Professor Emeritus

Rudolf Scitovski
5 (1st floor)
School of Applied Mathematics and Informatics

Josip Juraj Strossmayer University of Osijek

Research Interests

Numerical and applied mathematics – especially least squares and least absolute deviationsons problems, clustering and applications, global optimization

Applications: solving parameter identification problems in mathematical problems (medicine, agriculture, economy, marketing, electrical engineering, food industry), smoothing the data (electrical engineering, medicine), surface generating on the basis of experimental data (electrical engineering, selection in livestock industry, civil engineering), clustering (multiple geometrical objects detection (lines, circles, ellipses, generalized circles), earthquake zoning, medicine, short-term and long-term energy products prediction, short-term and long-term water level prediction, acceptable definition of constituencies, image and signal analysis)


Journal Publications

  1. 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 objavljivanje
    The 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.
  2. 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.
  3. V. Ivić, M. Zjalić, S. Blažetić, M. Fenrich, I. Labak, R. Scitovski, K.F. Szűcs, E. Ducza, T. Tábi, F. Bagamery, É. Szökő, R. Vuković, A. Rončević, D. Mandić, Ž. Debeljak, M. Berecki, M. Balog, A. Seres-Bokor, A. Sztojkov-Ivanov, J. Hajagos-Tóth, S. Gajović, A. Imširović, M. Bakula, S. Mahiiovych, R. Gaspar, S.G. Vari, M. Heffer, Elderly rats fed with a high-fat high-sucrose diet developed sex-dependent metabolic syndrome regardless of long-term metformin and liraglutide treatment, Frontiers in endocrinology (Lausanne) 14 (2023)
    Aim/Introduction: The study aimed to determine the effectiveness of early antidiabetic therapy in reversing metabolic changes caused by high-fat and high-sucrose diet (HFHSD) in both sexes. Methods: Elderly Sprague–Dawley rats, 45 weeks old, were randomized into four groups: a control group fed on the standard diet (STD), one group fed the HFHSD, and two groups fed the HFHSD along with long-term treatment of either metformin (HFHSD+M) or liraglutide (HFHSD+L). Antidiabetic treatment started 5 weeks after the introduction of the diet and lasted 13 weeks until the animals were 64 weeks old. Results: Unexpectedly, HFHSD-fed animals did not gain weight but underwent significant metabolic changes. Both antidiabetic treatments produced sex-specific effects, but neither prevented the onset of prediabetes nor diabetes. Conclusion: Liraglutide vested benefits to liver and skeletal muscle tissue in males but induced signs of insulin resistance in females.
  4. D. Dumančić, A. Stupin, M. Kožul, V. Šerić, A. Kibel, N. Goswami, B. Brix, Ž. Debeljak, R. Scitovski, I. Drenjančević, Increased cerebral vascular resistance underlies preserved cerebral blood flow in response to orthostasis in humans on a high-salt diet, European Journal of Applied Physiology 123/4 (2023), 923-933
    Cerebral blood flow autoregulation protects brain tissue from blood pressure variations and maintains cerebral perfusion pressure by changes in vascular resistance. High salt (HS) diet impairs endothelium-dependent vasodilation in many vascular beds, including cerebral microcirculation, and may affect vascular resistance. The aim of present study was to determine if 7-day HS diet affected the reactivity of middle cerebral artery (MCA) to orthostatic challenge in healthy human individuals, and if autoregulatory mechanisms and sympathetic neural regulation were involved in this phenomenon.Twenty-seven persons participated in study (F:21, M:6, age range 19-24). Participants consumed 7-day low-salt (LS) diet (< 2.3 g kitchen salt/day) and afterwards 7-day HS diet (> 11.2 g kitchen salt/day). Blood and urine analysis and anthropometric measurements were performed after each diet. Arterial blood pressure, heart rate and heart rate variability, and cerebral and systemic hemodynamic parameters were recorded simultaneously with transcranial Doppler ultrasound and The Task Force® Monitor in response to orthostatic test.Participants remained normotensive during HS diet. Following both, the LS and HS dietary protocols, mean cerebral blood flow (CBF), as well as the velocity time integral and diastolic blood pressure decreased, and cerebral pulsatility index increased after rising up. Importantly, cerebrovascular resistance significantly increased in response to orthostasis only after HS diet. Urine concentration of noradrenaline and vanillylmandelic acid, baroreflex sensitivity (BRS), and sympathetic neural control was significantly decreased in HS diet.Results suggest that CBF in response to orthostatic test was preserved in HS condition due to altered vascular reactivity of MCA, with increased cerebrovascular resistance and blunted BRS and sympathetic activity.
  5. 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.
  6. K. Sabo, R. Scitovski, Š. Ungar, Multiple spheres detection problem—Center based clustering approach, Pattern Recognition Letters 176 (2023), 34-41
    In 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.
  7. K. Sabo, R. Scitovski, Nova metoda za definiranje izbornih jedinica u Hrvatskoj, Hrvatska i komparativna javna uprava (2023), prihvaćen za objavljivanje
    U radu predlažemo novu metodu za definiranje konfiguracije izbornih jedinica primjenom metode spektralnog klasteriranja. Metoda pronalazi konfiguracije izbornih jedinica koje zadovoljavaju neku unaprijed zadanu toleranciju ujednačenosti težina biračkih glasova te pritom čuva granice županija. Također u metodu se prirodno može uključiti i razina političke/socijalne/gospodarske povezanosti županija. Nadalje, navodimo popis poznatih metoda za određivanje broja zastupničkih mjesta po izbornim jedinicama, koje se temelje na principu razmjernosti broja birača i broja zastupničkih mjesta. U radu dajemo pregled indeksa iz literature kojima se može mjeriti ujednačenost težina biračkih glasova. Primjene tih indeksa ilustriramo na najnovijem prijedlogu Hrvatske vlade, vlastitim prijedlozima, kao i na nekoliko primjera konfiguracija izbornih jedinica od kojih su neki već predstavljeni javnosti.
  8. 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), 1-8
    In 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.
  9. R. Scitovski, K. Sabo, A combination of k-means and DBSCAN algorithm for solving the multiple generalized circle detection problem, Advances in Data Analysis and Classification 15 (2021), 83-89
    Motivated by the problem of identifying rod-shaped particles (e.g. bacilliform bacterium), in this paper we consider the multiple generalized circle detection problem. We propose a method for solving this problem that is based on center-based clustering, where cluster-centers are generalized circles. An efficient algorithm is proposed which is based on a modification of the well-known $k$-means algorithm for generalized circles as cluster-centers. In doing so, it is extremely important to have a good initial approximation. For the purpose of recognizing detected generalized circles, a verb|QAD|-indicator is proposed. Also a new verb|DBC|-index is proposed, which is specialized for such situations. The recognition process is intitiated by searching for a good initial partition using the verb|DBSCAN|-algorithm. If verb|QAD|-indicator shows that generalized circle-cluster-center does not recognize searched generalized circle for some cluster, the procedure continues searching for corresponding initial generalized circles for these clusters using the Incremental algorithm. After that, corresponding generalized circle-cluster-centers are calculated for obtained clusters. This will happen if a data point set stems from intersected or touching generalized circles. The method is illustrated and tested on different artificial data sets coming from a number of generalized circles and real images.
  10. R. Scitovski, S. Majstorović Ergotić, K. Sabo, A combination of RANSAC and DBSCAN methods for solving the multiple geometrical object detection problem, Journal of Global Optimization 79/3 (2021), 669-686
    In this paper we consider the multiple geometrical object detection problem. On the basis of the set $A$ of data points coming from and scattered among a number of geometrical objects not known in advance, we should reconstruct or detect thosegeometrical objects. A new very efficient method for solving this problem based on avery popular RANSAC method using parameters from DBSCAN method is proposed.Thereby, instead of using classical indexes for recognizing the most appropriatepartition, we use parameters from DBSCAN method which define the necessaryconditions proven to be far more efficient.Especially, the method is applied to solving multiple circle detection problem. In this case, we give both the conditions for the existence of the best circle as arepresentative of the data set and the explicit formulas for the parameters of the bestcircle. In the illustrative example we consider the multiple circle detection problem for the datapoint set $A$ coming from $5$ intersected circles not known in advance. Using Wolfram Mathematica, the proposed method needed between 0.5 - 1 sec to solve this problem.


  • 1986 – 1990 – head of project task ( „Operationalization of categories and relationships of value laws“ that was carried out within project (2.08.01) „Fundamental research in economy“ (Ministry of Science, Technology and Computing)
  • 1991-1995 – principal investigator of scientific project (1-01-129) „Application of numerical and finite mathematics“ (Ministry of Science, Technology and Computing)
  • 1996 – 2000 – principal investigator of scientific project (165021) „Parameter identification problems in mathematical models“ (Department of Mathematics, University of Osijek – Ministry of Science and Technology)
  • 2002 – 2006 – principal investigator of scientific project (023501) “Parameter estimation in mathematical models“ (Department of Mathematics, University of Osijek – Ministry of Science and Technology)
  • 2007 – 2013 scientific project (235-2352818-1034) “Nonlinear parameter estimation problems in mathematical models“ (Ministry of Science, Education and Sports), investigator
  • 1-1-2017 – member of research project IP-2016-06-8350 (Principal investigator: Marijana Zekić Sušac, Faculty of Economics, University of Osijek) “Methodological Framework for Efficient Energy Management by Intelligent Data Analytics” (Croatian Science Foundation)
  • 1-3-2017 – principal investigator of scientific project IP-2016-06-6545 “The optimization and statistical models and methods in recognizing properties of data sets measured with errors” (Croatian Science Foundation)

Professional Activities

Editorial Boards

Mathematical Communications (since 1996)

Osječki matematički list (since 2003)

Croatian Operational Research Review (since 2013)

Committee Memberships

  • Chairman of the Organizing Committee of the VII Conference on Applied Mathematics, Osijek, September 13-15, 1989
  • Member of the Programming Committee and chairman or deputy chairman of the Organizing Committee of the 6th-10th International Conference on Operational Research, which were organized by the Croatian Operational Research Society
  • Member of the Scientific Committee of the  “International Conference on Operational Research“, Croatian Operational Research Society (since 1996 – )
  • Since 1999 – 2011: member of the Scientific or Organizing Committee of the International Conference  on Applied Mathematics and Scientific Computing
  • Member of the Scientific Committee of the scientific-professional conference PrimMath, 2001
  • Member of the Scientific Committee of the 2nd (Zagreb, 2000), 3rd (Split, 2004), 4th (Osijek, 2008), 5th (Rijeka, 2012), Croatian Congress of Mathematics
  • President of the Scientific Committee of the 4th Croatian Congress of Mathematics, Osijek, June 17-20, 2008




Service Activities

  • 2017 – 2019 – full profesor at the Department of Mathematics,  University of  Osijek
  • 2013 – 2017 – Vice-Rector for Science, Technology, Projects and International Cooperation, University of  Osijek
  • 1999 – 2003 and  2009 – 2013 Head of the Department of Mathematics,   University of  Osijek
  • since 2008 – chairman of the Seminar for optimization and applications
  • 1994 – 2000 – Head of the Mathematical Colloquium in Osijek
  • 2003 – 2010 – member of the Managing Board of The National Foundation for Science, Higher Education and Technological Development of Republic of Croatia
  • 2001 – 2008 – member of the National Council for Higher Education of the Republic of Croatia
  • 2006. – 2009 – member of the National Scientific Field Committee for Natural Sciences
  • 1997 – 1998 and 2001 – 2005 – member of the   National  Commission for Mathematics
  • 2003 – 2007 –   vice-head of the Department of Mathematics, University of Osijek
  • 1998 – 1999 – dean of the  Faculty of Electrical Engineering in Osijek
  • 1997 – vice-dean for science at the  Faculty of Electrical Engineering in Osijek
  • since 2008 – chairman of the Osijek Students Center Council
  • 2002 – 2008 – chairman of the Osijek Students Center Recovery Council
  • 1998 – 2001 – member and chairman of the Managing Board of the City and University Library in Osijek


Nastavne aktivnosti u zimskom semestru Akademske 2019./2020.


Nastavne aktivnosti u ljetnom semestru Akademske 2019./2020.

  • Grupiranje podataka i primjene
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