A topological method for time series complexity analysis

In this paper we propose a method based on algebraic topology for the analysis of collections of simultaneous time series. Each time series corresponds to a variable. The correlation matrix of the variables and the network with weights on the edges associated to this matrix are constructed. By varying a parameter p between 0 and 1, a simplicial complex filtering is obtained. The analysis of the cavities of the simplicial complex is performed by studying the changes of the Betti numbers of the complexes of the filtration. By means of computational experiments, we obtain results suggesting that in the case when the correlation matrix comes from simultaneous time series modeling a natural phenomenon, the complexity of the Betti number variations is much lower than when the correlation matrix is random.