### Danijel Grahovac

School of Applied Mathematics and Informatics

Josip Juraj Strossmayer University of Osijek

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

- PhD in mathematics, Faculty of Natural Science, Department of Mathematics, University of Zagreb, 2015
- MSc in Mathematics, Financial and Business Mathematics, Department of Mathematics, University of Osijek, Croatia, 2010
- MSc in Mathematics and Computer Science Education, Department of Mathematics, University of Osijek, Croatia, 2016
- BSc in Mathematics, Department of Mathematics, University of Osijek, Croatia, 2008

### Publications

Journal Publications

- R. Scitovski, K. Sabo, D. Grahovac, Š. Ungar, Minimal distance index — A new clustering performance metrics, Information Sciences
**640**/119046 (2023)We define a new index for measuring clustering performance called the Minimal Distance Index. The index is based on representing clusters by characteristic objects containing the majority of cluster points. It performs well for both spherical and ellipsoidal clusters. This method can recognize all acceptable partitions with well-separated clusters. Among such partitions, our minimal distance index may identify the most appropriate one. The proposed index is compared with other most frequently used indexes in numerous examples with spherical and ellipsoidal clusters. It turned out that our proposed minimal distance index always recognizes the most appropriate partition, whereas the same cannot be said for other indexes found in the literature. Furthermore, among all acceptable partitions, the one with the largest number of clusters, not necessarily the most appropriate ones, has a special significance in image analysis. Namely, following Mahalanobis image segmentation, our index recognizes partitions that might not be the most appropriate ones but are the ones using colors that significantly differ from each other. The minimal distance index recognizes partitions with dominant colors, thus making it possible to select specific details in the image. We apply this approach to some real-world applications such as the plant rows detection problem, painting analysis, and iris detection. This may also be useful for medical image analysis. - D. Grahovac, N.N. Leonenko, M. Taqqu, Intermittency and Multiscaling in Limit Theorems, Fractals
**30**/7 (2022), 1-18 - D. Grahovac, Intermittency in the small-time behavior of Levy processes, Statistics & Probability Letters
**187**/109507 (2022), 1-8 - 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 - K. Sabo, D. Grahovac, R. Scitovski, Incremental method for multiple line detection problem - iterative reweighted approach, Mathematics and Computers in Simulation
**178**(2020), 588-602In 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. - D. Grahovac, Multifractal processes: Definition, properties and new examples, Chaos, Solitons & Fractals
**134**/109735 (2020) - 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 - 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 - 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 - 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 - 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 - D. Grahovac, N.N. Leonenko, Bounds on the support of the multifractal spectrum of stochastic processes, Fractals
**26**/04 (2018), 1-21 - 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 - D. Grahovac, N.N. Leonenko, M. Taqqu, Intermittency of trawl processes, Statistics & Probability Letters
**137**(2018), 235-242 - R. Grbić, D. Grahovac, R. Scitovski, A method for solving the multiple ellipses detection problem, Pattern Recognition
**60**(2016), 824-834In 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. - 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 - 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 - D. Grahovac, N. Šuvak, Heavy-tailed modeling of CROBEX, Financial Theory and Practice
**39**/4 (2015), 411-430Classical 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. - 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 - D. Grahovac, N.N. Leonenko, Detecting multifractal stochastic processes under heavy-tailed effects, Chaos, Solitons & Fractals
**66**(2014), 78-89

Refereed Proceedings

- 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

- D. Grahovac, L. Grgić, Dugoročna zavisnost, Osječki matematički list
**19**/1 (2019), 15-29 - D. Grahovac, A. Leko, Modeliranje rizika u osiguranju, Osječki matematički list
**15**/2 (2015), 113-129 - D. Grahovac, Two-dimensional interpolation spline, Osječki matematički list
**10**/1 (2010), 59-69

Books

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

Software

- 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

**Konzultacije (Office Hours):**četvrtak 11:45, u vrijeme ispitnih rokova po dogovoru**Teme diplomskih radova 2023/2024**(pdf)**Courses in Winter semester 2023/2024:****Courses in Summer semester 2022/2023:****Courses in Winter semester 2022/2023:****Past courses:**Databases, Statistics, Stochastic Processes, Numerical Mathematics, Mathematics (Faculty of Economics), Statistics (PTFOS), Statistical Lab, Mathematical lab