Time series irreversibility analysis using Jensen–Shannon divergence calculated by permutation pattern

An important feature of the time series in the real world is that its distribution has different degrees of asymmetry, which is what we call irreversibility. In this paper, we propose a new method named permutation pattern (PP) to calculate the Kullback–Leibler divergence ( $${D}_{\mathrm{KL}}$$ ) and the Jensen–Shannon divergence ( $${D}_{\mathrm{JS}}$$ ) to explore the irreversibility of time series. Meanwhile, we improve $${D}_{\mathrm{JS}}$$ and obtain a complete mean divergence ( $${D}_{\mathrm{m}}$$ ) through averaging $${D}_{\mathrm{KL}}$$ of a time series and its inverse time series. The variation trend of $${D}_{\mathrm{m}}$$ is similar to $${D}_{\mathrm{JS}}$$ , but the value of $${D}_{\mathrm{m}}$$ is slightly larger and the description of irreversibility is more intuitive. Furthermore, we compare $${D}_{\mathrm{JS}}$$ and $${D}_{\mathrm{m}}$$ calculated by PP with those calculated by the horizontal visibility graph, and discuss their respective characteristics. Then, we investigate the advantages of $$\mathrm{D}_{\mathrm{JS}}$$ and $${D}_{\mathrm{m}}$$ through length variation, dynamic time variation, multiscale and so on. It is worth mentioning that we introduce Score and variance to analyze the practical significance of stock irreversibility.