Submitted Successfully!
To reward your contribution, here is a gift for you: A free trial for our video production service.
Thank you for your contribution! You can also upload a video entry or images related to this topic.
Version Summary Created by Modification Content Size Created at Operation
1 This method enables meaningful quantitative comparison of clustering patterns at various spatial scales and among different time spots, and it is a useful metric for understanding and quantifying the change in clustering through time.
G.-Fivos Sargentis
 + 2196 word(s) 2196 2020-09-28 11:48:12 |
2 format correct
Conner Chen
 Meta information modification 2196 2020-10-14 06:09:15 | |
3 Details added.
G.-Fivos Sargentis
 + 6 word(s) 2202 2020-10-15 08:03:13 | |
4 Format is changed
G.-Fivos Sargentis
 + 6 word(s) 2202 2020-10-15 08:05:22 | |
5 Format is changed
G.-Fivos Sargentis
 + 6 word(s) 2202 2020-10-15 08:06:52 | |
6 Format is changed
G.-Fivos Sargentis
 Meta information modification 2202 2020-10-15 18:50:32 | |
7 Type-writing errors
G.-Fivos Sargentis
 -2 word(s) 2200 2020-10-15 20:22:34 | |
8 format correct
Conner Chen
 Meta information modification 2200 2020-10-30 11:03:17 | |
9 format correct
Nora Tang
 Meta information modification 2200 2020-11-10 09:35:39 | |
10 format correct
Conner Chen
 Meta information modification 2200 2020-12-23 07:50:12 | |
11 format correct
Conner Chen
 Meta information modification 2200 2020-12-23 07:50:44 | |
12 Change the title to make it the same as the published entry
Dean Liu
 + 4 word(s) 2204 2021-10-28 08:13:51 | |
13 update the definition
Dean Liu
 + 49 word(s) 2249 2021-10-28 08:16:52 | |
14 format correct
Conner Chen
 -1772 word(s) 477 2022-04-14 07:42:43 |

Video Upload Options

We provide professional Video Production Services to translate complex research into visually appealing presentations. Would you like to try it?
No, upload directly Yes

Confirm

Are you sure to Delete?
Yes No
Cite
If you have any further questions, please contact Encyclopedia Editorial Office.
Sargentis, G.; Iliopoulou, T.; Sigourou, S.; Dimitriadis, P.; Koutsoyiannis, D. Spatial Hurst–Kolmogorov Clustering. Encyclopedia. Available online: https://encyclopedia.pub/entry/2568 (accessed on 23 November 2024).
Sargentis G, Iliopoulou T, Sigourou S, Dimitriadis P, Koutsoyiannis D. Spatial Hurst–Kolmogorov Clustering. Encyclopedia. Available at: https://encyclopedia.pub/entry/2568. Accessed November 23, 2024.
Sargentis, G.-Fivos, Theano Iliopoulou, Stavroula Sigourou, Panayiotis Dimitriadis, Demetris Koutsoyiannis. "Spatial Hurst–Kolmogorov Clustering" Encyclopedia, https://encyclopedia.pub/entry/2568 (accessed November 23, 2024).
Sargentis, G., Iliopoulou, T., Sigourou, S., Dimitriadis, P., & Koutsoyiannis, D. (2020, October 14). Spatial Hurst–Kolmogorov Clustering. In Encyclopedia. https://encyclopedia.pub/entry/2568
Sargentis, G.-Fivos, et al. "Spatial Hurst–Kolmogorov Clustering." Encyclopedia. Web. 14 October, 2020.
Peer Reviewed
Spatial Hurst–Kolmogorov Clustering
This is a peer-reviewed entry published in Encyclopedia Journal, full entry can be found at 10.3390/encyclopedia1040077

The stochastic analysis in the scale domain (instead of the traditional lag or frequency domains) is introduced as a robust means to identify, model and simulate the Hurst–Kolmogorov (HK) dynamics, ranging from small (fractal) to large scales exhibiting the clustering behavior (else known as the Hurst phenomenon or long-range dependence). The HK clustering is an attribute of a multidimensional (1D, 2D, etc.) spatio-temporal stationary stochastic process with an arbitrary marginal distribution function, and a fractal behavior on small spatio-temporal scales of the dependence structure and a power-type on large scales, yielding a high probability of low- or high-magnitude events to group together in space and time. This behavior is preferably analyzed through the second-order statistics, and in the scale domain, by the stochastic metric of the climacogram, i.e., the variance of the averaged spatio-temporal process vs. spatio-temporal scale.

stochastic method clustering evolution of clustering

Clustering in nature has been first identified by H.E. Hurst (1951) [1] (Figure 1a) while studying the long-term behaviour in a variety of scales of the discharge timeseries of the Nile River in the framework of developing engineering projects in its basin.

Particularly, H.E. Hurst discovered a tendency of high-discharge years to cluster into high-flow periods, and low-discharge years to cluster into low-flow periods. This behaviour, also known as the Hurst phenomenon or Joseph effect (Mandelbrot, 1977) [2], has been verified in a variety of hydrological [3], hydrometeorological and turbulent processes [4] [5] and in other geophysical and alternate fields such as finance, medicine [6], and art [7] [8] [9].

All these processes are characterized by long-term persistence (LTP), which leads to high unpredictability in long-term scales due to the clustering of events as compared to the purely random process, i.e. white-noise (e.g. as in a fair dice game [10]), or other short-range dependence models (e.g., Markov).

(a) (b)

Figure 1: (a) In 1951 H.E. Hurst discovered the clustering behaviour in nature (b) A.N.Kolmogorov proposed a decade before a stochastic process that describes this clustering behaviour. 

The mathematical description of the Hurst phenomenon is attributed to A.N. Kolmogorov (Figure 1b) who developed it while studying turbulence in 1940 [11] (Figure 1b), inspiring D. Koutsoyiannis [12] to name the general behaviour of the Hurst phenomenon as Hurst-Kolmogorov (HK) dynamics (Figure 2), to give credit to both contributing scientists and to distinguish it from the Gaussian LTP processes (e.g., fractional-Gaussian-noise [13]), and to incorporate alternate short-range dependence (e.g., Markov-behaviour [14]). 


Figure 2. Hurst-Kolmogorov (HK) dynamics and the perpetual change of Earth’s climate

The HK dynamics has been recently also linked to the entropy maximization principle, and thus, to robust physical justification [15]. The stochastic simulation of the HK dynamics has been a mathematical challenge since it requires the explicit preservation of high-order moments  in a vast range of scales, affecting both the intermittent behaviour in small scales [16]  and the dependence in extremes [17] as well as the trends often appearing in geophysical timeseries [18].

References

  1. Hurst, H.E.. Long term storage capacities of reservoirs; Trans. Am. Soc. Civil Eng: USA, 1951; pp. 116.
  2. Mandelbrot B. B. . Fractals: Form, Chance and Dimension; W. H. Freeman and Co.: US, 1977; pp. 365.
  3. Koutsoyiannis D.; The Hurst phenomenon and fractional Gaussian noise made easy. Hydrological Sciences Journal 2002, 47, 573-595, 10.1080/02626660209492961.
  4. Koutsoyiannis D.; Dimitriadis P.; Lombardo F.; Stevens S., From fractals to stochastics: Seeking theoretical consistency in analysis of geophysical data, Advances in Nonlinear Geosciences, edited by A.A. Tsonis, 237–278, doi:10.1007/978-3-319-58895-7_14, Springer, 2018.
  5. Dimitriadis, P. Hurst-Kolmogorov Dynamics in Hydrometeorological Processes and in the Microscale of Turbulence. Ph.D. Thesis, Department of Water Resources and Environmental Engineering—National Technical University of Athens, Athens, Greece, 2017.
  6. Mandelbrot B.; Van Ness J.W.; Fractional Brownian motions, fractional noises and applications. J. Soc. Ind. Appl. Math.l 1968, 10, 422-437.
  7. 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, 10.3390/heritage3020017.
  8. Sargentis G.-F.; Ioannidis R.; Meletopoulos I.T.; Dimitriadis P.; Koutsoyiannis; D. ; Aesthetical issues with stochastic evaluation, European Geosciences Union General Assembly 2020, Geophysical Research Abstracts, Vol. 22, Online, EGU2020- 19832, https://doi.org/10.5194/egusphere-egu2020-19832, European Geosciences Union, 2020.
  9. Sargentis G.-F.; Dimitriadis P.; Iliopoulou T.; Ioannidis R.; Koutsoyiannis D.; Stochastic investigation of the Hurst-Kolmogorov behaviour in arts, European Geosciences Union General Assembly 2018, Geophysical Research Abstracts, Vol. 20, Vienna, EGU2018-17082, European Geosciences Union, 2018.
  10. Dimitriadis P.; Koutsoyiannis D.; Tzouka K.; Predictability in dice motion: how does it differ from hydrometeorological processes?. Hydrological Sciences Journal 2016, 61, 1611-1622, 10.1080/02626667.2015.1034128.
  11. Kolmogorov, A.N.. Wiener spirals and some other interesting curves in a Hilbert space. Selected Works of A. N. Kolmogorov ; Kluwer, Dordrecht, Eds.; Mathematics and Mechanics, Tikhomirov, V. M. : The Netherlands, 1991; pp. 303-307.
  12. Koutsoyiannis D.; A random walk on water. Hydrology and Earth System Sciences 2010, 14, 585–601.
  13. Mandelbrot, B. B.; A Fast Fractional Gaussian Noise Generator. Water Resour. Res. 1971, 7(3), 543–553..
  14. Gneiting T.; Schlather M.; Stochastic Models That Separate Fractal Dimension and the Hurst Effect. SIAM Review 2004, 46, 269-282, 10.1137/s0036144501394387.
  15. Koutsoyiannis D.; Hurst–Kolmogorov dynamics as a result of extremal entropy production. Physica A: Statistical Mechanics and its Applications 2011, 390, 1424-1432, 10.1016/j.physa.2010.12.035.
  16. Dimitriadis P.; Koutsoyiannis D.; Stochastic synthesis approximating any process dependence and distribution. Stochastic Environmental Research and Risk Assessment 2018, 32, 1493-1515, 10.1007/s00477-018-1540-2.
  17. Iliopoulou T.; Koutsoyiannis D.; Revealing hidden persistence in maximum rainfall records. Hydrological Sciences Journal 2019, 64, 1673-1689, 10.1080/02626667.2019.1657578.
  18. Iliopoulou T.; Koutsoyiannis D.; Projecting the future of rainfall extremes: Better classic than trendy. Journal of Hydrology 2020, 588, 125005, 10.1016/j.jhydrol.2020.125005.
More
© Text is available under the terms and conditions of the Creative Commons Attribution (CC BY) license; additional terms may apply. By using this site, you agree to the Terms and Conditions and Privacy Policy.
Upload a video for this entry
Information
Subjects: Statistics & Probability
Contributors MDPI registered users' name will be linked to their SciProfiles pages. To register with us, please refer to https://encyclopedia.pub/register :
G.-Fivos Sargentis
,
Theano Iliopoulou
,
Stavroula Sigourou
,
Panayiotis Dimitriadis
,
Demetris Koutsoyiannis
View Times: 1.4K
Online Date: 14 Oct 2020
Table of Contents
    1000/1000
    Hot Most Recent
    ScholarVision Creations
    Feedback