A time-scale analysis of financial crises similarity via Earth Mover's Distance

This paper proposes a novel approach aimed at identifying similarities between portions of financial time series during crisis periods. A pattern observed in a short time interval during a specific crisis may resemble that of another crisis period in a longer time interval. To capture these similarities at different scales, the energy distribution of the wavelet transform of the time series is analyzed within limited portions of the scalogram at different scales. This is achieved by introducing Generalized Heisenberg Boxes (GHBs), which are boxes larger than, yet proportional to, the corresponding classical Heisenberg Boxes. Specifically, since time-scale trajectories of the wavelet transform modulus maxima characterize signal singularities, each GHB is described in terms of intra-and inter-scale relationships of the internal maxima. According to the Wavelet Atoms Approximation Theory, the similarity between GHBs covering a set of singularities is expected to persist across successive scales as long as it is not affected by other singularities. The comparison between GHBs is achieved by means of the Earth Mover's Distance, which allows comparing rectangles of different size. Experimental results on the S&P 500 stock market index, as well as Gold and Crude Oil historical data, have revealed some interesting similarities between well-known financial crises at different scales, confirming the potential of the proposed approach in providing a more in-depth analysis of financial time series.