I. Tsoukalas, S.M. Papalexiou, A. Efstratiadis, and C. Makropoulos, A cautionary note on the reproduction of dependencies through linear stochastic models with non-Gaussian white noise, *Water*, 10 (6), 771, doi:10.3390/w10060771, 2018.

[doc_id=1848]

[English]

Since the prime days of stochastic hydrology back in 1960s, autoregressive (AR) and moving average (MA) models (as well as their extensions) have been widely used to simulate hydrometeorological processes. Initially, AR(1) or Markovian models with Gaussian noise prevailed due to their conceptual and mathematical simplicity. However, the ubiquitous skewed behavior of most hydrometeorological processes, particularly at fine time scales, necessitated the generation of synthetic time series to also reproduce higher-order moments. In this respect, the former schemes were enhanced to preserve skewness through the use of non-Gaussian white noise— a modification attributed to Thomas and Fiering (TF). Although preserving higher-order moments to approximate a distribution is a limited and potentially risky solution, the TF approach has become a common choice in operational practice. In this study, almost half a century after its introduction, we reveal an important flaw that spans over all popular linear stochastic models that employ non-Gaussian white noise. Focusing on the Markovian case, we prove mathematically that this generating scheme provides bounded dependence patterns, which are both unrealistic and inconsistent with the observed data. This so-called “envelope behavior” is amplified as the skewness and correlation increases, as demonstrated on the basis of real-world and hypothetical simulation examples.

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http://www.mdpi.com/2073-4441/10/6/771

**Our works referenced by this work:**

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

2. | D. Koutsoyiannis, Optimal decomposition of covariance matrices for multivariate stochastic models in hydrology, Water Resources Research, 35 (4), 1219–1229, doi:10.1029/1998WR900093, 1999. |

3. | D. Koutsoyiannis, C. Onof, and H. S. Wheater, Multivariate rainfall disaggregation at a fine timescale, Water Resources Research, 39 (7), 1173, doi:10.1029/2002WR001600, 2003. |

4. | 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. |

5. | 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. |

6. | 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. |

7. | A. Efstratiadis, Y. Dialynas, S. Kozanis, and D. Koutsoyiannis, A multivariate stochastic model for the generation of synthetic time series at multiple time scales reproducing long-term persistence, Environmental Modelling and Software, 62, 139–152, doi:10.1016/j.envsoft.2014.08.017, 2014. |

8. | S.M. Papalexiou, and D. Koutsoyiannis, A global survey on the seasonal variation of the marginal distribution of daily precipitation, Advances in Water Resources, 94, 131–145, doi:10.1016/j.advwatres.2016.05.005, 2016. |

9. | I. Tsoukalas, C. Makropoulos, and A. Efstratiadis, Stochastic simulation of periodic processes with arbitrary marginal distributions, 15th International Conference on Environmental Science and Technology (CEST2017), Rhodes, Global Network on Environmental Science and Technology, 2017. |

10. | I. Tsoukalas, A. Efstratiadis, and C. Makropoulos, Stochastic periodic autoregressive to anything (SPARTA): Modelling and simulation of cyclostationary processes with arbitrary marginal distributions, Water Resources Research, 54 (1), 161–185, WRCR23047, doi:10.1002/2017WR021394, 2018. |

**Our works that reference this work:**

1. | 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. |

2. | P. Kossieris, and C. Makropoulos, Exploring the statistical and distributional properties of residential water demand at fine time scales, Water, 10 (10), 1481, doi:10.3390/w10101481, 2018. |

3. | 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. |

4. | 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. |

5. | D. Nikolopoulos, G. Moraitis, D. Bouziotas, A. Lykou, G. Karavokiros, and C. Makropoulos, Cyber-physical stress-testing platform for water distribution networks, Journal of Environmental Engineering, 146 (7), 04020061, doi:10.1061/(ASCE)EE.1943-7870.0001722, 2020. |

6. | I. Tsoukalas, P. Kossieris, and C. Makropoulos, Simulation of non-Gaussian correlated random variables, stochastic processes and random fields: Introducing the anySim R-Package for environmental applications and beyond, Water, 12 (6), 1645, doi:10.3390/w12061645, 2020. |

7. | H. Elsayed, S. Djordjević, D. Savic, I. Tsoukalas, and C. Makropoulos, The Nile water-food-energy nexus under uncertainty: Impacts of the Grand Ethiopian Renaissance Dam, Journal of Water Resources Planning and Management - ASCE, 146 (11), 04020085, doi:10.1061/(ASCE)WR.1943-5452.0001285, 2020. |

8. | G. Moraitis, I. Tsoukalas, P. Kossieris, D. Nikolopoulos, G. Karavokiros, D. Kalogeras, and C. Makropoulos, Assessing cyber-physical threats under water demand uncertainty, Environmental Sciences Proceedings, 21 (1), 18, doi:10.3390/environsciproc2022021018, October 2022. |

9. | A. Zisos, G.-K. Sakki, and A. Efstratiadis, Mixing renewable energy with pumped hydropower storage: Design optimization under uncertainty and other challenges, Sustainability, 15 (18), 13313, doi:10.3390/su151813313, 2023. |

10. | A. Efstratiadis, I. Tsoukalas, and P. Kossieris, Improving hydrological model identifiability by driving calibration with stochastic inputs, Advances in Hydroinformatics: Machine Learning and Optimization for Water Resources, edited by G. A. Corzo Perez and D. P. Solomatine, doi:10.1002/9781119639268.ch2, American Geophysical Union, 2024. |

**Other works that reference this work (this list might be obsolete):**

1. | Papalexiou, S. M., Y. Markonis, F. Lombardo, A. AghaKouchak, and E. Foufoula‐Georgiou, Precise temporal Disaggregation Preserving Marginals and Correlations (DiPMaC) for stationary and non‐stationary processes, Water Resources Research, 54(10), 7435-7458, doi:10.1029/2018WR022726, 2018. |

2. | Cheng, Y., P. Feng, J. Li, Y. Guo, and P. Ren, Water supply risk analysis based on runoff sequence simulation with change point under changing environment, Advances in Meteorology, 9619254, doi:10.1155/2019/9619254, 2019. |

3. | Marković, D., S. Ilić, D. Pavlović, J. Plavšić, and N. Ilich, Multivariate and multi-scale generator based on non-parametric stochastic algorithms, Journal of Hydroinformatics, 21(6), 1102–1117, doi:10.2166/hydro.2019.071, 2019. |

4. | Nazemi, A., M. Zaerpour, and E. Hassanzadeh, Uncertainty in bottom-up vulnerability assessments of water supply systems due to regional streamflow generation under changing conditions, Journal of Water Resources Planning and Management, 146(2), doi:10.1061/(ASCE)WR.1943-5452.0001149, 2020. |

5. | Wang, Q., J. Zhou, K. Huang, L. Dai, B. Jia, L. Chen, and H. Qin, A procedure for combining improved correlated sampling methods and a resampling strategy to generate a multi-site conditioned streamflow process, Water Resources Management, 35, 1011-1027, doi:10.1007/s11269-021-02769-8, 2021. |

6. | Zounemat-Kermani, M., A. Mahdavi-Meymand, and A. Hinkelmann, A comprehensive survey on conventional and modern neural networks: application to river flow forecasting, Earth Science Informatics, 14, 893-911, doi:10.1007/s12145-021-00599-1, 2021. |

7. | Pouliasis, G., G. A. Torres-Alves, and O. Morales-Napoles, Stochastic modeling of hydroclimatic processes using vine copulas, Water, 13(16), 2156, doi:10.3390/w13162156, 2021. |

8. | Jia, B., J. Zhou, Z. Tang, Z. Xu, X. Chen, and W. Fang, Effective stochastic streamflow simulation method based on Gaussian mixture model, Journal of Hydrology, 605, 127366, doi:10.1016/j.jhydrol.2021.127366, 2022. |

**Tagged under:**
Stochastics