Towards fusing sophisticated mathematical knowledge and informal expert knowledge: an arbitrary metric can be naturally interpreted in fuzzy terms

In many practical situations, we are faced with a necessity to combine sophisticated mathematical knowledge about the analyzed systems with informal expert knowledge. To make this combination natural, it is desirable to reformulate the abstract mathematical knowledge in understandable intuitive terms. In this paper, we show how this can be done for an abstract metric. 1 For Data Fusion, It Is Desirable to Express Abstract Mathematical Notions in Natural Terms In many practical situations, we are faced with a necessity to combine sophisticated mathematical knowledge about the analyzed systems with informal expert knowledge. To make this combination natural, it is desirable to reformulate the abstract mathematical knowledge in understandable intuitive terms. In this paper, we show how this can be done for a specific mathematical notion: the notion of a metric. One way to define a metric is to pick certain properties PI , . . . , P,, and to define a simi2 Some Metrics Are Natural, But Are All larity between two objects x and y as the deMetric Natural? gree to which PI ( x ) is similar to 4 (y) and P2 ( x ) is similar to P2k) etc. The distance between two points is a particular examSimilarity is naturally described by 1 Idl d21 (we can use robustness arguments to get this expression). Since we can have infinitely many properties, we should use min for "and". The distance is then 1 -this similarity. The resulting metrics are "natural". It seems, at first glance, that not all metrics are natural in this sense. Interestingly, an arbitrary continuous metric can be thus described. Similarly, we can thus describe all "kinematic metrics" (space-time analogues of metrics), while probabilistic explanation is difficult.