153 research outputs found
Mumford dendrograms and discrete p-adic symmetries
In this article, we present an effective encoding of dendrograms by embedding
them into the Bruhat-Tits trees associated to -adic number fields. As an
application, we show how strings over a finite alphabet can be encoded in
cyclotomic extensions of and discuss -adic DNA encoding. The
application leads to fast -adic agglomerative hierarchic algorithms similar
to the ones recently used e.g. by A. Khrennikov and others. From the viewpoint
of -adic geometry, to encode a dendrogram in a -adic field means
to fix a set of -rational punctures on the -adic projective line
. To is associated in a natural way a
subtree inside the Bruhat-Tits tree which recovers , a method first used by
F. Kato in 1999 in the classification of discrete subgroups of
.
Next, we show how the -adic moduli space of
with punctures can be applied to the study of time series of
dendrograms and those symmetries arising from hyperbolic actions on
. In this way, we can associate to certain classes of dynamical
systems a Mumford curve, i.e. a -adic algebraic curve with totally
degenerate reduction modulo .
Finally, we indicate some of our results in the study of general discrete
actions on , and their relation to -adic Hurwitz spaces.Comment: 14 pages, 6 figure
Degenerating families of dendrograms
Dendrograms used in data analysis are ultrametric spaces, hence objects of
nonarchimedean geometry. It is known that there exist -adic representation
of dendrograms. Completed by a point at infinity, they can be viewed as
subtrees of the Bruhat-Tits tree associated to the -adic projective line.
The implications are that certain moduli spaces known in algebraic geometry are
-adic parameter spaces of (families of) dendrograms, and stochastic
classification can also be handled within this framework. At the end, we
calculate the topology of the hidden part of a dendrogram.Comment: 13 pages, 8 figure
Fast redshift clustering with the Baire (ultra) metric
The Baire metric induces an ultrametric on a dataset and is of linear
computational complexity, contrasted with the standard quadratic time
agglomerative hierarchical clustering algorithm. We apply the Baire distance to
spectrometric and photometric redshifts from the Sloan Digital Sky Survey
using, in this work, about half a million astronomical objects. We want to know
how well the (more cos\ tly to determine) spectrometric redshifts can predict
the (more easily obtained) photometric redshifts, i.e. we seek to regress the
spectrometric on the photometric redshifts, and we develop a clusterwise
nearest neighbor regression procedure for this.Comment: 14 pages, 6 figure
Consensus clustering approach to group brain connectivity matrices
A novel approach rooted on the notion of consensus clustering, a strategy
developed for community detection in complex networks, is proposed to cope with
the heterogeneity that characterizes connectivity matrices in health and
disease. The method can be summarized as follows:
(i) define, for each node, a distance matrix for the set of subjects by
comparing the connectivity pattern of that node in all pairs of subjects; (ii)
cluster the distance matrix for each node; (iii) build the consensus network
from the corresponding partitions; (iv) extract groups of subjects by finding
the communities of the consensus network thus obtained.
Differently from the previous implementations of consensus clustering, we
thus propose to use the consensus strategy to combine the information arising
from the connectivity patterns of each node. The proposed approach may be seen
either as an exploratory technique or as an unsupervised pre-training step to
help the subsequent construction of a supervised classifier. Applications on a
toy model and two real data sets, show the effectiveness of the proposed
methodology, which represents heterogeneity of a set of subjects in terms of a
weighted network, the consensus matrix
A -adic RanSaC algorithm for stereo vision using Hensel lifting
A -adic variation of the Ran(dom) Sa(mple) C(onsensus) method for solving
the relative pose problem in stereo vision is developped. From two 2-adically
encoded images a random sample of five pairs of corresponding points is taken,
and the equations for the essential matrix are solved by lifting solutions
modulo 2 to the 2-adic integers. A recently devised -adic hierarchical
classification algorithm imitating the known LBG quantisation method classifies
the solutions for all the samples after having determined the number of
clusters using the known intra-inter validity of clusterings. In the successful
case, a cluster ranking will determine the cluster containing a 2-adic
approximation to the "true" solution of the problem.Comment: 15 pages; typos removed, abstract changed, computation error remove
ICA model order selection of task co-activation networks
Independent component analysis (ICA) has become a widely used method for extracting functional networks in the brain during rest and task. Historically, preferred ICA dimensionality has widely varied within the neuroimaging community, but typically varies between 20 and 100 components. This can be problematic when comparing results across multiple studies because of the impact ICA dimensionality has on the topology of its resultant components. Recent studies have demonstrated that ICA can be applied to peak activation coordinates archived in a large neuroimaging database (i.e., BrainMap Database) to yield whole-brain task-based co-activation networks. A strength of applying ICA to BrainMap data is that the vast amount of metadata in BrainMap can be used to quantitatively assess tasks and cognitive processes contributing to each component. In this study, we investigated the effect of model order on the distribution of functional properties across networks as a method for identifying the most informative decompositions of BrainMap-based ICA components. Our findings suggest dimensionality of 20 for low model order ICA to examine large-scale brain networks, and dimensionality of 70 to provide insight into how large-scale networks fractionate into sub-networks. We also provide a functional and organizational assessment of visual, motor, emotion, and interoceptive task co-activation networks as they fractionate from low to high model-orders
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