Analysing Imperfect Temporal Information in GIS Using the Triangular Model 1. Introduction

Every entity has an extent in time, such as the lifetime of an object or the duration of an event. These temporal extents are usually described by crisp time intervals bounded by a welldefined start and end. However, under some circumstances, the temporal extent of an entity is imperfect, and cannot be adequately modelled by a crisp time interval. On the one hand, some events may start or end gradually and therefore their start and end cannot be pinned to exact time stamps. For example, it is difficult to decide when the Industrial Revolution started and finished. Intervals of this kind of events are usually modelled by fuzzy sets (Zadeh, 1965) through the quantification of the graded truth of whether a time point is in the interval, bring the concept of fuzzy time interval. On the other hand, in some other cases, it is only known that the start and end of a crisp interval are within certain ranges, but no extra information or assumptions about the distribution of the start and the end is available. Modelling this kind of imperfect time intervals with fuzzy sets would induce extra overhead and unnecessary complexity. In these cases, the alternative approach, i.e. rough sets (Pawlak, 1982), can excellently suit the modelling and handling of time intervals. Currently, a lot of disciplines are faced with the issue of imperfect time intervals, which is reflected in many contributions in modelling intervals by fuzzy sets (De Caluwe et al., 1999, De Caluwe et al., 1997, Nagypál and Motik, 2003, Ohlbach, 2004, Schockaert et al., 2008, Garrido et al., 2009) and rough sets (Bittner, 2002, Bassiri et al., 2009). However, while most of this work focuses on modelling and reasoning about imperfect time intervals, techniques and tools for visualising and analysing imperfect time intervals are still lacking. This probably stems from the conventional representation of time intervals (e.g. Gantt chart, time table and historical timeline) which represents time intervals as linear segments along a one-dimensional axis. The second dimension is often exploited merely to differentiate intervals of entities and has no temporal meaning. In this representation, the visual distribution of intervals is variable, according to the application of different ordering rules in the second dimension, for example, ordering intervals from the shortest to the longest, or from the earliest started to the latest started. The characteristics of the distribution of intervals cannot be observed in one single display. This is not convenient for visual pattern detection of time intervals, let alone imperfect time intervals with more complex structure.