Associate Professor

Danijel Grahovac

dgrahova@mathos.hr
+385-31-224-821
18 (1st floor)
Google Scholar Profile
School of Applied Mathematics and Informatics

Josip Juraj Strossmayer University of Osijek

Research Interests

  • Limit theorems
  • Scaling properties of stochastic processes (self-similarity, multifractal processes)
  • Heavy-tailed distributions
  • Applied probability and statistics

Degrees

Publications

Journal Publications

  1. D. Grahovac, P. Kevei, Tail Behavior and Almost Sure Growth Rate of Superpositions of Ornstein–Uhlenbeck-type Processes, Journal of Theoretical Probability 38/1 (2025)
    Abstract
    In this paper, we consider sample path growth of superpositions of Ornstein–Uhlenbeck-type processes (supOU). SupOU processes are stationary infinitely divisible processes defined as integrals with respect to a random measure. They allow marginal distributions and correlations to be modeled independently. Our results show that the almost sure behavior is primarily governed by the tail of the marginal distribution. In particular, we obtain a general integral test for the sample path growth that covers both heavy-tailed and light-tailed scenarios. We also investigate the tail behavior of the marginal distributions in connection with the characteristics of the underlying random measure.
  2. R. Scitovski, K. Sabo, D. Grahovac, F. Martínez-Álvarez, Š. Ungar, A partitioning incremental algorithm using adaptive Mahalanobis fuzzy clustering and identifying the most appropriate partition, Pattern Analysis and Applications 28/3 (2024), 1-14
    Abstract
    This paper deals with the problem of determining the most appropriate number of clusters in a fuzzy Mahalanobis partition. First, a new fuzzy Mahalanobis incremental algorithm is constructed to search for an optimal fuzzy Mahalanobis partition with clusters. Among these partitions, selecting the one with the most appropriate number of clusters is based on appropriately modified existing fuzzy indexes. In addition, the Fuzzy Mahalanobis Minimal Distance index is defined as a natural extension of the recently proposed Mahalanobis Minimal Distance index for non-fuzzy clustering. The new fuzzy Mahalanobis incremental algorithm was tested on several artificial data sets and the color image segmentation problems from real-world applications: art images, nature photography images, and medical images. The algorithm includes multiple usage of the global optimization algorithm DIRECT. But unlike previously known fuzzy Mahalanobis indexes, the proposed Fuzzy Mahalanobis Minimal Distance index ensures accurate results even when applied to complex real-world applications. A possible disadvantage could be the need for longer CPU time. Furthermore, besides effective identification of the partition with the most appropriate number of clusters, it is shown how to use the proposed Fuzzy Mahalanobis Minimal Distance index to search for an acceptable partition, which proved particularly useful in the above-mentioned real-world applications.
  3. D. Grahovac, P. Kevei, Almost sure growth of integrated supOU processes, Bernoulli (2024), prihvaćen za objavljivanje
    Abstract
    Superpositions of Ornstein-Uhlenbeck processes allow a flexible dependence structure, including long range dependence for OU-type processes. Their complex asymptotics are governed by three effects: the behavior of the Lévy measure both at infinity and at zero, and the behavior at zero of the measure governing the dependence. We establish almost sure rates of growth depending on the characteristics of the process and prove a Marcinkiewicz–Zygmund type SLLN for the integrated process.
  4. 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.
  5. 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.
  6. D. Grahovac, N.N. Leonenko, M. Taqqu, Intermittency and Multiscaling in Limit Theorems, Fractals 30/7 (2022), 1-18
  7. D. Grahovac, Intermittency in the small-time behavior of Levy processes, Statistics & Probability Letters 187/109507 (2022), 1-8
  8. D. Grahovac, N.N. Leonenko, M. Taqqu, Intermittency and infinite variance: the case of integrated supOU processes, Electronic Journal of Probability 26 (2021), 1-31
  9. 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.
  10. D. Grahovac, Multifractal processes: Definition, properties and new examples, Chaos, Solitons & Fractals 134/109735 (2020)
  11. D. Grahovac, N.N. Leonenko, M. Taqqu, The multifaceted behavior of integrated supOU processes: The infinite variance case, Journal of Theoretical Probability 33 (2020), 1801-1831
  12. F. Avram, D. Grahovac, C. Vardar-Acar, The W,Z scale functions kit for first passage problems of spectrally negative Lévy processes, and applications to control problems, ESAIM: Probability and Statistics 24 (2020), 454-525
  13. D. Grahovac, N.N. Leonenko, M. Taqqu, Limit theorems, scaling of moments and intermittency for integrated finite variance supOU processes, Stochastic Processes and their Applications 129/12 (2019), 5113-5150
  14. D. Grahovac, N.N. Leonenko, A. Sikorskii, M. Taqqu, The unusual properties of aggregated superpositions of Ornstein-Uhlenbeck type processes, Bernoulli 25/3 (2019), 2029-2050
  15. F. Avram, D. Grahovac, C. Vardar-Acar, The W,Z/ν,δ Paradigm for the First Passage of Strong Markov Processes without Positive Jumps, Risks 7/1-18 (2019), 1-15
  16. D. Grahovac, N.N. Leonenko, Bounds on the support of the multifractal spectrum of stochastic processes, Fractals 26/04 (2018), 1-21
  17. D. Grahovac, Densities of Ruin-Related Quantities in the Cramér-Lundberg Model with Pareto Claims, Methodology and Computing in Applied Probability 20/1 (2018), 273-288
  18. D. Grahovac, N.N. Leonenko, M. Taqqu, Intermittency of trawl processes, Statistics & Probability Letters 137 (2018), 235-242
  19. 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.
  20. D. Grahovac, N.N. Leonenko, A. Sikorskii, I. Tešnjak, Intermittency of superpositions of Ornstein-Uhlenbeck type processes, Journal of Statistical Physics 165/2 (2016), 390-408
  21. D. Grahovac, M. Jia, N.N. Leonenko, E. Taufer, Asymptotic properties of the partition function and applications in tail index inference of heavy-tailed data, Statistics - a Journal of Theoretical and Applied Statistics 49/6 (2015), 1221-1242
  22. D. Grahovac, N. Šuvak, Heavy-tailed modeling of CROBEX, Financial Theory and Practice 39/4 (2015), 411-430
    Abstract
    Classical continuous-time stochastic models for log-returns of risky assets, such as the Black-Scholes model, usually assume independence and normality of distributions of log-returns. However, empirical properties of log-returns often show a specific correlation structure and a deviation from normality, in most cases suggesting that their distribution exhibits heavy tails. A natural alternative for modeling log-returns in continuous time would be a stochastic process incorporating a weak form of dependence and a heavy-tailed distribution that is in some way close to the normal distribution. The Student's distribution with small value of tail index (number of degrees of freedom) is the logical choice for such heavy-tailed distribution. Therefore we suggest an alternative continuous-time model for log-returns, a diffusion process with Student's marginal distributions and exponentially decaying autocorrelation structure. This model depends on several unknown parameters that need to be estimated. The tail index is estimated by the method based on the empirical scaling function, while the parameters describing the mean, the variance and the correlation structure of the model are estimated by the generalized method of moments. The model is applied to the CROBEX stock market index, meaning that the estimation of model parameters is based on the CROBEX log-returns. Quality of the proposed model is assessed by the means of simulations, specifically by comparing CROBEX log-returns with the simulated trajectories of the Student's diffusion depending on estimated parameter values.
  23. D. Grahovac, N.N. Leonenko, M. Taqqu, Scaling properties of the empirical structure function of linear fractional stable motion and estimation of its parameters, Journal of Statistical Physics 158/1 (2015), 105-119
  24. D. Grahovac, N.N. Leonenko, Detecting multifractal stochastic processes under heavy-tailed effects, Chaos, Solitons & Fractals 66 (2014), 78-89


Refereed Proceedings

  1. D. Grahovac, M. Jia, N.N. Leonenko, E. Taufer, On the tail index inference based on the scaling function method, 18th European Young Statisticians Meeting, Osijek, Croatia, 2013, 39-45


Others

  1. M. Benšić, D. Grahovac, D. Mihalčić, Linearna regresija kroz primjer, Osječki matematički list 23/2 (2023), 115-131
    Abstract
    U radu je prezentirana osnovna ideja linearne regresije. Glavni pojmovi objašnjeni su intuitivno. Naglasak je na konkretnom primjeru gdje je cijena automobila modelirana na temelju njegovih osnovnih karakteristika. Model je izrađen na temelju stvarnih podataka o rabljenim automobilima u Hrvatskoj.
  2. D. Grahovac, D. Mihalčić, Slučajni procesi definirani integralom u odnosu na slučajnu mjeru, Osječki matematički list 23/1 (2023), 31-44
    Abstract
    U ovom radu najprije je objašnjen pojam beskonačne djeljivosti za slučajne varijable i procese. Kao poseban oblik slučajnog procesa, definirane su slučajne mjere te su dani najvažniji primjeri. Integral u odnosu na slučajnu mjeru najprije je definiran za jednostavne, a potom za općenite funkcije. Navedeni su nužni i dovoljni uvjeti na podintegralnu funkciju koji osiguravaju da je integral dobro definiran te eksplicitni izrazi za karakterističnu trojku beskonačno djeljive slučajne varijable definirane integralom. U zadnjem dijelu dani su primjeri procesa definiranih integralom i njihova najvažnija svojstva.
  3. D. Grahovac, L. Grgić, Dugoročna zavisnost, Osječki matematički list 19/1 (2019), 15-29
  4. D. Grahovac, A. Leko, Modeliranje rizika u osiguranju, Osječki matematički list 15/2 (2015), 113-129
  5. D. Grahovac, Two-dimensional interpolation spline, Osječki matematički list 10/1 (2010), 59-69


Books

  1. R. Scitovski, K. Sabo, D. Grahovac, Globalna optimizacija, Sveučilište Josipa Jurja Strossmayera u Osijeku, Odjel za matematiku, Osijek, 2017.


Software

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


Projects

  • Scaling in stochastic models
    Project runs since December 15, 2023. Project leader: Danijel Grahovac. Project funded by Croatian Science Foundation
  • Limiting behavior of intermittent processes and diffusions
    Project runs in academic year 2019/2020. Project leader: Danijel Grahovac. Project funded by University of Osijek.
  • Stochastic models with long-range dependence
    Project runs in academic year 2017/2018. Project leader: Nenad Šuvak. Project funded by University of Osijek.
  • Fractional Pearson Diffusions
    Project run in academic year 2014/2015. Project leader: Nenad Šuvak. Project funded by University of Osijek.
  • Development and Application of Growth Potential Prediction Models for Small and Medium Enterprises in Croatia
    Project runs since June 2014. Project leader: Nataša Šarlija. Project funded by Croatian Science Foundation.
  • Statistical aspects of estimation problem in nonlinear parametric models
    Project run in 2007-2013. Project leader: Mirta Benšić. Project funded by Croatian Ministry of Science, Education and Sports.

Professional Activities

  • Editorial Boards
    • Co-Editor of Croatian Operational Research Review
  • Committee Memberships
    • Member of the Local Organizing Comitee of the 27th Young Statisticians Meeting, 31.9.2023. – 1.10.2023. Osijek, Croatia
    • Member of the Organizing Comitee of the Statistical Conference in Croatia – ISCCRO’18, 10.5.2018. – 11.5.2018., Opatija, Croatia
    • Member of the Local Organizing Comitee of the 18th European Young Statisticians Meeting, 26-30 August 2013, Osijek, Croatia
  • Service Activities
    • Secretary of the Mathematical Colloquium in Osijek, 2013 – 2017
    • Moderator of Optimization and applications seminar 2017 –

Teaching

 

Teaching materials (in Croatian):

Personal

Married, three children