Approximate Period Detection and Correction

Abstract

Abstract. Periodicity has been historically well studied and has numerous applications. In nature, however, few cyclic phenomena have an exact period.This paper surveys some recent results in approximate periodicity: concept definition, discovery or recovery, techniques and efficient algorithms. We will also show some interesting connections between error correction codes and periodicity.We will try to pinpoint the issues involved, the context in the literature, and possible future research directions.

Optimal Square Detection Over General Alphabets

“…After this procedure terminates, we have computed all runs, except for possibly some of the runs that were properly contained in a tail in list L. We may have reported some duplicate runs, which we filter out as follows. The number of runs reported so far is r = O(n log σ) 4 . We sort them in additional O(n + r) = O(n log σ) time, e.g., by using radix sort, and remove duplicates.…”

“…and the periodicity lemma states that if p and q are both such periods and p + q ≤ n + gcd(p, q) then gcd(p, q) is also a period [32]. This was generalised in a myriad of ways, for strings [17,49,74], partial words (words with don't cares) [11-13, 47, 50, 69, 70], Abelian periods [14,20], parametrized periods [6], order-preserving periods [42,64], approximate periods [2][3][4]. Now, a square can be defined as a fragment of length twice its period.…”

Improved Periodicity Mining in Time Series Databases

“…We characterize exact periodicity using the patterns κ = 0 and γ = 0. Following [7,8] a periodic string can be defined as follows: Let T be a string of length n = |T |. T is called periodic if T = P i pref (P ), where i ∈ N and i ≥ 2, P is a substring of T such that |P | ≤ n/2, P i is the concatenation of P to itself i times and pref (P ) is a prefix of P .…”