22,162 research outputs found
Real time clustering of time series using triangular potentials
Motivated by the problem of computing investment portfolio weightings we
investigate various methods of clustering as alternatives to traditional
mean-variance approaches. Such methods can have significant benefits from a
practical point of view since they remove the need to invert a sample
covariance matrix, which can suffer from estimation error and will almost
certainly be non-stationary. The general idea is to find groups of assets which
share similar return characteristics over time and treat each group as a single
composite asset. We then apply inverse volatility weightings to these new
composite assets. In the course of our investigation we devise a method of
clustering based on triangular potentials and we present associated theoretical
results as well as various examples based on synthetic data.Comment: AIFU1
Performance Analysis of Spectral Clustering on Compressed, Incomplete and Inaccurate Measurements
Spectral clustering is one of the most widely used techniques for extracting
the underlying global structure of a data set. Compressed sensing and matrix
completion have emerged as prevailing methods for efficiently recovering sparse
and partially observed signals respectively. We combine the distance preserving
measurements of compressed sensing and matrix completion with the power of
robust spectral clustering. Our analysis provides rigorous bounds on how small
errors in the affinity matrix can affect the spectral coordinates and
clusterability. This work generalizes the current perturbation results of
two-class spectral clustering to incorporate multi-class clustering with k
eigenvectors. We thoroughly track how small perturbation from using compressed
sensing and matrix completion affect the affinity matrix and in succession the
spectral coordinates. These perturbation results for multi-class clustering
require an eigengap between the kth and (k+1)th eigenvalues of the affinity
matrix, which naturally occurs in data with k well-defined clusters. Our
theoretical guarantees are complemented with numerical results along with a
number of examples of the unsupervised organization and clustering of image
data
How to Round Subspaces: A New Spectral Clustering Algorithm
A basic problem in spectral clustering is the following. If a solution
obtained from the spectral relaxation is close to an integral solution, is it
possible to find this integral solution even though they might be in completely
different basis? In this paper, we propose a new spectral clustering algorithm.
It can recover a -partition such that the subspace corresponding to the span
of its indicator vectors is close to the original subspace in
spectral norm with being the minimum possible ( always).
Moreover our algorithm does not impose any restriction on the cluster sizes.
Previously, no algorithm was known which could find a -partition closer than
.
We present two applications for our algorithm. First one finds a disjoint
union of bounded degree expanders which approximate a given graph in spectral
norm. The second one is for approximating the sparsest -partition in a graph
where each cluster have expansion at most provided where is the eigenvalue of
Laplacian matrix. This significantly improves upon the previous algorithms,
which required .Comment: Appeared in SODA 201
Hierarchical Pancaking: Why the Zel'dovich Approximation Describes Coherent Large-Scale Structure in N-Body Simulations of Gravitational Clustering
To explain the rich structure of voids, clusters, sheets, and filaments
apparent in the Universe, we present evidence for the convergence of the two
classic approaches to gravitational clustering, the ``pancake'' and
``hierarchical'' pictures. We compare these two models by looking at agreement
between individual structures -- the ``pancakes'' which are characteristic of
the Zel'dovich Approximation (ZA) and also appear in hierarchical N-body
simulations. We find that we can predict the orientation and position of N-body
simulation objects rather well, with decreasing accuracy for increasing
large- (small scale) power in the initial conditions. We examined an N-body
simulation with initial power spectrum , and found that a
modified version of ZA based on the smoothed initial potential worked well in
this extreme hierarchical case, implying that even here very low-amplitude long
waves dominate over local clumps (although we can see the beginning of the
breakdown expected for ). In this case the correlation length of the
initial potential is extremely small initially, but grows considerably as the
simulation evolves. We show that the nonlinear gravitational potential strongly
resembles the smoothed initial potential. This explains why ZA with smoothed
initial conditions reproduces large-scale structure so well, and probably why
our Universe has a coherent large-scale structure.Comment: 17 pages of uuencoded postscript. There are 8 figures which are too
large to post here. To receive the uuencoded figures by email (or hard copies
by regular mail), please send email to: [email protected]. This is
a revision of a paper posted earlier now in press at MNRA
Magnification bias as a novel probe for primordial magnetic fields
In this paper we investigate magnetic fields generated in the early Universe.
These fields are important candidates at explaining the origin of astrophysical
magnetism observed in galaxies and galaxy clusters, whose genesis is still by
and large unclear. Compared to the standard inflationary power spectrum,
intermediate to small scales would experience further substantial matter
clustering, were a cosmological magnetic field present prior to recombination.
As a consequence, the bias and redshift distribution of galaxies would also be
modified. Hitherto, primordial magnetic fields (PMFs) have been tested and
constrained with a number of cosmological observables, e.g. the cosmic
microwave background radiation, galaxy clustering and, more recently, weak
gravitational lensing. Here, we explore the constraining potential of the
density fluctuation bias induced by gravitational lensing magnification onto
the galaxy-galaxy angular power spectrum. Such an effect is known as
magnification bias. Compared to the usual galaxy clustering approach,
magnification bias helps in lifting the pathological degeneracy present amongst
power spectrum normalisation and galaxy bias. This is because magnification
bias cross-correlates galaxy number density fluctuations of nearby objects with
weak lensing distortions of high-redshift sources. Thus, it takes advantage of
the gravitational deflection of light, which is insensitive to galaxy bias but
powerful in constraining the density fluctuation amplitude. To scrutinise the
potentiality of this method, we adopt a deep and wide-field spectroscopic
galaxy survey. We show that magnification bias does contain important
information on primordial magnetism, which will be useful in combination with
galaxy clustering and shear. We find we shall be able to rule out at 95.4% CL
amplitudes of PMFs larger than 0.0005 nG for values of the PMF power spectral
index ~0.Comment: 21 pages, 9 figures; published on JCA
- …