438 research outputs found
Modeling and Forecasting of Realized Covariance Matrices of Asset Returns using State-Space Models
This thesis comprises three self-contained essays on the modeling and prediction of realized covariance
matrices of asset returns using state-space models
Variational methods in simultaneous optimum interpolation and initialization
The duality between optimum interpolation and variational objective analysis, is reviewed. This duality is used to set up a variational approach to objective analysis which uses prior information concerning the atmospheric spectral energy distribution, in the variational problem. In the wind analysis example, the wind field is partitioned into divergent and nondivergent parts, and a control parameter governing the relative energy in the two parts is estimated from the observational data being analyzed by generalized cross validation, along with a bandwidth parameter. A variational approach to combining objective analysis and initialization in a single step is proposed. In a simple example of this approach, data, forecast, and prior information concerning atmospheric energy distribution is combined into a single variational problem. This problem has (at least) one bandwidth parameter, one partitioning parameter governing the relative energy in fast slow modes, and one parameter governing the relative weight to be given to observational and forecast data
Point spread function approximation of high rank Hessians with locally supported non-negative integral kernels
We present an efficient matrix-free point spread function (PSF) method for
approximating operators that have locally supported non-negative integral
kernels. The method computes impulse responses of the operator at scattered
points, and interpolates these impulse responses to approximate integral kernel
entries. Impulse responses are computed by applying the operator to Dirac comb
batches of point sources, which are chosen by solving an ellipsoid packing
problem. Evaluation of kernel entries allows us to construct a hierarchical
matrix (H-matrix) approximation of the operator. Further matrix computations
are performed with H-matrix methods. We use the method to build preconditioners
for the Hessian operator in two inverse problems governed by partial
differential equations (PDEs): inversion for the basal friction coefficient in
an ice sheet flow problem and for the initial condition in an
advective-diffusive transport problem. While for many ill-posed inverse
problems the Hessian of the data misfit term exhibits a low rank structure, and
hence a low rank approximation is suitable, for many problems of practical
interest the numerical rank of the Hessian is still large. But Hessian impulse
responses typically become more local as the numerical rank increases, which
benefits the PSF method. Numerical results reveal that the PSF preconditioner
clusters the spectrum of the preconditioned Hessian near one, yielding roughly
5x-10x reductions in the required number of PDE solves, as compared to
regularization preconditioning and no preconditioning. We also present a
numerical study for the influence of various parameters (that control the shape
of the impulse responses) on the effectiveness of the advection-diffusion
Hessian approximation. The results show that the PSF-based preconditioners are
able to form good approximations of high-rank Hessians using a small number of
operator applications
Convergence and Error Propagation Results on a Linear Iterative Unfolding Method
Unfolding problems often arise in the context of statistical data analysis.
Such problematics occur when the probability distribution of a physical
quantity is to be measured, but it is randomized (smeared) by some well
understood process, such as a non-ideal detector response or a well described
physical phenomenon. In such case it is said that the original probability
distribution of interest is folded by a known response function. The
reconstruction of the original probability distribution from the measured one
is called unfolding. That technically involves evaluation of the non-bounded
inverse of an integral operator over the space of L^1 functions, which is known
to be an ill-posed problem. For the pertinent regularized operator inversion,
we propose a linear iterative formula and provide proof of convergence in a
probability theory context. Furthermore, we provide formulae for error
estimates at finite iteration stopping order which are of utmost importance in
practical applications: the approximation error, the propagated statistical
error, and the propagated systematic error can be quantified. The arguments are
based on the Riesz-Thorin theorem mapping the original L^1 problem to L^2
space, and subsequent application of ordinary L^2 spectral theory of operators.
A library implementation in C of the algorithm along with corresponding error
propagation is also provided. A numerical example also illustrates the method
in operation.Comment: 27 pages, 1 figur
A Geometric Variational Approach to Bayesian Inference
We propose a novel Riemannian geometric framework for variational inference
in Bayesian models based on the nonparametric Fisher-Rao metric on the manifold
of probability density functions. Under the square-root density representation,
the manifold can be identified with the positive orthant of the unit
hypersphere in L2, and the Fisher-Rao metric reduces to the standard L2 metric.
Exploiting such a Riemannian structure, we formulate the task of approximating
the posterior distribution as a variational problem on the hypersphere based on
the alpha-divergence. This provides a tighter lower bound on the marginal
distribution when compared to, and a corresponding upper bound unavailable
with, approaches based on the Kullback-Leibler divergence. We propose a novel
gradient-based algorithm for the variational problem based on Frechet
derivative operators motivated by the geometry of the Hilbert sphere, and
examine its properties. Through simulations and real-data applications, we
demonstrate the utility of the proposed geometric framework and algorithm on
several Bayesian models
- …