Characterizing and modelling cyclic behaviour in non-stationary time series through multi-resolution analysis

Abstract

A method based on wavelet transform is developed to characterize variations at multiple scales in non-stationary time series. We consider two different financial time series, SP this corroborates correlated behaviour observed earlier in financial time series through random matrix studies. Analysis is carried out through Haar, Daubechies-4 and continuous Morlet wavelets for studying the character of fluctuations at different scales and show that cyclic variations emerge at intermediate time scales. It is found that Daubechies family of wavelets can be effectively used to capture cyclic variations since these are local in nature. To get an insight into the occurrence of cyclic variations, we then proceed to model these wavelet coefficients using genetic programming (GP) approach and using the standard embedding technique in the reconstructed phase space. It is found that the standard methods (GP as well as artificial neural networks) fail to model these variations because of poor convergence. A novel interpolation approach is developed that overcomes this difficulty. The dynamical model equations have, primarily, linear terms with additive Pade-type terms. It is seen that the emergence of cyclic variations is due to an interplay of a few important terms in the model. Very interestingly GP model captures smooth variations as well as bursty behaviour quite nicely.