Journal Publications

 

[24.] K. Sabo,   Center-based $l_1$-clustering method,   International Journal of Applied Mathematics and Computer Science, 24(2014), 151-163

ABSTRACT:  In this paper, we consider the $l_1$-clustering problem for a data-points set $\mathcal{A}=\{a^i\in\R^n\colon i=1,\dots,m\}$ which should be partitioned into $k$ disjoint nonempty subsets $\pi_1,\dots,\pi_k$, $1\leq k\leq m$. In that case, the objective function does not have to be either convex or differentiable and generally it may have many local or global minima. Therefore, it becomes a complex global optimization problem.A method for searching for a locally optimal solution is proposed in the paper, convergence of the corresponding iterative process is proved and a corresponding algorithm is also given.The method is illustrated by and compared with some other clustering methods, especially with the $l_2-$clustering method, which is also known in literature as a smooth $k-$means method, on a few typical situations, such as the presence of outliers among the data and clustering of incomplete data. Numerical experiments show in this case that the proposed $l_1-$clustering algorithm is faster and gives significantly better results than the $l_2-$clustering algorithm.

 

[23.] 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 $\mathcal{A}\subset\R^n$ 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.

 

[22.] K. Sabo, R. Scitovski,  Interpretation and optimization of the k-means algorithm,   Applications of Mathematics, 2013

ABSTRACT:  The paper gives a new interpretation and a possible optimization of the well-known $k$-means algorithm for searching for a locally optimal partition of the set $\mathcal{A}=\{a_i\in\R^n:i=1,\dots,m\}$ which consists of $k$ disjoint nonempty subsets $\pi_1,\dots,\pi_k$, $1\leq k\leq m$. For this purpose, a new \emph{divided $k$-means algorithm} was constructed as a limit case of the known \emph{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 \emph{divided $k$-means algorithm} there are data points lying in the Voronoi diagram, it is shown that the obtained result can be improved further.

[21.] K. Sabo, R. Scitovski, I. Vazler,   One-dimensional center-based $l_1$-clustering method,     Optimization Letters, 7 (2013), 5-22

cluster1ABSTRACT:  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).
 		  

          

[20.] R. Grbić, K. Scitovski, K. Sabo, R. Scitovski,  Approximating surfaces by the moving least absolute deviations method,  Applied Mathematics and Computations, 219(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     

 

[19.] T. Marošević, K. Sabo, P. Taler,  A mathematical model for uniform distribution voters per constituencies,   Croatian Operational Research Review,  4(2012); 53-64.

ABSTRACT:  This paper presents two different approaches on the basis of which it is possible to generate constituencies. The first one is based on
cluster analysis by means of which one can get compact constituencies having an approximately equal number of voters. An optimal number of constituencies can be obtained by using this method. The second approach is based on partitioning the country to several areas with respect to territorial integrity of bigger administrative units. Natural units obtained in this way will represent constituencies which do not necessarily have to have an approximately equal number of voters. Each constituency is associated with a number of representatives that is proportional to its number of voters, so the problem is reduced to the integer approximation problem. Finally, these two approaches are combined and applied on the Republic of Croatia.

 

  [18.] 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 Tierzucht / Archives Animal Breeding, 55(2012), 4, 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.
[17.] K. Sabo, R. Scitovski, P. Taler, Uniform distribution of the number of voters per constituency on the basis of a mathematical model, Hrvatska i komparativna javna uprava, (14) 2012, 229-249
izborneABSTRACT:  This paper presents a mathematical model on the basis of which it is possible to define maximum compact well-separated constituencies, which can vary with respect to the number of voters by at most 5%. The model is set up so as not to favor any political option and it is  based on the application of cluster analysis, taking into account the rule according to which constituencies should have roughly the same number of voters. The method is illustrated on the example of the available data from 2007, sothat distribution of constituencies obtained in this way should not be taken as the final solution proposal, but only as a demonstration of the possibilities offered by this methodology. Under the current law, the elections in Croatia are carried out in 10 constituencies. In this paper, several approaches known from the literature are proposed on the basis of which it is possible to determine the appropriate number of constituencies, which retain the property of maximum internal compactness and satisfactory well-separation.     

[16.] I. Svalina, K. Sabo, G. Šimunović, Machined Surface Quality Prediction Models Based on Moving Least Squares and Moving Least Absolute Deviations Methods, International Journal of Advanced Manufacturing Technology, 57(2011), 1099-1106
hrapavaostABSTRACT:  Surface roughness is often taken as an indicator of the quality of machined work pieces. Achieving the desired surface quality is of great importance for the product function. The paper  analyses the influence of the cutting depth, feed rate and number of revolutions on surface roughness.  The obtained results of experimental research conducted on the work piece "diving manifold", were  used to determine the coefficients by different numerical methods of the same prediction model. The results of surface roughness provided by the prediction functions generated in this work were  compared with the results of surface roughness obtained by using neural networks. The assessment of  surface roughness provided by models and neural networks can facilitate the work of less experienced  technologists and thus shorten the time of production technology preparation. The obtained results  show that the total mean square deviation in models obtained by the application of the moving linear least squares and the moving linear least absolute deviations methods is nevertheless considerably  higher than by the application of neural network method.     
 
[15.] 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 Resarch, (20)2012, 719-733
growthABSTRACT: 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.    
[14.]  K. Sabo,  R. Scitovski, I. Vazler,  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), 2,  293-314
systemABSTRACT: (pdf)    
[13.]  I. Vazler, K. Sabo, R. Scitovski,  Weighted median of the data in solving least absolute deviations problems,  Communications in Statistics – Theory and Methods, (41) 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.
 		  
[12.] K. Sabo, R. Scitovski, I. Vazler, M. Zekić-Sušac, Mathematical models of natural gas consumption,  Energy Conversion and Management, 52(2011), 1721-1727.
gasABSTRACT:  In this paper we consider the problem of natural gas consumption hourly forecast 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 to natural gas consumption forecast 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. The results show that most acceptable forecast is provided by mathematical models in which natural gas consumption and temperature are related explicitly.  
[11.]  M. Benšić, K. Sabo,  Estimating a uniform distribution when data are measured with a normal additive error with unknown variance.  Statistics - A Journal of Theoretical and Applied Statistics, 44 (2010), 235-246.

ABSTRACT:  The problem of estimating the width of a symmetric uniform distribution on the line together with the error variance, when data are measured with normal additive error, is considered. The main purpose is to analyze the maximum likelihood estimator and to compare it with the moment method estimator. It is shown that this two-parameter model is regular so that the maximum likelihood estimator is asymptotically efficient. Necessary and sufficient conditions are given for the existence of the maximum likeli- hood estimator. As numerical problems are known to frequently occur while computing the maximum likelihood estimator in this model, useful suggestions for computing the maximum likelihood estimator are also given.

  
[10.] 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) , 3; 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.

  
[9.] K. Sabo, M. Benšić, Border estimation of a disc observed with random errors solved in two steps, Journal of Computational and Applied Mathematics. 229 (2009) , 1; 16-26

ABSTRACT: The problem of estimating the boundary of a uniform distribution on a disc is considered when data are measured with normally distributed additive random error. The problem is solved in two steps. In the first step the domain is subdivided into thin slices and the endpoints of slices are obtained within the framework of a corresponding one-dimensional problem. For the estimations implemented in that step the moment method and the maximum likelihood method are used. As there are numerical problems with calculating the variance of the estimator in the maximum likelihood approach, its good approximation is also given. In the second step the obtained endpoints are used to estimate the boundary using the total least-squares curve fitting procedure. A necessary and sufficient condition for the existence of the total least-squares solution is also given. Finally, simulation results are presented.

  
[8.] K. Sabo, R. Scitovski, The best least absolute deviations line-properties and two efficient methods for its derivation, ANZIAM Journal. 50 (2008) , 2; 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

  
[7.] M. Benšić, K. Sabo, Border Estimation of a Two-dimensional Uniform Distribution if Data are Measured with Additive Error,  Statistics - A Journal of Theoretical and Applied Statistics. 41 (2007) , 4; 311-319

ABSTRACT: The paper considers estimation of the boundary of an elliptical domain when the data without a measurement error are distributed uniformly on this domain but are superimposed by random errors. The problem is solved in two phases. In the first phase the domain is subdivided into thin slices and the endpoints of these slices are estimated within the framework of a corresponding one-dimensional problem. In the second phase the estimated endpoints are used to estimate the boundary using the total least squares curve fitting procedure.

  
[6.] K. P. Hadeler, D. Jukić, K. Sabo, Least squares problems for Michaelis Menten kinetics.  Mathematical Methods in the Applied Sciencies. 30 (2007) , 11; 1231-1241

ABSTRACT: The Michaelis-Menten kinetics is fundamental in chemical and physiological reaction theory. The problem of parameter identification, which is not well-posed for arbitrary data, is shown to be closely related to the Chebyshev sum inequality. This inequality yields sufficient conditions for existence of feasible solutions both for non-linear and for linear least squares problems. The conditions are natural and practical as they are satisfied if the data show the expected monotone and concave behavior.

  
[5.] D. Jukić, K. Sabo, R. Scitovski,  Total least squares fitting Michaelis-Menten enzyme kinetic model function. Journal of Computational and Applied Mathematics. 201 (2007) , 1; 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 $(p_i, x_i, y_i)$, $i=1, \ldots, m$, $m\geq 3$, 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.

  
[4.] M. Benšić, K. Sabo,  Estimating the width of a uniform distribution when data are measured with additive normal errors with known variance. Computational Statistics & Data Analysis. 51 (2007) , 9; 4731-4741

ABSTRACT: The problem of estimating the width of the symmetric uniform distribution on the line when data are measured with normal additive error is considered. The main purpose is to discuss the efficiency of the maximum likelihood estimator and the moment method estimator. It is shown that the model is regular and that the maximum likelihood estimator is more efficient than the moment method estimator. A sufficient condition is also given for the existence of both estimators.

  
[3.]  D. Jukić, R. Scitovski, K. Sabo,  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.

  
[2.] 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) , 2; 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 .

  
[1.] K. Sabo, A. Baumgartner,  One method for searching the best discrete TL_p approximation. Mathematical Communications - Supplement. 1 (2001) , 1; 63-68.

ABSTRACT:  On the basis of the given data we will show how efficiently the best TL_p natural cubic spline can be found. Cases p=1, 2 will be especially considered. The best TL_1 spline is of special interest because it is insensitive to the so called outliers, although for its constuction it is necessary to carry out a multidimensional minimization of an undifferetiable function. For that purpose Nelder-Meads Downhill Simplex Method is used. For the calculation of the distance from the data-point to the spline the Brent Method for onedimensional minimization is used. Also, based on the described methods we will show generating of the optimal curve of the second order on the basis of the given data. The method is illustrated with examples developed on the basis of our own programs written in the C programming language

  

 

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