New irreversibility measure and complexity analysis based on singular value decomposition

Visibility graph algorithm is a powerful tool for visualization and analysis of complex dynamical system. Kullback-Leibler divergence(KLD), based on the horizontal visibility graph algorithm, has received largely extensive attention in the field of irreversibility analysis. In this paper, we will propose a new irreversibility measure built on the multi-scale theory, i.e. the KLD based on singular value decomposition(KLD-SVD). Furthermore, since matrix singular value can reflect the basic characteristics of the complex system, we further bring forward the definition of Shannon entropy with singular value decomposition(SE-SVD) after considering that the Shannon entropy has been well defined as a complexity measure. In order to show the advantages of these two new measures in detecting the complexity of systems, several simulation and real data experiments are chosen to examine the performance of them. Through these results, we find that the average and variance of KLD-SVD corresponding to each time series is less than that of the KLD. Moreover, KLD-SVD's volatility is significantly weaker than KLD's. As there is a conceptual link between predictability and irreversibility, we can argue that KLD-SVD algorithm implies the higher predictability and lower confusion. On the other hand, SE-SVD can do well in revealing the internal complexity of the different time series. Meanwhile, it is useful to mention that SE-SVD is robust against noise efficiently.