Multi-scale transition matrix approach to time series

Statistical and structural characteristics provide us with a multi-dimensional picture of time series. Rooting the properties in a unified scheme is the preliminary step to develop a model that can reproduce most part or even the whole of the picture. In this paper, we proposed a concept called multi-scale transition matrix, which is a series of transition matrices describing the jumping probabilities between states after specified numbers of jumps each. The eigenvectors corresponding to the unitary eigenvalues are identical with the probability distribution function. The second largest eigenvalues depict the upper-boundary curve of the persistence. The change speed of the eigenvector corresponding to the second largest eigenvalue along with time scale displays the relaxation behavior of the time series. The multi-scale matrix maintains the structure of autocorrelation in the original time series and its evolution, which are merged by the averaging procedure in the statistical properties. These predictions are confirmed by using the series generated with the Auto-Regressive Conditional Heteroskedasticity Model, the Auto-Regression Model and the fractional Brownian motion, and the empirical records for Shenzhen Component index in Mainland China and the word length series of the novel entitled Remembrance of Things Past written by Marcel Proust. Hence, the concept is a good candidate of bridges between the multi-dimensional picture and the dynamical models of the time series.