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

[doc_id=2000]

[English]

Much is said and written about hydroclimatic hazards: storms, floods, droughts. Such hazards have existed and will always exist, while the usual scaremongering on them is of little help to avoid them. Instead, what is needed is a cool look at risk, based on measurement data, using scientific methodology, and ultimately employing technology in the service of reducing hazards and their consequences.

This is attempted in the book. Much of it is devoted to the theory of stochastics —the mathematical language for analysing extremes. Stochastics is a scientific area broader than statistics —according to the definition adopted in the book, statistics is part of stochastics. Another part is the theory of stochastic processes, in which time has a hypostasis that is typically absent in statistics. Thus, statistics is in relation to stochastic what statics are in relation to dynamics. The commonly used classical statistics (based on the assumption of independence) is a special case of stochastics and, as the book proves, is inappropriate for the subject. This does not mean that statistics are abandoned or underrepresented in the book. On the contrary, several new developments are presented —most notably the new tool of knowable moments, which have two relevant characteristics: they are closely connected to extremes and their estimation is unbiased in the framework of classical statistics or involves small (and determinable) bias in stochastic processes with dependence in time, whilst the bias in the estimation of classical statistical moments can be huge.

The new theoretical analyses are supported by mathematical proofs, which, to improve readability, are contained in a number of appendices in each of the 10 main chapters of the book. Along with the development of the theory, the book is oriented to the application, which is supported by a variety of examples, usually standing out as parenthetical sections or Digressions, as well as by tabulations of mathematical formulae that are used for each task.

**
Full text
**
(13157 KB)

**Additional material:**

- Cover (1532 KB)
- Cover (high resolution, jpeg) (5594 KB)
- Draft text (Edition 0, 2020) (8621 KB)

**Remarks:**

- Official publisher's site: https://repository.kallipos.gr/handle/11419/6522?locale=en
- Internet Archive's site: https://archive.org/details/stochastics-of-extremes-1
- ResearchGate's site: https://www.researchgate.net/publication/351081149
- Itia's site: http://www.itia.ntua.gr/2000/

**Our works referenced by this work:**

1. | D. Koutsoyiannis, A stochastic disaggregation method for design storm and flood synthesis, Journal of Hydrology, 156, 193–225, doi:10.1016/0022-1694(94)90078-7, 1994. |

2. | D. Koutsoyiannis, and A. Manetas, Simple disaggregation by accurate adjusting procedures, Water Resources Research, 32 (7), 2105–2117, doi:10.1029/96WR00488, 1996. |

3. | D. Koutsoyiannis, Statistical Hydrology, Edition 4, 312 pages, doi:10.13140/RG.2.1.5118.2325, National Technical University of Athens, Athens, 1997. |

4. | D. Koutsoyiannis, D. Kozonis, and A. Manetas, A mathematical framework for studying rainfall intensity-duration-frequency relationships, Journal of Hydrology, 206 (1-2), 118–135, doi:10.1016/S0022-1694(98)00097-3, 1998. |

5. | D. Koutsoyiannis, A probabilistic view of Hershfield's method for estimating probable maximum precipitation, Water Resources Research, 35 (4), 1313–1322, doi:10.1029/1999WR900002, 1999. |

6. | D. Koutsoyiannis, and Th. Xanthopoulos, Engineering Hydrology, Edition 3, 418 pages, doi:10.13140/RG.2.1.4856.0888, National Technical University of Athens, Athens, 1999. |

7. | D. Koutsoyiannis, A generalized mathematical framework for stochastic simulation and forecast of hydrologic time series, Water Resources Research, 36 (6), 1519–1533, doi:10.1029/2000WR900044, 2000. |

8. | D. Koutsoyiannis, Coupling stochastic models of different time scales, Water Resources Research, 37 (2), 379–391, doi:10.1029/2000WR900200, 2001. |

9. | D. Koutsoyiannis, The Hurst phenomenon and fractional Gaussian noise made easy, Hydrological Sciences Journal, 47 (4), 573–595, doi:10.1080/02626660209492961, 2002. |

10. | 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. |

11. | D. Koutsoyiannis, and A. Economou, Evaluation of the parameterization-simulation-optimization approach for the control of reservoir systems, Water Resources Research, 39 (6), 1170, doi:10.1029/2003WR002148, 2003. |

12. | D. Koutsoyiannis, Statistics of extremes and estimation of extreme rainfall, 1, Theoretical investigation, Hydrological Sciences Journal, 49 (4), 575–590, doi:10.1623/hysj.49.4.575.54430, 2004. |

13. | D. Koutsoyiannis, Statistics of extremes and estimation of extreme rainfall, 2, Empirical investigation of long rainfall records, Hydrological Sciences Journal, 49 (4), 591–610, doi:10.1623/hysj.49.4.591.54424, 2004. |

14. | D. Koutsoyiannis, Uncertainty, entropy, scaling and hydrological stochastics, 1, Marginal distributional properties of hydrological processes and state scaling, Hydrological Sciences Journal, 50 (3), 381–404, doi:10.1623/hysj.50.3.381.65031, 2005. |

15. | D. Koutsoyiannis, Uncertainty, entropy, scaling and hydrological stochastics, 2, Time dependence of hydrological processes and time scaling, Hydrological Sciences Journal, 50 (3), 405–426, doi:10.1623/hysj.50.3.405.65028, 2005. |

16. | D. Koutsoyiannis, Nonstationarity versus scaling in hydrology, Journal of Hydrology, 324, 239–254, doi:10.1016/j.jhydrol.2005.09.022, 2006. |

17. | D. Koutsoyiannis, An entropic-stochastic representation of rainfall intermittency: The origin of clustering and persistence, Water Resources Research, 42 (1), W01401, doi:10.1029/2005WR004175, 2006. |

18. | D. Koutsoyiannis, and A. Georgakakos, Lessons from the long flow records of the Nile: determinism vs indeterminism and maximum entropy, 20 Years of Nonlinear Dynamics in Geosciences, Rhodes, Greece, doi:10.13140/RG.2.2.10996.14727, 2006. |

19. | D. Koutsoyiannis, A. Efstratiadis, and K. Georgakakos, Uncertainty assessment of future hydroclimatic predictions: A comparison of probabilistic and scenario-based approaches, Journal of Hydrometeorology, 8 (3), 261–281, doi:10.1175/JHM576.1, 2007. |

20. | D. Koutsoyiannis, A critical review of probability of extreme rainfall: principles and models, Advances in Urban Flood Management, edited by R. Ashley, S. Garvin, E. Pasche, A. Vassilopoulos, and C. Zevenbergen, 139–166, doi:10.1201/9780203945988.ch7, Taylor and Francis, London, 2007. |

21. | D. Koutsoyiannis, and A. Montanari, Statistical analysis of hydroclimatic time series: Uncertainty and insights, Water Resources Research, 43 (5), W05429, doi:10.1029/2006WR005592, 2007. |

22. | D. Koutsoyiannis, A. Efstratiadis, N. Mamassis, and A. Christofides, On the credibility of climate predictions, Hydrological Sciences Journal, 53 (4), 671–684, doi:10.1623/hysj.53.4.671, 2008. |

23. | D. Koutsoyiannis, Probability and statistics for geophysical processes, doi:10.13140/RG.2.1.2300.1849/1, National Technical University of Athens, Athens, 2008. |

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

25. | D. Koutsoyiannis, Y. Markonis, A. Koukouvinos, S.M. Papalexiou, N. Mamassis, and P. Dimitriadis, Hydrological study of severe rainfall in the Kephisos basin, Greece, Study of the management of Kephisos , Commissioner: General Secretariat of Public Works – Ministry of Environment, Planning and Public Works, Contractors: Exarhou Nikolopoulos Bensasson, Denco, G. Karavokiris, et al., 154 pages, Athens, 2010. |

26. | G. G. Anagnostopoulos, D. Koutsoyiannis, A. Christofides, A. Efstratiadis, and N. Mamassis, A comparison of local and aggregated climate model outputs with observed data, Hydrological Sciences Journal, 55 (7), 1094–1110, doi:10.1080/02626667.2010.513518, 2010. |

27. | G. Di Baldassarre, A. Montanari, H. F. Lins, D. Koutsoyiannis, L. Brandimarte, and G. Blöschl, Flood fatalities in Africa: from diagnosis to mitigation, Geophysical Research Letters, 37, L22402, doi:10.1029/2010GL045467, 2010. |

28. | I. Mimiyianni, Stochastic investigation of the utility of multiple hydrological records for improving the reliability of estimation, Postgraduate Thesis, Department of Water Resources and Environmental Engineering – National Technical University of Athens, 2010. |

29. | H. Tyralis, and D. Koutsoyiannis, Simultaneous estimation of the parameters of the Hurst-Kolmogorov stochastic process, Stochastic Environmental Research & Risk Assessment, 25 (1), 21–33, 2011. |

30. | D. Koutsoyiannis, and A. Langousis, Precipitation, Treatise on Water Science, edited by P. Wilderer and S. Uhlenbrook, 2, 27–78, doi:10.1016/B978-0-444-53199-5.00027-0, Academic Press, Oxford, 2011. |

31. | D. Koutsoyiannis, Hurst-Kolmogorov dynamics and uncertainty, Journal of the American Water Resources Association, 47 (3), 481–495, doi:10.1111/j.1752-1688.2011.00543.x, 2011. |

32. | D. Koutsoyiannis, A. Paschalis, and N. Theodoratos, Two-dimensional Hurst-Kolmogorov process and its application to rainfall fields, Journal of Hydrology, 398 (1-2), 91–100, doi:10.1016/j.jhydrol.2010.12.012, 2011. |

33. | D. Koutsoyiannis, Hurst-Kolmogorov dynamics as a result of extremal entropy production, Physica A: Statistical Mechanics and its Applications, 390 (8), 1424–1432, doi:10.1016/j.physa.2010.12.035, 2011. |

34. | S.M. Papalexiou, and D. Koutsoyiannis, Entropy based derivation of probability distributions: A case study to daily rainfall, Advances in Water Resources, 45, 51–57, doi:10.1016/j.advwatres.2011.11.007, 2012. |

35. | D. Koutsoyiannis, Clausius-Clapeyron equation and saturation vapour pressure: simple theory reconciled with practice, European Journal of Physics, 33 (2), 295–305, doi:10.1088/0143-0807/33/2/295, 2012. |

36. | A. Montanari, and D. Koutsoyiannis, A blueprint for process-based modeling of uncertain hydrological systems, Water Resources Research, 48, W09555, doi:10.1029/2011WR011412, 2012. |

37. | H. Tyralis, D. Koutsoyiannis, and S. Kozanis, An algorithm to construct Monte Carlo confidence intervals for an arbitrary function of probability distribution parameters, Computational Statistics, 28 (4), 1501–1527, doi:10.1007/s00180-012-0364-7, 2013. |

38. | Y. Markonis, and D. Koutsoyiannis, Climatic variability over time scales spanning nine orders of magnitude: Connecting Milankovitch cycles with Hurst–Kolmogorov dynamics, Surveys in Geophysics, 34 (2), 181–207, doi:10.1007/s10712-012-9208-9, 2013. |

39. | S.M. Papalexiou, and D. Koutsoyiannis, Battle of extreme value distributions: A global survey on extreme daily rainfall, Water Resources Research, 49 (1), 187–201, doi:10.1029/2012WR012557, 2013. |

40. | D. Koutsoyiannis, Hydrology and Change, Hydrological Sciences Journal, 58 (6), 1177–1197, doi:10.1080/02626667.2013.804626, 2013. |

41. | D. Koutsoyiannis, Physics of uncertainty, the Gibbs paradox and indistinguishable particles, Studies in History and Philosophy of Modern Physics, 44, 480–489, doi:10.1016/j.shpsb.2013.08.007, 2013. |

42. | C. Pappas, S.M. Papalexiou, and D. Koutsoyiannis, A quick gap-filling of missing hydrometeorological data, Facets of Uncertainty: 5th EGU Leonardo Conference – Hydrofractals 2013 – STAHY 2013, Kos Island, Greece, doi:10.13140/RG.2.2.22772.96641, European Geosciences Union, International Association of Hydrological Sciences, International Union of Geodesy and Geophysics, 2013. |

43. | 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. |

44. | H. Tyralis, and D. Koutsoyiannis, A Bayesian statistical model for deriving the predictive distribution of hydroclimatic variables, Climate Dynamics, 42 (11-12), 2867–2883, doi:10.1007/s00382-013-1804-y, 2014. |

45. | G. Tsekouras, and D. Koutsoyiannis, Stochastic analysis and simulation of hydrometeorological processes associated with wind and solar energy, Renewable Energy, 63, 624–633, doi:10.1016/j.renene.2013.10.018, 2014. |

46. | D. Koutsoyiannis, Entropy: from thermodynamics to hydrology, Entropy, 16 (3), 1287–1314, doi:10.3390/e16031287, 2014. |

47. | D. Koutsoyiannis, Random musings on stochastics (Lorenz Lecture), AGU 2014 Fall Meeting, San Francisco, USA, doi:10.13140/RG.2.1.2852.8804, American Geophysical Union, 2014. |

48. | A. Montanari, and D. Koutsoyiannis, Modeling and mitigating natural hazards: Stationarity is immortal!, Water Resources Research, 50 (12), 9748–9756, doi:10.1002/2014WR016092, 2014. |

49. | A. Sikorska, A. Montanari, and D. Koutsoyiannis, Estimating the uncertainty of hydrological predictions through data-driven resampling techniques, Journal of Hydrologic Engineering (ASCE), 20 (1), doi:10.1061/(ASCE)HE.1943-5584.0000926, 2015. |

50. | D. Koutsoyiannis, and A. Montanari, Negligent killing of scientific concepts: the stationarity case, Hydrological Sciences Journal, 60 (7-8), 1174–1183, doi:10.1080/02626667.2014.959959, 2015. |

51. | 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. |

52. | P. Dimitriadis, and D. Koutsoyiannis, Application of stochastic methods to double cyclostationary processes for hourly wind speed simulation, Energy Procedia, 76, 406–411, doi:10.1016/j.egypro.2015.07.851, 2015. |

53. | E. Volpi, A. Fiori, S. Grimaldi, F. Lombardo, and D. Koutsoyiannis, One hundred years of return period: Strengths and limitations, Water Resources Research, doi:10.1002/2015WR017820, 2015. |

54. | 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. |

55. | 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. |

56. | P.E. O’Connell, D. Koutsoyiannis, H. F. Lins, Y. Markonis, A. Montanari, and T.A. Cohn, The scientific legacy of Harold Edwin Hurst (1880 – 1978), Hydrological Sciences Journal, 61 (9), 1571–1590, doi:10.1080/02626667.2015.1125998, 2016. |

57. | Y. Markonis, and D. Koutsoyiannis, Scale-dependence of persistence in precipitation records, Nature Climate Change, 6, 399–401, doi:10.1038/nclimate2894, 2016. |

58. | D. Tsaknias, D. Bouziotas, and D. Koutsoyiannis, Statistical comparison of observed temperature and rainfall extremes with climate model outputs in the Mediterranean region, ResearchGate, doi:10.13140/RG.2.2.11993.93281, 2016. |

59. | 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. |

60. | D. Koutsoyiannis, and S.M. Papalexiou, Extreme rainfall: Global perspective, Handbook of Applied Hydrology, Second Edition, edited by V.P. Singh, 74.1–74.16, McGraw-Hill, New York, 2017. |

61. | H. Tyralis, and D. Koutsoyiannis, On the prediction of persistent processes using the output of deterministic models, Hydrological Sciences Journal, 62 (13), 2083–2102, doi:10.1080/02626667.2017.1361535, 2017. |

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

63. | 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. |

64. | 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. |

65. | 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. |

66. | 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. |

67. | T. Iliopoulou, C. Aguilar , B. Arheimer, M. Bermúdez, N. Bezak, A. Ficchi, D. Koutsoyiannis, J. Parajka, M. J. Polo, G. Thirel, and A. Montanari, A large sample analysis of European rivers on seasonal river flow correlation and its physical drivers, Hydrology and Earth System Sciences, 23, 73–91, doi:10.5194/hess-23-73-2019, 2019. |

68. | 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. |

69. | E. Volpi, A. Fiori, S. Grimaldi, F. Lombardo, and D. Koutsoyiannis, Save hydrological observations! Return period estimation without data decimation, Journal of Hydrology, doi:10.1016/j.jhydrol.2019.02.017, 2019. |

70. | 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. |

71. | 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. |

72. | A. Efstratiadis, N. Mamassis, and I. Tsoukalas, Synoptic report on the evaluation of the flood risk for areas affected by the ongoing spilling of the Hylike-Paralimni system, Modernization of the management of the water supply system of Athens - Update, Contractor: Department of Water Resources and Environmental Engineering – National Technical University of Athens, 25 pages, March 2019. |

73. | F. Lombardo, F. Napolitano, F. Russo, and D. Koutsoyiannis, On the exact distribution of correlated extremes in hydrology, Water Resources Research, 55 (12), 10405–10423, doi:10.1029/2019WR025547, 2019. |

74. | K. Glynis, Stochastic investigation of the behavior of land surface air temperature on global scale, Diploma thesis, 159 pages, Athens, October 2019. |

75. | 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. |

76. | G. Papacharalampous, D. Koutsoyiannis, and A. Montanari, Quantification of predictive uncertainty in hydrological modelling by harnessing the wisdom of the crowd: Methodology development and investigation using toy models, Advances in Water Resources, 136, 103471, doi:10.1016/j.advwatres.2019.103471, 2020. |

77. | G. Papacharalampous, H. Tyralis, D. Koutsoyiannis, and A. Montanari, Quantification of predictive uncertainty in hydrological modelling by harnessing the wisdom of the crowd: A large-sample experiment at monthly timescale, Advances in Water Resources, 136, 103470, doi:10.1016/j.advwatres.2019.103470, 2020. |

78. | D. Koutsoyiannis, A voyage in climate, hydrology and life on a 4.5-billion-years old planet, Self-organized lecture, doi:10.13140/RG.2.2.27000.26883, School of Civil Engineering – National Technical University of Athens, Athens, 20 July 2020. |

79. | D. Koutsoyiannis, Revisiting the global hydrological cycle: is it intensifying?, Hydrology and Earth System Sciences, 24, 3899–3932, doi:10.5194/hess-24-3899-2020, 2020. |

80. | T. Iliopoulou, and D. Koutsoyiannis, Projecting the future of rainfall extremes: better classic than trendy, Journal of Hydrology, 588, doi:10.1016/j.jhydrol.2020.125005, 2020. |

81. | Z. W. Kundzewicz, I. Pińskwar, and D. Koutsoyiannis, Variability of global mean annual temperature is significantly influenced by the rhythm of ocean-atmosphere oscillations, Science of the Total Environment, 747, 141256, doi:10.1016/j.scitotenv.2020.141256, 2020. |

82. | D. Koutsoyiannis, and Z. W. Kundzewicz, Atmospheric temperature and CO₂: Hen-or-egg causality?, Sci, 2 (4), 83, doi:10.3390/sci2040083, 2020. |

83. | 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. |

84. | D. Koutsoyiannis, Rethinking climate, climate change, and their relationship with water, Water, 13 (6), 849, doi:10.3390/w13060849, 2021. |

**Our works that reference this work:**

1. | T. Iliopoulou, and D. Koutsoyiannis, Projecting the future of rainfall extremes: better classic than trendy, Journal of Hydrology, 588, doi:10.1016/j.jhydrol.2020.125005, 2020. |

2. | 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. |

3. | D. Koutsoyiannis, Rethinking climate, climate change, and their relationship with water, Water, 13 (6), 849, doi:10.3390/w13060849, 2021. |

4. | G.-F. Sargentis, T. Iliopoulou, P. Dimitriadis, N. Mamassis, and D. Koutsoyiannis, Stratification: An entropic view of society's structure, World, 2, 153–174, doi:10.3390/world2020011, 2021. |

5. | P. Dimitriadis, D. Koutsoyiannis, T. Iliopoulou, and P. Papanicolaou, A global-scale investigation of stochastic similarities in marginal distribution and dependence structure of key hydrological-cycle processes, Hydrology, 8 (2), 59, doi:10.3390/hydrology8020059, 2021. |

**
Works that cite this document:
**
View on
Google Scholar
or
ResearchGate

**Tagged under:**
Hurst-Kolmogorov dynamics,
Course bibliography: Stochastic methods,
Climate stochastics,
Determinism vs. stochasticity,
Entropy,
Extremes,
Most recent works,
Stochastics,
Uncertainty