197,824 research outputs found

    Procrustes analysis for high-dimensional data

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    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

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    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

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    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

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    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

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    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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