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## User: dan-schonfeld

### Title: Mapping Equivalence for Symbolic Sequences: Theory and Applications

Processing of symbolic sequences represented by mapping of symbolic data into
numerical signals is commonly used in various applications. It is a
particularly popular approach in genomic and proteomic sequence analysis.
Numerous mappings of symbolic sequences have been proposed for various
applications. It is unclear however whether the processing of symbolic data
provides an artifact of the numerical mapping or is an inherent property of the
symbolic data. This issue has been long ignored in the engineering and
scientific literature. It is possible that many of the results obtained in
symbolic signal processing could be a byproduct of the mapping and might not
shed any light on the underlying properties embedded in the data. Moreover, in
many applications, conflicting conclusions may arise due to the choice of the
mapping used for numerical representation of symbolic data. In this paper, we
present a novel framework for the analysis of the equivalence of the mappings
used for numerical representation of symbolic data. We present strong and weak
equivalence properties and rely on signal correlation to characterize
equivalent mappings. We derive theoretical results which establish conditions
for consistency among numerical mappings of symbolic data. Furthermore, we
introduce an abstract mapping model for symbolic sequences and extend the
notion of equivalence to an algebraic framework. Finally, we illustrate our
theoretical results by application to DNA sequence analysis.

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The first result in this paper provides a very general $\epsilon$-removal argument for the multilinear restriction estimate. The second result …