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.

[doc_id=22]

[English]

A new method is proposed for decomposing covariance matrices that appear in the parameter estimation phase of all multivariate stochastic models in hydrology. This method applies not only to positive definite covariance matrices (as do the typical methods of the literature) but to indefinite matrices, too, that often appear in stochastic hydrology. It is also appropriate for preserving the skewness coefficients of the model variables as it accounts for the resulting coefficients of skewness of the auxiliary (noise) variables used by the stochastic model, given that the latter coefficients are controlled by the decomposed matrix. The method is formulated in an optimization framework with the objective function being composed of three components aiming at (1) complete preservation of the variances of variables, (2) optimal approximation of the covariances of variables, in the case that complete preservation is not feasible due to inconsistent (i.e., not positive definite) structure of the covariance matrix, and (3) preservation of the skewness coefficients of the model variables by keeping the skewness of the auxiliary variables as low as possible. Analytical expressions of the derivatives of this objective function are derived, which allow the development of an effective nonlinear optimization algorithm using the steepest descent or the conjugate gradient methods. The method is illustrated and explored through a real world application, which indicates a very satisfactory performance of the method.

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**Additional material:**

- Preprint (390 KB)

**See also:**
http://dx.doi.org/10.1029/1998WR900093

**Remarks:**

**Our works referenced by this work:**

1. | D. Koutsoyiannis, A nonlinear disaggregation method with a reduced parameter set for simulation of hydrologic series, Water Resources Research, 28 (12), 3175–3191, doi:10.1029/92WR01299, 1992. |

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. | I. Nalbantis, and D. Koutsoyiannis, A parametric rule for planning and management of multiple reservoir systems, Water Resources Research, 33 (9), 2165–2177, doi:10.1029/97WR01034, 1997. |

**Our works that reference this work:**

1. | 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. |

2. | D. Koutsoyiannis, A. Efstratiadis, and G. Karavokiros, A decision support tool for the management of multi-reservoir systems, Journal of the American Water Resources Association, 38 (4), 945–958, doi:10.1111/j.1752-1688.2002.tb05536.x, 2002. |

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, G. Karavokiros, A. Efstratiadis, N. Mamassis, A. Koukouvinos, and A. Christofides, A decision support system for the management of the water resource system of Athens, Physics and Chemistry of the Earth, 28 (14-15), 599–609, doi:10.1016/S1474-7065(03)00106-2, 2003. |

5. | D. Koutsoyiannis, and A. Efstratiadis, Experience from the development of decision support systems for the management of large-scale hydrosystems of Greece, Proceedings of the Workshop "Water Resources Studies in Cyprus", edited by E. Sidiropoulos and I. Iakovidis, Nikosia, 159–180, Water Development Department of Cyprus, Aristotle University of Thessaloniki, Thessaloniki, 2003. |

6. | A. Efstratiadis, D. Koutsoyiannis, and D. Xenos, Minimizing water cost in the water resource management of Athens, Urban Water Journal, 1 (1), 3–15, doi:10.1080/15730620410001732099, 2004. |

7. | A. Langousis, and D. Koutsoyiannis, A stochastic methodology for generation of seasonal time series reproducing overyear scaling behaviour, Journal of Hydrology, 322, 138–154, 2006. |

8. | D. Koutsoyiannis, H. Yao, and A. Georgakakos, Medium-range flow prediction for the Nile: a comparison of stochastic and deterministic methods, Hydrological Sciences Journal, 53 (1), 142–164, doi:10.1623/hysj.53.1.142, 2008. |

9. | C. Ioannou, G. Tsekouras, A. Efstratiadis, and D. Koutsoyiannis, Stochastic analysis and simulation of hydrometeorological processes for optimizing hybrid renewable energy systems, Proceedings of the 2nd Hellenic Concerence on Dams and Reservoirs, Athens, Zappeion, doi:10.13140/RG.2.1.3787.0327, Hellenic Commission on Large Dams, 2013. |

10. | 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. |

11. | A. Efstratiadis, I. Nalbantis, and D. Koutsoyiannis, Hydrological modelling of temporally-varying catchments: Facets of change and the value of information, Hydrological Sciences Journal, 60 (7-8), 1438–1461, doi:10.1080/02626667.2014.982123, 2015. |

12. | 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. |

13. | 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. |

14. | 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. |

15. | 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. |

16. | 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. |

17. | 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. |

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12. | #Alparslan, C., and M. Byukyildiz, Comparison of monthly streamflow forecasting techniques, EWRA 9th World Congress - Water Resources Management In A Changing World, Istanbul, Turkey, 2015. |

**Tagged under:**
Course bibliography: Stochastic methods,
Stochastics