Estimating Good Discrete Partitions from Observed Data: Symbolic False Nearest Neighbors

Abstract

A symbolic analysis of observed time series data requires making a discrete partition of a continuous state space containing observations of the dynamics. A particular kind of partition, called "generating", preserves all dynamical information of a deterministic map in the symbolic representation, but such partitions are not obvious beyond one dimension, and existing methods to find them require significant knowledge of the dynamical evolution operator or the spectrum of unstable periodic orbits. We introduce a statistic and algorithm to refine empirical partitions for symbolic state reconstruction. This method optimizes an essential property of a generating partition: avoiding topological degeneracies. It requires only the observed time series and is sensible even in the presence of noise when no truly generating partition is possible. Because of its resemblance to a geometrical statistic frequently used for reconstructing valid time-delay embeddings, we call the algorithm "symbolic false nearest neighbors". Why might one want to represent observed time series of dynamical systems as sequences of low-precision discrete symbols? In this representation, there are often interesting techniques-often (but not exclusively) derived from information theory and its associated technologywhich may illuminate data in novel ways [1]. The initial step for all these methods requires making a partition: a coloring of the state space [2], x ∈ R d , into non-overlapping regions and associated symbols so that any x is assigned a unique symbol s in a discrete alphabet. The symbol may be represented as an integer in the set 0, 1, . . . A − 1. A partition P defines a discretization of the observed sequence x i , i = 1 . . . N into a symbolic sequence, s i , i = 1 . . . N .What partitions are "good"? Which discretizations retain the full structure of the original dynamics in the x space in the sequence of symbols? Unfortunately, the situation is unlike the remarkable time-delay embedding method for continuous dynamics: simple partitions are not generically satisfactory. The mathematics of symbolic dynamics specifies what we want: a "generating partition" (GP), where symbolic orbits uniquely identify one continuous space orbit, and thus the symbolic dynamics is fully equivalent to the continuous space dynamics.Unfortunately there is no satisfactory mathematical theory about how to find a GP as a general procedure (except for one dimensional dynamics (d = 1), where partitioning at the critical points works). Are the ad-hoc partitions often used still satisfactory? Unfortunately they are often not so. Bollt et al [3] examined the degradation in the symbolic dynamics which results from the frequently used "histogram partition", as opposed to a GP. A less optimal partition will induce improper projections or degeneracies, where a given symbolic segment may correspond to more than one topologically distinct state space orbit. This resulted in finding the wrong topological entropy. Chaotic communication with symbolic targeting works most sa...

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