Josie McCulloch

Zack Ellerby, Josie McCulloch, Melanie Wilson et al.

Subjective judgements from experts provide essential information when assessing and modelling threats in respect to cyber-physical systems. For example, the vulnerability of individual system components can be described using multiple factors, such as complexity, technological maturity, and the availability of tools to aid an attack. Such information is useful for determining attack risk, but much of it is challenging to acquire automatically and instead must be collected through expert assessments. However, most experts inherently carry some degree of uncertainty in their assessments. For example, it is impossible to be certain precisely how many tools are available to aid an attack. Traditional methods of capturing subjective judgements through choices such as \emph{high}, \emph{medium} or \emph{low} do not enable experts to quantify their uncertainty. However, it is important to measure the range of uncertainty surrounding responses in order to appropriately inform system vulnerability analysis. We use a recently introduced interval-valued response-format to capture uncertainty in experts' judgements and employ inferential statistical approaches to analyse the data. We identify key attributes that contribute to hop vulnerability in cyber-systems and demonstrate the value of capturing the uncertainty around these attributes. We find that this uncertainty is not only predictive of uncertainty in the overall vulnerability of a given system component, but also significantly informs ratings of overall component vulnerability itself. We propose that these methods and associated insights can be employed in real world situations, including vulnerability assessments of cyber-physical systems, which are becoming increasingly complex and integrated into society, making them particularly susceptible to uncertainty in assessment.

Josie McCulloch, Christian Wagner, Uwe Aickelin

Reasoning with fuzzy sets can be achieved through measures such as similarity and distance. However, these measures can often give misleading results when considered independently, for example giving the same value for two different pairs of fuzzy sets. This is particularly a problem where many fuzzy sets are generated from real data, and while two different measures may be used to automatically compare such fuzzy sets, it is difficult to interpret two different results. This is especially true where a large number of fuzzy sets are being compared as part of a reasoning system. This paper introduces a method for combining the results of multiple measures into a single measure for the purpose of analysing and comparing fuzzy sets. The combined measure alleviates ambiguous results and aids in the automatic comparison of fuzzy sets. The properties of the combined measure are given, and demonstrations are presented with discussions on the advantages over using a single measure.

Josie McCulloch, Christian Wagner, Uwe Aickelin

The measure of distance between two fuzzy sets is a fundamental tool within fuzzy set theory. However, current distance measures within the literature do not account for the direction of change between fuzzy sets; a useful concept in a variety of applications, such as Computing With Words. In this paper, we highlight this utility and introduce a distance measure which takes the direction between sets into account. We provide details of its application for normal and non-normal, as well as convex and non-convex fuzzy sets. We demonstrate the new distance measure using real data from the MovieLens dataset and establish the benefits of measuring the direction between fuzzy sets.

Josie McCulloch, Christian Wagner, Uwe Aickelin

Similarity measures provide one of the core tools that enable reasoning about fuzzy sets. While many types of similarity measures exist for type-1 and interval type-2 fuzzy sets, there are very few similarity measures that enable the comparison of general type-2 fuzzy sets. In this paper, we introduce a general method for extending existing interval type-2 similarity measures to similarity measures for general type-2 fuzzy sets. Specifically, we show how similarity measures for interval type-2 fuzzy sets can be employed in conjunction with the zSlices based general type-2 representation for fuzzy sets to provide measures of similarity which preserve all the common properties (i.e. reflexivity, symmetry, transitivity and overlapping) of the original interval type-2 similarity measure. We demonstrate examples of such extended fuzzy measures and provide comparisons between (different types of) interval and general type-2 fuzzy measures.