# Development of the Algorithmic Basis of the FCAZ Method for Earthquake-Prone Area Recognition
## 5 Conclusions  
Discrete mathematical analysis is a discrete data analysis method that uses scenarios of classical continuous mathematics, in which the fundamentals are replaced by fuzzy models of their discrete analogs. From a practical point of view, DMA is a new approach to data analysis focused on an expert and occupying an intermediate position between rigorous mathematical methods and soft combinatorial ones.  
The solution to the problem within the DMA framework consists of two parts. The first one is informal, as it analyzes the logic of the expert, introduces the necessary concepts, and explains the scenarios and principles of the solution. The second has a formal characteristic. With the help of the DMA apparatus, all concepts receive rigorous definitions within the framework of fuzzy mathematics and fuzzy logic, and the schemes and principles become algorithms.  
This article provides three such solutions:  
* The SDPS algorithm is a DMA response to the empirical definition of a cluster as a connected set, the measure of the presence of which at each of its points is higher than at any other. The answer is a rigorous DPS set theory and DPS clustering based on it.
* The Ext algorithm is a DMA formalization of the scanning ability and proximity based on an empirical understanding with the help of Kolmogorov means (one of the main technical means of DMA).
* The Int algorithm is a DMA formalization of smoothness (insideness) based on the empirical logic of the circle with the help of Kolmogorov means and standard linear algebra.  
It should be noted that DMA has all the necessary tools to generate unsupervised topological filtering and classification algorithms. Based on fuzzy sets and fuzzy logic, DMA can convey expert ideas about the spatial distributions of objects. DMA makes it possible to implement a system approach to the analysis of geophysical data in the problem of adequate seismic hazard assessment studied in the article.  
In conclusion, we emphasize that based on the mathematical apparatus of the FCAZ method, which has been substantially developed based on DMA, in this work, new zones prone to earthquakes with \(M\geq 6.5\) in California were constructed. A distinctive feature of the new FCAZ recognition system is the combination of a number of smaller zones located at a relatively small distance from each other into large, single, connected potential high-seismicity zones. Note that after applying the Int algorithm, the area of the FCAZ zones increased by only \(6\%\).  
The results presented in the present and previous works indicate the high reliability of the interpretation of FCAZ zones as earthquake-prone areas. The FCAZ method makes it possible to effectively recognize possible locations of future earthquakes solely from seismological data. We also note that the results of the FCAZ studies indicate that weak seismicity can actually \"manifest\" the properties of geophysical fields, which are used directly in the form of characteristics of recognition objects in other (similar to FCAZ) methodologies.  
All authors contributed to the study's conception and design. Conceptualization and original draft preparation, S.M.A., B.A.D. and S.R.B.; conceptualization, methodology, review and editing and validation, D.A.K. and B.V.D.; material preparation, formal analysis, data curation, and algorithm development, S.R.B. and I.O.B. All authors have read and agreed to the published version of the manuscript.  
This work was conducted in the framework of budgetary funding of the Geophysical Center of RAS, adopted by the Ministry of Science and Higher Education of the Russian Federation.  
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Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.  
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The authors declare no conflict of interest.