Fitting a superposition of Ornstein–Uhlenbeck process to time series of discharge in a perennial river environment

Abstract

Classical Ornstein–Uhlenbeck (ou) processes are Lévy-driven linear stochastic models with exponentially decaying autocorrelation functions which do not always fit more slowly decaying real time series data. A superposition of ou processes (known as a supou process) is proposed to overcome this issue for application to river discharge time series data. The discharge data has a sub-exponential autocorrelation function and this is captured by the supou process based on the mean reversion speed generated by a Gamma distribution. All the parameters of the supou process are identified by matching the autocorrelation and the first to fourth statistical moments of the discharge data. The empirical and modelled histograms of the discharge data are comparable with each other. References O. E. Barndorff-Nielsen. Superposition of Ornstein–Uhlenbeck type processes. Theory Prob. Appl. 45.2 (2001), pp. 175–194. doi: 10.1137/S0040585X97978166 O. E. Barndorff-Nielsen, F. E. Benth, and A. E. D. Veraart. Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes. Bernoulli 19.3 (2013), pp. 803–845. doi: 10.3150/12-BEJ476 O. E. Barndorff-Nielsen and N. N. Leonenko. Burgers’ turbulence problem with linear or quadratic external potential. J. Appl. Prob. 42.2 (2001), pp. 550–565. url: http://www.jstor.org/stable/30040809 J. Beran, Y. Feng, S. Ghosh, and R. Kulik. Long-Memory Processes. Springer-Verlag, Berlin, Heidelberg, 2016. doi: 10.1007/978-3-642-35512-7 F. Fuchs and R. Stelzer. Mixing conditions for multivariate infinitely divisible processes with an application to mixed moving averages and the supOU stochastic volatility model. ESAIM: Prob. Stat. 17 (2013), pp. 455–471. doi: 10.1051/ps/2011158 Y. Kabanov and S. Pergamenshchikov. Ruin probabilities for a Lévy-driven generalised Ornstein–Uhlenbeck process. Fin. Stoch. 24.1 (2020), pp. 39–69. doi: 10.1007/s00780-019-00413-3 R. Kawai and H. Masuda. On simulation of tempered stable random variates. J. Comput. Appl. Math. 235.8 (2011), pp. 2873–2887. doi: 10.1016/j.cam.2010.12.014 S. Pelacani and F. G. Schmitt. Scaling properties of the turbidity and streamflow time series at two different locations of an intra-Apennine stream: Case study. J. Hydro. 603.B (2021), p. 126943. doi: 10.1016/j.jhydrol.2021.126943 R. Stelzer, T. Tosstorff, and M. Wittlinger. Moment based estimation of supOU processes and a related stochastic volatility model. Stat. Risk Model. 32.1 (2015), pp. 1–24. doi: 10.1515/strm-2012-1152 S. Suweis, E. Bertuzzo, G. Botter, A. Porporato, I. Rodriguez-Iturbe, and A. Rinaldo. Impact of stochastic fluctuations in storage-discharge relations on streamflow distributions. Water Resource. Res. 46.3 (2010), W03517. doi: 10.1029/2009WR008038 M. Tamborrino and P. Lansky. Shot noise, weak convergence and diffusion approximations. Physica D: Nonlinear Phenomena 418 (2021), p. 132845. doi: 10.1016/j.physd.2021.132845 E. Taufer and N. Leonenko. Simulation of Lévy-driven Ornstein–Uhlenbeck processes with given marginal distribution. In: Comput. Stat. Data Anal. 53.6 (2009), pp. 2427–2437. doi: 10.1016/j.csda.2008.02.026 C. Van Den Broeck. On the relation between white shot noise, Gaussian white noise, and the dichotomic Markov process. J. Stat. Phys. 31 (1983), pp. 467–483. doi: 10.1007/BF01019494 H. Yoshioka and Y. Yoshioka. Designing cost-efficient inspection schemes for stochastic streamflow environment using an effective Hamiltonian approach. Opt. Eng. (2021), pp. 1–33. doi: 10.1007/s11081-021-09655-7

Statistical evaluation of a long‐memory process using the generalized entropic value‐at‐risk

“…This means that the reversion measure should be sufficiently regular near the origin r = 0. The Gamma-type measure is widely used as a convenient model (Fasen & Klüppelberg, 2007;Yoshioka, 2021) because it leads to closed-form statistical moments and autocorrelation function with subexponential decay that are efficient to use in applications. Further, it fits well to the real data as demonstrated later in Section 4:…”

“…First, the stationarity of the supOU process (3) is satisfied under (1). Moreover, the stationary statistics are analytically obtained as follows with (Yoshioka, 2021): the mean…”

“…First, the stationarity of the supOU process (3) is satisfied under (1). Moreover, the stationary statistics are analytically obtained as follows with $$$ {M}_k={\int}_0^{+\infty }{z}^kv\left(\mathrm{d}z\right) $$$ ($$$ k\in \mathrm{\mathbb{N}} $$$) if it exists (Yoshioka, 2021): the mean $$$$ \mathbbm{E}\left[{X}_t\right]={M}_1R, $$$$ variance $$$$ \mathbbm{E}\left[{\left({X}_t-\mathbbm{E}\left[{X}_t\right]\right)}^2\right]=\frac{M_2R}{2}, $$$$ (unnormalized) skewness $$$$ \mathbbm{E}\left[{\left({X}_t-\mathbbm{E}\left[{X}_t\right]\right)}^3\right]=\frac{M_3R}{3}, $$$$ and (unnormalized) kurtosis …”

“…Hurst (1956) found the long memory in the data of Nile River. The hourly data of the streamflow discharge in mountainous rivers have a long memory (Yoshioka, 2021; Yoshioka et al, 2023). The long memory of a streamflow discharge has been explained as the presence of watershed runoff processes with multiple time scales, which emerge as a superposition of exponential correlations with different relaxation rates (Mudelsee, 2007; Ranjram & Craig, 2022).…”

“…This means that the reversion measure should be sufficiently regular near the origin $$$ r=0 $$$. The Gamma‐type measure is widely used as a convenient model (Fasen & Klüppelberg, 2007; Yoshioka, 2021) because it leads to closed‐form statistical moments and autocorrelation function with subexponential decay that are efficient to use in applications. Further, it fits well to the real data as demonstrated later in Section 4: $$$$ \pi \left(\mathrm{d}r\right)=\frac{1}{\Gamma \left(\alpha \right){\beta}^{\alpha }}{r}^{\alpha -1}\exp \left(-\frac{r}{\beta}\right)\mathrm{d}r,r>0 $$$$ where $$$ \alpha >1 $$$ and $$$ \beta >0 $$$, and $$$ \Gamma $$$ is the Gamma function.…”