Matching strings in encoded sequences

Abstract

We investigate the length of the longest common substring for encoded sequences and its asymptotic behaviour. The main result is a strong law of large numbers for a re-scaled version of this quantity, which presents an explicit relation with the Rényi entropy of the source. We apply this result to the zero-inflated contamination model and the stochastic scrabble. In the case of dynamical systems, this problem is equivalent to the shortest distance between two observed orbits and its limiting relationship with the correlation dimension of the pushforward measure. An extension to the shortest distance between orbits for random dynamical systems is also provided.keywords: string matching, Rényi entropy, shortest distance, correlation dimension, random dynamical systems, coding MSC: 60F15, 60Axx, 60C05, 37A50, 37A25, 37Hxx, 37C45, 94A17, 68P30

Matching of observations of dynamical systems, with applications to sequence matching

“…We could also consider, as in the first section of the paper, a non-trivial f . In the context of sequence matching, f is called the encoding function (or encoder) and can model different treatments of the original source of information [14]. The clustering structure is however in this case too complex to yield such a general result.…”

“…Closely related problems have gained interest in recent years. The case of real, unobserved trajectories was considered in [19] and [11], using EVT techniques, while asymptotic results for the shortest distance between two orbits were obtained in [7] and then generalized to multiple orbits [8] and finally to observed orbits [14].…”

Rényi entropy and pattern matching for run-length encoded sequences

“…If the process is α-mixing with an exponential decay (or ψ-mixing with polynomial decay) and if for cylinders C n of length n in B n , their preimage f −1 C n is of length at most h(n) with h(n) = o(n γ ) for some γ > 0, one could use the ideas of Coutinho et al (2020) to get a lower bound. Nevertheless, the run-length encoder does not satisfy this last necessary assumption since preimage of cylinders under f can have arbitrary length.…”

“…In Coutinho et al (2020), the authors wondered if the above mentioned result holds if the sequences are transformed following certain rules of modification. Thus, if f is a measurable function (called an encoder) transforming a sequence x into another sequence f (x), they studied the behaviour of M n (f (x), f (y)) and obtain a relation with the Rényi entropy of the pushforward measure f * µ.…”

“…A natural question would be to ask if we could apply the results presented in Coutinho et al (2020) when the encoder is a compression algorithm and in particular the run-length encoder. Unfortunately, to obtain their main result, the authors needed that the encoder does not compress too much the sequences, an hypothesis which is not satisfied by the run-length encoder.…”