P. Dimitriadis, and D. Koutsoyiannis, Climacogram versus autocovariance and power spectrum in stochastic modelling for Markovian and Hurst–Kolmogorov processes, *Stochastic Environmental Research & Risk Assessment*, 29 (6), 1649–1669, doi:10.1007/s00477-015-1023-7, 2015.

[doc_id=1514]

[English]

Three common stochastic tools, the climacogram i.e. variance of the time averaged process over averaging time scale, the autocovariance function and the power spectrum are compared to each other to assess each one’s advantages and disadvantages in stochastic modelling and statistical inference. Although in theory, all three are equivalent to each other (transformations one another expressing second order stochastic properties), in practical application their ability to characterize a geophysical process and their utility as statistical estimators may vary. In the analysis both Markovian and non Markovian stochastic processes, which have exponential and power-type autocovariances, respectively, are used. It is shown that, due to high bias in autocovariance estimation, as well as effects of process discretization and finite sample size, the power spectrum is also prone to bias and discretization errors as well as high uncertainty, which may misrepresent the process behaviour (e.g. Hurst phenomenon) if not taken into account. Moreover, it is shown that the classical climacogram estimator has small error as well as an expected value always positive, well-behaved and close to its mode (most probable value), all of which are important advantages in stochastic model building. In contrast, the power spectrum and the autocovariance do not have some of these properties. Therefore, when building a stochastic model, it seems beneficial to start from the climacogram, rather than the power spectrum or the autocovariance. The results are illustrated by a real world application based on the analysis of a long time series of high-frequency turbulent flow measurements.

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**See also:**
http://dx.doi.org/10.1007/s00477-015-1023-7

**Our works referenced by this work:**

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2. | D. Koutsoyiannis, Climate change, the Hurst phenomenon, and hydrological statistics, Hydrological Sciences Journal, 48 (1), 3–24, doi:10.1623/hysj.48.1.3.43481, 2003. |

3. | D. Koutsoyiannis, A random walk on water, Hydrology and Earth System Sciences, 14, 585–601, doi:10.5194/hess-14-585-2010, 2010. |

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6. | P. Dimitriadis, D. Koutsoyiannis, and Y. Markonis, Spectrum vs Climacogram, European Geosciences Union General Assembly 2012, Geophysical Research Abstracts, Vol. 14, Vienna, EGU2012-993, doi:10.13140/RG.2.2.27838.89920, European Geosciences Union, 2012. |

7. | S.M. Papalexiou, D. Koutsoyiannis, and C. Makropoulos, How extreme is extreme? An assessment of daily rainfall distribution tails, Hydrology and Earth System Sciences, 17, 851–862, doi:10.5194/hess-17-851-2013, 2013. |

8. | D. Koutsoyiannis, Encolpion of stochastics: Fundamentals of stochastic processes, doi:10.13140/RG.2.2.10956.82564, Department of Water Resources and Environmental Engineering – National Technical University of Athens, Athens, 2013. |

9. | D. Koutsoyiannis, Climacogram-based pseudospectrum: a simple tool to assess scaling properties, European Geosciences Union General Assembly 2013, Geophysical Research Abstracts, Vol. 15, Vienna, EGU2013-4209, doi:10.13140/RG.2.2.18506.57284, European Geosciences Union, 2013. |

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11. | F. Lombardo, E. Volpi, D. Koutsoyiannis, and S.M. Papalexiou, Just two moments! A cautionary note against use of high-order moments in multifractal models in hydrology, Hydrology and Earth System Sciences, 18, 243–255, doi:10.5194/hess-18-243-2014, 2014. |

**Our works that reference this work:**

1. | D. Koutsoyiannis, Generic and parsimonious stochastic modelling for hydrology and beyond, Hydrological Sciences Journal, 61 (2), 225–244, doi:10.1080/02626667.2015.1016950, 2016. |

2. | P. Dimitriadis, D. Koutsoyiannis, and K. Tzouka, Predictability in dice motion: how does it differ from hydrometeorological processes?, Hydrological Sciences Journal, 61 (9), 1611–1622, doi:10.1080/02626667.2015.1034128, 2016. |

3. | P. Dimitriadis, D. Koutsoyiannis, and P. Papanicolaou, Stochastic similarities between the microscale of turbulence and hydrometeorological processes, Hydrological Sciences Journal, 61 (9), 1623–1640, doi:10.1080/02626667.2015.1085988, 2016. |

4. | I. Deligiannis, P. Dimitriadis, Ο. Daskalou, Y. Dimakos, and D. Koutsoyiannis, Global investigation of double periodicity οf hourly wind speed for stochastic simulation; application in Greece, Energy Procedia, 97, 278–285, doi:10.1016/j.egypro.2016.10.001, 2016. |

5. | C. Pappas, M.D. Mahecha, D.C. Frank, F. Babst, and D. Koutsoyiannis, Ecosystem functioning is enveloped by hydrometeorological variability, Nature Ecology & Evolution, 1, 1263–1270, doi:10.1038/s41559-017-0277-5, 2017. |

6. | M. Chalakatevaki, P. Stamou, S. Karali, V. Daniil, P. Dimitriadis, K. Tzouka, T. Iliopoulou, D. Koutsoyiannis, P. Papanicolaou, and N. Mamassis, Creating the electric energy mix in a non-connected island, Energy Procedia, 125, 425–434, doi:10.1016/j.egypro.2017.08.089, 2017. |

7. | G. Koudouris, P. Dimitriadis, T. Iliopoulou, N. Mamassis, and D. Koutsoyiannis, Investigation on the stochastic nature of the solar radiation process, Energy Procedia, 125, 398–404, 2017. |

8. | E. Moschos, G. Manou, P. Dimitriadis, V. Afendoulis, D. Koutsoyiannis, and V. Tsoukala, Harnessing wind and wave resources for a Hybrid Renewable Energy System in remote islands: a combined stochastic and deterministic approach, Energy Procedia, 125, 415–424, doi:10.1016/j.egypro.2017.08.084, 2017. |

9. | D. Koutsoyiannis, Entropy production in stochastics, Entropy, 19 (11), 581, doi:10.3390/e19110581, 2017. |

10. | T. Iliopoulou, S.M. Papalexiou, Y. Markonis, and D. Koutsoyiannis, Revisiting long-range dependence in annual precipitation, Journal of Hydrology, 556, 891–900, doi:10.1016/j.jhydrol.2016.04.015, 2018. |

11. | P. Dimitriadis, and D. Koutsoyiannis, Stochastic synthesis approximating any process dependence and distribution, Stochastic Environmental Research & Risk Assessment, 32 (6), 1493–1515, doi:10.1007/s00477-018-1540-2, 2018. |

12. | D. Koutsoyiannis, P. Dimitriadis, F. Lombardo, and S. Stevens, 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. |

13. | Y. Markonis, Y. Moustakis, C. Nasika, P. Sychova, P. Dimitriadis, M. Hanel, P. Máca, and S.M. Papalexiou, Global estimation of long-term persistence in annual river runoff, Advances in Water Resources, 113, 1–12, doi:10.1016/j.advwatres.2018.01.003, 2018. |

14. | I. Tsoukalas, C. Makropoulos, and D. Koutsoyiannis, Simulation of stochastic processes exhibiting any-range dependence and arbitrary marginal distributions, Water Resources Research, 54 (11), 9484–9513, doi:10.1029/2017WR022462, 2018. |

15. | E. Klousakou, M. Chalakatevaki, P. Dimitriadis, T. Iliopoulou, R. Ioannidis, G. Karakatsanis, A. Efstratiadis, N. Mamassis, R. Tomani, E. Chardavellas, and D. Koutsoyiannis, A preliminary stochastic analysis of the uncertainty of natural processes related to renewable energy resources, Advances in Geosciences, 45, 193–199, doi:10.5194/adgeo-45-193-2018, 2018. |

16. | G. Koudouris, P. Dimitriadis, T. Iliopoulou, N. Mamassis, and D. Koutsoyiannis, A stochastic model for the hourly solar radiation process for application in renewable resources management, Advances in Geosciences, 45, 139–145, doi:10.5194/adgeo-45-139-2018, 2018. |

17. | I. Tsoukalas, A. Efstratiadis, and C. Makropoulos, Building a puzzle to solve a riddle: A multi-scale disaggregation approach for multivariate stochastic processes with any marginal distribution and correlation structure, Journal of Hydrology, 575, 354–380, doi:10.1016/j.jhydrol.2019.05.017, 2019. |

18. | P. Dimitriadis, K. Tzouka, D. Koutsoyiannis, H. Tyralis, A. Kalamioti, E. Lerias, and P. Voudouris, Stochastic investigation of long-term persistence in two-dimensional images of rocks, Spatial Statistics, 29, 177–191, doi:10.1016/j.spasta.2018.11.002, 2019. |

19. | D. Koutsoyiannis, Knowable moments for high-order stochastic characterization and modelling of hydrological processes, Hydrological Sciences Journal, 64 (1), 19–33, doi:10.1080/02626667.2018.1556794, 2019. |

20. | D. Koutsoyiannis, Time’s arrow in stochastic characterization and simulation of atmospheric and hydrological processes, Hydrological Sciences Journal, 64 (9), 1013–1037, doi:10.1080/02626667.2019.1600700, 2019. |

21. | P. Kossieris, I. Tsoukalas, C. Makropoulos, and D. Savic, Simulating marginal and dependence behaviour of water demand processes at any fine time scale, Water, 11 (5), 885, doi:10.3390/w11050885, 2019. |

22. | T. Iliopoulou, and D. Koutsoyiannis, Revealing hidden persistence in maximum rainfall records, Hydrological Sciences Journal, 64 (14), 1673–1689, doi:10.1080/02626667.2019.1657578, 2019. |

23. | R. Ioannidis, T. Iliopoulou, C. Iliopoulou, L. Katikas, A. Petsou, M.-E. Merakou, M.-E. Asimomiti, N. Pelekanos, G. Koudouris, P. Dimitriadis, C. Plati, E. Vlahogianni, K. Kepaptsoglou, N. Mamassis, and D. Koutsoyiannis, Solar-powered bus route: introducing renewable energy into a university campus transport system, Advances in Geosciences, 49, doi:10.5194/adgeo-49-215-2019, 2019. |

24. | D. Koutsoyiannis, Simple stochastic simulation of time irreversible and reversible processes, Hydrological Sciences Journal, 65 (4), 536–551, doi:10.1080/02626667.2019.1705302, 2020. |

25. | G.-F. Sargentis, P. Dimitriadis, and D. Koutsoyiannis, Aesthetical issues of Leonardo Da Vinci’s and Pablo Picasso’s paintings with stochastic evaluation, Heritage, 3 (2), 283–305, doi:10.3390/heritage3020017, 2020. |

26. | G.-F. Sargentis, T. Iliopoulou, S. Sigourou, P. Dimitriadis, and D. Koutsoyiannis, Evolution of clustering quantified by a stochastic method — Case studies on natural and human social structures, Sustainability, 12 (19), 7972, doi:10.3390/su12197972, 2020. |

27. | D. Koutsoyiannis, Stochastics of Hydroclimatic Extremes - A Cool Look at Risk, ISBN: 978-618-85370-0-2, 333 pages, Kallipos, Athens, 2021. |

28. | N. Mamassis, A. Efstratiadis, P. Dimitriadis, T. Iliopoulou, R. Ioannidis, and D. Koutsoyiannis, Water and Energy, Handbook of Water Resources Management: Discourses, Concepts and Examples, edited by J.J. Bogardi, T. Tingsanchali, K.D.W. Nandalal, J. Gupta, L. Salamé, R.R.P. van Nooijen, A.G. Kolechkina, N. Kumar, and A. Bhaduri, Chapter 20, 617–655, doi:10.1007/978-3-030-60147-8_20, Springer Nature, Switzerland, 2021. |

29. | G.-F. Sargentis, R. Ioannidis, T. Iliopoulou, P. Dimitriadis, and D. Koutsoyiannis, Landscape planning of infrastructure through focus points’ clustering analysis. Case study: Plastiras artificial lake (Greece), Infrastructures, 6 (1), 12, doi:10.3390/infrastructures6010012, 2021. |

30. | G.-F. Sargentis, P. Dimitriadis, T. Iliopoulou, and D. Koutsoyiannis, A stochastic view of varying styles in art paintings, Heritage, 4, 21, doi:10.3390/heritage4010021, 2021. |

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32. | L. Katikas, P. Dimitriadis, D. Koutsoyiannis, T. Kontos, and P. Kyriakidis, A stochastic simulation scheme for the long-term persistence, heavy-tailed and double periodic behavior of observational and reanalysis wind time-series, Applied Energy, 295, 116873, doi:10.1016/j.apenergy.2021.116873, 2021. |

33. | S. Vavoulogiannis, T. Iliopoulou, P. Dimitriadis, and D. Koutsoyiannis, Multiscale temporal irreversibility of streamflow and its stochastic modelling, Hydrology, 8 (2), 63, doi:10.3390/hydrology8020063, 2021. |

34. | D. Koutsoyiannis, and P. Dimitriadis, Towards generic simulation for demanding stochastic processes, Sci, 3, 34, doi:10.3390/sci3030034, 2021. |

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Scaling,
Stochastics