An Interval Temporal Logic for Time Series Specification and Data Integration

Abstract

The analysis of temporal series—in particular, analysis of multisensor data—is a complex problem. It depends on the application domain, the way the data have to be used, and sensors available, among other factors. Various models, algorithms, and technologies have been designed for this goal. Temporal logics are used to describe temporal properties of systems. The properties may specify the occurrence and the order of events in time, recurring patterns, complex behaviors, and processes. In this paper, a new interval logic, called duration calculus for functions (DC4F), is proposed for the specification of temporal series corresponding to multisensor data. DC4F is a natural extension of the well-known duration calculus, an interval temporal logic for the specification of process duration. The adequacy of the proposed logic is analyzed in the case of multisensor data concerning volcanic eruption monitoring. It turns out that the relevant behavior concerns time intervals, not only accumulated history as it is described in other kinds of temporal logics. The examples analyzed demonstrate that a description language is required to specify time series of various kind relative to time intervals. The duration calculus cannot be successfully applied for this task. The proposed calculus allows one to specify temporal series and complex interval-dependent behaviors, and to evaluate the corresponding data within a unifying logical framework. It allows to formulate hypotheses concerning volcano eruption phenomena. However, the expressivity of DC4F comes at the cost of its decidability.

Temporal Analysis and Classification of Sensor Signals

“…In this section, we present the basic ideas of Duration Calculus for Functions (DC4F) (see [ 8 ] for details), an extension of Duration Calculus (DC). DC allows one to specify and reason only about durations of Boolean-valued functions.…”

“…The time domain considered here consists of non-negative real numbers, . The discrete time model consisting of natural numbers can be treated as a special case (see [ 8 ] for details). Consequently, we can handle discrete functions, or discrete time series, as if they were defined for the whole time domain.…”

“…Formulas are obtained from functional expressions using logical operators. The semantics of formulas are outlined here as well (see [ 8 ] for details).…”

“…Formula is satisfied in interval I if there is an n such that we can split I into subintervals , , …, , such that F is satisfied in each subinterval (cf. [ 7 ], Section 4 and [ 8 ]). Below, we present examples of iteration formulas.…”

“…Recently, we have proposed a new interval temporal logic called Duration Calculus for Functions (DC4F) [ 8 ]. It allows one to integrate information specified by various means, such as diagrams, curves, tables, and textual descriptions in one formal framework suitable for specification, validation, and reasoning processes.…”