### Kristian Sabo

Josip Juraj Strossmayer University of Osijek

### Research Interests

Applied and Numerical Mathematics (Curve Fitting, Parameter Estimation, Data Cluster Analysis) with applications in Agriculture, Economy, Chemistry, Politics, Electrical Engineering, Medicine, Food Industry, Mechanical Engineering.

### Degrees

- 2007 PhD in Numerical Mathematics, Department of Mathematics, University of Zagreb
- 2003 MSc in Mathematics, Department of Mathematics, University of Zagreb
- 1999 BSc in Mathematics and Computer Science, Department of Mathematics, University of Osijek, Croatia

### Publications

- K. Sabo, R. Scitovski, Š. Ungar, Z. Tomljanović, A method for searching for a globally optimal k-partition of higher-dimensional datasets, Journal of Global Optimization
**89**(2024), 633-653The problem with finding a globally optimal k-partition of a set A is a very intricate optimization problem for which in general, except in the case of one-dimensional data, i.e., for data with one feature (A\subset\R), there is no method to solve. Only in the one-dimensional case there exist efficient methods that are based on the fact that the search for a globally optimal partition is equivalent to solving a global optimization problem for a symmetric Lipschitz-continuous function using the global optimization algorithm DIRECT. In the present paper, we propose a method for finding a globally optimal k-partition in the general case (A\subset \R^n, n\geq 1), generalizing an idea for solving the Lipschitz global optimization for symmetric functions. To do this, we propose a method that combines a global optimization algorithm with linear constraints and the k-means algorithm. The first of these two algorithms is used only to find a good initial approximation for the $k$-means algorithm. The method was tested on a number of artificial datasets and on several examples from the UCI Machine Learning Repository, and an application in spectral clustering for linearly non-separable datasets is also demonstrated. Our proposed method proved to be very efficient. - A. Morales-Esteban, R. Scitovski, K. Sabo, D. Grahovac, Š. Ungar, Earthquake analysis of clusters of the most appropriate partition, Journal of Seismology (2024), prihvaćen za objavljivanjeIn our paper, we propose the most appropriate partition to depict the seismogenic zones of an active seismic region.To do so,the earthquake data considered are the location and magnitude. To determine three ellipsoidal layers of shallow, intermediate, and deep earthquakes, we switch from the geoid to a solid ball model and solve an appropriate multiple concentric sphere detection problem. Considering the Iberian Peninsula region, by using the Mahalanobis incremental algorithm with the help of the Mahalanobis area index and Mahalanobis minimal distance index, we first determine the most appropriate partition of earth quake positions, consisting of as compact and mutually separated clusters as possible. The result shows four clusters representing the main seismogenic zones of that area. In each of these clusters, we analyze some important earthquake properties, notably the hypocentral depths—a less researched property. Furthermore, we show how to generate a smooth surface best fitting the hypocenters in the considered area, and since the data contain many outliers, for that purpose we use the moving least absolute deviation method. In addition, for each cluster of the most appropriate partition, we ponder the question of estimating the Gutenberg–Richter’s b-value. To avoid the known drawbacks mentioned in the literature for estimating the b-parameter in the Gutenberg–Richter law, we propose the estimation of parameters a and b by using the least absolute deviation method. We also found that the hypocenters are notably deeper in the southwestern Iberian Peninsula and the Azores-Gibraltar fault zone, where the largest earthquakes take place. Finally, one should emphasizethat the hypocenters study proposed in this research demonstrated that the most hazardous zone encompasses the most deep focuses. The CPU-time required for all calculations has been moderate. The methodology, used in this work, could easily be applied to other seismological areas, for which we list our freely available Mathematica-modules.
- R. Scitovski, K. Sabo, P. Nikić, S. Majstorović Ergotić, A new efficient method for solving the multiple ellipse detection problem, Expert systems with applications
**222**/119853 (2023)In this paper, we consider the multiple ellipse detection problem based on data points coming from a number of ellipses in the plane not known in advance. In so doing, data points are usually contaminated with some noisy errors. In this paper, the multiple ellipse detection problem is solved as a center-based problem from cluster analysis. Therefore, an ellipse is considered a Mahalanobis circle. In this way, we easily determine a distance from a point to the ellipse and also an ellipse as the cluster center. In the case when the number of ellipses is known in advance, an optimal partition is searched for on the basis of the -means algorithm that is modified for this case. Hence, a good initial approximation for M-circle-centers is searched for as unit circles with the application of a few iterations of the well-known DIRECT algorithm for global optimization. In the case when the number of ellipses is not known in advance, optimal partitions with clusters for the case when cluster-centers are ellipses are determined by using an incremental algorithm. Among them, the partition with the most appropriate number of clusters is selected. For that purpose, a new Geometrical Objects-index (GO-index) is defined. Numerous test-examples point to high efficiency of the proposed method. Many algorithms can be found in the literature that recognize ellipses with clear edges well, but that do not recognize ellipses with unclear or noisy edges. On the other hand, our algorithm is specifically used for recognition of ellipses with unclear or noisy edges. - R. Scitovski, K. Sabo, D. Grahovac, Š. Ungar, Minimal distance index — A new clustering performance metrics, Information Sciences
**640**/119046 (2023)We define a new index for measuring clustering performance called the Minimal Distance Index. The index is based on representing clusters by characteristic objects containing the majority of cluster points. It performs well for both spherical and ellipsoidal clusters. This method can recognize all acceptable partitions with well-separated clusters. Among such partitions, our minimal distance index may identify the most appropriate one. The proposed index is compared with other most frequently used indexes in numerous examples with spherical and ellipsoidal clusters. It turned out that our proposed minimal distance index always recognizes the most appropriate partition, whereas the same cannot be said for other indexes found in the literature. Furthermore, among all acceptable partitions, the one with the largest number of clusters, not necessarily the most appropriate ones, has a special significance in image analysis. Namely, following Mahalanobis image segmentation, our index recognizes partitions that might not be the most appropriate ones but are the ones using colors that significantly differ from each other. The minimal distance index recognizes partitions with dominant colors, thus making it possible to select specific details in the image. We apply this approach to some real-world applications such as the plant rows detection problem, painting analysis, and iris detection. This may also be useful for medical image analysis. - K. Sabo, R. Scitovski, Š. Ungar, Multiple spheres detection problem—Center based clustering approach, Pattern Recognition Letters
**176**(2023), 34-41In this paper, we propose an adaptation of the well-known -means algorithm for solving the multiple spheres detection problem when data points are homogeneously scattered around several spheres. We call this adaptation the -closest spheres algorithm. In order to choose good initial spheres, we use a few iterations of the global optimizing algorithm DIRECT , resulting in the high efficiency of the proposed -closest spheres algorithm. We present illustrative examples for the case of non-intersecting and for the case of intersecting spheres. We also show a real-world application in analyzing earthquake depths.

### Projects

The optimization and statistical models and methods in recognizing properties of data sets measured with errors (Member of the scientific project entitled above. Project started on March 1, 2017. Principal investigator is professor Rudolf Scitovski from Department of Mathematics, University of Osijek. Project was supported by Croatian Science Foundation.)

### Professional Activities

**Editorial Board**

Since 2012 member of the Editorial board of the Journal Osječki matematički list

Since 2017 member of the Editorial board of the Journal Croatian Operational Research Review

2001-2012 Editor in Chief of the Journal Osječki matematički list

**Service Activities**

2017-2023 Head of Department of Mathematics, University of Osijek

2013-2017 president of Osijek Mathematical Society

2001-2013 secretary of Osijek Mathematical Society

### Teaching

2023/2024

Numerical mathematics

Machine Learning

Nonlinear Optimization