Measures and Metrics of Biological Signals

The concept of biological signals is becoming broader. Some of the challenges are: searching for inner and structural characteristics; selecting appropriate modeling to enhance perceived properties in the signals; extracting the representative components, identifying their mathematical correspondents; and performing necessary transformations in order to obtain form for subtle analysis, comparisons, derived recognition, and classification. There is that unique moment when we correspond the adequate mathematical structures to the observed phenomena. It allows application of various mathematical constructs, transformations and reconstructions. Finally, comparisons and classifications of the newly observed phenomena often lead to enrichment of the existing models with some additional structurality. For a specialized context the modeling takes place in a suitable set of mathematical representations of the same kind, a set of models M, where the mentioned transformations take place. They are used for determination of structures M, where mathematical finalization processes are preformed. Normalized representations of the initial content are measured in order to determine the key invariants (characterizing characteristics). Then, comparisons are preformed for specialized or targeted purposes. The process converges to the measures and distance measurements in the space M. Thus, we are dealing with measure and metric spaces, gaining opportunities that have not been initially available. Obviously, the different aspects in the research or diagnostics will demand specific spaces. In our practice we faced a large variety of problems in analysis of biological signals with very rich palette of measures and metrics. Even when a unique phenomena are observed for slightly different aspects of their characteristics, the corresponding measurements differ, or are refinements of the initial structures. Certain criteria need to be fulfilled. Namely, characterization and semantic stability. The small changes in the structures have to induce the small changes in measures and metrics. We offer a collection of the models that we have been involved in, together with the problems we met and their solutions, with representative visualizations.

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