195,038 research outputs found
Measuring Cluster Stability for Bayesian Nonparametrics Using the Linear Bootstrap
Clustering procedures typically estimate which data points are clustered
together, a quantity of primary importance in many analyses. Often used as a
preliminary step for dimensionality reduction or to facilitate interpretation,
finding robust and stable clusters is often crucial for appropriate for
downstream analysis. In the present work, we consider Bayesian nonparametric
(BNP) models, a particularly popular set of Bayesian models for clustering due
to their flexibility. Because of its complexity, the Bayesian posterior often
cannot be computed exactly, and approximations must be employed. Mean-field
variational Bayes forms a posterior approximation by solving an optimization
problem and is widely used due to its speed. An exact BNP posterior might vary
dramatically when presented with different data. As such, stability and
robustness of the clustering should be assessed.
A popular mean to assess stability is to apply the bootstrap by resampling
the data, and rerun the clustering for each simulated data set. The time cost
is thus often very expensive, especially for the sort of exploratory analysis
where clustering is typically used. We propose to use a fast and automatic
approximation to the full bootstrap called the "linear bootstrap", which can be
seen by local data perturbation. In this work, we demonstrate how to apply this
idea to a data analysis pipeline, consisting of an MFVB approximation to a BNP
clustering posterior of time course gene expression data. We show that using
auto-differentiation tools, the necessary calculations can be done
automatically, and that the linear bootstrap is a fast but approximate
alternative to the bootstrap.Comment: 9 pages, NIPS 2017 Advances in Approximate Bayesian Inference
Worksho
The growth of structure in the Szekeres inhomogeneous cosmological models and the matter-dominated era
This study belongs to a series devoted to using Szekeres inhomogeneous models
to develop a theoretical framework where observations can be investigated with
a wider range of possible interpretations. We look here into the growth of
large-scale structure in the models. The Szekeres models are exact solutions to
Einstein's equations that were originally derived with no symmetries. We use a
formulation of the models that is due to Goode and Wainwright, who considered
the models as exact perturbations of an FLRW background. Using the Raychaudhuri
equation, we write for the two classes of the models, exact growth equations in
terms of the under/overdensity and measurable cosmological parameters. The new
equations in the overdensity split into two informative parts. The first part,
while exact, is identical to the growth equation in the usual linearly
perturbed FLRW models, while the second part constitutes exact non-linear
perturbations. We integrate numerically the full exact growth rate equations
for the flat and curved cases. We find that for the matter-dominated era, the
Szekeres growth rate is up to a factor of three to five stronger than the usual
linearly perturbed FLRW cases, reflecting the effect of exact Szekeres
non-linear perturbations. The growth is also stronger than that of the
non-linear spherical collapse model, and the difference between the two
increases with time. This highlights the distinction when we use general
inhomogeneous models where shear and a tidal gravitational field are present
and contribute to the gravitational clustering. Additionally, it is worth
observing that the enhancement of the growth found in the Szekeres models
during the matter-dominated era could suggest a substitute to the argument that
dark matter is needed when using FLRW models to explain the enhanced growth and
resulting large-scale structures that we observe today (abridged)Comment: 18 pages, 4 figures, matches PRD accepted versio
Monte Carlo Algorithm for Simulating Reversible Aggregation of Multisite Particles
We present an efficient and exact Monte Carlo algorithm to simulate
reversible aggregation of particles with dedicated binding sites. This method
introduces a novel data structure of dynamic bond tree to record clusters and
sequences of bond formations. The algorithm achieves a constant time cost for
processing cluster association and a cost between and
for processing bond dissociation in clusters with bonds.
The algorithm is statistically exact and can reproduce results obtained by the
standard method. We applied the method to simulate a trivalent ligand and a
bivalent receptor clustering system and obtained an average scaling of
for processing bond dissociation in acyclic
aggregation, compared to a linear scaling with the cluster size in standard
methods. The algorithm also demands substantially less memory than the
conventional method.Comment: 8 pages, 3 figure
Using the Zeldovich dynamics to test expansion schemes
We apply various expansion schemes that may be used to study gravitational
clustering to the simple case of the Zeldovich dynamics. Using the well-known
exact solution of the Zeldovich dynamics we can compare the predictions of
these various perturbative methods with the exact nonlinear result and study
their convergence properties. We find that most systematic expansions fail to
recover the decay of the response function in the highly nonlinear regime.
``Linear methods'' lead to increasingly fast growth in the nonlinear regime for
higher orders, except for Pade approximants that give a bounded response at any
order. ``Nonlinear methods'' manage to obtain some damping at one-loop order
but they fail at higher orders. Although it recovers the exact Gaussian
damping, a resummation in the high-k limit is not justified very well as the
generation of nonlinear power does not originate from a finite range of
wavenumbers (hence there is no simple separation of scales). No method is able
to recover the relaxation of the matter power spectrum on highly nonlinear
scales. It is possible to impose a Gaussian cutoff in a somewhat ad-hoc fashion
to reproduce the behavior of the exact two-point functions for two different
times. However, this cutoff is not directly related to the clustering of matter
and disappears in exact equal-time statistics such as the matter power
spectrum. On a quantitative level, the usual perturbation theory, and the
nonlinear scheme to which one adds an ansatz for the response function with
such a Gaussian cutoff, are the two most efficient methods. These results
should hold for the gravitational dynamics as well (this has been checked at
one-loop order), since the structure of the equations of motion is identical
for both dynamics.Comment: 29 pages, published in A&
Pressure wave propagation studies for oscillating cascades
The unsteady flow field around an oscillating cascade of flat plates is studied using a time marching Euler code. Exact solutions based on linear theory serve as model problems to study pressure wave propagation in the numerical solution. The importance of using proper unsteady boundary conditions, grid resolution, and time step is demonstrated. Results show that an approximate non-reflecting boundary condition based on linear theory does a good job of minimizing reflections from the inflow and outflow boundaries and allows the placement of the boundaries to be closer than cases using reflective boundary conditions. Stretching the boundary to dampen the unsteady waves is another way to minimize reflections. Grid clustering near the plates does a better job of capturing the unsteady flow field than cases using uniform grids as long as the CFL number is less than one for a sufficient portion of the grid. Results for various stagger angles and oscillation frequencies show good agreement with linear theory as long as the grid is properly resolved
Nonlinear Gravitational Clustering: dreams of a paradigm
We discuss the late time evolution of the gravitational clustering in an
expanding universe, based on the nonlinear scaling relations (NSR) which
connect the nonlinear and linear two point correlation functions. The existence
of critical indices for the NSR suggests that the evolution may proceed towards
a universal profile which does not change its shape at late times. We begin by
clarifying the relation between the density profiles of the individual halo and
the slope of the correlation function and discuss the conditions under which
the slopes of the correlation function at the extreme nonlinear end can be
independent of the initial power spectrum. If the evolution should lead to a
profile which preserves the shape at late times, then the correlation function
should grow as [in a universe] een at nonlinear scales. We
prove that such exact solutions do not exist; however, ther e exists a class of
solutions (``psuedo-linear profiles'', PLP's for short) which evolve as
to a good approximation. It turns out that the PLP's are the correlation
functions which arise if the individual halos are assumed to be isothermal
spheres. They are also configurations of mass in which the nonlinear effects of
gravitational clustering is a minimum and hence can act as building blocks of
the nonlinear universe. We discuss the implicatios of this result.Comment: 32 Pages, Submitted to Ap
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