K. Sabo, R. Scitovski, Š. Ungar, Z. Tomljanović, A method for searching for a globally optimal k-partition of higher-dimensional datasets, Journal of Global Optimization (2024), prihvaćen za objavljivanje
The problem with finding a globally optimal k-partition of a set A is a very intricate
optimization problem for which in general, except in the case of one-dimensional data,
i.e., for data with one feature (A\\subset\\R), there is no method to solve. Only in the
one-dimensional case there exist efficient methods that are based on the fact that the
search for a globally optimal partition is equivalent to solving a global optimization
problem for a symmetric Lipschitz-continuous function using the global optimization
algorithm DIRECT. In the present paper, we propose a method for finding a globally
optimal k-partition in the general case (A\\subset \\R^n, n\\geq 1), generalizing an idea
for solving the Lipschitz global optimization for symmetric functions. To do this, we
propose a method that combines a global optimization algorithm with linear constraints
and the k-means algorithm. The first of these two algorithms is used only to find a
good initial approximation for the $k$-means algorithm. The method was tested on a
number of artificial datasets and on several examples from the UCI Machine Learning
Repository, and an application in spectral clustering for linearly non-separable datasets
is also demonstrated. Our proposed method proved to be very efficient.
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.
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.
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.
K. Sabo, R. Scitovski, Nova metoda za definiranje izbornih jedinica u Hrvatskoj, Hrvatska i komparativna javna uprava (2023), prihvaćen za objavljivanje
U radu predlažemo novu metodu za definiranje konfiguracije izbornih jedinica primjenom metode spektralnog klasteriranja. Metoda pronalazi konfiguracije izbornih jedinica koje zadovoljavaju neku unaprijed zadanu toleranciju ujednačenosti težina biračkih glasova te pritom čuva granice županija. Također u metodu se prirodno može uključiti i razina političke/socijalne/gospodarske povezanosti županija. Nadalje, navodimo popis poznatih metoda za određivanje broja zastupničkih mjesta po izbornim jedinicama, koje se temelje na principu razmjernosti broja birača i broja zastupničkih mjesta. U radu dajemo pregled indeksa iz literature kojima se može mjeriti ujednačenost težina biračkih glasova. Primjene tih indeksa ilustriramo na najnovijem prijedlogu Hrvatske vlade, vlastitim prijedlozima, kao i na nekoliko primjera konfiguracija izbornih jedinica od kojih su neki već predstavljeni javnosti.
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.
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.
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.
In this paper, we give a necessary and sufficient criterion for the existence of the $ell_p-$norm estimate
for the nonlinear $ell_p-$norm fitting problem. Our criterion is based on the existence level that describes
the behavior of the objective function as its argument approaches the extended boundary of the parameter space.
R. Scitovski, K. Sabo, DBSCAN-like clustering method for various data densities, Pattern Analysis and Applications 23 (2020), 541-554
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.
In this paper we consider the problem of estimating the support of a uniform distribution under symmetric additive errors. The maximum likelihood (ML) estimator is of our primary interest, but we also analyze the method of moments (MM) estimator, when it exists. Under some regularity conditions, the ML estimator is consistent and asymptotically efficient. Errors with Student's t distribution are shown to be a good choice for robustness issues.
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.
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.
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.
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
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.
The relationship between the norm square of the standardized cumulative
distribution and the chi-square statistic is examined using the form
of the covariance matrix as well as the projection perspective. This investigation
enables us to give uncorrelated components of the chi-square statistic
and to provide interpretation of these components as innovations standardizing
the cumulative distribution values. The norm square of the standardized
difference between empirical and theoretical cumulative distributions is also
examined as an objective function for parameter estimation. Its relationship
to the chi-square distance enables us to discuss the large sample properties
of these estimators and a difference in their properties in the cases that the
distribution is evaluated at fixed and random points.
In many applications we are faced with the problem of estimating object dimensions
from a noisy image. Some devices like a fluorescent microscope, X-ray
or ultrasound machines, etc., produce imperfect images. Image noise comes from a
variety of sources. It can be produced by the physical processes of imaging, or may
be caused by the presence of some unwanted structures (e.g. soft tissue captured
in images of bones ). In the proposed models we suppose that the data are drawn
from uniform distribution on the object of interest, but contaminated by an additive
error. Here we use two one-dimensional parametric models to construct confidence
intervals and statistical tests pertaining to the object size and suggest the possibility
of application in two-dimensional problems. Normal and Laplace distributions
are used as error distributions. Finally, we illustrate ability of the R-programs we
created for these problems on a real-world example.
The growth curve clustering problem is analyzed and its connec-
tion with the spectral relaxation method is described. For a given set of growth
curves and similarity function, a similarity matrix is defined, from which the
corresponding similarity graph is constructed. It is shown that a nearly op-
timal growth curve partition can be obtained from the eigendecomposition
of a specific matrix associated with a similarity graph. The results are illus-
trated and analyzed on the set of synthetically generated growth curves. One
real-world problem is also given.
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.
This paper describes an R package LeArEst that can be used for estimating object dimensions
from a noisy image. The package is based on the simple parametric model for data that are drawn
from uniform distribution contaminated by an additive error. Our package is able to estimate the
length of the object of interest on a given straight line that intersects it as well as to estimate the object
area if it is elliptically shaped. The input data may be a numerical vector as well as an image in JPEG
format. In this paper, background statistical models and methods for the package are summarized,
and algorithms and key functions implemented are described. Also, examples that demonstrate its
usage are provided.
In this paper, we consider the problem of the existence of a least absolute deviations estimator for the Michaelis--Menten model function. We give necessary and sufficient conditions under which the least absolute deviations problem has a solution. In order to illustrate the usefulness of such conditions we give several numerical examples.
A one-dimensional problem of a uniform distribution width estimation from
data observed with a Laplace additive error is analyzed. The error variance is
considered as a nuisance parameter and it is supposed to be known or consistently
estimated before. It is proved that the maximum likelihood estimator
in the described model is consistent and asymptotically efficient and sufficient
conditions for its existence are given. The method of moment estimator
is also analyzed in this model and compared with the maximum likelihood
estimator theoretically and in simulations. Finally, one real-world example
illustrates the possibility for applications in two-dimensional problems.
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.
M. Duspara, K. Sabo, A. Stoić, Acoustic emision as tool wear monitoring, Tehnički vjesnik 21/5 (2014), 1097-1101
This paper describes a fast
Fourie transformation and its application to monitoring tool wear.
Tool wear monitoring is a difficult task because many machining processes
are non-linear time-variant systems, which makes them difficult to model and
the signals obtained from sensors are dependent on a number of other factors,
such as machining conditions. Indirect method of tool condition monitoring is
based on the acquisition of measured values of process variables (such as
cutting force, temperature, vibration, acoustic emission, surface roughness)
and the relationship between tool wear and these values.
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.
In this paper, we consider the $l_1$-clustering problem for a data-points set $mathcal{A}={a^iinR^ncolon i=1,dots,m}$ which should be partitioned into $k$ disjoint nonempty subsets $pi_1,dots,pi_k$, $1leq kleq m$. 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.
Motivated by the method for color image segmentation based on intensity and hue clustering proposed in [26] we give some theoretical explanations for this method that directly follows from the natural connection between the maximum likelihood approach and Least Squares or Least Absolute Deviations clustering optimality criteria. The method is tested and illustrated on a few typical situations, such as the presence of outliers among the data.
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 $\mathcal{A}=\{a_i\in\R^n:i=1,\dots,m\}\subset \R^n$ which consists of $k$ disjoint nonempty subsets $\pi_1,\dots,\pi_k$, $1\leq k\leq m$. 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.
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.
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
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.
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).
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
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.
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
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.
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.
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.
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.
We consider the problem of searching for a best LAD-solution of an overdetermined system of linear equations $\mathbf{Xa}=\mathbf{z}$, $\mathbf{X}\in\mathbb{R}^{m\times n}$, $m\geq n$, $\mathbf{a}\in\mathbb{R}^n, \mathbf{z}\in\mathbb{R}^m$. This problem is equivalent to the problem
of determining a best LAD-hyperplane $\mathbf{x}\mapsto \mathbf{a}^T\mathbf{x}$, $\mathbf{x}\in\mathbb{R}^n$ on the basis of given data $(\mathbf{x}_i,z_i),\,\mathbf{x}_i=(x_1^{(i)},\ldots,x_n^{(i)})^T\in\mathbb{R}^n,\,z_i\in\mathbb{R},\,i=1,\ldots,m$, whereby the minimizing functional is of the form
\[
F(\mathbf{a})=\|\mathbf{z}-\mathbf{Xa}\|_1=\sum_{i=1}^m|z_i-\mathbf{a}^T\mathbf{x}_i|.
\]
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 $\mathbf{a}^*\in\mathbb{R}^n$ is the point of a global minimum of the functional $F$ if and only if $\mathbf{0}\in\partial F(\mathbf{a}^*)$.
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 $(x_i,y_i,z_i),\,i=1,\ldots,m$.
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 e± ; cient. Necessary and su± ; cient 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.
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.
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.
K. Sabo, R. Scitovski, The best least absolute deviations line-properties and two efficient methods for its derivation, ANZIAM Journal 50 (2008), 185-198
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.
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
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.
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), 311-319
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
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), 4731-4741
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.
K. Hadeler, D. Jukić, K. Sabo, Least squares problems for Michaelis Menten kinetics, Mathematical Methods in the Applied Sciencies 30 (2007), 1231-1241
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.
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
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.
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
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.
K. Sabo, A. Baumgartner, One method for searching the best discrete TL_p approximation, Mathematical Communications - Supplement 1 (2001), 63-68
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.
In this paper, we give a necessary and sufficient criterion for the existence of the least squares estimate
for the nonlinear sum of squares. Our criterion is based on the existence level that describes the behavior of the sum of squares
as its argument approaches the extended boundary of the parameter space.
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
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.
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
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
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
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
D. Jukić, K. Sabo, G. Bokun, Least squares problem for the Hubbert function, 9th International Conference on Operational Research KOI 2002, Trogir, 2002, 37-46
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
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.
Others
T. Milas, M. Ribičić Penava, K. Sabo, Primjene gama funkcije, Osječki matematički list 21/1 (2021), 1-18
U ovom je radu ukratko opisana gama funkcija te je stavljen naglasak na neke njezine primjene u različitim područjima matematike.
Opisane su primjene u geometriji, vjerojatnosti (momenti nekih neprekidnih slučajnih varijabli te gama distribucija) te u teoriji Laplaceovih transformacija.
A. Jovičić, K. Sabo, Formule za udaljenost točke do pravca u ravnini, u smislu $l_p-$udaljenosti, $1leq p leq infty$, Math.e : hrvatski matematički elektronski časopis 26 (2016)
Formula za euklidsku udaljenost točke do pravca u ravnini dobro je poznata učenicima završnih razreda srednjih škola. U ovom radu promatramo općenitije probleme udaljenosti točke do pravca u ravnini, u smislu lp−udaljenosti, 1≤p≤∞. Pokazat ćemo da se i u tim slučajevima, također, mogu izvesti analogne formule za računanje udaljenosti točke do pravca.
M. Zec, K. Sabo, Kvadratne interpolacijske metode za jednodimenzionalnu bezuvjetnu lokalnu optimizaciju, Matematički kolokvijum (MAT-KOL) 22 (2016), 5-19
U radu je opisana klasa kvadratnih interpolacijskih metoda za
jednodimenzionalnu lokalnu optimizaciju. U ovu se klasu metoda ubrajaju Newtonova metoda, Metoda dvije tocke te Metoda tri tocke. Za svaku od ovih metoda dana je geometrijska motivacija, izvod, odgovarajuci algoritam te su navedeni rezultati o konvergenciji i brzini konvergencije. Spomenute metode su
jednostavne te je za njihovo razumijevanje dovoljno osnovno znanje Diferencijalnog racuna. U svrhu ilustracije ekasnosti metoda, dan je jedan numericki primjer izraden u programskom paketu Mathematica.
L. Grgić, K. Sabo, Nelder-Meadova metoda: lokalna metoda direktne bezuvjetne optimizacije, Osječki matematički list 15 (2015), 131-143
U radu je opisana poznata Nelder–Meadova metoda, koja se smatra
jednom od najpopularnijih lokalnih metoda direktne bezuvjetne optimizacije.
Zbog jednostavnosti, analiziran je specijalni slucaj optimizacije
u R^2, jer se tada Nelder–Meadova metoda svodi na niz elementarnih
geometrijskih transformacija u ravnini te je za njezino potpuno
razumijevanje dovoljno znanje srednjoškolske matematike. U svrhu
ilustracije metode, dano je nekoliko numerickih primjera koji su izrađeni u programskom paketu Mathematica.
K. Sabo, S. Scitovski, Lokacija objekata u ravnini, KoG (Scientific and Professional Journal of Croatian Society for Geometry and Graphics) 15 (2011), 57-62
U radu razmatramo izravni i obratni problem lokacije objekata u ravnini uz korištenje različitih kvazimetričkih funkcija s odgovarajucim ilustracijama. Dano je nekoliko primjera iz razlicitih podrucja primjena.
K. Sabo, R. Scitovski, I. Vazler, Grupiranje podataka: klasteri, Osječki matematički list 10 (2010), 149-176
U ovom radu razmatramo problem grupiranja elemenata skupa A u disjunktne neprazne podskupove - klastere, pri cemu pretpostavljamo da su elementi skupa A odreeni 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 meu tim kriterijima. Izraena je odgovarajuca programska podrška s ciljem da zainteresirani strucnjaci u svom znanstvenom ili strucnom radu mogu olakšano koristiti ovu metodologiju i pristup.
M. Meštrović, K. Sabo, Grafička ilustracija pogreške u rješenju sustava linearnih jednadžbi, Osječki matematički list 4 (2004), 87-94
U članku se razmatra problem pogreške u rješenju sustava linearnih jednadžbi koja nastaje zbog promjena u desnoj strani sustava. Za ilustraciju tog problema koriste se mogućnosti programskog paketa Mathematica.
K. Sabo, R. Scitovski, Prosti brojevi, Osječki matematički list 3 (2003), 13-20
U članku se opisuju neka važna svojstva prostih brojeva. Pored danih primjera i zadataka, navodi se i nekoliko neriješenih problema vezanih uz proste brojeve.
K. Sabo, Minimizacija realne funkcije realne varijable, Osječki matematički list 1 (2001), 109-118
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.
Package provides methods for estimating borders of uniform distribution on the interval (one-dimensional) and on the elliptical domain (two-dimensional) under measurement errors. For one-dimensional case, it also estimates the length of underlying uniform domain and tests the hypothesized length against two-sided or one-sided alternatives. For two-dimensional case, it estimates the area of underlying uniform domain. It works with numerical inputs as well as with pictures in JPG format.
Standardizing the empirical distribution function yields a statistic with norm
square that matches the chi-square test statistic. To show this one may use the
covariance matrix of the empirical distribution which, at any finite set of points, is
shown to have an inverse which is tridiagonal. Moreover, a representation of the
inverse is given which is a product of bidiagonal matrices corresponding to a representation
of the standardization of the empirical distribution via a linear combination
of values at two consecutive points. These properties are discussed also in the context
of minimum distance estimation
Scientifically branded Pork(Member of the scientific project entitled above. Project started on June 1, 2014. Principal investigator is professor Goran Kušec from Faculty of Agriculture in Osijek, University of Osijek. Project was supported by Croatian Science Foundation.)
2014., 2015. Večer matematike – manifestacija popularizacije matematike u organizaciji Udruge matematičara Osijeku i Hrvatskog matematičkog društva - Član Programskog i Organizacijskog odbora
2013.-2016. Matematičke pripreme za učenike srednjih škola • Programski i Organizacijski koordinator
2000.-2016. Zimska matematička škola za učenike srednjih škola • Član Programskog i Organizacijskog odbora
2000.-2016. Zimska matematička škola za učenike osnovnih škola • Član Programskog i Organizacijskog odbora
2006.-2016. Stručni kolokvij Udruge matematičara Osijek • Član Programskog i Organizacijskog odbora
travanj, 2014. Geometrijska škola Stanko Bilinski, Našice: Predavanje za nastavnike: „Funkcija udaljenosti i odgovarajuća geometrija“, Radionica za učenike: “Neki optimizacijski problemi u geometriji“
travanj, 2014. Festival znanosti Sveučilišta Josipa Jurja Strossmayera u Osijeku Predavanje: „Što su optimalne izborne (upravne) jedinice i kako ih odrediti“. Član Programskog i Organizacijskog odbora
listopad, 2012. Stručni skup: Nastava matematike i izazovi moderne tehnologije u organizaciji Udruge Normala - Predavanje: „Zaglađivanje podataka: metode, pristupi i primjene“
