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Pretopology is a mathematical model developed from a weakening of the topological axiomatic. It was initially used in economic, social, and biological sciences and next in pattern recognition and image analysis. More recently, it has been applied to the analysis of complex networks. Pretopology enables to work in a mathematical framework with weak properties, and its nonidempotent operator called pseudo-closure permits to implement iterative algorithms. It proposes a formalism that generalizes graph theory concepts and allows to model problems universally. In this paper, the authors will extend this mathematical model to analyze complex data with spatiotemporal dimensions. The authors define the notion of a temporal pretopology based on a temporal function. They give an example of temporal function based on a binary relation and construct a temporal pretopology. They define two new notions of temporal substructures which aim at representing evolution of substructures. They propose algorithms to extract these substructures. They experiment the proposition on two data and two economic real data.

Concept of Temporal Pretopology for the Analysis for Structural Changes: Application to Econometrics