Research Projects

Entropy has many applications in various fields such as image and signal processing, finance, statistical mechanics, biology, statistics etc. Artificial intelligence and machine learning are becoming very popular day by day due to their usefulness is solving real life problems. Entropy also finds its applications in these areas of research. Recently, Lad et al. (2015, Statistical Science) proposed a dual measure of entropy called extropy which possesses similar properties like entropy. For example, like entropy, extropy is also maximized for uniform distribution, it is continuous in its arguments and it is invariant under monotone transformations and permutations of its mass function. One major difference is that the scale of maximum entropy is unbounded as the number of sample increases however, the scale of maximum extropy is bounded by one. Both entropy and extropy are measures of uncertainty. Because of its unique and interesting properties and easy implementations in problem solving, extropy became very popular. Numerous research on extropy is carried out in the last decade on its generalizations, estimations and applications. Now a days, the problems that a statistician mostly encounter is involved with big data and multivariate statistics. Multivariate entropy is an important measure and is extensively studied by researchers. But multivariate extropy is not even introduced to this date. In this project, we will work with multivariate extropy measure, study its properties, discuss its estimations and potential applications in statistics and machine learning. One very important property that we will study is the maximum extropy principle. We will try to obtain distributions that maximizes multivariate extropy measure under appropriate moment constraints. This will help us to develop multivariate goodness-of-fit tests for the maximum extropy distributions. Also, we are planning to develop non-parametric estimators for this measure using nearest-neighbor and k-nearest neighbor graphs for frequentist and Bayesian approaches. The basic of Bayesian inference is to select a prior distribution and get the posterior and work with the posterior distribution. In non-parametric Bayesian analysis Dirichlet process prior is widely used and it is a conjugate prior i.e. the posterior distribution will also be a Dirichlet distribution. We will work with Dirichlet process for non-parametric Bayesian estimator of multivariate extropy measure. As applications, we will develop multivariate goodness-of-fit tests. Relative extropy is a distance measure based on extropy which we will also discuss for multivariate data. Using relative extropy, we can develop equivalence testing for two multivariate densities. Also, using relative extropy a novel hierarchical clustering for multivariate data will be developed. The completion of this project will open a new brunch in multivariate information measures using the dual of entropy.