Stable and efficient cubature rules by metaheuristic optimization with application to Kalman filtering


In advanced mathematics, the Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function e⁻ˣ² over the entire real line. This integral has over the years been connected to numerous scientific research questions. Accordingly, the derivation of methods for multidimensional deterministic numerical integration, also called cubature, has been ascribed great importance in the past 60 years in the field of numerical mathematics. Following numerous studies, on matters concerning the weight functions and integration regions, notable progress and essential results have so far been obtained. In-depth research has also shown that many cubature approaches have weakness in that their stability decreases with rising degree of exactness and rising dimension due to the increasing influence of negative weights. In recent decades, numerous efficient cubature rules of high polynomial exactness for the case of high-dimensional Gaussian integrals have been proposed. In addition to the question of efficiency, however, the question of stability, which depends on the influence of negative weights, is also of crucial importance. In fact, for certain degrees of polynomial exactness, no stable rules are known.

In a recent publication, Dr. Dominik Ballreich from the Department of Economics, Chair of Applied Statistics and Empirical Social Research, Fernuniversität in Germany, presented a study where he proposed a novel approach based on metaheuristic optimization. The proposed approach was postulated to allow the computation of cubature rules which exceed comparable rules known from existing literature with respect to stability and efficiency. The motivation behind his publication was the fact that, for the mentioned non-linear and high-dimensional Gaussian integrals, the use of unstable cubature rules may lead to poor filtering results or, in the worst case, to the total divergence of the filtering process. His work is currently published in the research journal, Automatica.

In brief, Dr. Ballreich started by engaging in a thorough review of basic elements of the state-space methodology and the principles of cubature-based Kalman filtering. Next, an overview of properties of fully symmetric cubature rules was presented. He then described the proposed optimization algorithm and a selection of the results obtained. Lastly, two simulation studies in which the new cubature rules were compared to well-established rules known from the literature utilizing the Kalman filter were undertaken.

Exemplary optimization results were given for d = 5 to d = 12 and polynomial exactness m = 9 as well as d = 5 to d = 10 and polynomial exactness m = 11, i.e. OPT-rules. In fact, the arising OPT-rules on the average use 14.53% less abscissae than the (HW-rules) which, together with re-known Genz and Keister, have been reported to being the most efficient comparison rules known from existing literature.

In summary, a novel optimization procedure for the construction of stable and efficient cubature rules was presented. The researcher reported that, based on the two simulations undertaken, the necessity for the stable use cubature rules was motivated. Altogether, he demonstrated that unstable rules could have very negative effects on the filtering performance.

About the author

Dominik Ballreich studied economics (specialization in econometrics) and business mathematics at the Bergische Universität Wuppertal/Germany. After completing his studies, he began working as a research assistant at the Chair of Applied Statistics and Empirical Social Research at the FernUniversität in Hagen (Prof. Dr. Hermann Singer)/Germany. There he mainly conducted research in the fields of “nonlinear filtering theory” and “heuristic optimization”. His dissertation focused on the development of novel cubature methods that can be applied to the filtering of high-dimensional nonlinear dynamical systems. He is currently working at the it-consulting company codecentric AG in Solingen/Germany as a Data Scientist, where he has been involved in projects concerning “churn prediction” and “optimal bin packing”.

Here is the link to the repo where the cubature rules


Dominik Ballreich. Stable and efficient cubature rules by metaheuristic optimization with application to Kalman filtering. Automatica 101 (2019) page 157–165.

Go To Automatica

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