Abstract
Given a multivariate time series, possibly of high dimension, with unknown and time-varying joint distribution, it is of interest to be able to completely partition the time series into disjoint, contiguous subseries, each of which has different distributional or pattern attributes from the preceding and succeeding subseries. An additional feature of many time series is that they display self-affinity, so that subseries at one time scale are similar to subseries at another after application of an affine transformation. Such qualities are observed in time series from many disciplines, including biology, medicine, economics, finance, and computer science. This paper defines the relevant multiobjective combinatorial optimization problem with limited assumptions as a biobjective one, and a specialized evolutionary algorithm is presented which finds optimal self-affine time series partitionings with a minimum of choice parameters. The algorithm not only finds partitionings for all possible numbers of partitions given data constraints, but also for self-affinities between these partitionings and some fine-grained partitioning. The resulting set of Pareto-efficient solution sets provides a rich representation of the self-affine properties of a multivariate time series at different locations and time scales.
Highlights
Given a multivariate time series, possibly of high dimension, with unknown and time-varying joint distribution, it is of interest to be able to completely partition the time series into disjoint, contiguous subseries, each of which might be assumed for the purpose of further analysis to have different distributional or pattern attributes from the preceding and succeeding subseries
An additional feature of many time series is that they display self-affinity, which we may loosely define as the property that subseries at one time scale are similar to subseries at another after application of an affine transformation
The best results in terms of the first measure had a maximum error of 5 periods, i.e. an “error” of less than 1%, for any cut point . the results demonstrate that the evolutionary algorithms (EAs) can find the “correct” cut points with a good degree of accuracy, larger T and κ will lead to slower convergence or equivalently, lower accuracy for a given computational budget.One solution is illustrated in Figure V.1, where the solid lines indicate the cutpoints in the original data and the dashed lines the coarse-grained cutpoints found by the algorithm
Summary
Given a multivariate time series, possibly of high dimension, with unknown and time-varying joint distribution, it is of interest to be able to completely partition the time series into disjoint, contiguous subseries, each of which might be assumed for the purpose of further analysis to have different distributional or pattern attributes from the preceding and succeeding subseries. An additional feature of many time series is that they display self-affinity, which we may loosely define as the property that subseries at one time scale are similar to subseries at another after application of an affine transformation Such observations in the natural world date back at least to work in the mid-20th century on hydrography of the Nile [35] and the length of international borders [60], and a more general theory was advanced in [48].
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