We study the mean-field approximation for a general class of quantum Ising spin states from an information geometrical point of view. The states we consider are assumed to have at most second-order interactions with arbitrary but deterministic coupling coefficients. We call such a state a quantum Boltzmann machine (QBM) for the reason that it can be regarded as a quantum extension of the equilibrium distribution of a (classical) Boltzmann machine (CBM), which is a well-known stochastic neural network model. The totality of QBMs is then shown to form a quantum exponential family and thus can be considered as a smooth manifold having similar geometrical structures to those of CBMs. We elaborate on the significance and usefulness of information geometrical concepts, in particular the e- and m-projections, in studying the naive mean-field approximation for QBMs. We also discuss the higher-order corrections to the naive mean-field approximation based on the idea of the Plefka expansion in statistical physics. We elucidate the geometrical essence of the corrections and provide the expansion coefficients with expressions in terms of information geometrical quantities. Here, one may note this work as the information geometrical interpretation of (Plefka T 2006 Phys. Rev. E 73 016129) and as the quantum extension of (Tanaka T 2000 Neural Comput. 12 1951–68).
The monsoon rainfall variation and distribution have an impact on economical, ecological and sociological aspects of Sri Lanka. Therefore, this research aimed to study the spatialtemporal variation and distribution of the Southwest monsoon rainfall during the period of 1981-2010. Daily rainfall data were collected from 13 rain gauge stations which are located in Southwest part (wet zone) of Sri Lanka. Monsoon rainfall distribution was analyzed by comparing monthly average rainfall and rainy days. The monsoon rainfall variability was analyzed using drought indices: Rainfall Anomaly Index (RAI) and Standardized Precipitation Index (SPI). Trend analysis was carried out on both indices considering the Mann-Kendall test and Sen's slope estimator. Monthly average rainfall highlighted that the mid country experienced the highest amount of rainfall during the season and maximum number of rainy days occurred in the middle of the season. Analysis based on drought indices revealed that 14 years were dry. Both indices agreed on significant linear decreasing trends in rainfall at three rain gauge stations. Population growth, global warming, deforestation, etc., may be the reasons for these decreasing trends. These results create the need to discuss the economical, ecological and social impact of the rainfall which leads to better planning and rationalization.
In literature, one can find diverse applications of the power-law distribution to model naturally occurring phenomena in the sciences. With the emerging field of complex networks this applicability has been observed and emphasised more and more. In the present paper several interesting as well as important properties of this distribution have been explored. First, it is shown that the totality of power-law distributions form a one-parameter family or a statistical model which turns out to be a Riemannian differentiable manifold. Closed form expressions are obtained for several information theoretically important measures such as differential entropy, information divergence and Fisher information which are interpreted to have geometrical and statistical meanings. Next, it is shown that the statistical manifold of power-law distributions forms an exponential family which is a very important aspect in mathematical statistics and information geometry including a number of other fields in the sciences. The parameter estimation problem is addressed using both maximum likelihood and entropy methods. The close relationship of the two in the sense of Statistics is elucidated. Finally, an example of an exponential distribution having the same information divergence and Fisher information as that of power-law distributions is given, thus having the same lower bound in the Cramer-Rao inequality. In this case, an approximate structural similarity can be expected between the two statistical manifolds.
Image restoration has been a popular and active field of research for decades. Images are destroyed when exposed to 'noise' which can occur due to physical contact or electrical/electronic interference. Here, Bayesian statistical techniques and Markov random field (MRF) theory were used to restore a black and white binary image corrupted by additive Gaussian noise with zero mean and constant variance. The binary image was used as a Markov random field. An image is comprised of pixels and these pixels have a regular two-dimensional lattice structure. A matrix including ±1 values was generated randomly. It was represented and interpreted as an Ising model in Statistical Mechanics. The probability distribution of the Ising model was used as the prior distribution (a Gibbs or Boltzmann distribution). Likelihood function was obtained by using the random matrix and the observed corrupted image. Markov Chain Monte Carlo (MCMC) method was used to simulate posterior distribution which again turns out to be a Gibbs or Boltzmann distribution. More specifically, Metropolis-Hastings algorithm which is one of the popular MCMC algorithms was used in this simulation. In this study, Peak Signal to Noise Ratio (PSNR) and Mean Squared Error (MSE) methods were used to measure the quality of restored images. MATLAB (R2013a (8.1.0.604)) was used to construct the program in this research work. Finally, high quality images were restored which were almost similar to the original image. A decrease in restored image quality was observed with the increase in noise. When the image size was increased, a higher number of iterations was required to obtain an acceptable level of quality in the restored image.
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