One of the most computationally convenient nonredundant ways to describe the dependence between two variables is by describing the corresponding copula. In many applications, a special class of copulas—known as FGM copulas—turned out to be most successful in describing the dependence between quantities. The main result of this paper is that these copulas are the fastest to compute, and this explains their empirical success. As an auxiliary result, we also show that a similar explanation can be given in terms of fuzzy logic.
As a system becomes more complex, at first, its description and analysis becomes more complicated. However, a further increase in the system’s complexity often makes this analysis simpler. A classical example is Central Limit Theorem: when we have a few independent sources of uncertainty, the resulting uncertainty is very difficult to describe, but as the number of such sources increases, the resulting distribution gets close to an easy-to-analyze normal one—and indeed, normal distributions are ubiquitous. We show that such limit theorems often make analysis of complex systems easier—i.e., lead to blessing of dimensionality phenomenon—for all the aspects of these systems: the corresponding transformation, the system’s uncertainty, and the desired result of the system’s analysis.
In this paper, we propose a new technique—called Ellipsoidal and Gaussian Kalman filter—for state estimation of discrete-time nonlinear systems in situations when for some parts of uncertainty, we know the probability distributions, while for other parts of uncertainty, we only know the bounds (but we do not know the corresponding probabilities). Similarly to the usual Kalman filter, our algorithm is iterative: on each iteration, we first predict the state at the next moment of time, and then we use measurement results to correct the corresponding estimates. On each correction step, we solve a convex optimization problem to find the optimal estimate for the system’s state (and the optimal ellipsoid for describing the systems’s uncertainty). Testing our algorithm on several highly nonlinear problems has shown that the new algorithm performs the extended Kalman filter technique better—the state estimation technique usually applied to such nonlinear problems.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.