Towards Generic Simulation for Demanding Stochastic Processes
- Complex dependence structures that extend way beyond the Markov dependence, and incorporate long-range dependence and short-scale fractal (smoothness/roughness) behavior. This is achieved by using a symmetric moving average scheme, which can involve a large number of white noise terms, with their weights determined in an explicit analytical manner.
- Marginal distributions that extend beyond Gaussian and incorporate heavy tails, boundedness, and intermittence. This is achieved by using an appropriate number of cumulants, analytically determined from the distribution function, thus resulting in genuine simulation of the process (without a transformation).
- Time asymmetry (irreversibility), achieved by using a non-Gaussian distribution function, combined with an asymmetric moving average scheme, with the weights again determined in an explicit analytical manner.
2.2. Moments and Cumulants
- The meaning of the parameters is the following.
- (a) Dimensional parameters, with dimensions identical to those of the stochastic variable : μ: mean; : standard deviation; : scale parameter; a, b: lower and upper bound of .
- (b) Dimensionless parameters: : upper-tail index; : lower-tail index; : additional shape parameter, : probability.
- The meaning of constants and standard functions is this: γ: Euler constant; : Bernoulli number of order p; : Dirac delta function of ; : gamma function of ; : digamma function of ; : beta function of .
- Distributions named “half” have their “full” version whose density and tail function are obtained by dividing those given in the tables by 2. The “half” version given in the tables corresponds to , while in the “full” version . The moments of the “full” version is: (a) for even p, 0; (b) for odd p, equal to those of half version.
- All other distributions, defined for but not named “half”, can also be extended to the whole real line by replacing with and dividing by 2. Again, the moments of this extended version is: (a) for even p, 0; (b) for odd p, equal to those of original version.
2.3. Second Order Properties
- The generalized Cauchy-type FHK (FHK-C) with climacogram:
- The mixed Cauchy–Dagum-type FHK (FHK-CD) climacogram:
2.4. Stochastic Simulation
- Consists of real numbers, despite the expression in (31) involving complex numbers;
- Satisfies precisely Equation (30); and
- Is easy and fast to calculate using the fast Fourier transform (FFT).
- From the continuous-time stochastic model, expressed through its climacogram , we calculate its autocovariance function in discrete time (assuming time step ) by Equation (26). (This step is obviously omitted if the model is already expressed in discrete time through its autocovariance function).
- We choose an appropriate number of coefficients J that is a power of 2 and perform inverse FFT (using common software) to calculate the discrete-time power spectrum and the frequency function for an array of :
- We choose (see below) and we form the arrays (vectors) and , both of size 2J indexed as , with the superscripts R and I standing for the real and imaginary part of a vector of complex numbers, respectively:
- We perform FFT on the vector (using common software), and get the real part of the result, which is precisely the sequence of .
2.5. Distribution Function Approximation
3. Applications and Results
3.1. Simulating a Persistent Process with Exponential Distribution
3.2. Simulating a Persistent Process with Uniform Distribution
3.3. Simulating an Antipersistent Process with Uniform Distribution
3.4. Simulating the Precipitation Process at the Hourly Time Scale
- Pareto distribution with discontinuity at the origin for small time scales (Table 5, Equation (46), left). The tail index ξ is constant for all time scales k, while the probability wet, , and the state scale parameter, , are functions of the time scale k.
- Continuous PBF distribution, possibly with discontinuity at zero, for large time scales (Table 5, Equation (46), right). In this case, a new parameter is introduced, which is again a function of time scale. The Pareto distribution is a special case of the PBF for . In contrast to the Pareto distribution, whose density is a consistently decreasing function of , the PBF tends to be bell-shaped for increasing , a property consistent with empirical observation and reason.
- Constant mean of the time-averaged process.
- Climacogram of type FHK-CD (Equation (24)), where to reduce the number of parameters it is assumed that , thus getting Equation (48) in Table 5. By inspection of Equation (48), it is seen that, as , , which makes the process ergodic; for , , which is finite, as required for physical consistency.
- Probability wet and dry, , varying with time scale according to Equation (49) in Table 5. It is clarified that two different expressions are used for the small and the large scales, where the transition time scale from the Pareto to the PBF distribution is denoted as . In the Pareto case, can be determined directly from the climacogram and the mean (left column of Equation (49) in Table 5). For the PBF case, an additional equation is required, which has been derived based on maximum entropy considerations  and involves an additional parameter (. Continuity of the transition demands that .
- Mean intensity, μ = 0.0823 mm/h;
- Intensity scale parameters, ;
- Time scale parameter, = 8.74 h;
- Hurst parameter, H = 0.92; fractal (smoothness) parameter, ;
- Exponent of the expression of probability dry/wet, = 0.787;
- Upper tail index, ξ = 0.121.
3.5. Simulating the Precipitation Process at the Annual Time Scale
4. Discussion and Conclusions
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Comparison with a Conventional Approach
- Koutsoyiannis, D. Simple stochastic simulation of time irreversible and reversible processes. Hydrol. Sci. J. 2020, 65, 536–551. [Google Scholar] [CrossRef]
- Koutsoyiannis, D. Stochastics of Hydroclimatic Extremes—A Cool Look at Risk; Kallipos: Athens, Greece, 2021; ISBN 978-618-85370-0-2. [Google Scholar]
- Beven, K. Issues in generating stochastic observables for hydrological models. Hydrol. Process. 2021. [Google Scholar] [CrossRef]
- Box, G.E.; Jenkins, G.M. Time Series Models for Forecasting and Control; Holden Day: San Francisco, CA, USA, 1970. [Google Scholar]
- Stigler, S.M. Statistics on the Table: The History of Statistical Concepts and Methods; Harvard University Press: Cambridge, MA, USA, 2002. [Google Scholar]
- Whittle, P. Hypothesis Testing in Times Series Analysis. Ph.D. Thesis, Almqvist & Wiksells, Uppsala, Sweden, 1951. [Google Scholar]
- Whittle, P. Tests of fit in time series. Biometrika 1952, 39, 309–318. [Google Scholar] [CrossRef]
- Whittle, P. The analysis of multiple stationary time series. J. R. Stat. Soc. B 1953, 15, 125–139. [Google Scholar] [CrossRef]
- Dimitriadis, P.; Koutsoyiannis, D.; Iliopoulou, T.; Papanicolaou, P. A global-scale investigation of stochastic similarities in marginal distribution and dependence structure of key hydrological-cycle processes. Hydrology 2021, 8, 59. [Google Scholar] [CrossRef]
- Hosking, J.R.M. Fractional differencing. Biometrika 1981, 68, 165–176. [Google Scholar] [CrossRef]
- Koutsoyiannis, D. A generalized mathematical framework for stochastic simulation and forecast of hydrologic time series. Water Resour. Res. 2000, 36, 1519–1533. [Google Scholar] [CrossRef][Green Version]
- Koutsoyiannis, D. Generic and parsimonious stochastic modelling for hydrology and beyond. Hydrol. Sci. J. 2016, 61, 225–244. [Google Scholar] [CrossRef]
- Onof, C.; Chandler, R.E.; Kakou, A.; Northrop, P.; Wheater, H.S.; Isham, V. Rainfall modelling using Poisson-cluster processes: A review of developments. Stoch. Environ. Res. Risk Assess. 2000, 14, 384–411. [Google Scholar] [CrossRef]
- Cowpertwait, P.; Isham, V.; Onof, C. Point process models of rainfall: Developments for fine-scale structure. Proc. R. Soc. A 2007, 463, 2569–2588. [Google Scholar] [CrossRef]
- Kim, D.; Onof, C. A stochastic rainfall model that can reproduce important rainfall properties across the timescales from several minutes to a decade. J. Hydrol. 2020, 589, 125–150. [Google Scholar] [CrossRef]
- Kossieris, P.; Makropoulos, C.; Onof, C.; Koutsoyiannis, D. A rainfall disaggregation scheme for sub-hourly time scales: Coupling a Bartlett-Lewis based model with adjusting procedures. J. Hydrol. 2018, 556, 980–992. [Google Scholar] [CrossRef][Green Version]
- Koutsoyiannis, D.; Onof, C.; Wheater, H.S. Multivariate rainfall disaggregation at a fine timescale. Water Resour. Res. 2003, 39, 1173. [Google Scholar] [CrossRef]
- Northrop, P.J. A clustered spatial-temporal model of rainfall. Proc. R. Soc. Lond. Ser. A 1998, 454, 1875–1888. [Google Scholar] [CrossRef]
- Hoeffding, W. Scale-Invariant Correlation Theory. In The Collected Works of Wassily Hoeffding; Fisher, N.I., Sen, P.K., Eds.; Springer: New York, NY, USA, 1940; pp. 57–107. [Google Scholar]
- Frechet, M. Sur les tableaux de correlation dont les marges son donnees. Ann. Univ. Lyon 1951, 14, 53–77. [Google Scholar]
- Sklar, A. Fonctions de Repartition a n Dimensions et Leurs Marges; Publications de l’Institut de Statistique de l’Universite de Paris: Paris, France, 1959; Volume 8, pp. 229–231. [Google Scholar]
- Nelsen, R.B. An Introduction to Copulas, 2nd ed.; Springer: New York, NY, USA, 2006. [Google Scholar]
- Lebrun, R.; Dutfoy, A. An innovating analysis of the Nataf transformation from the copula viewpoint. Probabilistic Eng. Mech. 2009, 24, 312–320. [Google Scholar] [CrossRef]
- Tsoukalas, I.; Makropoulos, C.; Koutsoyiannis, D. Simulation of stochastic processes exhibiting any-range dependence and arbitrary marginal distributions. Water Resour. Res. 2018, 54, 9484–9513. [Google Scholar] [CrossRef]
- Eddington, A. The Nature of the Physical World; Cambridge University Press: Cambridge, UK, 1928. [Google Scholar]
- Koutsoyiannis, D. Time’s arrow in stochastic characterization and simulation of atmospheric and hydrological processes. Hydrol. Sci. J. 2019, 64, 1013–1037. [Google Scholar] [CrossRef]
- Vavoulogiannis, S.; Iliopoulou, T.; Dimitriadis, P.; Koutsoyiannis, D. Multiscale temporal irreversibility of streamflow and its stochastic modelling. Hydrology 2021, 8, 63. [Google Scholar] [CrossRef]
- Lombardo, F.; Napolitano, F.; Russo, F.; Koutsoyiannis, D. On the exact distribution of correlated extremes in hydrology. Water Resour. Res. 2019, 55, 10405–10423. [Google Scholar] [CrossRef][Green Version]
- Rozos, E.; Dimitriadis, P.; Mazi, K.; Koussis, A.D. A multilayer perceptron model for stochastic synthesis. Hydrology 2021, 8, 67. [Google Scholar] [CrossRef]
- Dimitriadis, P.; Koutsoyiannis, D. Stochastic synthesis approximating any process dependence and distribution. Stoch. Environ. Res. Risk Assess. 2018, 32, 1493–1515. [Google Scholar] [CrossRef]
- Koutsoyiannis, D. Knowable moments for high-order stochastic characterization and modelling of hydrological processes. Hydrol. Sci. J. 2019, 64, 19–33. [Google Scholar] [CrossRef][Green Version]
- Koutsoyiannis, D. Coupling stochastic models of different time scales. Water Resour. Res. 2001, 37, 379–391. [Google Scholar] [CrossRef][Green Version]
- Wold, H.O. A Study in the Analysis of Stationary Time-Series. Ph.D. Thesis, Almquist and Wicksell, Uppsala, Sweden, 1938. [Google Scholar]
- Wold, H.O. On prediction in stationary time series. Ann. Math. Stat. 1948, 19, 558–567. [Google Scholar] [CrossRef]
- Thiele, T.N. Forelaesninger over Almindelig Iagttagelseslaere: Sandsynlighedsregning og Mindste Kvadraters Methode. C.A. Reitzel, Kjøbenhavn, 1889. Available online: https://archive.org/details/forlaesingerove00thiegoog (accessed on 18 May 2021).
- Thiele, T.N. Om Iagttagelseslærens Halvinvarianter. Kgl. Dan. Vidensk. Selsk. Forh. 1899, 3, 135–141. [Google Scholar]
- Hald, A. The Early History of the Cumulants and the Gram-Charlier Series. Int. Stat. Rev. 2000, 68, 137–153. [Google Scholar] [CrossRef]
- Fisher, R. Statistical Methods for Research Workers; Oliver and Boyd: Edinburgh, UK, 1932. [Google Scholar]
- Hotelling, H. Review of statistical methods for research workers, by RA Fisher. J. Am. Stat. Assoc. 1933, 28, 374–375. [Google Scholar] [CrossRef]
- Smith, P.J. A recursive formulation of the old problem of obtaining moments from cumulants and vice versa. Am. Stat. 1995, 49, 217–218. [Google Scholar]
- Koutsoyiannis, D. A random walk on water. Hydrol. Earth Syst. Sci. 2010, 14, 585–601. [Google Scholar] [CrossRef][Green Version]
- Kolmogorov, A.N. Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. Dokl. Akad. Nauk SSSR 1940, 26, 115–118. [Google Scholar]
- Kolmogorov, A.N. Wiener spirals and some other interesting curves in a Hilbert space. In Selected Works of A. N. Kolmogorov—Volume 1, Mathematics and Mechanics; Tikhomirov, V.M., Ed.; Kluwer: Dordrecht, The Netherlands, 1991; pp. 303–307. [Google Scholar]
- Hurst, H.E. Long term storage capacities of reservoirs. Trans. Am. Soc. Civ. Eng. 1951, 116, 776–808. [Google Scholar]
- Koutsoyiannis, D. Entropy production in stochastics. Entropy 2017, 19, 581. [Google Scholar] [CrossRef][Green Version]
- Fernandez, B.; Salas, J.D. Periodic gamma autoregressive processes for operational hydrology. Water Resour. Res. 1986, 22, 1385–1396. [Google Scholar] [CrossRef]
- Koutsoyiannis, D. Climate change, the Hurst phenomenon, and hydrological statistics. Hydrol. Sci. J. 2003, 48, 3–24. [Google Scholar] [CrossRef]
- Koutsoyiannis, D.; Montanari, A. Statistical analysis of hydroclimatic time series: Uncertainty and insights. Water Resour. Res. 2007, 43, W05429. [Google Scholar] [CrossRef]
- Dimitriadis, P.; Koutsoyiannis, D. Climacogram versus autocovariance and power spectrum in stochastic modelling for Markovian and Hurst–Kolmogorov processes. Stoch. Environ. Res. Risk Assess. 2015, 29, 1649–1669. [Google Scholar] [CrossRef]
- Koutsoyiannis, D. An entropic-stochastic representation of rainfall intermittency: The origin of clustering and persistence. Water Resour. Res. 2006, 42, W01401. [Google Scholar] [CrossRef][Green Version]
- McCullagh, P.; Kolassa, J. Cumulants. Scholarpedia 2009, 4, 4699. Available online: http://www.scholarpedia.org/article/Cumulants (accessed on 18 May 2021). [CrossRef]
- Kirby, W. Computer-oriented Wilson-Hilferty transformation that preserves the first three moments and the lower bound of the Pearson type 3 distribution. Water Resour. Res. 1972, 8, 1251–1254. [Google Scholar] [CrossRef]
|Operation||Mathematical Relationship||Eqn. no.|
|Shift of origin||(11)|
|Multiplication by a constant||(12)|
|Linear combination of independent variables||(13)|
|Name, Domain||Probability Density or Distribution Function||Moments,||Cumulants,|
|Finite number of impulses,|
|Name, Domain||Probability Density or Distribution Function||Moments,||Cumulants,|
|Extended half-normal (Chi),|
|Extreme value type I (EV1),|
|Name, Domain||Probability Density or Distribution Function|
|Pareto-Burr-Feller (PBF) |
|Extreme value type II (EV2)|
|Half extended Student|
|Generalized beta prime (GBP)|
|Quantity and Symbol||(Pareto)||(PBF)||Eqn. no.|
|Lower tail index (inverse),||(50)|
|Upper tail index,||(51)|
|Scale parameter (inverse),||(52)|
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Koutsoyiannis, D.; Dimitriadis, P. Towards Generic Simulation for Demanding Stochastic Processes. Sci 2021, 3, 34. https://doi.org/10.3390/sci3030034
Koutsoyiannis D, Dimitriadis P. Towards Generic Simulation for Demanding Stochastic Processes. Sci. 2021; 3(3):34. https://doi.org/10.3390/sci3030034Chicago/Turabian Style
Koutsoyiannis, Demetris, and Panayiotis Dimitriadis. 2021. "Towards Generic Simulation for Demanding Stochastic Processes" Sci 3, no. 3: 34. https://doi.org/10.3390/sci3030034