# The use of spatial information in entropy measures

The concept of entropy, firstly introduced in information theory, rapidly became popular in many applied sciences via Shannon's formula to measure the degree of heterogeneity among observations. A rather recent research field aims at accounting for space in entropy measures, as a generalization when the spatial location of occurrences ought to be accounted for. The main limit of these developments is that all indices are computed conditional on a chosen distance. This work follows and extends the route for including spatial components in entropy measures. Starting from the probabilistic properties of Shannon's entropy for categorical variables, it investigates the characteristics of the quantities known as residual entropy and mutual information, when space is included as a second dimension. This way, the proposal of entropy measures based on univariate distributions is extended to the consideration of bivariate distributions, in a setting where the probabilistic meaning of all components is well defined. As a direct consequence, a spatial entropy measure satisfying the additivity property is obtained, as global residual entropy is a sum of partial entropies based on different distance classes. Moreover, the quantity known as mutual information measures the information brought by the inclusion of space, and also has the property of additivity. A thorough comparative study illustrates the superiority of the proposed indices.

**Comments:**33 pages, 13 figures

## Similar Publications

This article reviews the application of advanced Monte Carlo techniques in the context of Multilevel Monte Carlo (MLMC). MLMC is a strategy employed to compute expectations which can be biased in some sense, for instance, by using the discretization of a associated probability law. The MLMC approach works with a hierarchy of biased approximations which become progressively more accurate and more expensive. Read More

The multivariate linear regression model is an important tool for investigating relationships between several response variables and several predictor variables. The primary interest is in inference about the unknown regression coefficient matrix. We propose multivariate bootstrap techniques as a means for making inferences about the unknown regression coefficient matrix. Read More

In the probabilistic topic models, the quantity of interest---a low-rank matrix consisting of topic vectors---is hidden in the text corpus matrix, masked by noise, and Singular Value Decomposition (SVD) is a potentially useful tool for learning such a low-rank matrix. However, the connection between this low-rank matrix and the singular vectors of the text corpus matrix are usually complicated and hard to spell out, so how to use SVD for learning topic models faces challenges. We overcome the challenge by revealing a surprising insight: there is a low-dimensional $\textit{simplex}$ structure which can be viewed as a bridge between the low-rank matrix of interest and the SVD of the text corpus matrix, and which allows us to conveniently reconstruct the former using the latter. Read More

Current statistical inference problems in areas like astronomy, genomics, and marketing routinely involve the simultaneous testing of thousands -- even millions -- of null hypotheses. For high-dimensional multivariate distributions, these hypotheses may concern a wide range of parameters, with complex and unknown dependence structures among variables. In analyzing such hypothesis testing procedures, gains in efficiency and power can be achieved by performing variable reduction on the set of hypotheses prior to testing. Read More

The ensemble Kalman filter (EnKF) is a computational technique for approximate inference on the state vector in spatio-temporal state-space models. It has been successfully used in many real-world nonlinear data-assimilation problems with very high dimensions, such as weather forecasting. However, the EnKF is most appropriate for additive Gaussian state-space models with linear observation equation and without unknown parameters. Read More

Variable clustering is one of the most important unsupervised learning methods, ubiquitous in most research areas. In the statistics and computer science literature, most of the clustering methods lead to non-overlapping partitions of the variables. However, in many applications, some variables may belong to multiple groups, yielding clusters with overlap. Read More

We study the problem of testing for structure in networks using relations between the observed frequencies of small subgraphs. We consider the statistics \begin{align*} T_3 & =(\text{edge frequency})^3 - \text{triangle frequency}\\ T_2 & =3(\text{edge frequency})^2(1-\text{edge frequency}) - \text{V-shape frequency} \end{align*} and prove a central limit theorem for $(T_2, T_3)$ under an Erd\H{o}s-R\'{e}nyi null model. We then analyze the power of the associated $\chi^2$ test statistic under a general class of alternative models. Read More

A new recalibration post-processing method is presented to improve the quality of the posterior approximation when using Approximate Bayesian Computation (ABC) algorithms. Recalibration may be used in conjunction with existing post-processing methods, such as regression-adjustments. In addition, this work extends and strengthens the links between ABC and indirect inference algorithms, allowing more extensive use of misspecified auxiliary models in the ABC context. Read More

We provide compact algebraic expressions that replace the lengthy symbolic-algebra-generated integrals I6 and I8 in Part I of this series of papers [1]. The MRSE entries of Part I, Table 4.3 are thus updated to simpler algebraic expressions. Read More

Distributional approximations of (bi--) linear functions of sample variance-covariance matrices play a critical role to analyze vector time series, as they are needed for various purposes, especially to draw inference on the dependence structure in terms of second moments and to analyze projections onto lower dimensional spaces as those generated by principal components. This particularly applies to the high-dimensional case, where the dimension $d$ is allowed to grow with the sample size $n$ and may even be larger than $n$. We establish large-sample approximations for such bilinear forms related to the sample variance-covariance matrix of a high-dimensional vector time series in terms of strong approximations by Brownian motions. Read More