Linguistic summaries for periodicity detection based on mathematical morphology

Abstract

The paper presents a methodology to evaluate the periodicity of a temporal data series, neither relying on assumption about the series form nor requiring expert knowledge to set parameters. It exploits tools from mathematical morphology to compute a periodicity degree and a candidate period, as well as the fuzzy set theory to generate a natural language sentence, improving the result interpretability. Experiments on both artificial and real data illustrate the relevance of the proposed approach.

Interpretability of fuzzy linguistic summaries

“…This approach for instance leads to protoforms of the form "regularly y's are A", where the adverb "regularly" indeed describes the extent to which "y's are A" applies considering a specific temporal adjustment [17,18]. Such a protoform can be enriched with information about the period, as "Adv every p units, y's are A", where Adv is an adverb as "roughly" or "exactly", p is an approximation of the period and "unit" a unit considered the most appropriate to express the period, as detailed in Section 3.3.…”

“…For instance, instead of saying "the game lasted 1 hour and 7 minutes", one would rather say "the game lasted approximately 1 hour", adding the adverb to indicate that this period is not exact. An implementation of these principles is proposed in [17] to generate relevant linguistic expressions.…”

“…In the case of fuzzy temporal references, several aspects of the time formulation can be considered, as proposed in [17] for the case of regularity assessment: a numerical candidate period is automatically computed from the data, then transformed into a linguistic representation emulating the way humans usually express time, with respect to time unit, time approximation, and approximation characterisation through an adverb. The choice of the unit is based on the general observation that speakers usually choose a convenient unit to express time with small numbers.…”

“…In the case of temporal protoforms based on the "regularly" mode, i.e. of the form "regularly, y's are A", the quality criterion can be interpreted along the same lines as a two-step process, namely individual evaluation and aggregation, although the steps differ [17,18]: first for each time stamp individually, a binary decision is made to decide whether it is considered as supporting A or not, several methods being presented in [18]. In the second step, aggregation measures the regularity of the episodes of the two types of groups and their alternation, based on a dispersion measure of their respective durations.…”

“…This is for instance the case for the protoform "Adv every p units y's are A" [17], where the quality is evaluated only for the most relevant p values.…”