Structure of memory in time series: A one-dimensional atomic chain model

In current time series analysis, the structure of memory is lost. A concept called memory structure network is proposed to preserve the memory details. From the time series, one extracts all the possible segments, maps them each to nodes, and assigns the dependence between the node pairs to their corresponding edges as weights. The adjacency matrix for the resulting network is identical with the Hamiltonian for an electron walking along a one-dimensional lattice, with the help of which the memory is represented in a multi-dimensional space spanned by the spectral properties for the adjacency matrix. The behaviors for the index series of the Shanghai Stock Exchange Composite are shown and compared with that for the fractional Brownian motion (fBm) and the Autoregressive Conditional Heteroskedasticity (ARCH) models. The spectral distribution follows perfectly the beta-distribution and its skew increases monotonically with the Hurst exponent, which can (cannot) be reproduced with the fBm (ARCH) model. For the empirical and modeling series, the full spectra expose stronger repulsion effect due to the intrinsic mirror symmetry, while the symmetry-resolved sub-spectra obey the Weibull distribution and the repulsion effect increases monotonically with the long-term persistence and the fluctuation of variance. In addition, the rigidities for the symmetry-resolved sub-spectra are between the Poisson-like and the Gaussian orthogonal ensemble ones, while that for the full spectra are not sensitive to the difference in time series. This approach may answer the questions such as what a specific value of H means for the memory from the viewpoint of spectral behaviors.