### Šime Ungar

sime@mathos.hr

School of Applied Mathematics and Informatics

Josip Juraj Strossmayer University of Osijek

Josip Juraj Strossmayer University of Osijek

### Research Interests

- Geometric and Algebraic Topology
- Shape theory

### Degrees

- PhD in mathematics, Department of Mathematics, University of Zagreb , 1977.
- MSc in mathematics, Department of Mathematics, University of Zagreb , 1972.
- BSc in mathematics, Department of Mathematics, University of Zagreb, 1969.

### Publications

Journal 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 (2024), prihvaćen za objavljivanjeThe 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.
- 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. - R. Scitovski, K. Sabo, Š. Ungar, A method for forecasting the number of hospitalized and deceased based on the number of newly infected during a pandemic, Scientific Reports - Nature
**12**/4773 (2022), 1-8In this paper we propose a phenomenological model for forecasting the numbers of deaths and of hospitalized persons in a pandemic wave, assuming that these numbers linearly depend, with certain delays τ>0 for deaths and δ>0 for hospitalized, on the number of new cases. We illustrate the application of our method using data from the third wave of the COVID-19 pandemic in Croatia, but the method can be applied to any new wave of the COVID-19 pandemic, as well as to any other possible pandemic. We also supply freely available Mathematica modules to implement the method. - J. Pečarić, Š. Ungar, On the two-point Ostrowski inequality, Mathematical Inequalities and Applications
**13**/2 (2010), 339-347We prove the <i>L<sub>p</sub></i>-version of an inequality similar to the two-point Ostrowski inequality of Matić and Pečarić.