197,824 research outputs found
Procrustes analysis for high-dimensional data
The Procrustes-based perturbation model \citep{Goodall} allows to minimize
the Frobenius distance between matrices by similarity transformation. However,
it suffers from non-identifiability, critical interpretation of the transformed
matrices, and non-applicability in high-dimensional data. We provide an
extension of the perturbation model focused on the high-dimensional data
framework, called the ProMises (Procrustes von Mises-Fisher) model. The
ill-posed and interpretability problems are solved by imposing a proper prior
distribution for the orthogonal matrix parameter, i.e., the von Mises-Fisher
distribution, which is a conjugate prior, resulting in a fast estimation
process. Furthermore, we present the Efficient ProMises model for the
high-dimensional framework, useful in neuroimaging, where the problem has much
more than three dimensions. We found a great improvement in functional Magnetic
Resonance Imaging connectivity analysis since the ProMises model permits to
incorporate topological brain information in the alignment's estimation
process.Comment: 20 pages, 7 figure
Conjugate field and fluctuation-dissipation relation for the dynamic phase transition in the two-dimensional kinetic Ising model
The two-dimensional kinetic Ising model, when exposed to an oscillating
applied magnetic field, has been shown to exhibit a nonequilibrium,
second-order dynamic phase transition (DPT), whose order parameter Q is the
period-averaged magnetization. It has been established that this DPT falls in
the same universality class as the equilibrium phase transition in the
two-dimensional Ising model in zero applied field. Here we study for the first
time the scaling of the dynamic order parameter with respect to a nonzero,
period-averaged, magnetic `bias' field, H_b, for a DPT produced by a
square-wave applied field. We find evidence that the scaling exponent,
\delta_d, of H_b at the critical period of the DPT is equal to the exponent for
the critical isotherm, \delta_e, in the equilibrium Ising model. This implies
that H_b is a significant component of the field conjugate to Q. A finite-size
scaling analysis of the dynamic order parameter above the critical period
provides further support for this result. We also demonstrate numerically that,
for a range of periods and values of H_b in the critical region, a
fluctuation-dissipation relation (FDR), with an effective temperature
T_{eff}(T, P, H_0) depending on the period, and possibly the temperature and
field amplitude, holds for the variables Q and H_b. This FDR justifies the use
of the scaled variance of Q as a proxy for the nonequilibrium susceptibility,
\partial / \partial H_b, in the critical region.Comment: revised version; 31 pages, 12 figures; accepted by Phys. Rev.
Second order adjoints for solving PDE-constrained optimization problems
Inverse problems are of utmost importance in many fields of science and engineering. In the
variational approach inverse problems are formulated as PDE-constrained optimization problems,
where the optimal estimate of the uncertain parameters is the minimizer of a certain cost
functional subject to the constraints posed by the model equations. The numerical solution
of such optimization problems requires the computation of derivatives of the model output
with respect to model parameters. The first order derivatives of a cost functional (defined
on the model output) with respect to a large number of model parameters can be calculated
efficiently through first order adjoint sensitivity analysis. Second order adjoint models
give second derivative information in the form of matrix-vector products between the Hessian
of the cost functional and user defined vectors. Traditionally, the construction of second
order derivatives for large scale models has been considered too costly. Consequently, data
assimilation applications employ optimization algorithms that use only first order derivative
information, like nonlinear conjugate gradients and quasi-Newton methods.
In this paper we discuss the mathematical foundations of second order adjoint sensitivity
analysis and show that it provides an efficient approach to obtain Hessian-vector products. We
study the benefits of using of second order information in the numerical optimization process
for data assimilation applications. The numerical studies are performed in a twin experiment
setting with a two-dimensional shallow water model. Different scenarios are considered with
different discretization approaches, observation sets, and noise levels. Optimization algorithms
that employ second order derivatives are tested against widely used methods that require
only first order derivatives. Conclusions are drawn regarding the potential benefits and the
limitations of using high-order information in large scale data assimilation problems
Practical Bayesian Modeling and Inference for Massive Spatial Datasets On Modest Computing Environments
With continued advances in Geographic Information Systems and related
computational technologies, statisticians are often required to analyze very
large spatial datasets. This has generated substantial interest over the last
decade, already too vast to be summarized here, in scalable methodologies for
analyzing large spatial datasets. Scalable spatial process models have been
found especially attractive due to their richness and flexibility and,
particularly so in the Bayesian paradigm, due to their presence in hierarchical
model settings. However, the vast majority of research articles present in this
domain have been geared toward innovative theory or more complex model
development. Very limited attention has been accorded to approaches for easily
implementable scalable hierarchical models for the practicing scientist or
spatial analyst. This article is submitted to the Practice section of the
journal with the aim of developing massively scalable Bayesian approaches that
can rapidly deliver Bayesian inference on spatial process that are practically
indistinguishable from inference obtained using more expensive alternatives. A
key emphasis is on implementation within very standard (modest) computing
environments (e.g., a standard desktop or laptop) using easily available
statistical software packages without requiring message-parsing interfaces or
parallel programming paradigms. Key insights are offered regarding assumptions
and approximations concerning practical efficiency.Comment: 20 pages, 4 figures, 2 table
Conjugate Projective Limits
We characterize conjugate nonparametric Bayesian models as projective limits
of conjugate, finite-dimensional Bayesian models. In particular, we identify a
large class of nonparametric models representable as infinite-dimensional
analogues of exponential family distributions and their canonical conjugate
priors. This class contains most models studied in the literature, including
Dirichlet processes and Gaussian process regression models. To derive these
results, we introduce a representation of infinite-dimensional Bayesian models
by projective limits of regular conditional probabilities. We show under which
conditions the nonparametric model itself, its sufficient statistics, and -- if
they exist -- conjugate updates of the posterior are projective limits of their
respective finite-dimensional counterparts. We illustrate our results both by
application to existing nonparametric models and by construction of a model on
infinite permutations.Comment: 49 pages; improved version: revised proof of theorem 3 (results
unchanged), discussion added, exposition revise
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