  Recognizing subtle historical patterns is central to modeling and forecasting
problems in time series analysis. Here we introduce and develop a new approach
to quantify deviations in the underlying hidden generators of observed data
streams, resulting in a new efficiently computable universal metric for time
series. The proposed metric is in the sense that we can compare and contrast
data streams regardless of where and how they are generated and without any
feature engineering step. The approach proposed in this paper is conceptually
distinct from our previous work on data smashing, and vastly improves
discrimination performance and computing speed. The core idea here is the
generalization of the notion of KL divergence often used to compare probability
distributions to a notion of divergence in time series. We call this the
sequence likelihood (SL) divergence, which may be used to measure deviations
within a well-defined class of discrete-valued stochastic processes. We devise
efficient estimators of SL divergence from finite sample paths and subsequently
formulate a universal metric useful for computing distance between time series
produced by hidden stochastic generators.
