Generalized Fisher information matrix in nonextensive systems with spatial correlation

Details Generalized Fisher information matrix in nonextensive systems with spatial correlation Generalized Fisher information matrix in nonextensive systems with spatial correlation

2009-02-11

arxiv.org/abs/0902.1787v2

Phys. Rev. E 80 (2009) 051125.

Physics

Condensed Matter

Statistical Mechanics

18 pages, 3 figures: revised version accepted in Phys. Rev. E

Scientific paper

By using the $q$-Gaussian distribution derived by the maximum entropy method for spatially-correlated $N$-unit nonextensive systems, we have calculated the generalized Fisher information matrix of $g_{\theta_n \theta_m}$ for $(\theta_1, \theta_2, \theta_3) = (\mu_q, \sigma_q^2$, $s$), where $\mu_q$, $\sigma_q^2$ and $s$ denote the mean, variance and degree of spatial correlation, respectively, for a given entropic index $q$. It has been shown from the Cram\'{e}r-Rao theorem that (1) an accuracy of an unbiased estimate of $\mu_q$ is improved (degraded) by a negative (positive) correlation $s$, (2) that of $\sigma_q^2$ is worsen with increasing $s$, and (3) that of $s$ is much improved for $s \simeq -1/(N-1)$ or $s \simeq 1.0$ though it is worst at $s = (N-2)/2(N-1)$. Our calculation provides a clear insight to the long-standing controversy whether the spatial correlation is beneficial or detrimental to decoding in neuronal ensembles. We discuss also a calculation of the $q$-Gaussian distribution, applying the superstatistics to the Langevin model subjected to spatially-correlated inputs.

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