Analysis of “mixing” combination rules and Smet’s combination rule

Keywords: hyperspectral satellite images, evidence theory, image classification, combination rules


The process of solution of different practical and ecological problems, using hyperspectral satellite images usually includes a procedure of classification. Classification is one of the most difficult and important procedures. Some image classification methods were considered and analyzed in this work. These methods are based on the theory of evidence. Evidence theory can simulate uncertainty and process imprecise and incomplete information. It were considered such combination rules in this paper: “mixing” combination rule (or averaging), convolutive x-averaging (or c-averaging) and Smet’s combination rule. It was shown, that these methods can process the data from multiple sources or spectral bands, that provide different assessments for the same hypotheses. It was noted, that the purpose of aggregation of information is to simplify data, whether the data is coming from multiple sources or different spectral bands. It was shown, that Smet’s rule is unnormalized version of Dempster rule, that applied in Smet’s Transferable Belief Model. It also processes imprecise and incomplete data. Smet’s combination rule entails a slightly different formulation of Dempster-Shafer theory. Mixing (or averaging) rule was considered in this paper too. It is the averaging operation that is used for probability distributions. This rule uses basic probability assignments from different sources (spectral bands) and weighs assigned according to the reliability of the sources. Convolutive x-averaging (or c-averaging) rule was considered in this paper too. This combination rule is a generalization of the average for scalar numbers. This rule is commutative and not associative. It also was noted, that convolutive x-averaging (c-averaging) rule can include any number of basic probability assignments. It were also considered examples, where these proposed combination rules were used. Mixing, convolutive x-averaging (c-averaging) rule and Smet’s combination rule can be applied for analysis of hyperspectral satellite images, in remote searching for minerals and oil, solving different environmental and thematic problems.


Alpert, М. І., Alpert, S. І. (2019). A new approach to the application of Jaccard coefficient and Cosine similarity in Hyperspectral Image Classification. XVIII-th International Conference on Geoinformatics — Theoretical and Applied Aspects. 1–5, Kiev.

Beynon, M. J., Curry, B., Morgan, P. (2000). The Dempster-Shafer theory of evidence: an alternative approach to multicriteria decision modeling. Omega: Int. Journal of Management Science, 28 (1), 37–50.

Chang, C. I. (2013). Hyperspectral Data Processing: Algorithm Design and Analysis. Hoboken, NJ: John Willey & Sons.

Ferson, S., Kreinovich, V. (2002). Representation, Propagation, and Aggregation of Uncertainty. SAND Report.

Gong, P. (1996). Integrated Analysis of Spatial Data from Multiple Sources: Using Evidential Reasoning and Artificial Neural Network Techniques for Geological Mapping. Photogrammetric Engineering and Remote Sensing, 62 (5), 513–523.

Lein, J. K. (2003). Applying evidential reasoning methods to agricultural land cover classification. Int. Journal of Remote Sensing, 24 (21), 4161–4180.

McCoy, R. M. (2005). Fields Methods in Remote Sensing, 150–160, New York: Guilford Press.

Mertikas, P., Zervakis, M.E. (2001). Exemplifying the Theory of Evidence in Remote Sensing Image Classification. Int. Journal of Remote Sensing, 22 (6), 1081–1095.

Popov, M., Alpert, S., Podorvan, V., Topolnytskyi, M., Mieshkov, S. (2015). Method of Hyperspectral Satellite Image Classification under Contaminated Training Samples Based on Dempster-Shafer’s Paradigm. Central European Researchers Journal, 1 (1), 86–97.

Shafer, G. (1990). Perspectives in the theory and practice of belief functions. Int. J. Approx. Reasoning, 4, 323–362.

Smarandache, F. (2006). An In-Depth Look at Information Fusion Rules and the Unification of Fusion Theories.The University of New Mexico 200 College Road Gallup, NM 87301, USA. Retrieved from:

Smets, P. (1992). The transferable belief model and random sets. Int. J. Intell. Syst, 7, 37–46.

Smets, P. (1993). Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem. Int. J. Approximate Reasoning, 9, 1–35.

Smets, P., Henrion, M., Shachter, R. D., Kanal, L.N., Lemmer, J. F. (1990). Constructing the pignistic probability function in a context of uncertainty. Uncertainty in Artificial Intelligence. North Holland, Amsterdam, 5, 29–40.

Yager, R. (1987). On the Dempster-Shafer Framework and New Combination Rules. Information Sciences, 41, 93–137.

Zhang, L., Yager, R. R., Kacprzyk, J., Fedrizzi, M. (1994). Representation, independence, and combination of evidence in the Dempster-Shafer theory. Advances in the Dempster-Shafer Theory of Evidence, 51–69, New York: John Wiley and Sons. Inc.

Techniques for Earth observation data acquisition, processing and interpretation