familiar two-sample testing strategies. Networks, however, are not Euclidean
objects, and hence classical methods do not directly apply. We address this
challenge by drawing on concepts and techniques from geometry, and
high-dimensional statistical inference. Our work is based on a precise
geometric characterization of the space of graph Laplacian matrices and a
nonparametric notion of averaging due to Fr\'echet. We motivate and illustrate
our resulting methodologies for testing in the context of networks derived from
functional neuroimaging data on human subjects from the 1000 Functional
Connectomes Project. In particular, we show that this global test is more
statistical powerful, than a mass-univariate approach. In addition, we have
also provided a method for visualizing the individual contribution of each edge
to the overall test statistic.Comment: 34 pages. 5 figure
On high moments of strongly diluted large Wigner random matrices
We consider a dilute version of the Wigner ensemble of nxn random matrices
and study the asymptotic behavior of their moments in the limit of
infinite , and , where is the dilution parameter. We show
that in the asymptotic regime of the strong dilution, the moments with
depend on the second and the fourth moments of the random entries
and do not depend on other even moments of . This fact can be
regarded as an evidence of a new type of the universal behavior of the local
eigenvalue distribution of strongly dilute random matrices at the border of the
limiting spectrum. As a by-product of the proof, we describe a new kind of
Catalan-type numbers related with the tree-type walks.Comment: 43 pages (version four: misprints corrected, discussion added, other
minor modifications
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