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)

Degrees

• PhD in mathematics, Department of Mathematics, University of Zagreb, 1984.
• MSc in mathematics, Department of Mathematics, University of Zagreb, 1977.
• BSc in mathematics, Department of Mathematics, University of Zagreb, 1974.

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 89 (2024), 633-653
   Abstract

   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. 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 objavljivanje
   Abstract

   In 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.
3. 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)
   Abstract

   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.
4. 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)
   Abstract

   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.
5. 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
   Abstract

   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.
6. R. Scitovski, K. Sabo, D. Grahovac, Š. Ungar, Minimal distance index — A new clustering performance metrics, Information Sciences 640/119046 (2023)
   Abstract

   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.
7. K. Sabo, R. Scitovski, Š. Ungar, Multiple spheres detection problem—Center based clustering approach, Pattern Recognition Letters 176 (2023), 34-41
   Abstract

   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.
8. K. Sabo, R. Scitovski, Nova metoda za definiranje izbornih jedinica u Hrvatskoj, Hrvatska i komparativna javna uprava 23/24 (2023), 642-672
   Abstract

   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.
9. 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
   Abstract

   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.
10. 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
    Abstract

    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.

11. 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
    Abstract

    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.
12. A. Morales-Esteban, F. Martínez-Álvarez, S. Scitovski, R. Scitovski, Mahalanobis clustering for the determination of incidence-magnitude seismic parameters for the Iberian Peninsula and the Republic of Croatia, Computers & Geosciences 156 (2021)
    Abstract

    The aim of this paper is to analyse the seismic activity of the Iberian Peninsula and a wide area of the Republic of Croatia. To do so, two incidence-magnitude seismic parameters have been defined. First, the areas have been divided into several ellipsoidal clusters using Mahalanobis clustering. Four generalised indexes (Mahalanobis Calinski Harabasz, Mahalanobis Davies–Bouldin, Mahalanobis Simplified Silhouette Width Criterion and Mahalanobis Area) have been used to determine the most appropriate number of ellipsoidal clusters, on the basis of which a partition with four and a partition with eleven clusters have been considered. For the wide area of the Republic of Croatia there are fourteen clusters and the five areas that just affect Croatia have been analysed in detail. Then, to analyse the seismic activity of the areas, two incidence-magnitude seismic parameters have been defined and calculated: a) Delta^{(4)} , that represents the minimal number of successive years in which at least one earthquake of magnitude between 4 and 5 has been registered; b) Delta^{(5)}, that shows the number of years in which at least one earthquake of magnitude larger than 5 occurred. The calculation of Delta^{(4)} for the South-west and the South-east of the Iberian Peninsula has provided two years for both. Regarding Delta^{(5)} , 10 and 12 years have been obtained for the South-west and the South-east of the Iberian Peninsula, respectively. The analysis of Croatia has shown that the Ston–Metković area has the highest seismic activity. The following results have been determined: 5 years for Delta^{(4)} and 22 for Delta^{(5)} . It should be mentioned that these results cannot be used for predicting earthquakes. However, data about the incidences of earthquake events and their magnitudes can certainly serve as useful information in civil engineering.
13. M. Balog, V. Ivić, R. Scitovski, I. Labak, K.F. Szűcs, R. Gaspar, S.G. Vari, M. Heffer, A mathematical model reveals sex-specific changes in glucose and insulin tolerance during rat puberty and maturation, Croatian medical journal 61/2 (2020), 107-118
    Abstract

    Aim To evaluate the effects of maturation and sex on glucose metabolism during glucose tolerance (GTT) and insulin tolerance tests (ITT) in young and adult male and female rats by using two different approaches – the conventional, which uses area under the curve and glucose curve, and mathematical modeling that identifies parameters necessary for determining the function that models glucose metabolism. Methods Male and female rats at 3.5 and 12 months of age underwent standard GTT and ITT after overnight fasting. The parameters were identified by using Mathematica-module NonlinearModelFit [ ] for experimentally obtained data. Results When data were statistically analyzed, both sexes and age groups had similar glucose and insulin tolerance. In the mathematical model of GTT, parameters describing the rate of glucose concentration increase G’(0) and decrease G’I multiplied with maturation, with a concomitant decrease in the time point (t max, tI ) of reaching maximum and minimum glucose concentration (Gmax, G0). The mathematical model of ITT for males was independent of age, unlike of that for females, which had increased G’(0) and G’I, and more quickly recovered from hypoglycemia after maturation. Conclusion The mathematical model revealed female susceptibility to large glucose excursions, which are better reflected by ITT in young animals and by GTT in adults.
14. R. Scitovski, K. Sabo, DBSCAN-like clustering method for various data densities, Pattern Analysis and Applications 23 (2020), 541-554
    Abstract

    In this paper, we propose a modification of the well-known DBSCAN algorithm, which recognizes clusters with various data densities in a given set of data points $A = {a^i in R^n : i = 1, ldots , m}$. First, we define the parameter $MinPts = floor ln |A| floor$ and after that, by using a standard procedure from DBSCAN algorithm, for each $a in A$ we determine radius $epsilon_a$ of the circle containing $MinPts$ elements from the set $A$. We group the set of all these radii into the most appropriate number $(t)$ of clusters by using Least Square distance-like function applying {tt SymDIRECT} or {tt SepDIRECT} algorithm. In that way we obtain parameters $epsilon_1 > · · · > epsilon_t$. Furthermore, for parameters ${MinPts, epsilon_1} we construct a partition starting with one cluster and then add new clusters for as long as the isolated groups of at least $MinPts$ data points in some circle with radius $epsilon_1$ exist. We follow a similar procedure for other parameters $epsilon_2, ldots, , epsilon_t$. After the implementation of the algorithm, a larger number of clusters appear than can be expected in the optimal partition. Along with defined criteria, some of them are merged by applying a merging process for which a detailed algorithm has been written. Compared to the standard DBSCAN algorithm, we show an obvious advantage for the case of data with various densities.
15. K. Sabo, D. Grahovac, R. Scitovski, Incremental method for multiple line detection problem - iterative reweighted approach, Mathematics and Computers in Simulation 178 (2020), 588-602
    Abstract

    In this paper we consider the multiple line detection problem by using the center-based clustering approach, and propose a new incremental method based on iterative reweighted approach. We prove the convergence theorem, and construct an appropriate algorithm which we test on numerous artificial data sets. Stopping criterion in the algorithm is defined by using the parameters from DBSCAN algorithm. We give necessary conditions for the most appropriate partition, which have been used during elimination of unacceptable center-lines that appear in the output of the algorithm. The algorithm is also illustrated on a real-world image coming from Precision Agriculture.
16. M. Heffer, V. Ivić, R. Scitovski, Parameter identification in the mathematical model of glucose and insulin tolerance test – the mathematical markers of diabetes, Croatian Operational Research Review 11 (2020), 121-133
    Abstract

    Glucose tolerance test (GTT) is standard diagnostic procedure that tests the efficiency of blood glucose-lowering hormones (insulin, incretins, leptin). Contrary, insulin tolerance test (ITT) is probing efficiency of blood glucose–rising hormones (glucagon, thyroxine, growth hormone, glucocorticoids, adrenalin, noradrenalin). These two hormone systems together maintain blood glucose levels in a narrow range. Various pathophysiological mechanisms give rise to a reversible condition – prediabetes which then progresses to an irreversible chronic disease – diabetes, both marked with deviation of blood glucose levels outside the set range. In diagnostic purpose, the patient is given glucose load, and blood glucose is measured right before and 2 hours after load. Measurements are more frequent after insulin injection (ITT) or if both tests are performed on experimental animals. In this paper we analyse the mathematical model for GTT and ITT. The obtained model function is an useful tool in describing the dynamics of blood glucose changes.
17. R. Scitovski, U. Radojičić, K. Sabo, A fast and efficient method for solving the multiple line detection problem, Rad HAZU, Matematičke znanosti. 23 (2019), 123-140
    Abstract

    In this paper, we consider the multiple line detection problem on the basis of a data points set coming from a number of lines not known in advance. A new and efficient method is proposed, which is based upon center-based clustering, and it solves this problem quickly and precisely. The method has been tested on $100$ randomly generated data sets. In comparison to the incremental algorithm, the method gives significantly better results. Also, in order to identify a partition with the most appropriate number of clusters, a new index has been proposed for the case of a cluster whose lines are cluster-centers. The index can also be generalized for other geometrical objects.
18. R. Scitovski, K. Sabo, Application of the DIRECT algorithm to searching for an optimal k-partition of the set $AsubsetR^n$ and its application to the multiple circle detection problem, Journal of Global Optimization 74/1 (2019), 63-77
    Abstract

    In this paper, we propose an efficient method for searching for a globally optimal k-partition of the set A subseteq R^n. Due to the property of the DIRECT global optimization algorithm to usually quickly arrive close to a point of global minimum, after which it slowly attains the desired accuracy, the proposed method uses the well-known k-means algorithm with a initial approximation chosen on the basis of only a few iterations of the DIRECT algorithm. In case of searching for an optimal k-partition of spherical clusters, the method is not worse than other known methods, but in case of solving the multiple circle detection problem, the proposed method shows remarkable superiority.
19. M. Zekić-Sušac, M. Knežević, R. Scitovski, Modeling the cost of energy in public sector buildings by linear regression and deep learning, Central European Journal of Operations Research 28 (2019), 1-16
    Abstract

    Modeling the cost of energy consumption of public buildings is vital for planning reconstruction measures in the public sector. The methods of predictive analytics have not been sufficiently exploited in this domain. This paper aimed to create a model for predicting the cost of the total energy consumption of a building based on deep learning (DL) and compare it to the standard linear regression (MLR), as well to identify key predictors that can significantly influence the cost of energy. An algorithm for modeling procedure is proposed which includes data pre-processing, variable reduction procedures, training and testing MLR and deep neural networks (DNN) and, finally, performance evaluation. Variable reduction in the MLR model was conducted by a backward procedure; while in DNN, the Olden method was used. The algorithm was tested on a high-dimensional real dataset of Croatian public buildings. The results showed that there is a statistically significant difference in the distribution of DNN predictions and distribution of actual values in the validation set, as opposed to distribution of MLR predictions and real values. However, DNN model had a lower normalized root mean square error, while the MLR model had a lower symmetric mean average error. Those findings reveal the potential of DL for solving this type of problems but also the need for more advanced algorithms adjusted to deal with large-range numeric outputs. The created models could be implemented in public sector business intelligence systems to support policy and decision makers in allocating resources for building reconstructions.
20. R. Scitovski, K. Sabo, The adaptation of the k-means algorithm to solving the multiple ellipses detection problem by using an initial approximation obtained by the DIRECT global optimization algorithm, Applications of Mathematics 64/6 (2019), 663-678
    Abstract

    In this paper, we consider the multiple ellipses detection problem on the basis of a data points set coming from a number of ellipses in the plane not known in advance, whereby an ellipse E is considered as a Mahalanobis circle with center S, radius r, and some positive definite matrix Sigma. A very efficient method for solving this problem is proposed. The method uses a modification of the k-means algorithm for Mahalanobis- circle centers. The initial approximation consists of the set of circles whose centers are determined by means of a smaller number of iterations of the DIRECT global optimization algorithm. Unlike other methods known from the literature, our method recognizes well not only ellipses with clear edges, but also ellipses with noisy edges. CPU-time necessary for running the corresponding algorithm is very short and this raises hope that, with appropriate software optimization, the algorithm could be run in real time. The method is illustrated and tested on 100 randomly generated data sets.
21. M. Zekić-Sušac, R. Scitovski, A. Has, Cluster analysis and artificial neural networks in predicting energy efficiency of public buildings as a cost-saving approach, Croatian Review of Economic, Business and Social Statistics (CREBSS) 4/2 (2018), 57-66
    Abstract

    Scientific efforts in predicting energy efficiency and consumption align with the European Commission directives about reducing greenhouse gas emissions, increasing energy efficiency and using 20% of energy from renewable resources until 2020. The largest individual energy consumer is the building sector which contains 40% of total primary energy consumption (Tommerup et al., 2007). That stresses out the importance of creating efficient models that will be able to extract features and predict the energy efficiency level of a building according to its characteristics and planned reconstruction measures. This work deals with creating such prediction model by using data from Croatian Agency for Legal Trade and Real Estate Brokerage (APN) which maintains the centralized information system of energy efficiency in public buildings (ISGE). In order to create prediction models, the machine learning methods were used such as clustering and artificial neural networks. The real dataset of Croatian public buildings with a large number of attributes which were reduced in pre- modelling phase. The aim of this research was to investigate the effect of a clustering procedure on artificial neural network model accuracy. For that purpose, the symmetric mean average percentage error (MAPE) of an initial prediction model obtained on the whole dataset is compared to the MAPE of separate models obtained on each cluster, and the results were discussed. Due to a lack of approaches which integrate clustering and neural networks in modelling energy efficiency, the scientific contribution of this paper is in providing such an approach and in analysing its effects to modeling accuracy. Guidelines for using the model as an approach for saving costs in public sector are also given.
22. R. Scitovski, M. Zekić-Sušac, A. Has, Searching for an optimal partition of incomplete data with application in modeling energy efficiency of public buildings, Croatian Operational Research Review 9/2 (2018), 255-268
    Abstract

    In this paper, we consider the problem of searching for an optimal partition with the most appropriate number of clusters for an incomplete data set in which several outliers might occur. Special attention is given to the application of the Least Squares distance-like function. The procedure of preparing the incomplete data set and the outlier elimination procedure are proposed such that the clustering process gives acceptable solutions. Appropriate justifications with proof are provided for these procedures. An incremental algorithm for searching for optimal partitions with 2, 3, ... clusters is applied on the prepared data set. After that, by using the Davies-Bouldin and the Calinski Harabasz index the most appropriate number of clusters is determined. The whole procedure is organized as an algorithm given in the paper. In order to illustrate its applicability, the above steps are applied on the real data set of public buildings and their energy efficiency data, providing clear clusters that could be used for further modeling procedures.
23. R. Scitovski, A new global optimization method for a symmetric Lipschitz continuous function and the application to searching for a globally optimal partition of a one-dimensional set, Journal of Global Optimization 68 (2017), 713-727
    Abstract

    In this paper, we consider a global optimization problem for a symmetric Lipschitz continuous function $g : [a, b]^k to mathbb{R}$ whose domain $[a, b]^k in mathbb{R}^k$ consists of k! hypertetrahedrons of the same size and shape, in which function g attains equal values. A global minimum can therefore be searched for in one hypertetrahedron only, but then this becomes a global optimization problem with linear constraints. Apart from that, some known global optimization algorithms in standard form cannot be applied to solving the problem. In this paper, it is shown how this global optimization problem with linear constraints can easily be transformed into a global optimization problem on hypercube $[0, 1]^k$, for the solving of which an applied DIRECT algorithm in standard form is possible. This approach has a somewhat lower efficiency than known global optimization methods for symmetric Lipschitz continuous functions (such as SymDIRECT or DISIMPL), but, on the other hand, this method allows for the use of publicly available and well developed computer codes for solving a global optimization problem on hypercube $[0, 1]^k$ (e.g. the DIRECT algorithm). The method is illustrated and tested on standard symmetric functions and very demanding center-based clustering problems for the data that have only one feature. An application to the image segmentation problem is also shown.
24. R. Scitovski, M. Vinković, K. Sabo, A. Kozić, A research project ranking method based on independent reviews by using the principle of the distance to the perfectly assessed project, Croatian Operational Research Review 8 (2017), 429-442
    Abstract

    The paper discusses the problem of ranking research projects based on the assessment obtained from n ≥ 1 independent blinded reviewers. Each reviewer assesses several project features, and the total score is defined as the weighted arithmetic mean, where the weights of features are determined according to the well-known AHP method. In this way, it is possible to identify each project by a point in n-dimensional space. The ranking is performed on the basis of the distance of each project to the perfectly assessed project. Thereby the application of different metric functions is analyzed. We believe it is inappropriate to use a larger number of decimal places if two projects are almost equidistant (according to some distance function) to the perfectly assessed project. In that case, it would be more appropriate to give priority to the project that has received more uniform ratings. This can be achieved by combining different distance functions. The method is illustrated by several simple examples and applied by ranking internal research projects at Josip Juraj Strossmayer University of Osijek, Croatia.
25. G. Asencio-Cortés, S. Scitovski, R. Scitovski, F. Martínez-Álvarez, Temporal analysis of Croatian seismogenic zones to improve eartquake magnitude prediction, Earth Science Informatics 10 (2017), 303-320
    Abstract

    The prediction of earthquakes is a task of utmost difficulty that has been addressed in many different ways. However, an initial definition of the area of interest is needed, with adequate catalogs. In this work, different seismogenic zones proposals in the Republic of Croatia are studied, in terms of predictability. Such zones have been characterized with widely used seismicity parameters. Later, studies based on training and test sets properties as well as the quality of the data involved are carried out. The studies presented in this work analyze the prediction performance across the earthquake magnitude time series of the target seismogenic zones. Results show that specific prediction techniques could be used in some zones to improve earthquake magnitude prediction.
26. R. Grbić, D. Grahovac, R. Scitovski, A method for solving the multiple ellipses detection problem, Pattern Recognition 60 (2016), 824-834
    Abstract

    In this paper, the multiple ellipses detection problem on the basis of a data points set coming from a number of ellipses in the plane not known in advance is considered. An ellipse is considered as a Mahalanobis circle with some positive definite matrix. A very efficient method for solving this problem is proposed. This method very successfully combines the well-known direct least squares method and the RANSAC algorithm with a realistic statistical model of multiple ellipses in the plane. The method is illustrated and tested on numerous synthetic and real-world applications. The method was also compared with other similar methods. In the case when a data points set comes from a number of ellipses with clear edges, the proposed method gives results similar to other known methods. However, when a data points set comes from a number of ellipses with noisy edges, the proposed method performs significantly better than the other methods. We should emphasize the advantage and utility of the proposed methods in a variety of applications such as: medical image analysis, ultrasound image segmentation, etc.
27. R. Scitovski, I. Vidović, D. Bajer, A new fast fuzzy partitioning algorithm, Expert systems with applications 51 (2016), 143-150
    Abstract

    In this paper, a new fast incremental fuzzy partitioning algorithm able to find either a fuzzy globally optimal partition or a fuzzy locally optimal partition of the set $AsubsetR^n$ close to the global one is proposed. This is the main impact of the paper, which could have an important role in applied research. Since fuzzy $k$-optimal partitions with $k=2,3,dots,k_{max}$ clusters are determined successively in the algorithm, it is possible to calculate corresponding validity indices for every obtained partition. The number $k_{max}$ is defined in such a way that the objective function value of optimal partition with $k_{max}$ clusters is relatively very close to the objective function value of optimal partition with $(k_{max}!-!1)$ clusters. Before clustering, the data are normalized and afterwards several validity indices are applied to partitions of the normalized data. Very simple relationships between used validity indices on normalized and original data are given as well. Hence, the proposed algorithm is able to find optimal partitions with the most appropriate number of clusters. The algorithm is tested on numerous synthetic data sets and several real data sets from the UCI data repository.
28. Ž. Turkalj, D. Markulak, S. Singer, R. Scitovski, Research project grouping and ranking by using adaptive Mahalanobis clustering, Croatian Operational Research Review 7 (2016), 81-96
    Abstract

    The paper discusses the problem of grouping and ranking of research projects submitted for a call. The projects are grouped into clusters based on the assessment obtained in the review procedure and by using the adaptive Mahalanobis clustering method as a special case of the Expectation Maximization algorithm. The cluster of projects assessed as best is specially analyzed and ranked. The paper outlines several possibilities for the use of data obtained in the review procedure, and the proposed method is illustrated with the example of internal research projects at the University of Osijek.
29. K. Sabo, R. Scitovski, An approach to cluster separability in a partition, Information Sciences 305 (2015), 208-218
    Abstract

    In this paper, we consider the problem of cluster separability in a minimum distance partition based on the squared Euclidean distance. We give a characterization of a well-separated partition and provide an operational criterion that gives the possibility to measure the quality of cluster separability in a partition. Especially, the analysis of cluster separability in a partition is illustrated by implementation of the $k$-means algorithm.
30. R. Scitovski, T. Marošević, Multiple circle detection based on center-based clustering, Pattern Recognition Letters 52 (2015), 9-16
    Abstract

    The multiple circle detection problem has been considered in the paper on the basis of given data point set $mathcal{A}subset  Rn$. It is supposed that all data points from the set $mathcal{A}$ come from $k$ circles that should be reconstructed or detected. The problem has been solved by the application of center-based clustering of the set $mathcal{A}$, i.e. an optimal $k$-partition is searched for, whose clusters are determined by corresponding circle-centers. Thereby, the algebraic distance from a point to the circle is used. First, an adaptation of the well-known $k$-means algorithm is given in the paper. Also, the incremental algorithm for searching for an approximate globally optimal $k$-partition is proposed. The algorithm locates either a globally optimal $k$-partition or a locally optimal k-partition close to the global one. Since optimal partitions with 2, 3, ... clusters are determined successively in the algorithm, several well-known indexes for determining an appropriate number of clusters in a partition are adopted for this case. Thereby, the Hausdorff distance between two circles is used and adopted. The proposed method and algorithm are illustrated and tested on several numerical examples.
31. T. Marošević, R. Scitovski, Multiple ellipse fitting by center-based clustering, Croatian Operational Research Review 6/1 (2015), 43-53
    Abstract

    This paper deals with the multiple ellipse fitting problem based on a given set of data points in a plane. The presumption is that all data points are derived from k ellipses that should be fitted. The problem is solved by means of center-based clustering, where cluster centers are ellipses. If the Mahalanobis distance-like function is introduced in each cluster, then the cluster center is represented by the corresponding Mahalanobis circle-center. The distance from a point a∈R^2 to the Mahalanobis circle is based on the algebraic criterion. The well-known k-means algorithm has been adapted to search for a locally optimal partition of the Mahalanobis circle-centers. Several numerical examples are used to illustrate the proposed algorithm.
32. A. Morales-Esteban, F. Martínez-Álvarez, S. Scitovski, R. Scitovski, A fast partitioning algorithm using adaptive Mahalanobis clustering with application to seismic zoning, Computers & Geosciences 73 (2014), 132-141
    Abstract

    In this paper we construct an efficient adaptive Mahalanobis k-means algorithm. In addition, we propose a new efficient algorithm to search for a globally optimal partition obtained by using the adoptive Mahalanobis distance-like function. The algorithm is a generalization of the previously proposed incremental algorithm [36]. It successively finds optimal partitions with k = 2, 3, . . . clusters. Therefore, it can also be used for the estimation of the most appropriate number of clusters in a partition by using various validity indexes. The algorithm has been applied to the seismic catalogues of Croatia and the Iberian Peninsula. Both regions are characterized by a moderate seismic activity. One of the main advantages of the algorithm is its ability to discover not only circular but also elliptical shapes, whose geometry fits the faults better. Three seismogenic zonings are proposed for Croatia and two for the Iberian Peninsula and adjacent areas, according to the clusters discovered by the algorithm.
33. I. Vidović, D. Bajer, R. Scitovski, A new fusion algorithm for fuzzy clustering, Croatian Operational Research Review 5/2 (2014), 149-159
    Abstract

    In this paper, we have considered the merging problem of two ellipsoidal clusters in order to construct a new fusion algorithm for fuzzy clustering. We have proposed a criterion for merging two ellipsoidal clusters ∏1, ∏2 with associated main Mahalanobis circles Ej(cj,σj), where cj is the centroid and σ^2j is the Mahalanobis variance of cluster ∏j . Based on the well-known Davies-Bouldin index, we have constructed a new fusion algorithm. The criterion has been tested on several data sets, and the performance of the fusion algorithm has been demonstrated on an illustrative example. 
34. R. Scitovski, K. Sabo, Analysis of the k-means algorithm in the case of data points occurring on the border of two or more clusters, Knowledge-Based Systems 57 (2014), 1-7
    Abstract

    In this paper, the well-known $k$-means algorithm for searching for a locally optimal partition of the set $A subset R^n$ is analyzed in the case if some data points occur on the border of two or more clusters. For this special case, a useful strategy by implementation of the $k$-means algorithm is proposed.
35. I. Vidović, R. Scitovski, Center-based clustering for line detection and application to crop rows detection, Computers and Electronics in Agriculture 109 (2014), 212-220
    Abstract

    This paper proposes a new efficient method for line detection based on 12 known incremental methods of searching for an approximate globally optimal partition 13 of a set of data points A and on the DIRECT algorithm for global optimization. The pro14 posed method was modified for solving the problem of detecting crop rows in agricultural 15 production. This modification can recognize crop rows with a high accuracy, and the 16 corresponding CPU-time is very acceptable. The method has been tested and compared 17 on synthetic data sets with the method based on Hough transformation. The efficiency of 18 this method might be significantly improved in direct application. The proposed method 19 has been used in this paper for the case of two or three crop rows. The generalization to 20 several crop rows is also given in the paper, but was not implemented. Also, the method 21 could be expanded in case when the number of crop rows is not known in advance.
36. K. Sabo, R. Scitovski, Interpretation and optimization of the k-means algorithm, Applications of Mathematics 59/4 (2014), 391-406
    Abstract

    The paper gives a new interpretation and a possible optimization of the well-known $k$-means algorithm for searching for the locally optimal partition of the set $mathcal{A}={a_iinR^n:i=1,dots,m}subset R^n$ which consists of $k$ disjoint nonempty subsets $pi_1,dots,pi_k$, $1leq kleq m$. For this purpose, a new Divided $k$-means Algorithm was constructed as a limit case of the well-known Smoothed k-means Algorithm. It is shown that the algorithm constructed in such way coincides with the $k$-means algorithm if during the iterative procedure no data points appear in the Voronoi diagram. If in the partition obtained by applying the Divided $k$-means Algorithm there are data points lying in the Voronoi diagram, it is shown that the obtained result can be improved further.
37. R. Scitovski, S. Scitovski, A fast partitioning algorithm and its application to earthquake investigation, Computers & Geosciences 59 (2013), 124-131
    Abstract

    In this paper a new fast partitioning algorithm able to find either a globally optimal partition or a locally optimal partition of the set A subset R^n close to the global one is proposed. The performance of the algorithm in terms of CPU time shows significant improvement in comparison with other incremental algorithms. Since optimal partitions with $2, 3, . . .$ clusters are determined successively in the algorithm, it is possible to calculate corresponding clustering validity indexes for every number of clusters in a partition. In that way the algorithm also proposes the appropriate number of clusters in a partition. The algorithm is illustrated and tested on several synthetic and seismic activity data from a wider area of the Republic of Croatia in order to locate the most intense seismic activity in the observed area.
38. R. Grbić, E.K. Nyarko, R. Scitovski, A modification of the DIRECT method for Lipschitz global optimization for a symmetric function, Journal of Global Optimization 57/4 (2013), 1193-1212
    Abstract

    In this paper, we consider a global optimization problem for a symmetric Lipschitz continuous function. An efficient modification of the well- known DIRECT (DIviding RECTangles) method called SymDIRECT is proposed for solving this problem. The method is illustrated and tested on several standard test functions. The application of this method to solving complex center-based clustering problems for the data having only one feature is particularly presented.
39. R. Grbić, K. Scitovski, K. Sabo, R. Scitovski, Approximating surfaces by the moving least absolute deviations method, Applied mathematics and computation 219/9 (2013), 4387-4399
    Abstract

    In this paper we are going to consider the problem of global data approximation on the basis of data containing outliers. For that purpose a new method entitled the moving least absolute deviations method is proposed. In the region of data in the network of knots weighted least absolute deviations local planes are constructed by means of which a global approximant is defined. The method is tested on the well known Franke’s function. An application in gridding of sonar data is also shown.
40. K. Sabo, R. Scitovski, I. Vazler, One-dimensional center-based $l_1$-clustering method, Optimization Letters 7/1 (2013), 5-22
    Abstract

    Motivated by the method for solving center-based Least Squares - clustering problem (Kogan(2007), Teboulle(2007)), we construct a very efficient iterative process for solving a one-dimensional center-based $l_1$ -clustering problem, on the basis of which it is possible to determine the optimal partition. We analyze the basic properties and convergence of our iterative process, which converges to a stationary point of the corresponding objective function for each choice of the initial approximation. Given is also a corresponding algorithm, which in only few steps gives a stationary point and the corresponding partition. The method is illustrated and visualized on the example of looking for an optimal partition with two clusters, where we check all stationary points of the corresponding minimizing functional. Also, the method is tested on the basis of large numbers of data points and clusters and compared with the method for solving the center-based Least Squares - clustering problem described in Kogan(2007) and Teboulle (2007).
41. D. Vincek, G. Kralik, G. Kušec, K. Sabo, R. Scitovski, Application of growth functions in the prediction of live weight of domestic animals, Central European Journal of Operations Research 20/4 (2012), 719-733
    Abstract

    We consider several most frequently used growth functions with the aim of predicting live weight of domestic animals. Special attention is paid to the possibility of estimating well the saturation level of animal weight and defining life cycle phases based on animal weight. Parameters of the growth function are most often estimated on the basis of measurement data by applying the Least Squares (LS) principle. These nonlinear optimization problems very often refer to a numerically very demanding and unstable process. In practice, it is also possible that among the data there might appear several measurement errors or poor measurement samples. Such data might lead not only to unreliable, but very often to wrong conclusions. The Least Absolute Deviations (LAD) principle can be successfully applied for the purpose of detecting and minorizing the effect of such data. On the other hand, by using known properties of LAD-approximation it is possible to significantly simplify the minimizing functional, by which parameters of the growth function are estimated. Implementation of two such possibilities is shown in terms of methodology
42. D. Vincek, K. Sabo, G. Kušec, G. Kralik, I. Đurkin, R. Scitovski, Modeling of pig growth by S-function - least absolute deviation approach for parameter estimation, Archiv für Tierzucht 55/4 (2012), 364-374
    Abstract

    The aim of this study was to determine a mathematical model which can be used to describe the growth of the pig. The study was conducted on 60 pigs (30 barrows and 30 gilts) in the interval between the age of 49 and 215 days. All animals were weighed at 49th day after birth. For the purpose of growth measurements pigs were weighted every 7th day during the experiment. Every 21th day four pigs were selected for the slaughter according to average live weight (LW). By applying the generalized logistic function, the growth of live weight and tissues were described. Thereby optimal parameters in the model were estimated on the basis of measurement data by means of the robust Least Absolute Deviations (LAD) principle. The prediction of optimum slaughter age/weight, on the basis of such model represent a contribution of this paper to the practice.
43. K. Sabo, R. Scitovski, P. Taler, Ravnomjerna raspodjela broja birača po izbornim jedinicama na bazi matematičkog modela, Hrvatska i komparativna javna uprava 14 (2012), 229-249
    Abstract

    U ovom radu predložen je matematički model, na osnovi kojeg je moguće definirati maksimalno kompaktne i dobro razdijeljene izborne jedinice, koje se po broju birača međusobno mogu razlikovati najviše za 5%. Model se temelji na primjeni klaster analize uz poštivanje zakonom propisanih pravila prema kojem izborne jedinice trebaju imati približno jednak broj birača. Metoda je ilustrirana na primjeru dostupnih podataka iz 2007. godine te tako dobivenu raspodjelu izbornih jedinica ne treba shvatiti kao konačni prijedlog rješenja, već isključivo kao prikaz mogućnosti koje nudi ova metodologija. Prema trenutno važećem zakonu izbori se u Republici Hrvatskoj provode u 10 izbornih jedinica. U radu je predloženo nekoliko pristupa poznatih iz literature na osnovi kojih je moguće definirati primjereni broj izbornih jedinica, koje zadržavaju svojstvo maksimalne unutrašnje kompaktnosti i dobre razdijeljenosti.
44. R. Scitovski, S. Maričić, S. Scitovski, Short-term and long-term water level prediction at one river measurement location, Croatian Operational Research Review 3 (2012), 80-90
    Abstract

    Global hydrological cycles mainly depend on climate changes whose occurrence is predominantly triggered by solar and terrestrial influence, and the knowledge of the high water regime is widely applied in hydrology. Regular monitoring and studying of river water level behavior is important from several perspectives. On the basis of the given data, by using modifications of general approaches known from literature, especially from investigation in hydrology, the problem of long- and short-term water level forecast at one river measurement location is considered in the paper. Long-term forecasting is considered as the problem of investigating the periodicity of water level behavior by using linear-trigonometric regression and short-term forecasting is based on the modification of the nearest neighbor method. The proposed methods are tested on data referring to the Drava River level by Donji Miholjac, Croatia, in the period between the beginning of 1900 and the end of 2012.
45. I. Vazler, K. Sabo, R. Scitovski, Weighted median of the data in solving least absolute deviations problems, Communications is Statistics - Theory and Methods 41/8 (2012), 1455-1465
    Abstract

    We consider the weighted median problem for a given set of data and analyze its main properties. As an illustration, an efficient method for searching for a weighted Least Absolute Deviations (LAD)-line is given, which is used as the basis for solving various linear and nonlinear LAD-problems occurring in applications. Our method is illustrated by an example of hourly natural gas consumption forecast.
46. K. Sabo, R. Scitovski, I. Vazler, M. Zekić-Sušac, Mathematical models of natural gas consumption, Energy conversion and management 52/3 (2011), 1721-1727
    Abstract

    In this paper we consider the problem of hourly forecast of natural gas consumption on the basis of hourly movement of temperature and natural gas consumption in the preceding period. There are various methods and approaches for solving this problem in the literature. Some mathematical models with linear and nonlinear model functions relating forecast of natural gas consumption with the past natural gas consumption data, temperature data and temperature forecast data, are mentioned. The methods are tested on concrete examples referring to temperature and natural gas consumption for the area of the city of Osijek (Croatia) from the beginning of the year 2008.
47. K. Sabo, I. Vazler, R. Scitovski, Searching for a best LAD-solution of an overdetermined system of linear equations motivated by searching for a best LAD-hyperplane on the basis of given data, Journal of optimization theory and applications 149 (2011), 293-314
    Abstract

    We consider the problem of searching for a best LAD-solution of an overdetermined system of linear equations $mathbf{Xa}=mathbf{z}$, $mathbf{X}inmathbb{R}^{mtimes n}$, $mgeq n$, $mathbf{a}inmathbb{R}^n, mathbf{z}inmathbb{R}^m$. This problem is equivalent to the problem of determining a best LAD-hyperplane $mathbf{x}mapsto mathbf{a}^Tmathbf{x}$, $mathbf{x}inmathbb{R}^n$ on the basis of given data $(mathbf{x}_i,z_i),,mathbf{x}_i=(x_1^{(i)},ldots,x_n^{(i)})^Tinmathbb{R}^n,,z_iinmathbb{R},,i=1,ldots,m$, whereby the minimizing functional is of the form [ F(mathbf{a})=|mathbf{z}-mathbf{Xa}|_1=sum_{i=1}^m|z_i-mathbf{a}^Tmathbf{x}_i|. ] An iterative procedure is constructed as a sequence of weighted median problems, which gives the solution in finitely many steps. A criterion of optimality follows from the fact that the minimizing functional $F$ is convex, and therefore the point $mathbf{a}^*inmathbb{R}^n$ is the point of a global minimum of the functional $F$ if and only if $mathbf{0}inpartial F(mathbf{a}^*)$. Motivation for the construction of the algorithm was found in a geometrically visible algorithm for determining a best LAD-plane $(x,y)mapsto alpha x+beta y$, passing through the origin of the coordinate system, on the basis of the data $(x_i,y_i,z_i),,i=1,ldots,m$.
48. J. Cumin, B. Grizelj, R. Scitovski, Numerical solving of ballistic flight equations for big bore air rifle, Tehnički vjesnik 16 (2009), 41-46
    Abstract

    This paper is about solving ballistic equations by means of numerical mathematics. Ballistic flight equations are applied for modern big bore air rifles, operated with high pressure carbon-dioxide (CO2) gas. Ballistic equations use air drag function for obtaining results. Today there are many complex commercial ballistic programs on the market, based on modified mass point model - "MPMM", or "6DOF" model. Big bore air rile commonly uses ball projectiles, and velocities of the projectiles are lower than the speed of sound. Therefore simplified models for quick calculations of ballistic trajectories, and projectile velocities can be used.
49. K. Scitovski, R. Scitovski, Rekonstrukcija meridijanske elipse na bazi eksperimentalnih mjerenja, Croatian Journal of Education 63/86 (2009), 19-35
    Abstract

    Na bazi novih metoda matematičke optimizacije analizira se stari problem rekonstrukcije Zemljina rotacijskog elipsoida na bazi podataka mjerenja, koji je već početkom 18. stoljeća promatrao francuski matematičar i astronom markiz Pierre-Simon de Laplace. Posebno je analizirana, ilustrirana i modificirana efikasna metoda iz rada (Sabo i Scitovski 2009) za traženje optimalnih parametara najbolje l1 linearne aproksimacije. U znanstvenoj literaturi problem procjene najbolje l1 aproksimacije obično se povezuje s imenom hrvatskog znanstvenika J. R. Boškovića. Metode za traženje najbolje l1 aproksimacije u posljednje vrijeme dobivaju novi zamah u različitim tehničkim primjenama zbog svojstva robusnosti i mogućnosti implementacije u realnom vremenu na suvremenim računalima.
50. R. Cupec, R. Grbić, K. Sabo, R. Scitovski, Three points method for searching the best least absolute deviations plane, Applied mathematics and computation 215 (2009), 983-994
    Abstract

    In this paper a new method for estimation of optimal parameters of a best least absolute deviations plane is proposed, which is based on the fact that there always exists a best least absolute deviations plane passing through at least three different data points. The proposed method leads to a solution in finitely many steps. Moreover, a modification of the aforementioned method is proposed that is especially adjusted to the case of a large number of data and the need to estimate parameters in real time. Both methods are illustrated by numerical examples on the basis of simulated data and by one practical example from the field of robotics.
51. G. Kušec, G. Kralik, R. Scitovski, D. Vincek, B. Ulrich, Growth characteristics of hybrid pig predicted by means of asymmetric s-curve, Poljoprivreda 14 (2008), 55-61
    Abstract

    The present study was performed on 24 pigs distributed over two MHS-genotypes (NN, Nn) and two feeding regimes (intensive, restrictive). These pigs were investigated as the last of four trials in the experiment carried out by Kušec et al. (2005). The data on muscle and fatty tissue volumes were obtained by means of Magnetic Resonance Imaging (MRI). For the growth depiction and for the predictions of live weight as well as of muscle and fat tissue, four models developed by KuSec et al. (2007) were used. These models were set up on the basis of the general form of asymmetric S-function. This function showed to fit well to the data collected in the present study ; but the inaccuracy of the models appeared to increase with age. At the age of 124 days, the estimated values of live weight, muscle and fat volumes were close to the actual values measured in the group of intensively fed pigs ; in the group of restrictively fed pigs live weight and muscle volume were underestimated while the volume of fat was overestimated. At the time of the last MR imaging (154 days of age), live weight and muscle volume was to some extent overestimated in the pigs fed intensively and underestimated in those fed restrictively. The volume of fatty tissue was estimated with favourable accuracy in all investigated groups of pigs. For further predictions, two approaches were used. In the first approach to predict individual live weights, at the age of 124 days, the predictions were fairly accurate for most of the pigs, falling in the range within one week ; 4 of them being estimated falsely by more than ± 7 days. Generally, when expressed as absolute value, the average misestimate was ~4 days in all groups, except for the restrictively fed MHS-gene carrier pigs which were estimated with 6 days difference on average. At the age of 154 days, the predictions were to some extent less accurate ; 7 pigs (in total) were incorrectly predicted by more than a week. The prediction accuracy was lower in the intensively fed pigs, ~6 days on average ; in the restrictive group of pigs, misestimates of live weight predictions were on average ~4 and ~5 days for NN and Nn genotypes, respectively. Finally, in the second approach, the time that particular group of pigs would need to achieve a predetermined live weight of 100 kg was calculated. The closest prediction was achieved in the group of intensively fed MHS-homozygous negative pigs (4 days) while other groups were estimated with 6 days divergence from the actual age at 100 kg live weight. It was concluded that the predictions made by means of asymmetric S-function were fair enough for a pig producer to make a fattening plan or some other important decisions, e.g. in selection on growth traits.
52. J. Sertić, D. Kozak, R. Scitovski, LU-decomposition for solving sparse band matrix systems and its application in thin plate bending, Transactions of FAMENA 32 (2008), 41-48
    Abstract

    In this paper algorithm for solving sparse band system matrices is proposed. Algorithm is based on LU-decomposition, therefore has good numerical properties. Proposed algorithm is applied within the finite difference method (FDM) on solving thin plates bend problem. In order to compare the solution accuracy obtained by proposed algorithm, the same example has been solved by finite element method.
53. D. Jukić, M. Benšić, R. Scitovski, On the existence of the nonlinear weighted least squares estimate for a three-parameter Weibull distribution, Computational Statistics & Data Analysis 52/9 (2008), 4502-4511
    Abstract

    The problem of nonlinear weighted least squares fitting of the three-parameter Weibull distribution to the given data (wi,ti,yi), i=1,…,n, is considered. The part wi>0 of the data stands for the data weights. It is shown that the best least squares estimate exists provided that the data satisfy just the following two natural conditions: (i) 0<t1<t2<⋯<tn and (ii) 0<y1<y2<⋯<yn<1. To support this, an illustrative numerical example is given.
54. K. Sabo, R. Scitovski, The best least absolute deviations line-properties and two efficient methods for its derivation, ANZIAM Journal 50 (2008), 185-198
    Abstract

    Given a set of points in the plane, the problem of existence and finding the least absolute deviations line is considered. The most important properties are stated and proved and two efficient methods for finding the best absolute deviations line are proposed. Compared to other known methods, our proposed methods proved to be considerably more efficient.
55. D. Jukić, K. Sabo, R. Scitovski, A review of existence criteria for parameter estimation of the Michaelis-Menten regression model, Annali dell'Universita' di Ferrara 53 (2007), 281-291
    Abstract

    In this paper we consider the least squares (LS) and total least squares (TLS) problems for a Michaelis-Menten enzyme kinetic model $f(x ; a, b)=ax/(b+x)$, $a, b>0$. In various applied research such as biochemistry, pharmacology, biology and medicine there are lots of different applications of this model. We will systematize some of our results pertaining to the existence of the LS and TLS estimate, which were proved in papers [16] and [17]. Finally, we suggest a choice of good initial approximation and give one numerical example.
56. T. Marošević, R. Scitovski, An application of a few inequalities among sequences in electoral systems, Applied mathematics and computation 194 (2007), 480-485
    Abstract

    We look at the concept of ‘favouring large states’ for divisor methods in proportional electoral systems, which is based on the comparison of the ratios of divisors. We show that it is possible in the ordered way to insert new divisor methods between any two divisor methods which have the property that one ‘favours large states’ over the other. It follows from a few sequences’ inequalities of the harmonic, geometric, arithmetic and quadratic means.
57. D. Jukić, R. Scitovski, K. Sabo, Total least squares fitting Michaelis-Menten enzyme kinetic model function, Journal of Computational and Applied Mathematics, 201 (2007), 230-246
    Abstract

    The Michaelis-Menten enzyme kinetic model $f(x ; a, b)=ax/(b+x)$, $a, b>0$, is widely used in biochemistry, pharmacology, biology and medical research. Given the data $(p_i, x_i, y_i)$, $i=1, ldots, m$, $mgeq 3$, we consider the total least squares (TLS) problem for the Michaelis-Menten model. We show that it is possible that the TLS estimate does not exist. As the main result, we show that the TLS estimate exists if the data satisfy some natural conditions. Some numerical examples are included.
58. R. Scitovski, G. Kralik, K. Sabo, T. Jelen, A mathematical model of controlling the growth of tissue in pigs, Applied mathematics and computation 181 (2006), 1126-1138
    Abstract

    A mathematical model of controlling the growth of tissues in pigs is described in this paper. In that sense, a method is given by which it is possible to periodically and very accurately estimate live pig weight of backfat based upon measurements done by ultrasound. These estimations will be used for the purpose of predicting growth of backfat in live pigs. Backfat weight is estimated on the basis of measurements done by using the Moving Total Least Squares Method, whereas estimation of live pig backfat growth is done by using a generalized logistic function, whose parameters are estimated by means of the Least Squares Method. Since thereby the Hessian of the corresponding minimizing function is very close to a singular matrix, an additional problem analysis was necessary.
59. D. Jukić, R. Scitovski, Least squares fitting Gaussian type curve, Applied mathematics and computation 167/1 (2005), 286-298
    Abstract

    Given the data (pi, ti, yi), i = 1, …, m, m ⩾ 3, we give necessary and sufficient conditions which guarantee the existence of the weighted least squares estimate for a Gaussian type function. To this end, we suggest a choice of the suitable initial approximation for an iterative minimization, and give some numerical examples.
60. G. Kralik, R. Scitovski, Z. Škrtić, Model procjene rasta teškog hibrida pura, Krmiva 47 (2005), 195-205
    Abstract

    U radu je istražena primjena generalizirane logističke funkcije u modeliranju rasta teškog hibrida pura. Istraživanje je trajalo 19 tjedana, tijekom kojih je kontroliran rast 20 muških i 20 ženskih pura Nicholas 700 provenijencije. Tijekom čitavog razdoblja tova razlike u živim težinama između purana i pura bile su statistički vrlo visoko značajne (P<0, 001). Na kraju istraživanja purani su u prosjeku bili teški 15, 17 kg, a pure 10, 88 kg. Purani su od pura bili teži za otprilike 4, 3 kg ili 39, 53%. Prosječni tjedni prirasti kod purana bili su 795 g, a kod pura 577 g, dok su prosječne stope rasta u tovu teškog hibrida bile 12, 45% (mužjaci) i 12, 11% (ženke). Procjene točke infleksije i pojedinih faza rasta po spolu pura obavljene su primjenom generalizirane logističke funkcije: Koeficijenti asimetrije bili su γ =0, 2 (purani) i γ =0, 3 (pure). Biološki maksimum za purane bio je 18, 57 kg, a za pure 12, 63 kg. Prema izračunatim parametrima točka infleksije kod pura je početkom, a kod purana sredinom 10. tjedna tova kada su teški 5, 27 kg (pure) i 7, 46 kg (purani). Na temelju rezultata primijenjenog modela (generalizirana logistička funkcija) u opisivanju rasta pura teškog hibrida može se istaknuti kako ženke završavaju intenzivnu fazu rasta skoro tjedan dana ranije nego mužjaci, a točka infleksije kod pura nastupa ranije nego kod purana.
61. D. Jukić, G. Kralik, R. Scitovski, Least squares fitting Gompertz curve, Journal of Computational and Applied Mathematics, 169/2 (2004), 359-375
    Abstract

    In this paper we consider the least-squares (LS) fitting of the Gompertz curve to the given nonconstant data (pi,ti,yi), i=1,…,m, m⩾3. We give necessary and sufficient conditions which guarantee the existence of the LS estimate, suggest a choice of a good initial approximation and give some numerical examples.
62. E.K. Nyarko, R. Scitovski, Solving the parameter identification problem of mathematical model using genetic algorithms, Applied mathematics and computation 153 (2004), 651-658
63. D. Jukić, R. Scitovski, Solution of the least-squares problem for logistic function, Journal of Computational and Applied Mathematics, 156/1 (2003), 159-177
    Abstract

    Given the data (pi,ti,fi), i=1,…,m, m>3, we consider the best least-squares approximation of parameters for the logistic function . We give necessary and sufficient conditions which guarantee the existence of such optimal parameters.
64. R. Scitovski, M. Meler, Solving Parameter Estimation Problem in New Product Diffusion Models, Applied mathematics and computation 127 (2002), 45-63
    Abstract

    In this paper we propose a new method for the parameter identification in new product diffusion models. The method can be used regardless of whether the analytical solution of the differential equation describing the model is known or not. This is the main advantage of our method over the known methods. Another advantage of this method is the simplicity of its implementation and negligible computing time. The method is based on finite differences method and the moving least squares method for data smoothing.
65. R. Scitovski, G. Kralik, G. Kušec, Modeli procjene maksimuma rasta kod provenijencije tovnih pilića, Poljoprivreda 7 (2001), 37-41
    Abstract

    Cilj rada bio je istražiti matematički model koji daje najbolje rezultate prilikom procjene biološkog maksimuma rasta pilića u tovu. Upotrebljeni su: opća eksponencijalna funkcija, Von Bertalanffy model, logistička funkcija i generalizirana logistička funkcija. U istraživanju specifičnosti rasta korištene su Avian 24 K, Arbor Acres i Ross 208 provenijencije tovnih pilića. Rezultati istraživanja pokazuju da najbolje podatke za procjenu biološkog maksimuma rasta pilića, s obzirom na spol, omogućava generalizirana logistička funkcija uz gama = 0, 18 (asimetrična S-funkcija) . Utvrđena vrijednost za asimptotu A kod muških pilića, prema naprijed navedenom redosljedu, iznosi: 5, 72 kg, 5, 90 kg i 4, 47 kg, a kod ženskih pilića 4, 06 kg, 4, 75 kg i 4, 46 kg. Mogućnost procjene težina pilića kod odvojitog tova prema spolu u određenoj dobi od velikog je značenja za optimalno iskorištavanje genetskog potencijala pilića i postizanje veće uniformnosti jata prilikom isporuke i klanja pilića.
66. R. Scitovski, D. Jukić, I. Urbiha, Solving the parameter identification problem by using $TL_p$ spline, Mathematical Communications - Supplement 1/1 (2001), 81-91
67. D. Jukić, R. Scitovski, The best least squares approximation problem for a 3-parametric exponential regression model, ANZIAM Journal 42/2 (2000), 254-266
    Abstract

    Given the data (pi, ti, fi), i = 1,…,m, we consider the existence problem for the best least squares approximation of parameters for the 3-parametric exponential regression model. This problem does not always have a solution. In this paper it is shown that this problem has a solution provided that the data are strongly increasing at the ends.
68. R. Scitovski, D. Jukić, Analysis of solutions of the least squares problem, Mathematical Communications 4/1 (1999), 53-61
    Abstract

    For the given data $(p_i,t_i,f_i),$ $i=1,ldots,m$, we consider the existence problem of the best parameter approximation of the exponential model function in the sense of ordinary least squares and total least squares. Results related to that problem which have been obtained and published by the authors so far are given in the paper, as well as some new results on nonuniqueness of the best parameter approximation.
69. G. Kralik, T. Jelen, R. Scitovski, G. Kušec, Analysis of phenotypic expression and growth of gilts using asymmetric S-function, Krmiva 41 (1999), 159-165
    Abstract

    Research on growth capacity of different genotypes of gilts was carried out at the station for performance test of gilts. Two groups of gilts, German Landrace and Swedish Landrace, 15 each were included in the experiment, while 3rd group contained 14 crossed gilts (Swedish Landrace x Large White). Using asymmetric S-function, differences were observed in growth characteristics between genotypes of gilts. Growth curves for gilts were depicted by following models: German Landrace f(t) Swedish Landrace f(t) Crosses (SL x LW) f(t) Pigs with higher average daily gain reach maximal growth rates (inflexion point) in earlier age, and stage of progressive growth lasts shorter time than in animals with lower average daily gain. Inflexion point was reach in 166.59 days in German Landrace gilts, 171.14 days in Swedish Landrace and 168.53 days in crossed gilts (SL x LW).
70. G. Kralik, G. Kušec, R. Scitovski, Z. Škrtić, A. Petričević, Growth and carcass quality of broilers, Czech J. Animal Science 44 (1999), 233-237
    Abstract

    This research was carried out on 115 Ross 208 female broilers. Data on live weights were collected from 91 broilers during 7 weeks of fattening which was the basis for modelling the growth curve using asymmetric S-function. The same function was used for modelling growth of Ross 208 female broilers on the basis of data obtained from technological procedure. The parameters of the model was established as well as the stages of growth and inflection point. By the model, 99.924 percent and 99.954 percent of variance was explained for live weights of female broilers from experiment and from technological procedure, respectively. This means that asymmetric S-function can be used as the model for description of broilers’ growth. Using this function it was possible to predict the future weights of boilers. It is also possible to predict the time in which broilers reach certain live weight. The birds from experiment would cease useful growth at weeks. Other 24 Ross 208 female broilers were slaughtered when they reached determined weights from 2200 to 3400 g in the period from 7th to 12th week. The data obtained by dissection of those broilers on the main parts (breast, thighs with drumstick, wings and back) were used for the analysis of relative growth by allometric equation. Growth rates (b) for the most valuable parts and abdominal fat were: breast - 1.158; thighs with drumstick - 0.869; wings - 0.725; back - 1.063 and fat - 1.519. Allometric coefficients indicate that only fat production was intensive in the studied period. This confirms the conclusion that the best age for the slaughter of broilers is that established by asymmetric S-function.
71. D. Jukić, R. Scitovski, H. Späth, Partial linearization of one class of the nonlinear total least squares problem by using the inverse model function, Computing 62/2 (1999), 163-178
    Abstract

    In this paper we consider a special nonlinear total least squares problem, where the model function is of the form f(x;a,b)=ϕ−1(ax+b) . Using the fact that after an appropriate substitution, the model function becomes linear in parameters, and that the symmetry preserves the distances, this nonlinear total least squares problem can be greatly simplified. In this paper we give the existence theorem, propose an efficient algorithm for searching the parameters and give some numerical examples.
72. R. Scitovski, Š. Ungar, D. Jukić, Approximating surfaces by moving total least squares method, Applied mathematics and computation 93/2-3 (1998), 219-232
    Abstract

    We suggest a method for generating a surface approximating the given data (xi, yi, zi) ϵ R^3, i = 1, …. m, assuming that the errors can occur both in the independent variables x and y, as well as in the dependent variable z. Our approach is based on the moving total least squares method, where the local approximants (local planes) are determined in the sense of total least squares. The parameters of the local approximants are obtained by finding the eigenvector, corresponding to the smallest eigenvalue of a certain symmetric matrix. To this end, we develop a procedure based on the inverse power method. The method is tested on several examples.
73. D. Jukić, T. Marošević, R. Scitovski, Discrete total lp-norm approximation problem for the exponential function, Applied mathematics and computation 94/2-3 (1998), 137-143
    Abstract

    In this paper we consider the total lp-norm (p > 0) approximation problem for the exponential function. We give sufficient conditions which guarantee the existence of such optimal parameters.
74. D. Jukić, R. Scitovski, Existence results for special nonlinear total least squares problem, Journal of Mathematical Analysis and Applications 226/2 (1998), 348-363
    Abstract

    In this paper we prove an existence theorem for a special nonlinear total least squares problem. We show that the optimal parameters of the generalized logistic function exist in the sense of total least squares, provided the data satisfy the Chebyshev's inequality.
75. R. Scitovski, Analysis of parameter identification problem, Applied mathematics and computation 82 (1997), 39-55
    Abstract

    We consider a special parameter identification problem occurring in marketing research, and give conditions on the empirical data which guarantee the existence of the solution to the problem. Furthermore, we analyze several different approaches and methods for solving this parameter identification problem.
76. D. Jukić, R. Scitovski, Existence of optimal solution for exponential model by least squares, Journal of Computational and Applied Mathematics, 78/2 (1997), 317-328
    Abstract

    In this paper we prove the existence theorem for the best least squares approximation of the optimal parameters for the exponential model function. We give sufficient conditions which guarantee the existence of such optimal parameters. Using these results and methods, we are able to localize a sufficiently narrow area where one can choose a good initial approximation.
77. R. Scitovski, Surface generating on the basis of experimental data, Mathematical Communications 2 (1997), 59-67
    Abstract

    The problem of surface generating on the basis of experimental data is presented in this lecture. Special attention is given to the implementation of moving ordinary least squares and moving total least squares. Some results done in the Institute for Applied Mathematics in Osijek are mentioned which were published in the last several years.
78. D. Jukić, R. Scitovski, The existence of the optimal parameters of the generalized logistic function, Applied mathematics and computation 77/2-3 (1996), 281-294
    Abstract

    The estimation of optimal parameters in a mathematical model described by the generalized logistic function with saturation level A and the asymmetry coefficient γ is a nonlinear least squares problem. In this paper we prove the existence of optimal parameters under considerably weaker conditions than those required in [1].
79. R. Scitovski, D. Jukić, A method for solving the parameter identification problem for ordinary differential equations of the second order, Applied mathematics and computation 74/2-3 (1996), 273-291
    Abstract

    We give a method for solving the parameter identification problem for ordinary differential equations of the second order using a noninterpolated moving least squares method. The method is tested in two practical examples.
80. R. Scitovski, Minima over an interval, The Mathematica Journal 6 (1996), 20-21
81. R. Scitovski, D. Jukić, Total least squares problem for exponential function, Inverse Problems 12/3 (1996), 341-349
    Abstract

    Given the data (p_i,t_i,f_i), , i = 1,...,m, we consider the existence problem for the optimal parameters for the exponential function approximating these data in the sense of total least squares. We give sufficient conditions which guarantee the existence of such optimal parameters.
82. R. Scitovski, A special nonlinear least squares problem, Journal of Computational and Applied Mathematics, 53 (1994), 323-331
    Abstract

    In this paper we consider the existence of the solution of a special nonlinear least-squares problem. We find the necessary conditions on the data, which insure the existence of the optimal parameters for the asymmetric S-function in the sense of the least squares.
83. B. Dukić, D. Francišković, D. Jukić, R. Scitovski, M. Benšić, Strategije otplate zajma, Financijska teorija i praksa (1994), 15-26
84. G. Kralik, R. Scitovski, Đ. Senčić, Application of Asymetric S-Function for analysis and Valuation of the Growth of Boars, Stočarstvo 47 (1993), 425-433
    Abstract

    Growth capacity of the pig is a subject of permanent research. The growth rate of pigs has different quantitative and qualitative course. Profitability of pig rearing is in use of progessive growth stage, i.e. phase. This study with the curves of growth of the Large White breed based on the gain intensity of the animals tested with the body weights 30 to 100 kg. According to the average daily gains the boars were devided into three groups (0.81-0.90 kg, 0.91-1.1 kg and 1.01 to 1.10 kg). For description of the growth rate asymmetric S-function was used modelled as: [f(t)=frac{; ; ; A}; ; ; {; ; ; /(1+be^{; ; ; -c gamma t}; ; ; )^{; ; ; 1/gamma}; ; ; }; ; ; .] Regarding the neceessary time for achievement of certain mass, the stages are separated as: preparations $(t=t_B)$, growth-both progressive and degressive $(t=t_B=t_C)$ and phase saturation $(t=t_C)$. The analysis showed that boars, entering the test in younger age, showed higher growth intensity than those beginning the test in older age (77.4 ; 83.4 and 82.6 days). In these boars the stage of progressive and degressive growth occurred earlier and lasted shorter period time. The asymmetric S-function could be therefore used for valuation of the boars growth rate in older age based on the starting control measurements of animals.
85. G. Kralik, R. Scitovski, Istraživanje značajki rasta brojlera pomoću asimetrične S-funkcije, Stočarstvo 47 (1993), 207-213
    Abstract

    U radu su istraživane fenotipske značajke rasta Arbor Acres pilića do 6. tjedna odnosno 10. tjedna. U istraživanja je uključeno po 30 pilića ženskog odnosno muškog spola. U modeliranju krivulje rasta uprotrebljena je asimetrična S-funkcija. Pomoću navedene funkcije utvrđene su karakteristične faza rasta i točke infleksije. Istraživanje je pokazalo da je asimetrična S-funkcija prikladna i za prognozu životne mase pilića do 10. tjedna.
86. R. Scitovski, V. Redžep, Primjena grafičkih mogućnosti PC računala prilikom rješavanja nelinearnih problema najmanjih kvadrata, Ekonomski vjesnik 2 (1989), 277-284
    Abstract

    U ekonomskim istraživanjima često je potrebno, na osnovu zadanih podataka i pretpostavljenog modela-funkcije, riješiti odgovarajući nelinearni problem najmanjih kvadrata. Ovakvi problemi nemaju uvijek rješenja, a ako su rješivi, tada numerički iterativni proces, kojim se rješavaju, znatno ovisi o izboru početne aproksimacije. Ovim radom želi se ukazati na neke grafičke mogućnosti primjene PC računala prilikom definiranja početne aproksimacije u ovim iterativnim procesima. Izborom različitih ekonomskih primjera prikazane su široke mogućnosti primjene ovakvih modela-funkcija u ekonomskim istraživanjima.
87. S. Kosanović, R. Scitovski, Matematičko modeliranje ekonomskih pojava koje teže zasićenju, Ekonomski vjesnik 1 (1988), 49-58
    Abstract

    U ekonomskim istraživanjimma često je potrebno neku pojavu opisati funkcijom čija se vrijednost asimptotski približava razini zasićenja. U radu se promatra klasa tzv. S-funkcija, koje služe za kratkoročne i dugoročne prognoze. Poseban naglasak dat je na analizu klasa funkcija, koje ovisno o parametru, mogu imati pozitivnu ili negativnu asimetriju ili se mogu preobraziti u poznatu logističku funkciju. Procjenjivanje parametara ovakvih funkcija je složen nelinearni problem najmanjih kvadrata, koji ne mora uvijek imati rješenje. Ekonomska interpretacija svojstava tih funkcija je izuzetno značajna za strategijsko upravljanje ekonomskim sistemima.
88. R. Scitovski, Some special nonlinear least squares problems, Sarajevo Journal of Mathematics 4 (1988), 279-298
    Abstract

    In this paper we consider the property of preponderant increase and decrease of data and the existence of the solution of a special nonlinear least squares problems. We find the necessary conditions on the data, which insure the existence of the optimal parameters for the exponential function in the sense of the least squares. It is also shown how to choose the initial approximation to ge optimal parameters with some of the known iterative methods.
89. R. Scitovski, Ispodgodišnje ukamaćivanje, Economic analysis and worker's management 21 (1987), 243-257
90. R. Scitovski, S. Kosanović, Rate of change in economics research, Economic analysis and worker's management 19 (1985), 65-75
    Abstract

    Certain economic phenomena are often considered in research through rate of change. Rates of change represent information which enable researchers to come to conclusions about tendencies and the intensities of observed phenomena. In applications the computation of rate of change is mostly carried out by using the geometric mean of ratios of data. The basic disadvantage of such an approach is the elimination of the influence of all data (except two of them), because the conclusions is carried out on the basis of the first and last datum. Besides, using the geometric mean, the rate of change can be computed only for the equidistant distributed data, and this also represents a limitation in its application. This paper points out the disadvantages of such an approach, introduces precise definition of the rate of change and proposes the manner in which this important economic inductor can be more accurately determined.
91. R. Scitovski, Searching method and existence of solution of special nolinear least squares problems, Glasnik matematički 20 (1985), 451-467
    Abstract

    In the paper the nonlinear least squares problems for the logistic function (0.3) and the generalized logistic function (0.4) are considered. First, the conditions for the existence of optimal parameters a and c of logistic function are determined, and then an algorithm is proposed for determination of optimal parameters for the generalized logistic function (0.4). The proposed algorithms are compared with the known nonlinear least squares methods.
92. R. Scitovski, Kvantitativna analiza vremenskih nizova s obzirom na periodične utjecaje, Privreda 27 (1983), 69-73
93. R. Scitovski, Nelinearni problemi najmanjih kvadrata gdje se varijable mogu separirati, Statistička revija 32 (1982), 221-232
    Abstract

    U članku se opisuje pristup rješavanju nelinearnih problema najamnjih kavdrata za funkciju koja se može prikazati kao linearna kombinacija nelinearnih funkcija. Posebno je obrađen slučaj kada su te nelinearne funkcije istog tipa. Predložen je efikasan algoritam za rješavanje ovakvih problema najmanjih kvadrata. U primjeru na kraju članka prikazano je rješavanje problema najmanjih kvadrata za periodičnu funkciju i ukazano na mogućnost analize periodičnih utjecaja u praktičnim istraživanjima.
94. S. Kosanović, M. Medić, M. Meler, R. Scitovski, S. Šandrk, Operacionalizacija kategorija i mjerila za raspodjelu zajedničkog prihoda u nekoj reprodukcijskoj cjelini na osnovama Zakona o udruženom radu, Privreda 10 (1981), 36-45
95. B. Schön, R. Scitovski, Prognoza broja stanovnika i prirodnog priraštaja SFRJ, Statistička revija 31 (1981), 1-4
    Abstract

    U članku je primjenjena Newtonova iterativna metoda na prognozu broja stanovnika SFRJ pomoću logističke funkcije f(t ; A, b, c)=A/(1+be^{;-ct};).
96. R. Scitovski, Linearne aproksimacije u smislu metode najmanjih kvadrata, Statistička revija 30 (1980), 207-220
    Abstract

    U ovom radu razmatraju se linearne model funkcije kao linearne kombinacije potencija, trigonometrijskih funkcija ili eksponencijalnih funkcija. Pokazano je pod kojim uvjetima se koeficijenti ovih funkcija na jedinstven način mogu odrediti i kako se pri tome može primjeniti metoda najmanjih kvadrata na odgovarajući empirijski statistički materijal. Također priložen je odgovarajući FORTRAN kod.
97. B. Schön, R. Scitovski, Pokušaj prilagođavanja logističke funkcije zadanom empirijskom statističkom materijalu, Statistička revija 27 (1977), 178-194

1. S. Scitovski, R. Scitovski, Cluster analysis of the data on unit circle, 1st Virtual International Conference in Advanced Research in Scientific Areas, Slovakia, 2012, 1574-1577
   Abstract

   The problem of data clustering on the unit circle is considered in this paper. For that purpose, the metric on the unit circle is introduced and the problem is reduced to a one-dimensional center-based clustering problem. For solving this problem, an efficient method is proposed. The method is illustrated on seismic activity data from a wider area of the town of Dubrovnik in the Republic of Croatia since 1973. In this way, the moments in the year with most intensive seismic activity in this area are detected.
2. G. Kralik, K. Sabo, R. Scitovski, I. Vazler, Solving parameter identification problem by the moving least absolute deviations method, 12th International Conference on Operational Research, Pula, Croatia, 2010, 297-307
   Abstract

   On the basis of measured data, among which a significant number of outliers might appear, we introduce one method for parameter identification in a mathematical model given by the ordinary differential equation of the first order. The method consists of two steps. In the first step, we construct a smooth function by applying the moving least absolute deviations method. In the second step, by applying the least absolute deviations method we estimate unknown parameters of mathematical models. The method is applied to and tested on the problem of estimating saturation level and asymmetry coefficient in the mathematical model with saturation. The mathematical model described by a generalized Verhulst differential equation [frac{;dy(t)};{;dt};= c, y(t)left(1-left(frac{;y(t)};{;A};right)^gammaright), quad c, , gamma, , A > 0, ] is considered especially. In this case the parameter estimation problem is reduced to the nonlinear least absolute deviations problem for a 3-parametric exponential regression model. For solving this problem an efficient method is developed. The method is tested on real measurement data of weights of 60 pigs in the period of 26 weeks.
3. I. Kuzmanović Ivičić, G. Kušec, K. Sabo, R. Scitovski, A new method for searching an L_1 solution of an overdetermined system of linear equations and applications, 12th International Conference on Operational Research, Pula, Croatia, 2008, 309-319
4. I. Kuzmanović Ivičić, R. Scitovski, K. Sabo, I. Vazler, The least absolute deviation linear regression: properties and two efficient methods, Aplimat 2008, Bratislava, 2008, 227-240
5. B. Dukić, R. Scitovski, Analiza učinaka jednostavnog i složenog ispodgodišnjeg ukamaćivanja kod obračuna zakonskih zateznih kamata u RH, 10th International Conference on Operational Research KOI 2004, Trogir, 2005, 249-259
   Abstract

   Potaknuta problemima velikih potraživanja vjerovnika od dužnika po sudskim presudama u ovršnom postupku Vlada RH je 30. travnja 2004. godine donijela Vjerodostojno tumačenje čl. 3. st. 1. Zakona o zateznim kamatama objavljenog u NN br. 28/96 kojim je praktično aplicirala primjenu metode jednostavnog ukamaćivanja na potraživanja dulja od godinu dana. S ovakvom mjerom, uz ZOO-om propisani nominalistički pristup, kada se radi o potraživanjima koja su podložna monetarnim promjenama koje su se zbivale krajem 80-tih i početkom 90-tih godina prošlog stoljeća, što je analizom i utvrđeno, Vlada je praktično dovela vjerovnike u vrlo težak položaj obezvrjeđujući njihova potraživanja u ovršnom postupku. Sagledavajući i tumačeći efekte primjene metoda jednostavnog i složenog ispodgodišnjeg ukamaćivanja, u ovom istraživanju pokušat će se ukazati na bit problema, te sugerirati rješenje za otklanjanje problema koje kumulira navedeno Vjerodostojno tumačenje. Za potrebe analize načinjeni su algoritmi koji su aplicirani u vidu računalnih programa kojima je testiran učinak jednostavnog i složenog ispodgodišnjeg ukamaćivanja.
6. D. Jukić, R. Scitovski, A. Baumgartner, K. Sabo, Localization of the least squares estimate for two-parametric regression models, 10th International Conference on Operational Research KOI 2004, Trogir, 2005, 165-174
7. G. Kralik, S. Ivanković, R. Scitovski, Effects of dietary PBE oil on performance of broilers, XXII World's Poultry Congress, Istambul, 2004, 1-4
   Abstract

   The research was carried out on 225 one-day-old Ross 208 male broilers, who were divided into five groups. Chickens in each group were fed from 1st to 21st day with starter mixture that contained 22.67 % of crude proteins and 14.19 MJ/kg ME. From 22nd to 42nd day, chickens were fed with finisher mixtures containing 20.43% of crude proteins and 14.18 MJ/kg ME. Besides basic feedstuffs such as corn, soybean meal, fish meal, vitamin-mineral additive and methionine, finisher compositions differed in the content of poultry fat and PBE preparation (5.0%/0%, 4.5%/0.5%, 4.0%/1.0%, 3.5%/1.5% and 3.0%/2.0%). Fattening process lasted for 42 days, after which the chickens reach weight of 2479.52 +/- 123.31, 2369.76 +/- 204.96 g, 2175.76 +/- 191.99 g, 2210.68 +/- 135.93 g and 2250.00 +/- 108.77 g, respectively. Chicken growth during the fattening period is described by asymmetric S-function. According to the parameters of asymmetric S-function, it can be concluded that the larger portions of PBE preparation in mixtures stimulate the intensification of the progressive growth stage, but it finally results in decreased live chicken weight. Biological maximum of growth by chicken groups was 3.80 kg, 3.53 kg, 3.10 kg, 3.29 kg and 3.29 kg. PUFA n-6/PUFA n-3 ratio in the lipids of breast muscles was 2.95 +/- 0.48, 2.68 +/- 0.71, 1.93 +/- 0.40, 1.63 +/- 0.41 and 1.22 +/- 0.40, and in the lipids of thigh muscles 5.35 +/- 0.81, 4.05 +/- 0.68, 2.96 +/- 0.47, 2.10 +/- 0.28 and 1.60 +/- 0.37. Research results showed that the larger percentage of PBE preparation in mixtures, as a supplement for the poultry fat, effects on decrease of live chicken weight growth, but it creates more favourable PUFA n-6/PUFA n-3 ratio in the lipids of muscular tissue.
8. G. Kralik, S. Ivanković, R. Scitovski, Growth characteristics and performances of male broilers, XVIth European Symposium on the Quality of Poultry Meat, Saint-Brieuc, Francuska, 2003, 178-184
   Abstract

   This research aimed at investigation of growth phases of male broilers and establishment of qualitative characteristics of carcasses, as well as of chemical content of breast muscles and thighs. The research was carried out on 45 male broilers of Ross 208 provenience, which were divided into 3 groups. Broilers were fed with starter and finisher diets containing 22.67%/20.43% of crude proteins and 14.19 MJ/kg/14.18 MJ/kg ME, respectively. After 35 days of fattening broilers gained weight of 2065.75 ± ; ; ; 100.36 g. Asymmetric S-function was used for estimation of the growth phases, and parameters of the used model were the following: A=3.8 kg, b=1.41969, c=2.69601 and g=0.18. Average weight of carcasses was 1815± ; ; ; 25 g. Portions in the whole carcass were as follows: breasts 33.75%, thighs with drumsticks 30.35%, wings 11.15% and backs 22.40%. Breast muscles differed statistically (P<0.05) than thigh muscles in the content of proteins (23.03% : 19.42%), fats (1.32% : 4.51%) and ashes (1.23% : 1.10%). Breast muscles contained more SFA (39.48% : 33.83%) and PUFA n-3 (6.02% : 3.71%), but less MUFA (29.61% : 35.63%) and PUFA n-6 (17.70% : 19.85%) when compared to the thigh muscles. Ratio PUFA n-6 / PUFA n-3 was better in the breast muscles than in the thighs.
9. R. Scitovski, I. Kuzmanović Ivičić, Jedna metoda procjene parametara u smislu minimizacije sume L_p ortogonalnih udaljenosti, Programski sustav Mathematica u znanosti, tehnologiji i obrazovanju. PrimMath[2003]. , Zagreb, Hrvatska, 2003, 147-157
10. D. Jukić, R. Scitovski, K. Sabo, Total least squares problem for the Hubbert function, Conference on Applied Mathematics and Scientific Computing, Brijuni, 2003, 217-234
11. R. Scitovski, G. Kralik, D. Jukić, R. Galić, Estimation of the saturation level and asymmetry coefficient of the generalized logistic model, 9th International Conference on Operational Research KOI 2002, Trogir, 2002, 57-66
12. R. Scitovski, LU dekompozicija trodijagonalne matrice i primjena na konstrukciju spline-a, Prvi znanstveno-stručni skup: Programski sustav Mathematica u znanosti, tehnologiji i obrazovanju, Zagreb, Hrvatska, 2002, 245-265
    Abstract

    Razrađen je program načinjen u Mathematici za konstrukciju spline-a pri čemu se koristi LU dekompozicija trodijagonalne vrpčaste matrice.
13. R. Scitovski, Smoothing the data method for solving parameter identification problem, Bosnian-Croatian Analysis Meeting, Bihać, BiH, 2001
    Abstract

    We consider the smoothing a data method for solving the parameter identification problem in the mathematical model described by one differential equation $frac{;dy};{;dt};=f(t, y(t), p)$, where $p=(p_1, ldots, p_n)^T$ is the vector of unknown parameters. We assume that the solution of the differential equation cannot be represented by elementary functions. Also, we assume that the additive errors can appear in the measured values of the independent variable $t$ and in the measured values of the dependent variable $y$, as well as among the data so-called outliers can appear. Therefore, for smoothing the data will use the so-called total $L_1$ natural cubic spline, i.e. in general, the total $L_p$, ($pgeq 1$) natural cubic spline. For the construction of the best total $L_p$ natural cubic spline we use a new approach of our own. The method will be shown on several illustrative examples.
14. D. Jukić, R. Scitovski, Š. Ungar, The best total least squares line in R^3, 7th International Conference on Operational Research KOI 1998, Rovinj, 1998, 311-316
15. R. Scitovski, R. Šalić, K. Petrić, K. Sabo, Optimal allocation of nodes for surface generating, 19th International Conference ITI 1997, Pula, Hrvatska, 1997, 383-408
    Abstract

    The problem of generating a smooth surface on the basis of experimental data is considered in this paper. Special attention is given to the problem of the optimal number and optimal allocation of nodes at which local approximants will be generated. Therefore, it is possible to generate local paraboloids at chosen nodes instead of local planes, thereby even reducing the computing time, and having the obtained surface sufficiently smooth.
16. D. Jukić, M. Crnjac, R. Scitovski, Surfaces generated by the noninterpolating moving least squares method, 4th International Symposium on Operations Research in Slovenia SOR’97, Preddvor, 1997, 189-194
17. R. Scitovski, Š. Ungar, D. Jukić, M. Crnjac, Moving total least squares for parameter identification in mathematical model, Symposium on Operations Research SOR '95, Passau, 1996, 196-201
18. M. Benšić, R. Scitovski, Određivanje intervala povjerenja za neke specijalne nelinearne regresije, 6th International Conference on Operational Research KOI 1996, Rovinj, 1996, 87-92
    Abstract

    U radu se analiziraju rezultati primjene metode linearizacije na određivanje intervala i područja povjerenja za nepoznate parametre logističke i 3-parametarske eksponencijalne regresije.
19. R. Scitovski, M. Meler, R. Galić, Solving parameter identification problem in new product diffusion models, 6th International Conference on Operational Research KOI 1996, Rovinj, 1996, 113-118
    Abstract

    Solving the parameter identification problems in applied research is usually an unstable numerical process. Special (cumulative) structure of the data in new product diffusion models gives the possibility of defining an efficient method for solving the parameter identification problem in this case. The method can be used regardless of whether the analytical solution of the differential equation describing the model is known or not. Another advantage of this method is the simplicity of its implementation and negligible computing time.
20. D. Jukić, R. Scitovski, The exponential growth model, 6th International Conference on Operational Research KOI 1996, Rovinj, 1996, 107-112
21. R. Scitovski, D. Jukić, I. Bašić, Application of the moving least squares method in surface generating, 17th Int. Conf. Information Technology Interfaces, Cavtat, 1995, 469-474
22. R. Scitovski, T. Marošević, D. Jukić, Estimation of the optimal initial conditions in mathematical model, 17th Int. Conf. Information Technology Interfaces, Cavtat, 1995, 475-480
23. M. Meler, R. Scitovski, Matematički modeli difuzije novog proizvoda, 5th Conference on Operational Research KOI 1995, Rab, 1995, 194-203
    Abstract

    U radu se analiziraju neki poznati modeli difuzije novog proizvoda u odnosu na pretpostavke o kretanju stope rasta difuzije. Na toj osnovi predlaže se jedan novi model difuzije novog proizvoda. Za navedene modele analiziraju se također broj i vrsta potrošačkih kategorija.
24. D. Jukić, R. Scitovski, M. Crnjac, Primjena metode potpunih najmanjih kvadrata za procjenu parametara u matematičkom modelu, 5th Conference on Operational Research KOI 1995, Rab, 1995, 99-110
25. R. Galić, R. Scitovski, T. Marošević, D. Jukić, Problem optimalnih početnih uvjeta u matematičkom modelu, 5th Conference on Operational Research KOI 1995, Rab, 1995, 62-71
26. R. Scitovski, R. Galić, I. Kolund, I. Bašić, D. Jukić, Procjena rasprostiranja slojeva po dubini sondažnog profila, 5th Conference on Operational Research KOI 1995, Rab, 1995, 111-120
27. M. Crnjac, D. Jukić, R. Scitovski, Nove strategije otplate zajma, 4th Conference on Operational Research KOI 1994, Rab, 1994, 147-154
28. R. Galić, R. Scitovski, T. Marošević, Primjena pomične metode najmanjih kvadrata za rješavanje problema identifikacije parametara u matematičkom modelu, 4th Conference on Operational Research KOI 1994, Rab, 1994, 181-191
29. D. Fischer, F. Jović, R. Scitovski, Feasibility of Tools for Solving a Linear Over-determined System, 15th International Conference ITI 1993, Pula, Hrvatska, 1993, 315-319
    Abstract

    Over-determined linear system is presented. Four types of feasible solutions are given: mean value of all determined subsystems, mean value of determined subsystem which solutions are in the confidence interval, minimum square values for absolute differences and minimum relative square values of all equations. The feasibility of solutions is discussed.
30. R. Scitovski, R. Galić, Jedna generalizacija zakona rasta, 3rd Conference on Operational Research KOI 1993, Rovinj, Hrvatska, 1993, 313-322
    Abstract

    U radu razmatramo jednu generalizaciju zakona rasta sa zasićenjem. Pokazano je kako se na osnovi zadanih podatka, primjenom Eulerove metode za rješavanje diferencijalnih jednadžbi i metode najmanjih kvadrata mogu dobiti optimalni parametri u modelu.
31. S. Kosanović, R. Scitovski, Ž. Turkalj, Nelinearni problemi najmanjih kvadrata u poljoprivredi, Primjena kompjutora u poljoprivredi, Osijek, 1990, 246-250
    Abstract

    U radu su prikazani matematički modeli koji služe za opisivanje ponašanja prirodnih zakona u različitim primjenjenim istraživanjima. Matematički gledano, to su nelinearni problemi najmanjih kvadrata. Korisnički program koji podržava ove modele omogućava raznovrsnu primjenu u poljoprivredi, kao i u drugim područjima.
32. Ž. Turkalj, R. Scitovski, T. Jagnjić, Primjena kompjutora u poljoprivredi, Primjena kompjutora u poljoprivredi, Osijek, 1990, 205-209
    Abstract

    U radu je prikazan model planiranja i kontrole tehnološkog procesa u mesnim industrijama. Na osnovu zadane količine poznate ulazne sirovine i zadanog načina rasjecanja, a iz poznatih tehnoloških normativa ustanoviti konačno dobivene vrste i količine pojedinih finalnih proizvoda rasjeka.
33. Ž. Turkalj, R. Scitovski, T. Jagnjić, Primjena PC računala u optimizaciji proizvodnje mesne industrije, Primjena kompjutora u poljoprivredi, Osijek, 1990, 210-216
    Abstract

    U radu je prikazan model optimalnog proizvodnog programa mesne industrije na osnovi kojeg je izrađen i odgovarajući korisnički software. Model ne daje kompleksno rješenje za definiranje optimalnog proizvodnog programa u mesnoj industriji, ali može predstavljati važan instrument u donošenju brzih poslovnih odluka.
34. R. Scitovski, Problemi najmanjih kvadrata u primjenama, Transfer razvojnog znanja u regionalna središta, Osijek, Hrvatska, 1989, 141-148
    Abstract

    U radu se razmatraju problemi najmanjih kvadrata. Dan je pregled matematičkih modela koji se najčešće javljaju u različitim znanstvenim područjima. Ukazano je na najvažnije metode za rješavanje i osnovne uvjete vezane uz egzistenciju rješenja ovih problema. Na kraju je detaljnije analizirana jedna klasa matematičkih modela koji se javljaju u širokom području znanstvenih istraživanja.
35. R. Scitovski, Condition of Preponderant Increase and Tchebycheff's Inequality, VI. Conference on Applied Mathematics, Tara, Yugoslavia, 1988, 189-194
    Abstract

    In searching the conditions on the data $(p_i, t_i, f_i)$, $i=1, ldots, m$, which ensure the solvability of some nonlinear least squares problem, the conditions of the so called preponderant increase often appers. In this paper we show that this condition is equivalent to a certain Tchebycheff's inequality. The class of data with the property of prenderant incerase is analysed in order to get the solvability of some special least square problems.
36. R. Scitovski, Das korrekte Verfahren bei unterjährlicher Verzinsung, X Symposium Pforzheim, Osijek, 1987, 12-19
37. R. Scitovski, Die Magnetisierungskurve, II Internationales wissenschaftliches Kolloquium ”Wissenschaft für die Praxis”, Bremen, 1986, 47-53
38. R. Scitovski, Nelinearna optimizacija, Zbornik radova nastavnika Studija elektrostrojarstva 1979-1984, Osijek, 1985, 105-120
39. R. Scitovski, An Approach in Solving Nonlinear Least Squares Problems, IV.Conference on Applied Mathematics, Split, Hrvatska, 1984, 109-113
    Abstract

    A new iterative method for solving nonlinear least squares problems is presented in this paper. It is shown that step direction vector is always acceptable. On examples from reality this method has been proved to be more efficient than other similar methods.
40. R. Scitovski, Egzistencija najbolje aproksimacije u smislu metode najmanjih kvadrata pomoću klase funkcija (t; b; c) ->b exp(-ct), XIV godišnji sastanak Saveza statističkih društava Jugoslavije, Novi Sad, Srbija, 1981

1. P. Corn, R. Scitovski, Jedan problem globalne optimizacije, Osječki matematički list 14/2 (2014), 93-103
   Abstract

   U radu se promatra sljedeći problem globalne optimizacije [ {argminlimits_{ainmathbb{R}^n}F(a)},quad F(a)=intlimits_0^{+infty}e^{-x}left(1+a_1x+cdots+a_nx^nright)^2dx. ] Pokazano je da ovaj problem ima jedinstveno rješenje, koje se može odrediti rješavanjem odgovarajućeg problema najmanjih kvadrata ili kao specijalni slučaj jednog općenitijeg problema najbolje aproksimacije u unitarnom vektorskom prostoru. U drugom slučaju primijenjeni su Laguerrovi ortogonalni polinomi. Rješavanje problema ilustrirano je s nekoliko numeričkih primjera.
2. K. Sabo, R. Scitovski, I. Vazler, Grupiranje podataka: klasteri, Osječki matematički list 10 (2010), 149-176
   Abstract

   U ovom radu razmatramo problem grupiranja elemenata skupa A u disjunktne neprazne podskupove - klastere, pri cemu pretpostavljamo da su elementi skupa A odreženi s jednim ili dva obilježja. Za rješavanje problema koristi se kriterij najmanjih kvadrata te kriterij najmanjih apsolutnih udaljenosti. Naveden je niz primjera koji ilustriraju razlike mežu tim kriterijima. Izražena je odgovarajuca programska podrška s ciljem da zainteresirani strucnjaci u svom znanstvenom ili strucnom radu mogu olakšano koristiti ovu metodologiju i pristup.
3. K. Sabo, R. Scitovski, Prosti brojevi, Osječki matematički list 3 (2003), 13-20
   Abstract

   U članku se opisuju neka važna svojstva prostih brojeva. Pored danih primjera i zadataka, navodi se i nekoliko neriješenih problema vezanih uz proste brojeve.
4. D. Vukojević, R. Scitovski, Matematika na šahovskoj ploči, Matka 4 (1996), 17-25
   Abstract

   U članku se navode neke zanimljive pravilnosti i matematička svojstva vezana uz šahovsku ploču i samu igru, kao primjerice magični kvadrati na šahovskoj ploči, dokaz Pitagorinog teorema na šahovskoj ploči, Pascalov trokut itd.
5. D. Galić, R. Scitovski, Neka geometrijska svojstva balističkih parabola u vakuumu, Matematičko fizički list 46 (1996), 129-133
   Abstract

   Promatra se najjednostavniji problem eksterne balistike za vakuum, bez utjecaja specijalnih efekata (promjena sile teže o visini, rotacija Zemlje, Coriolisova sila, utjecaj Zemljinog magnetskog polja itd.). Daju se neka svojstva sustava balističkih parabola u slučaju ako je početna brzina konstantna a varira se izlazni kut ili ako je izlazni kut konstantan, a varira se početna brzina.
6. R. Scitovski, K. Scitovski, Nemoguće figure, Matka 3 (1995)
   Abstract

   U radu se opisuju i daju konstrukcije nekih osnovnih nemogućih figura, primjerice nemogućeg trokuta.
7. R. Scitovski, Ž. Turkalj, T. Jagnjić, B. Guljaš, V. Redžep, Definiranje optimalnog proizvodnog programa mesne industrije, Ekonomski vjesnik 2 (1989), 309-316
   Abstract

   Brze promjene na tržištu proizvoda od mesa, s jedne strane, te široka primjena personalnih računala i njihovo prisustvo u gotovo svim radnim organizacijama, s druge strane, motivaciju su za pokušaj izrade matematičkog modela na osnovu kojeg bi se u glavnim pravcima definirao optimalni proizvodni program mesne industrije. Model je koncipiran tako da ga je moguće realizirati na PC računalu. U tom smislu, od ovog modela ne treba očekivati kompletan prijedlog za definiranje optimalnog proizvodnog programa, već ga treba shvatiti kao instrument za donošenje poslovnih odluka.
8. R. Scitovski, M. Šilac, D. Francišković, Problemi i nesporazumi u primjeni financijske matematike, Privreda 33 (1989), 243-257
9. R. Scitovski, Krivulja magnetiziranja, Elektrotehnika 31 (1988), 31-34
   Abstract

   U radu je pokazano kako se može odrediti krivulja magnetiziranja nekoga magnetskog materijala. Pritom je posebna pažnja posvećena dobivanju što jednostavnijega analitičkog izraza odgovarajuće funkcije, što je važna pretpostavka prilikom korištenja u projektiranju i u analizi rada električnih strojeva primjenom magnetskih ekvivalentnih shema.

1. R. Scitovski, K. Sabo, F. Martínez-Álvarez, Š. Ungar, Cluster Analysis and Applications, Springer, Cham, 2021.
   Abstract

   Clear and precise definitions of basic concepts and notions in clustering, and analysis of their properties. Analysis and implementation of most important methods for searching for optimal partitions. Covers different primitives in clustering, such as points, lines, multiple lines, circles, and ellipses. A new efficient principle of choosing optimal partitions with the most appropriate number of clusters. Detailed description and analysis of several important applications.
2. R. Scitovski, K. Sabo, Klaster analiza i prepoznavanje geometrijskih objekata, Sveucilište Josipa Jurja Strossmayera u Osijeku, Odjel za matematiku, Osijek, 2020.
3. R. Scitovski, K. Sabo, D. Grahovac, Globalna optimizacija, Sveučilište Josipa Jurja Strossmayera u Osijeku, Odjel za matematiku, Osijek, 2017.
4. D. Jukić, R. Scitovski, Matematika I, Sveučilište Josipa Jurja Strossmayera u Osijeku, Odjel za matematiku, Osijek, 2017.
5. R. Scitovski, M. Briš Alić, Grupiranje podataka, Sveučilište Josipa Jurja Strossmayera u Osijeku, Odjel za matematiku, Osijek, 2016.
6. R. Scitovski, Numerička matematika - izmijenjeno i dopunjeno izdanje, Sveučilište Josipa Jurja Strossmayera u Osijeku, Odjel za matematiku, Osijek, 2015.
   Abstract

   Sadržaj: 1. Pogreške 2. Interpolacija. Spline interpolacija 3. Rješavanje sustava linearnih jednadžbi 4. Rješavanje nelinearnih jednadžbi 5. Aproksimacija funkcija 6. Problemi najmanjih kvadrata 7. Numerička integracija 8. Numeričko rješavanje običnih diferencijalnih jednadžbi 9. Mathematica-moduli
7. R. Scitovski, N. Truhar, Z. Tomljanović, Metode optimizacije, Svučilište Josipa Jurja Strossmayera u Osijeku, Odjel za matematiku, Osijek, 2014.
   Abstract

   Namjena ovog teksta je upoznati čitatelja s glavnim metodama jednodimenzionalne i višedimenzionalne minimizacije sa i bez ograničnja. Posebna je pozornost posvećena metodama minimizacije nediferencijabilnih funkcija. Pri tome je izbjegavano dokazivanje zahtjevnih teorema, osim u slučajevima konstruktivnih dokaza koji sami po sebi upućuju na izgradnju ideja ili metoda. Navedeni optimizacijski problemi imaju veliku primjenu u raznim dijelovima života. Na primjer, često se javljaju problemi poput optimalnog oblikovanja odredenih mehaničkih sustava (oblikovanje dijelova automobilskih motora, nosivih konstrukcija u gradjevinarstvu, . . .), problem modeliranja ponašsanja tržišta, problemi iz teorije upravljanja (smirivanje sustava, optimalno upravljanje, . . . ) i mnogi drugi. Upravo činjenica da se razni problemi optimizicije pojavljuju u raznim dijelovima ljudske djelatnosti osigurava ovom tekstu široku primjenu.
8. R. Scitovski, Numerička matematika, Sveučilište Josipa Jurja Strossmayera u Osijeku, Elektrotehnički fakultet, Osijek, 1999.
   Abstract

   Sadržaj: 1. Pogreške. 2. Interpolacija. Spline interpolacija. 3. Rješavanje sustava linearnih jednadžbi. 4. Rješavanje nelinearnih jednadžbi. 5. Aproksimacija funkcija. 6. Problemi najmanjih kvadrata. 7. Numerička integracija. 8. Numeričko rješavanje običnih diferencijalnih jednadžbi. 9. Numeričko rješavanje parcijalnih diferencijalnih jednadžbi.
9. M. Crnjac, D. Jukić, R. Scitovski, Matematika, Sveučilište Josipa Jurja Strossmayera u Osijeku, Ekonomski fakultet u Osijeku, Osijek, 1994.
10. R. Scitovski, R. Galić, M. Šilac-Benšić, Numerička analiza. Vjerojatnost i statistika, Elektrotehnički fakultet Osijek, Osijek, 1993.
    Abstract

    Sažetak: Greške. Rješavanje jednadžbe f(x)=0. Interpolacija. Spline aproksimacije. Rješavanje sustava linearnih jednadžbi. Problemi najmanjih kvaddrata. Numerička integracija. Numeričko rješavanje običnih diferencijalnih jednadžbi. Kombinatorika. Vjerojatnost događaja. Slučajne varijable. Diskretna slučajna varijabla. Kontinuirana slučajna varijabla. Višedimenzionalna slučajna varijabla. Granični teoremi slučajne vjerojatnosti. Uzorak. Procjenjivanje parametara. Testiranje statističkih hipoteza.

1. R. Scitovski, K. Sabo, D. Grahovac, Globalna optimizacija (2017)

Projects

• 1986 – 1990 – head of project task (2.08.01.03.02) „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 6^th-10^th 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 2^nd (Zagreb, 2000), 3^rd (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

Refereeing/Reviewing

Refereeing:

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

Teaching

Nastavne aktivnosti u zimskom semestru Akademske 2019./2020.

• Linearna algebra I
• Matematički praktikum

Nastavne aktivnosti u ljetnom semestru Akademske 2019./2020.

• Grupiranje podataka i primjene
• Grupiranje podataka: pristupi, metode i primjene

Konzultacije (Office Hours): Srijeda (Wed) 12:00am.