# Order and Metric Compatible Symbolic Sequence Processing

- Published
- Accepted

- Subject Areas
- Bioinformatics, Computational Biology, Data Mining and Machine Learning, Data Science
- Keywords
- metric linear space, signal processing, GSP, genomic signal processing, genomic sequence processing, symbolic sequence processing

- Copyright
- © 2016 Greenhoe
- Licence
- This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction and adaptation in any medium and for any purpose provided that it is properly attributed. For attribution, the original author(s), title, publication source (PeerJ Preprints) and either DOI or URL of the article must be cited.

- Cite this article
- 2016. Order and Metric Compatible Symbolic Sequence Processing. PeerJ Preprints 4:e2052v1 https://doi.org/10.7287/peerj.preprints.2052v1

## Abstract

A traditional random variable X is a function that maps from a stochastic process to the real line (X,<=,d,+,.), where R is the set of real numbers, <= is the standard linear order relation on R, d(x,y)=|x-y| is the usual metric on R, and (R, +, .) is the standard field on R. Greenhoe(2015b) has demonstrated that this definition of random variable is often a poor choice for computing statistics when the stochastic process that X maps from has structure that is dissimilar to that of the real line. Greenhoe(2015b) has further proposed an alternative statistical system, that rather than mapping a stochastic process to the real line, instead maps to a weighted graph that has order and metric geometry structures similar to that of the underlying stochastic process. In particular, ideally the structure X maps from and the structure X maps to are, with respect to each other, both isomorphic and isometric.Mapping to a weighted graph is useful for analysis of a single random variable.for example the expectation EX of X can be defined simply as the center of its weighted graph. However, the mapping has limitations with regards to a sequence of random variables in performing sequence analysis (using for example Fourier analysis or wavelet analysis), in performing sequence processing (using for example FIR filtering or IIR filtering), in making diagnostic measurements (using a post-transform metric space), or in making goptimalh decisions (based on gdistanceh measurements in a metric space or more generally a distance space). Rather than mapping to a weighted graph, this paper proposes instead mapping to an ordered distance linear space Y=(R^n,<=,d,+,.,R,+,x), where (R,+,x) is a field, + is the vector addition operator on R^n x R^n, and . is the scalar-vector multiplication operator on R x R^n. The linear space component of Y provides a much more convenient (as compared to the weighted graph) framework for sequence analysis and processing. The ordered set and distance space components of Y allow one to preserve the order structure and distance geometry inherent in the underlying stochastic process, which in turn likely provides a less distorted (as compared to the real line) framework for analysis, diagnostics, and optimal decision making.

## Author Comment

This paper is a kind of followup to another paper entitled “Order and metric geometry compatible stochastic processing”

that was submitted to PeerJ on 2015 February 19 (almost 15 months ago) and is available here:

https://peerj.com/preprints/844/

The 2015 paper presents a traditional random variable

## Supplemental Information

### C++ source code

C++ source code (written by the author of the paper) for the prgrogram ssp.exe, which was used to generate TeX files for the 128 or so data plots presented in the paper.