On Comparing and Selecting Approaches to Model Interval-Valued Data as Fuzzy Sets

The capture of interval-valued data is becoming an increasingly common approach in data collection (from survey based research to the collation of sensor data) as an efficient method of obtaining information about uncertainty associated with the data in question. To best utilise this data, several methods of aggregating intervals into fuzzy sets have been proposed in the fuzzy set literature, particularly within the field of Computing with Words. Two key examples are the Interval Approach and the Interval Agreement Approach and their respective extensions. Each method takes a fundamentally different approach to constructing fuzzy sets, making different assumptions in respect to the nature and the reliability of the data. The result is noticeably different fuzzy sets that do not share the same statistical properties (such as central-tendency and standard deviation). This begs the question of how these techniques differ in respect to the relationship between the original interval-valued data and the fuzzy sets produced - and thus when and why each of the methods is the most appropriate. This paper compares the results of both methods of constructing fuzzy sets from interval-valued data. Statistical moments of the fuzzy sets are compared against the interval-valued data to evaluate how well key properties of the fuzzy sets match those of the data; for example, does the standard deviation of the fuzzy set represent the standard deviation of the raw interval-valued data? We use comparisons on real-world data to demonstrate how the methods differ and which is more appropriate given the assumptions of the data.