35,142 research outputs found
Evaluating functions of positive-definite matrices using colored noise thermostats
Many applications in computational science require computing the elements of
a function of a large matrix. A commonly used approach is based on the the
evaluation of the eigenvalue decomposition, a task that, in general, involves a
computing time that scales with the cube of the size of the matrix. We present
here a method that can be used to evaluate the elements of a function of a
positive-definite matrix with a scaling that is linear for sparse matrices and
quadratic in the general case. This methodology is based on the properties of
the dynamics of a multidimensional harmonic potential coupled with colored
noise generalized Langevin equation (GLE) thermostats. This "thermostat"
(FTH) approach allows us to calculate directly elements of functions of a
positive-definite matrix by carefully tailoring the properties of the
stochastic dynamics. We demonstrate the scaling and the accuracy of this
approach for both dense and sparse problems and compare the results with other
established methodologies.Comment: 8 pages, 4 figure
Clustering of discretely observed diffusion processes
In this paper a new dissimilarity measure to identify groups of assets
dynamics is proposed. The underlying generating process is assumed to be a
diffusion process solution of stochastic differential equations and observed at
discrete time. The mesh of observations is not required to shrink to zero. As
distance between two observed paths, the quadratic distance of the
corresponding estimated Markov operators is considered. Analysis of both
synthetic data and real financial data from NYSE/NASDAQ stocks, give evidence
that this distance seems capable to catch differences in both the drift and
diffusion coefficients contrary to other commonly used metrics
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