Visualization of Fuzzy Clustering Result in Metric Space

AbstractThis paper presents a visualization of a result of fuzzy clustering. The feature of fuzzy clustering is to obtain the degree of belongingness of objects to fuzzy clusters so the result will be more commensurate with reality. In addition, the number of clusters requires less and the solution of the result will be more robust when compared with conventional hard clustering. In contrast, the fuzzy clustering result interpretation tends to be more complicated. Therefore, measuring the similarity (or dissimilarity) between a pair of fuzzy classification status of objects is important. In order to measure the similarity (or dissimilarity) mathematically, it is necessary to introduce a scale to the fuzzy clustering result. That is, the obtained solutions as a fuzzy clustering result must be in a metric space. In order to implement this, we have proposed multidimensional joint scale and cluster analysis. In this analysis, we exploit a scale obtained by multidimensional scaling. This paper clarifies that the multidimensional joint scale and cluster analysis introduces scale to the fuzzy clustering result and then the visualization of the fuzzy clustering result in the metric vector space has a theoretical mathematical meaning through the Euclidean distance structure. In this paper, this is shown by using several numerical comparisons with ordinary visualizations of the fuzzy clustering result

Footfall and the territorialisation of urban places measured through the rhythms of social activity

The UK high street is constantly changing and evolving in response to, for example, online sales, out-of-town developments, and economic crises. With over 10 years of hourly footfall counts from sensors across the UK, this study was an opportunity to perform a longitudinal and quantitative investigation to diagnose how these changes are reflected in the changing patterns of pedestrian activity. Footfall provides a recognised performance measure of place vitality. However, through a lack of data availability due to historic manual counting methods, few opportunities to contextualise the temporal patterns longitudinally have existed. This study therefore investigates daily, weekly, and annual footfall patterns, to diagnose the similarities and differences between places as social activity patterns from UK high streets evolve over time. Theoretically, footfall is conceptualised within the framework of Territorology and Assemblage Theory, conceptually underpinning a quantitative approach to represent the collective meso-level (street and town-centre) patterns of footfall (social) activity. To explore the data, the periodic signatures of daily, weekly, and annual footfall are extracted using STL (seasonal trend decomposition using Loess) algorithms and the outputs are then analysed using fuzzy clustering techniques. The analyses successfully identify daily, weekly, and annual periodic patterns and diagnose the varying social activity patterns for different urban place types and how places, both individually and collectively are changing. Footfall is demonstrated to be a performance measure of meso-scale changes in collective social activity. For place management, the fuzzy analysis provides an analytical tool to monitor the annual, weekly, and daily footfall signatures providing an evidence-based diagnostic of how places are changing over time. The place manager is therefore better able to identify place specific interventions that correspond to the usage patterns of visitors and adapt these interventions as behaviours change