# Development of the Algorithmic Basis of the FCAZ Method for Earthquake-Prone Area Recognition
This implements the transition from the classification of point objects into high- and low-seismicity zones to the original flat high-seismicity zones. Ext formalizes the construction of a unique mapping of DPS clusters into flat zones of nonzero measure inside and on the boundary of which an earthquake with \(M\geq M_{0}\) may occur [28]. FCAZ made it possible to move from simple pattern recognition to system analysis in the problem of earthquake-prone area recognition. In particular, with the help of FCAZ, it was possible to uniquely distinguish a subsystem of high-seismicity zones from a non-empty complement using an exact boundary. Previously, FCAZ was used to successfully recognize earthquake-prone areas in the Andean mountain belt of South America, on the Pacific coast of the Kamchatka Peninsula and the Kuril Islands in California, in the Baikal-Transbaikal and Altai-Sayan regions, in the Caucasus, and in the Crimean Peninsula and northwestern Caucasus. A detailed description of the FCAZ method, its mathematical apparatus, and the results obtained is given in [26]. It should be noted that at present, in parallel with the mathematical tools described here by the authors of earthquake-prone area recognition, methods for seismic hazard and subsequent seismic risk assessment are being created and developed based on other ideological foundations and mathematical solutions [2, 51, 52, 53, 54, 55, 56, 57]. Most of them are still part of PSHA, although in recent years, there have been more and more publications in which PSHA has been criticized [2]. Returning to DPS clustering, it should be noted that, conceptually, its initial concept is a fuzzy model of the fundamental mathematical \"limit\" property. It is called the density and is a non-negative function that depends on an arbitrary subset and any point in the initial space in which clustering is assumed. The value of the density should be understood as the strength of the connection between a subset and a point, as the degree of influence of a subset on a point, or dually as the degree of limiting a point to a subset. Nontrivial densities always exist in finite metric spaces (FMS). By fixing the density level \(\alpha\) and interpreting it as a limit level, we can introduce the notion of discrete perfection with a level \(\alpha\). The set in the initial space is called discretely perfect with a level \(\alpha\) (\(\alpha\)-DPS set or just DPS set) if it consists of all points of the original space's \(\alpha\) limit. A rigorous theory of DPS sets (DPS theory) was constructed within the framework of DMA, in which, in particular, it is shown that DPS sets have the properties of clusters. The currently developed DPS clustering algorithms (DPS algorithms) operate in finite metric spaces and depend on a number of parameters, the main ones of which are the density \(P\), its level \(\alpha\), and the local survey radius \(r\), and they have three stages [43, 46]. In the first stage, topological filtering of the original space is carried out, and its noise is cleared. DPS algorithms iteratively cut out from the original space the maximum \(\alpha\)-perfect subset, the existence and uniqueness of which is guaranteed by the DPS theory. In the second stage, the DPS algorithm splits the result of the first stage into \(r\)-connected components, which according to the DPS theory will be DPS sets. These are local DPS clusters. Due to the locality of the viewing radius, the division into \(r\)-connected components of the maximum \(\alpha\)-perfect subset at the second stage is often small, detailed, and needs to be enlarged. This is the essence of the third and final stage of DPS clustering. Its result will be the representation of the maximum DPS subset in the form of a disjunct union of groups of local DPS clusters, each of which is a fragmentary manifestation (edge) of the global anomalous entity behind it in the original FMS. A detailed description of all DPS stages will be given below using the example of the SDPS algorithm, which is the most famous of the DPS algorithms.