Dynamic fuzzy paths and cycles in multi-level directed graphs B. Petelin a,n , I. Kononenko b , V. Malačič a , M. Kukar b a Marine Biology Station Piran, National Institute of Biology, Fornače 41, 6330 Piran, Slovenia b Faculty of Computer and Information Science, University of Ljubljana, Tržaška cesta 25, 1000 Ljubljana, Slovenia article info Article history: Received 3 November 2013 Received in revised form 15 September 2014 Accepted 18 September 2014 Keywords: Dynamic fuzzy paths Dynamic fuzzy cycles Multi-level directed graphs Oceanography abstract In this paper we propose improved algorithms for the discovery of significant paths and cycles that dynamically evolve through time in a series of multi-level directed graphs. First, we search for the most probable paths and combine them into clusters based on similar edges. We combine paths into dynamic fuzzy paths. We also detect cycles in different paths and combine them into dynamic fuzzy cycles. We obtain dynamic fuzzy structures using the hierarchical clustering of individual structures. For paths, the clustering distance depends on common edges, while for cycles we calculate the distance on the basis of common vertices. We apply the developed algorithms to a time series of multi-level directed graphs obtained from the results from the numerical model Mediterranean Ocean Forecasting System during the period 1999–2011. We compare the results with known structures from the oceanographic literature. With our approach we find a high similarity between the resulting dynamic fuzzy paths and cycles and structures found by oceanographic experts. When comparing the cycles, the expert sees our results as a convex hull of the average of individual cycles. On the other hand, the method reveals undiscovered paths and gyres, which can be verified through observation. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction The amount of research in the field of spatial-temporal data mining in the earth sciences has significantly increased in recent years. If we limit ourselves only to data mining in oceanography and meteorology, a considerable amount of literature is already available. Scientists have developed and upgraded a variety of methods for spatial-temporal data mining. Significant examples, which are related to our work, are spatial-temporal association rules (Huang et al., 2007, 2008) and trajectory clustering (Camargo et al., 2005; Lee et al., 2007; Nanni and Pedreschi, 2006; Žabkar et al., 2008). In our previous work (Petelin et al., 2013), we developed a novel framework for spatial-temporal data mining which utilizes Lagrangian particle tracking, spatial-temporal association rules and multi-level directed graphs. Lagrangian trajectories occur in many areas, for example the tracking of vehicles, vessels and animals which are equipped with GPS devices, etc., which allows a great number of possibilities for mining the trajectory data. In order to mine the resulting graphs further, we developed an algorithm which detects cycles in these graphs. However, the resulting cycles and their corresponding graph apply only to a short period (one month). The reason for this is that the velocity field in the model changes continuously and therefore we need to aggregate, with the help of association rules, the daily model results in monthly multi-level directed graphs with constant weights of edges. Thus, it became necessary to upgrade this algo- rithm in order to reveal paths and cycles that evolve over longer periods, i.e. several months but less than one year. In addition, we needed to combine new paths and cycles into groups (clusters) that we call dynamic fuzzy paths and cycles. In this work, we construct dynamic fuzzy paths and cycles from the most probable paths in given multi-level directed graphs. We call these structures dynamic because they are formed using different multi-level directed graphs over time, and fuzzy because we join these structures into clusters based on their inexact (fuzzy) similarity. We discuss dynamic fuzzy paths and dynamic fuzzy cycles separately. The former are constructed by hierarchical clustering using a distance based on common edges, while the latter are formed in a similar manner, yet using a distance roughly based on their spatial overlap. Definition 1. Dynamic fuzzy structures represent the most sig- nificant paths and cycles performed by Lagrangian particles in a given domain. They are obtained by walking along the most probable edges in a time series of multi-level directed graphs and their grouping with the hierarchical clustering. The structures are “dynamic” because the weights of edges in the graphs change over time, and “fuzzy” because we combine them into clusters, based on some distance between them. The paper is organized as follows. Section 2 is divided into sub- sections that cover two main concepts: dynamic fuzzy paths and Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/engappai Engineering Applications of Artificial Intelligence http://dx.doi.org/10.1016/j.engappai.2014.09.012 0952-1976/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author. Tel.: þ386 5 6712907; fax: þ386 5 6712902. E-mail addresses: petelin@mbss.org (B. Petelin), igor.kononenko@fri.uni-lj.si (I. Kononenko), malacic@mbss.org (V. Malačič), matjaz.kukar@fri.uni-lj.si (M. Kukar). Engineering Applications of Artificial Intelligence 37 (2015) 194–206