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Register a new userEfficient Computation of the Bergsma-Dassios Sign Covariance
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Added by
Luca Weihs
Publication date
2015
fields
Mathematical Statistics
and research's language is
English
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In an extension of Kendalls $tau$, Bergsma and Dassios (2014) introduced a covariance measure $tau^*$ for two ordinal random variables that vanishes if and only if the two variables are independent. For a sample of size $n$, a direct computation of $t^*$, the empirical version of $tau^*$, requires $O(n^4)$ operations. We derive an algorithm that computes the statistic using only $O(n^2log(n))$ operations.
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Bergsma and Dassios (2014) introduced an independence measure which is zero if and only if two random variables are independent. This measure can be naively calculated in $O(n^4)$. Weihs et al. (2015) showed that it can be calculated in $O(n^2 log n)$. In this note we will show that using the methods described in Heller et al. (2016), the measure can easily be calculated in only $O(n^2)$.
We gather several results on the eigenvalues of the spatial sign covariance matrix of an elliptical distribution. It is shown that the eigenvalues are a one-to-one function of the eigenvalues of the shape matrix and that they are closer together than the latter. We further provide a one-dimensional integral representation of the eigenvalues, which facilitates their numerical computation.
In this paper the computational aspects of probability calculations for dynamical partial sum expressions are discussed. Such dynamical partial sum expressions have many important applications, and examples are provided in the fields of reliability, product quality assessment, and stochastic control. While these probability calculations are ostensibly of a high dimension, and consequently intractable in general, it is shown how a recursive integration methodology can be implemented to obtain exact calculations as a series of two-dimensional calculations. The computational aspects of the implementaion of this methodology, with the adoption of Fast Fourier Transforms, are discussed.
Bayesian inference of Gibbs random fields (GRFs) is often referred to as a doubly intractable problem, since the likelihood function is intractable. The exploration of the posterior distribution of such models is typically carried out with a sophisticated Markov chain Monte Carlo (MCMC) method, the exchange algorithm (Murray et al., 2006), which requires simulations from the likelihood function at each iteration. The purpose of this paper is to consider an approach to dramatically reduce this computational overhead. To this end we introduce a novel class of algorithms which use realizations of the GRF model, simulated offline, at locations specified by a grid that spans the parameter space. This strategy speeds up dramatically the posterior inference, as illustrated on several examples. However, using the pre-computed graphs introduces a noise in the MCMC algorithm, which is no longer exact. We study the theoretical behaviour of the resulting approximate MCMC algorithm and derive convergence bounds using a recent theoretical development on approximate MCMC methods.
The consistency and asymptotic normality of the spatial sign covariance matrix with unknown location are shown. Simulations illustrate the different asymptotic behavior when using the mean and the spatial median as location estimator.
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