Order and Metric Compatible Symbolic Sequence Processing

A traditional random variable X is a function that maps from a stochastic process to the “real line” (R, ≤, 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 limitationswith regards to a sequence of randomvariables in performing sequence analysis (using for exampleFourier analysis orwavelet analysis), in performing sequenceprocessing (using for example FIR filtering or IIR filtering ), in making diagnostic measurements (using a post-transformmetric space), or in making “optimal” decisions (based on “distance” measurements in a metric space or more generally a distance space). Rather thanmapping to aweighted graph, this paper proposes insteadmapping to an ordereddistance linear space Y ≜ (R, ≤, d, +, ⋅, R, ∔, ⨰) , where (R,∔, ⨰) is a field, + is the vector addition operator on Rn ×Rn , and ⋅ is the scalar-vector multiplication operator on R×Rn . 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. 2010 Mathematics Subject Classification: 60B99,60G05 (primary); 60E15,60G20,62F03,62P10,92D20 (secondary) Universal Decimal Classification (UDC): 519.2+519.6+601