# Spatial Hurst–Kolmogorov Clustering

^{*}

## Definition

**:**

## 1. Introduction

#### 1.1. HK Clustering

#### 1.2. Stochastic Analysis in the Scale Domain

## 2. Methodology

## 3. Illustrative Applications

#### 3.1. One-Dimensional Turbulence

^{6}records, and after a data-homogenization procedure, we were able to analyze them as if they were recorded at the same location and with similar initial conditions (for more details on the homogenization, see [12]). In Figure 4, we show the standardized second-order metrics of the climacogram, autocovariance and variogram, i.e., c(h) − c(0), and power spectrum, where it can be seen that the variogram and the climacogram exhibit the smallest uncertainty at the long-range scales and, in particular, on the double logarithmic slope that is attributed to the strength of the long-range dependence (i.e., the magnitude of the Hurst parameter). However, since the variogram has the limitation that the double logarithmic slope tends to zero, only from the climacogram is it possible to robustly estimate the Hurst parameter (for comparison between the two metrics, see [32]).

#### 3.2. Benchmark Analysis of Two-Dimensional Art Paintings

#### 3.3. Two-Dimensional Rock-Formations

#### 3.4. Spatio-Temporal Wind Speed of a Hurricane

#### 3.5. Spatio-Temporal Evolution of Clustering

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Conflicts of Interest

## Entry Link on the Encyclopedia Platform

## References

- Hurst, H.E. Long term storage capacities of reservoirs. Trans. Am. Soc. Civ. Eng.
**1951**, 116, 770–799. [Google Scholar] [CrossRef] - Mandelbrot, B.B.; Wallis, J.R. Noah, Joseph and operational hydrology. Water Resour. Res.
**1968**, 4, 909–918. [Google Scholar] [CrossRef] - Kolmogorov, A.N. Wiener spirals and some other interesting curves in a Hilbert space. In Selected Works of A. N. Kolmogorov; Mathematics and Mechanics; Tikhomirov, V.M., Ed.; Kluwer: Dordrecht, The Netherlands, 1991; pp. 303–307. [Google Scholar]
- Koutsoyiannis, D. A random walk on water. Hydrol. Earth Syst. Sci.
**2010**, 14, 585–601. [Google Scholar] [CrossRef] - Papoulis, A.; Pillai, S.U. Stochastic Processes; McGraw-Hill: New York, NY, USA, 1991. [Google Scholar]
- Gneiting, T.; Schlather, M. Stochastic Models That Separate Fractal Dimension and the Hurst Effect. SIAM Rev.
**2004**, 46, 269–282. [Google Scholar] [CrossRef] - Dimitriadis, P.; Koutsoyiannis, D.; Iliopoulou, T.; Papanicolaou, P. A Global-Scale Investigation of Stochastic Similarities in Marginal Distribution and Dependence Structure of Key Hydrological-Cycle Processes. Hydrology
**2021**, 8, 59. [Google Scholar] [CrossRef] - Koutsoyiannis, D. Hurst–Kolmogorov dynamics as a result of extremal entropy production. Phys. A Stat. Mech. Appl.
**2011**, 390, 1424–1432. [Google Scholar] [CrossRef] - Koutsoyiannis, D.; Dimitriadis, P. Towards generic simulation for demanding stochastic processes. Science
**2021**, 3, 34. [Google Scholar] [CrossRef] - Koutsoyiannis, D. Generic and parsimonious stochastic modelling for hydrology and beyond. Hydrol. Sci. J.
**2016**, 61, 225–244. [Google Scholar] [CrossRef] - Beven, K. Issues in Generating Stochastic Observables for Hydrological Models. Hydrol. Process.
**2021**. [Google Scholar] [CrossRef] - Dimitriadis, P.; Koutsoyiannis, D. Stochastic synthesis approximating any process dependence and distribution. Stoch. Environ. Res. Risk Assess.
**2018**, 32, 1493–1515. [Google Scholar] [CrossRef] - Koutsoyiannis, D. Stochastics of Hydroclimatic Extremes—A Cool Look at Risk; Edition 0; National Technical University of Athens: Athens, Greece, 2021; 330p. [Google Scholar]
- Koutsoyiannis, D. Hurst-Kolmogorov Dynamics and Uncertainty. J. Am. Water Resour. Assoc.
**2011**, 47, 481–495. [Google Scholar] [CrossRef] - Beran, J.; Feng, Y.; Ghosh, S.; Kulik, R. Long-Memory Processes; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar] [CrossRef]
- O’Connell, P.; Koutsoyiannis, D.; Lins, H.F.; Markonis, Y.; Montanari, A.; Cohn, T. The scientific legacy of Harold Edwin Hurst (1880–1978). Hydrol. Sci. J.
**2016**, 61, 1571–1590. [Google Scholar] [CrossRef] - Graves, T.; Gramacy, R.; Watkins, N.; Franzke, C. A Brief History of Long Memory: Hurst, Mandelbrot and the Road to ARFIMA, 1951–1980. Entropy
**2017**, 19, 437. [Google Scholar] [CrossRef] - Koutsoyiannis, D.; Paschalis, A.; Theodoratos, N. Two-dimensional Hurst–Kolmogorov process and its application to rainfall fields. J. Hydrol.
**2011**, 398, 91–100. [Google Scholar] [CrossRef] - Paschalis, A.; Molnar, P.; Fatichi, S.; Burlando, P. A stochastic model for high-resolution space-time precipitation simulation. Water Resour. Res.
**2013**, 49, 8400–8417. [Google Scholar] [CrossRef] - Varouchakis, E.A.; Hristopulos, D.T. Comparison of spatiotemporal variogram functions based on a sparse dataset of ground-water level variations. Spat. Stat.
**2017**, 34, 100245. [Google Scholar] [CrossRef] - McGarvey, R.; Feenstra, J.E.; Mayfield, S.; Sautter, E.V. A diver survey method to quantify the clustering of sedentary invertebrates by the scale of spatial autocorrelation. Mar. Freshw. Res.
**2010**, 61, 153–162. [Google Scholar] [CrossRef] - Tachmazidou, I.; Verzilli, C.J.; De Iorio, M. Genetic Association Mapping via Evolution-Based Clustering of Haplotypes. PLoS Genet.
**2007**, 3, e111. [Google Scholar] [CrossRef] - Hütt, M.-T.; Neff, R. Quantification of spatiotemporal phenomena by means of cellular automata techniques. Phys. A Stat. Mech. Appl.
**2001**, 289, 498–516. [Google Scholar] [CrossRef] - Khater, I.M.; Nabi, I.R.; Hamarneh, G. A Review of Super-Resolution Single-Molecule Localization Microscopy Cluster Analysis and Quantification Methods. Patterns
**2020**, 1, 100038. [Google Scholar] [CrossRef] - Murase, K.; Kikuchi, K.; Miki, H.; Shimizu, T.; Ikezoe, J. Determination of arterial input function using fuzzy clustering for quantification of cerebral blood flow with dynamic susceptibility contrast-enhanced MR imaging. J. Magn. Reson. Imaging
**2001**, 13, 797–806. [Google Scholar] [CrossRef] - Mier, P.; Andrade-Navarro, M.A. FastaHerder2: Four Ways to Research Protein Function and Evolution with Clustering and Clustered Databases. J. Comput. Biol.
**2016**, 23, 270–278. [Google Scholar] [CrossRef] [PubMed] - McDermott, P.L.; Wikle, C.K. Bayesian Recurrent Neural Network Models for Forecasting and Quantifying Uncertainty in Spatial-Temporal Data. Entropy
**2019**, 21, 184. [Google Scholar] [CrossRef] - Papacharalampous, G.; Tyralis, H.; Koutsoyiannis, D. Comparison of stochastic and machine learning methods for multi-step ahead forecasting of hydrological processes. Stoch. Environ. Res. Risk Assess.
**2019**, 33, 481–514. [Google Scholar] [CrossRef] - Abe, S.; Suzuki, N. Dynamical evolution of clustering in complex network of earthquakes. Eur. Phys. J. B
**2007**, 59, 93–97. [Google Scholar] [CrossRef] - Ellam, L.; Girolami, M.; Pavliotis, G.A.; Wilson, A. Stochastic modelling of urban structure. Proc. R. Soc. A Math. Phys. Eng. Sci.
**2018**, 474, 20170700. [Google Scholar] [CrossRef] - Levine, N. Spatial Statistics and GIS: Software Tools to Quantify Spatial Patterns. J. Am. Plan. Assoc.
**1996**, 62, 381–391. [Google Scholar] [CrossRef] - Dimitriadis, P.; Tzouka, K.; Koutsoyiannis, D.; Tyralis, H.; Kalamioti, A.; Lerias, E.; Voudouris, P. Stochastic investigation of long-term persistence in two-dimensional images of rocks. Spat. Stat.
**2019**, 29, 177–191. [Google Scholar] [CrossRef] - Pandey, P.; Mitra, D. Clustering and energy spectra in two-dimensional dusty gas turbulence. Phys. Rev. E
**2019**, 100, 013114. [Google Scholar] [CrossRef] - Sargentis, G.F.; Ioannidis, R.; Chiotinis, M.; Dimitriadis, P.; Koutsoyiannis, D. Aesthetical Issues with Stochastic Evaluation. In Data Analytics for Cultural Heritage; Belhi, A., Bouras, A., Al-Ali, A.K., Sadka, A.H., Eds.; Springer: Cham, Switzerland, 2021. [Google Scholar] [CrossRef]
- Sargentis, G.-F.; Dimitriadis, P.; Koutsoyiannis, D. Aesthetical Issues of Leonardo Da Vinci’s and Pablo Picasso’s Paintings with Stochastic Evaluation. Heritage
**2020**, 3, 283–305. [Google Scholar] [CrossRef] - Sargentis, G.-F.; Dimitriadis, P.; Iliopoulou, T.; Koutsoyiannis, D. A Stochastic View of Varying Styles in Art Paintings. Heritage
**2021**, 4, 21. [Google Scholar] [CrossRef] - Sargentis, G.-F.; Dimitriadis, P.; Ioannidis, R.; Iliopoulou, T.; Koutsoyiannis, D. Stochastic Evaluation of Landscapes Transformed by Renewable Energy Installations and Civil Works. Energies
**2019**, 12, 2817. [Google Scholar] [CrossRef] - Sargentis, G.-F.; Ioannidis, R.; Iliopoulou, T.; Dimitriadis, P.; Koutsoyiannis, D. Landscape Planning of Infrastructure through Focus Points’ Clustering Analysis. Case Study: Plastiras Artificial Lake (Greece). Infrastructures
**2021**, 6, 12. [Google Scholar] [CrossRef] - Sargentis, G.-F.; Iliopoulou, T.; Sigourou, S.; Dimitriadis, P.; Koutsoyiannis, D. Evolution of clustering quantified by a stochastic method—Case studies on natural and human social structures. Sustainability
**2020**, 12, 7972. [Google Scholar] [CrossRef] - Dimitriadis, P.; Koutsoyiannis, D.; Onof, C. N-Dimensional generalized Hurst-Kolmogorov process and its application to wind fields. In Proceedings of the Facets of Uncertainty: 5th EGU Leonardo Conference—Hydrofractals 2013—STAHY 2013, Kos Island, Greece, 17–19 October 2013. [Google Scholar] [CrossRef]
- Lombardo, F.C.; Volpi, E.; Koutsoyiannis, D.; Papalexiou, S.M. Just two moments! A cautionary note against use of high-order moments in multifractal models in hydrology. Hydrol. Earth Syst. Sci.
**2014**, 18, 243–255. [Google Scholar] [CrossRef] - Geweke, J.; Porter-Hudak, S. The estimation and application of long memory time series models. J. Time Ser. Anal.
**1983**, 4, 221–238. [Google Scholar] [CrossRef] - Beran, J. Statistical Methods for Data with Long-Range Dependence. Stat. Sci.
**1992**, 7, 404–416. [Google Scholar] - Beran, J. Estimation, Testing and Prediction for Self-Similar and Related Processes. Ph.D. Thesis, ETH Zurich, Zurich, Switzerland, 1986. [Google Scholar]
- Smith, F.H. An empirical law describing heterogeneity in the yields of agricultural crops. Agric. Sci.
**1938**, 28, 1–23. [Google Scholar] [CrossRef] - Cox, D.R. Long-Range Dependence: A review, Statistics: An Appraisal. In Proceedings of the 50th Anniversary Conference; David, H.A., David, H.T., Eds.; Iowa State University Press: Ames, IA, USA, 1984; pp. 55–74. [Google Scholar]
- Beran, J. A Test of Location for Data with Slowly Decaying Serial Correlations. Biometrika
**1989**, 76, 261. [Google Scholar] [CrossRef] - Mandelbrot, B.B.; Van Ness, J.W. Fractional Brownian motions, fractional noises and applications. J. Soc. Ind. Appl. Math.
**1968**, 10, 422–437. [Google Scholar] [CrossRef] - Granger, C.W.J.; Joyeux, R. An Introduction to Long-memory Time Series, Models and Fractional Differencing. J. Time Ser. Anal.
**1980**, 1, 15–29. [Google Scholar] [CrossRef] - Beran, J. Statistical Aspects of Stationary Processes with Long-Range Dependence; Mimeo Series 1743; University of North Carolina: Chapel Hill, NC, USA, 1988. [Google Scholar]
- Cannon, M.J.; Percival, D.B.; Caccia, D.C.; Raymond, G.M.; Bassingthwaighte, J.B. Evaluating scaled windowed variance methods for estimating the Hurst coefficient of time series. Phys. A Stat. Mech. Appl.
**1997**, 241, 606–626. [Google Scholar] [CrossRef] - Montanari, A.; Rosso, R.; Taqqu, M.S. Fractionally differenced ARIMA models applied to hydrologic time series: Identification, estimation, and simulation. Water Resour. Res.
**1997**, 33, 1035–1044. [Google Scholar] [CrossRef] - Montanari, A.; Taqqu, M.S.; Teverovsky, V. Estimating long-range dependence in the presence of periodicity: An empirical study. Math. Comput. Model.
**1999**, 29, 217–228. [Google Scholar] [CrossRef] - Bloschl, G.; Sivapalan, M. Scale issues in hydrological modelling: A review. In Scale Issues in Hydrological Modelling; Kalma, J.D., Sivapalan, M., Eds.; John Wiley: New York, NY, USA, 1995; pp. 9–48. [Google Scholar]
- Koutsoyiannis, D. The Hurst phenomenon and fractional Gaussian noise made easy. Hydrol. Sci. J.
**2002**, 47, 573–595. [Google Scholar] [CrossRef] - Dimitriadis, P.; Koutsoyiannis, D. Climacogram versus autocovariance and power spectrum in stochastic modelling for Markovian and Hurst–Kolmogorov processes. Stoch. Environ. Res. Risk Assess.
**2015**, 29, 1649–1669. [Google Scholar] [CrossRef] - Koutsoyiannis, D. Climate change impacts on hydrological science: A comment on the relationship of the climacogram with Allan variance and variogram. ResearchGate
**2018**. [Google Scholar] [CrossRef] - Koutsoyiannis, D. Climate change, the Hurst phenomenon, and hydrological statistics. Hydrol. Sci. J.
**2003**, 48, 3–24. [Google Scholar] [CrossRef] - Tyralis, H.; Koutsoyiannis, D. Simultaneous estimation of the parameters of the Hurst–Kolmogorov stochastic process. Stoch. Environ. Res. Risk Assess.
**2010**, 25, 21–33. [Google Scholar] [CrossRef] - Mamassis, N.; Efstratiadis, A.; Dimitriadis, P.; Iliopoulou, T.; Ioannidis, R.; Koutsoyiannis, D. Water and Energy, Handbook of Water Resources Management: Discourses, Concepts and Examples; Bogardi, J.J., Tingsanchali, T., Nandalal, K.D.W., Gupta, J., Salamé, L., van Nooijen, R.R.P., Kolechkina, A.G., Kumar, N., Bhaduri, A., Eds.; Springer Nature: Cham, Switzerland, 2021; Chapter 20; pp. 617–655. [Google Scholar] [CrossRef]
- Papoulakos, K.; Pollakis, G.; Moustakis, Y.; Markopoulos, A.; Iliopoulou, T.; Dimitriadis, P.; Koutsoyiannis, D.; Efstratiadis, A. Simulation of water-energy fluxes through small-scale reservoir systems under limited data availability. Energy Procedia
**2017**, 125, 405–414. [Google Scholar] [CrossRef] - Koutsoyiannis, D. Time’s arrow in stochastic characterization and simulation of atmospheric and hydrological processes. Hydrol. Sci. J.
**2019**, 64, 1013–1037. [Google Scholar] [CrossRef] - Vavoulogiannis, S.; Iliopoulou, T.; Dimitriadis, P.; Koutsoyiannis, D. Multiscale Temporal Irreversibility of Streamflow and Its Stochastic Modelling. Hydrology
**2021**, 8, 63. [Google Scholar] [CrossRef] - Rozos, E.; Dimitriadis, P.; Mazi, K.; Koussis, A.D. A Multilayer Perceptron Model for Stochastic Synthesis. J. Hydrol.
**2021**, 8, 67. [Google Scholar] [CrossRef] - Zhang, H.; Fritts, J.E.; Goldman, S.A. An entropy-based objective evaluation method for image segmentation. In Proceedings of the SPIE 5307, Storage and Retrieval Methods and Applications for Multimedia, San Jose, CA, USA, 20 January 2004. [Google Scholar] [CrossRef]
- Kang, H.S.; Chester, S.; Meneveau, C. Decaying turbulence in an active-grid-generated flow and comparisons with large-eddy simulation. J. Fluid Mech.
**2003**, 480, 129–160. [Google Scholar] [CrossRef] - Evolution of the Universe. Available online: http://timemachine.cmucreatelab.org/wiki/Early_Universe (accessed on 13 October 2020).
- Di Matteo, T.; Colberg, J.; Springel, V.; Hernquist, L.; Sijacki, D. Direct cosmological simulations of the growth of black holes and galaxies. Astrophys. J.
**2008**, 676, 33–53. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) In 1951, H.E. Hurst discovered the clustering behavior in nature while (

**b**) A.N. Kolmogorov proposed a decade before a stochastic process that describes this clustering behavior. Source: Wikipedia.

**Figure 2.**Hurst–Kolmogorov (HK) dynamics present in the annual minimum water level of the Nile River as a result of the perpetual change of Earth’s climate, and as compared to a roulette timeseries resembling a white noise process. Source: [14] (supplementary material).

**Figure 3.**(

**a**,

**c**) Examples of data series with different statistical characteristics in one dimension. (

**b**,

**d**) Standardized climacograms of the data series shown in (

**a**,

**c**).

**Figure 4.**Standardized second-order stochastic metrics for the longitudinal velocity of the grid-turbulence dataset: (

**a**) climacogram; (

**b**) autocovariance; (

**c**) variogram; (

**d**) power spectrum. Source: [56].

**Figure 5.**Benchmark image analysis; (

**a**) white noise; (

**b**) image with clustering; (

**c**) art painting. Source: [35].

**Figure 6.**(

**a**) Climacograms of the benchmark images; (

**b**) standardized climacograms (i.e., divided by the variance of the image) of the benchmark images. Source: [35].

**Figure 7.**Images of sandstone as seen (

**a**) from the SEM (50 μm); (

**b**) from a polarizing microscope, (3.5 mm); (

**c**) from a hand specimen (with length approximately 5 cm); and (

**d**) from a field outcrop (1 m). For more information on the source, description and processing of the images, see [32].

**Figure 8.**Standardized climacograms of sandstone images depicted at four different scales and the fitted 2D HK model (source: [32]).

**Figure 9.**The image on the left shows Hurricane Sandy from a satellite view (source: www.nhc.noaa.gov/data; accessed on 15 January 2013), on which wind velocity and directions measurements are based. The images on the right represent the wind speed magnitude (in m/s) across a 150 × 150 grid, with 10 km resolution, centralized to the eye of the hurricane observed on (from upper left to lower right) 23, 25, 27 and 29 October 2012 (more details in [40]).

**Figure 10.**Climacograms (i.e., aggregated variance of brightness vs. spatial scale in units k × 10 km) for the (

**a**) 23rd, (

**b**) 25th, (

**c**) 27th and (

**d**) 29th day of October 2012. Source: [40].

**Figure 11.**Benchmark of image analysis, the evolution of the universe [68]: (

**a**) 500 million years after Big Bang image with faint clustering, average brightness 0.45; (

**b**) 5000 million years after Big Bang image with clustering, average brightness 0.37; (

**c**) 10,000 million years after Big Bang image with intense clustering, average brightness 0.33. Source: [39].

**Figure 12.**Example of the stochastic analysis of a 2D picture, in escalating spatial scales, as shown in red color on the left. The variability of the grouped pixels at different scales are used to construct the climacogram’s images; (

**a**–

**c**) correspond to different time frames as given in Figure 11. Source: [39].

**Figure 13.**(

**a**) Climacograms of the benchmark images; (

**b**) standardized climacograms of the benchmark images. A standardized climacogram is not helpful to evaluate the range of the evolution of clustering but is helpful to estimate the curves’ slope for further analysis. Source: [39].

**Figure 14.**(

**a**) Cumulative areas underneath each climacogram for each scale; (

**b**) rate of alteration. Source: [39].

**Figure 15.**Example of image analysis, evolution of the universe (zoomed view): Images of millions of years after the Big Bang (

**a**) 1000; (

**b**) 2000; (

**c**) 3000; (

**d**) 6000. Source: [39].

**Table 1.**Climacogram definition and expressions for a 2D continue process, a discretized one, a common estimator for the climacogram and the estimated value, based on this estimator.

Type | 2D Climacogram | |
---|---|---|

continuous | $\gamma \left({k}_{1},{k}_{2}\right):=\frac{\mathrm{Var}\left[{{\displaystyle \int}}_{{t}_{1}}^{{t}_{1}+{k}_{1}}{{\displaystyle \int}}_{{t}_{L}}^{{t}_{L}+{k}_{L}}\underset{\_}{x}\left({\xi}_{1,}{\xi}_{2}\right)\mathrm{d}{\xi}_{1}\mathrm{d}{\xi}_{2}\right]}{{\left({k}_{1}{k}_{2}\right)}^{2}}$ | (T1-1) |

discretized | ${\gamma}^{\left(\Delta \right)}\left({\kappa}_{1}{\Delta}_{1},{\kappa}_{2}{\Delta}_{2}\right):=\gamma \left({k}_{1},{k}_{2}\right)$ | (T1-2) |

typical estimator | $\underset{\_}{\widehat{\gamma}}\left({\kappa}_{1}{\Delta}_{1},{\kappa}_{2}{\Delta}_{2}\right)=\frac{{\kappa}_{1}{\kappa}_{2}}{{n}_{1}{n}_{2}}{\displaystyle {\displaystyle \sum}_{{r}_{2}=1}^{[{n}_{2}/{\kappa}_{2}]}}{\displaystyle {\displaystyle \sum}_{{r}_{1}=1}^{[{n}_{1}/{\kappa}_{1}]}}\left(\frac{1}{{\kappa}_{1}{\kappa}_{2}}\right({\displaystyle {\displaystyle \sum}_{{i}_{2}={\kappa}_{2}({r}_{2}-1)+1}^{{\kappa}_{2}{r}_{2}}}{\displaystyle {\displaystyle \sum}_{{i}_{1}={\kappa}_{1}({r}_{1}-1)+1}^{{\kappa}_{1}{r}_{1}}}{\underset{\_}{x}}_{{i}_{1},{i}_{2}}^{\left({\Delta}_{1},{\Delta}_{2}\right)})-\frac{{{\displaystyle \sum}}_{{i}_{2}=1}^{{n}_{2}}{{\displaystyle \sum}}_{{i}_{1}=1}^{{n}_{1}}{\underset{\_}{x}}_{{i}_{1},{i}_{2}}^{\left({\Delta}_{1},{\Delta}_{2}\right)}}{{n}_{1}{n}_{2}}),\mathrm{where}[{n}_{l}/{\kappa}_{l}]\mathrm{is}\mathrm{the}\mathrm{floor}\mathrm{of}{n}_{l}/{\kappa}_{l}$ | (T1-3) |

expected value of estimator | $\mathrm{E}\left[\underset{\_}{\widehat{\gamma}}\left({\kappa}_{1}{\Delta}_{1},{\kappa}_{2}{\Delta}_{2}\right)\right]=\gamma \left({\kappa}_{1}{\Delta}_{1},{\kappa}_{2}{\Delta}_{2}\right)-\gamma \left({n}_{1}{\Delta}_{1},{n}_{2}{\Delta}_{2}\right)$ | (T1-4) |

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**MDPI and ACS Style**

Dimitriadis, P.; Iliopoulou, T.; Sargentis, G.-F.; Koutsoyiannis, D.
Spatial Hurst–Kolmogorov Clustering. *Encyclopedia* **2021**, *1*, 1010-1025.
https://doi.org/10.3390/encyclopedia1040077

**AMA Style**

Dimitriadis P, Iliopoulou T, Sargentis G-F, Koutsoyiannis D.
Spatial Hurst–Kolmogorov Clustering. *Encyclopedia*. 2021; 1(4):1010-1025.
https://doi.org/10.3390/encyclopedia1040077

**Chicago/Turabian Style**

Dimitriadis, Panayiotis, Theano Iliopoulou, G.-Fivos Sargentis, and Demetris Koutsoyiannis.
2021. "Spatial Hurst–Kolmogorov Clustering" *Encyclopedia* 1, no. 4: 1010-1025.
https://doi.org/10.3390/encyclopedia1040077