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Mapping between tree metric spaces and time series: New constructions with applications to limit theorems for branch counts in partial Galton-Watson trees and signal processing
AuthorHaskell, Zoe Allegra
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Mappings between trees and piece-wise linear functions are well-known and used in combinatorics and probability literature to explore the properties of trees and stochastic processes. However, the usage of such representation in applied analysis of time series remains limited. This work examines a particular class of mappings between tree metric spaces (trees with edge lengths) and time series related to the Harris paths of these trees. Our ultimate goal is to adopt selected well-known properties in one space to make new inferences regarding the other. We introduce new objects, partial trees and partial Harris paths, which facilitate mapping between finite time series and rooted plane trees, and study relations between natural operations on partial trees and their time series counterparts. In particular, we establish the reciprocity of the level-set tree and partial Harris path operations on the space of partial trees and show that the Horton pruning of a partial tree (cutting leaves with subsequent series reduction) corresponds to taking the local minima of a time series that does not need to be an excursion. We use the correspondence between a critical binary Galton-Watson tree and a random walk to prove limit theorems for random trees. Specifically, we establish the Weak Law of Large Numbers and Central Limit Theorem for the branch counts in a partial critical binary Galton-Watson tree via counting the local extrema of the respective partial Harris path, which is known to be a random walk. The proposed technique greatly simplifies the proofs that exist for the analogs of these results in regular (non-partial) trees. We use the hierarchical structure of tree representation of a time series parameterized by the Horton-Strahler orders to develop new techniques of time series analysis. In particular, we present new pruning algorithms adopted for the analysis of `spiky' time series and of series with significant flat periods (plateaus). We illustrate the proposed techniques in the problem of noise reduction.