[go: up one dir, main page]
Skip to main content
SpringerLink
Log in

A Bayesian statistical model for deriving the predictive distribution of hydroclimatic variables

  • Published:
Climate Dynamics Aims and scope Submit manuscript

Abstract

Recent publications have provided evidence that hydrological processes exhibit a scaling behaviour, also known as the Hurst phenomenon. An appropriate way to model this behaviour is to use the Hurst-Kolmogorov stochastic process. The Hurst-Kolmogorov process entails high autocorrelations even for large lags, as well as high variability even at climatic scales. A problem that, thus, arises is how to incorporate the observed past hydroclimatic data in deriving the predictive distribution of hydroclimatic processes at climatic time scales. Here with the use of Bayesian techniques we create a framework to solve the aforementioned problem. We assume that there is no prior information for the parameters of the process and use a non-informative prior distribution. We apply this method with real-world data to derive the posterior distribution of the parameters and the posterior predictive distribution of various 30-year moving average climatic variables. The marginal distributions we examine are the normal and the truncated normal (for nonnegative variables). We also compare the results with two alternative models, one that assumes independence in time and one with Markovian dependence, and the results are dramatically different. The conclusion is that this framework is appropriate for the prediction of future hydroclimatic variables conditional on the observations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Bakker A, Hurk B (2012) Estimation of persistence and trends in geostrophic wind speed for the assessment of wind energy yields in Northwest Europe. Clim Dyn 39(3–4):767–782. doi:10.1007/s00382-011-1248-1

    Article  Google Scholar 

  • Berliner LM, Wikle CK, Cressie N (2000) Long-lead prediction of Pacific SSTs via Bayesian dynamic modeling. J Clim 13:3953–3968

    Article  Google Scholar 

  • Duan Q, Ajami NK, Gao X, Sorooshian S (2007) Multi-model ensemble hydrologic prediction using Bayesian model averaging. Adv Water Resour 30(5):1371–1386

    Article  Google Scholar 

  • Eaton ML (1983) Multivariate statistics: a vector space approach. Institute of Mathematical Statistics, Beachwood

    Google Scholar 

  • Falconer K, Fernadez C (2007) Inference on fractal processes using multiresolution approximation. Biometrica 94(2):313–334

    Article  Google Scholar 

  • Gamerman D, Lopes H (2006) Markov chain Monte Carlo, stochastic simulation for Bayesian inference, 2nd edn. Chapman & Hall/CRC, London

    Google Scholar 

  • Gelman A (1996) Inference and monitoring convergence. In: Gilks WR, Richardson S, Spiegelhalter DJ (eds) Markov chain Monte Carlo in practice. Chapman & Hall, New York, pp 131–143

    Chapter  Google Scholar 

  • Gelman A, Rubin DR (1992) A single series from the Gibbs sampler provides a false sense of security. In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics 4. Oxford University Press, Oxford, pp 625–632

    Google Scholar 

  • Gelman A, Carlin J, Stern H, Rubin D (2004) Bayesian data analysis, 2nd edn. Chapman & Hall/CRC, Boca Raton

    Google Scholar 

  • Heidelberger P, Welch PD (1983) Simulation run length control in the presence of an initial transient. Oper Res 31(6):1109–1144. doi:10.1287/opre.31.6.1109

    Article  Google Scholar 

  • Hemelrijk J (1966) Underlining random variables. Stat Neerl 20:1–7. doi:10.1111/j.1467-9574.1966.tb00488.x

    Article  Google Scholar 

  • Horrace W (2005) Some results on the multivariate truncated normal distribution. J Multivar Anal 94(1):209–221

    Article  Google Scholar 

  • Hurst HE (1951) Long term storage capacities of reservoirs. Trans Am Soc Civil Eng 116:776–808 (published in 1950 as Proceedings Separate no. 11)

    Google Scholar 

  • Koutsoyiannis D (2003) Climate change, the Hurst phenomenon, and hydrological statistics. Hydrol Sci J 48(1):3–24. doi:10.1623/hysj.48.1.3.43481

    Article  Google Scholar 

  • Koutsoyiannis D (2011) Hurst-Kolmogorov dynamics as a result of extremal entropy production. Phys A 390(8):1424–1432

    Article  Google Scholar 

  • Koutsoyiannis D, Efstratiadis A, Georgakakos KP (2007) Uncertainty assessment of future hydroclimatic predictions: a comparison of probabilistic and scenario-based approaches. J Hydrometeorol 8(3):261–281. doi:10.1175/JHM576.1

    Article  Google Scholar 

  • Koutsoyiannis D, Efstratiadis A, Mamassis N, Christofides A (2008) On the credibility of climate predictions. Hydrol Sci J 53(4):671–684. doi:10.1623/hysj.53.4.671

    Article  Google Scholar 

  • Kumar DN, Maity R (2008) Bayesian dynamic modeling for nonstationary hydroclimatic time series forecasting along with uncertainty quantification. Hydrol Process 22(17):3488–3499. doi:10.1002/hyp.6951

    Article  Google Scholar 

  • Maity R, Kumar DN (2006) Bayesian dynamic modeling for monthly Indian summer monsoon using El Nino-Southern Oscillation (ENSO) and Equatorial Indian Ocean Oscillation (EQUINOO). J Geophys Res 111:D07104. doi:10.1029/2005JD006539

    Google Scholar 

  • Markonis Y, Koutsoyiannis D (2013) Climatic variability over time scales spanning nine orders of magnitude: connecting Milankovitch cycles with Hurst-Kolmogorov dynamics. Surv Geophys 34(2):181–207

    Article  Google Scholar 

  • Martin A, Quinn K, Park JH (2011) MCMCpack: Markov chain Monte Carlo (MCMC). R package version 1.2–1, http://cran.r-project.org/web/packages/MCMCpack/index.html

  • Plummer M, Best N, Cowles K, Vines K (2011) coda: output analysis and diagnostics for MCMC. R package version 0.14–6, http://cran.r-project.org/web/packages/coda/index.html

  • Raftery AL, Lewis S (1992) How many iterations in the Gibbs sampler? In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics 4. Oxford University Press, Oxford, pp 763–774

    Google Scholar 

  • Robert C (2007) The Bayesian choice: from decision-theoretic foundations to computational implementation. Springer, New York

    Google Scholar 

  • Robert C, Casella G (2004) Monte Carlo statistical methods, 2nd edn. Springer-Verlag New York, Inc., Secaucus

    Book  Google Scholar 

  • Rozos E, Efstratiadis A, Nalbantis I, Koutsoyiannis D (2004) Calibration of a semi-distributed model for conjunctive simulation of surface and groundwater flows. Hydrol Sci J 49(5):819–842

    Google Scholar 

  • Smith B (2007) Boa: an R package for MCMC output convergence assessment and posterior inference. J Stat Softw 21(11):1–37

    Google Scholar 

  • Tyralis H, Koutsoyiannis D (2011) Simultaneous estimation of the parameters of the Hurst-Kolmogorov stochastic process. Stoch Environ Res Risk Assess 25(1):21–33. doi:10.1007/s00477-010-0408-x

    Article  Google Scholar 

  • Wei WWS (2006) Time series analysis, univariate and multivariate methods, 2nd edn. Pearson Addison Wesley, Chichester

    Google Scholar 

Download references

Acknowledgments

The authors wish to thank the eponymous reviewer Dr. Federico Lombardo and an anonymous reviewer for their encouraging and constructive comments which helped to improve the quality of the manuscript significantly.

Author information

Authors and Affiliations

  1. Department of Water Resources, Faculty of Civil Engineering, National Technical University, Athens Heroon Polytechneiou 5, 157 80, Zographou, Greece

    Hristos Tyralis & Demetris Koutsoyiannis

Authors
  1. Hristos Tyralis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Demetris Koutsoyiannis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hristos Tyralis.

Appendices

Appendix 1: Standard probability distributions

For easy reference, the details of the distribution functions used in this paper are summarized in Table 11

Table 11 Distributions used in the Bayesian framework
Full size table

Appendix 2: Mathematical proofs

In Appendix 2 the proofs of (8), (9), (10), (12), (13), (14) are given. It is easily shown that

$$ (\varvec{x}_{n} - \mu \varvec{e}_{n} )^{\text{T}} \varvec{R}_{n}^{ - 1} (\varvec{x}_{n} - \mu \varvec{e}_{n} ) \, = \varvec{e}_{n}^{\text{T}} \varvec{R} _{n}^{ - 1} \varvec{e}_{n} \mu^{ 2} - 2\varvec{x}_{n}^{\text{T}} \varvec{R}_{n}^{ - 1} \varvec{e}_{n} \mu + \varvec{x}_{n}^{\text{T}} \varvec{R}_{n}^{ - 1} \varvec{x}_{n} $$
(34)

After completing the squares the above expression becomes:

$$ \begin{aligned} \varvec{e}_{n}^{\text{T}} \varvec{R}_{n}^{ - 1} \varvec{e}_{n} \mu^{2} - 2\varvec{x}_{n}^{\text{T}} \varvec{R}_{n}^{ - 1} \varvec{e}_{n} \mu + \varvec{x}_{n}^{\text{T}} \varvec{R}_{n}^{ - 1} \varvec{x}_{n} &= \varvec{e}_{n}^{\text{T}} \varvec{R}_{n}^{ - 1} \varvec{e}_{n} \left[ {\mu - \left( {\varvec{x}_{n}^{\text{T}} \varvec{R}_{n}^{ - 1} \varvec{e}_{n} } \right) \, / \, \left( {\varvec{e}_{n}^{\text{T}} \varvec{R}_{n}^{ - 1} \varvec{e}_{n} } \right)} \right]^{2} \hfill \\ &\quad \varvec{ + } \left[ {\varvec{e}_{n}^{\text{T}} \varvec{R}_{n}^{ - 1} \varvec{e}_{n} \varvec{x}_{n}^{\text{T}} \varvec{R}_{n}^{ - 1} \varvec{x}_{n} - \left( {\varvec{x}_{n}^{\text{T}} \varvec{R}_{n}^{ - 1} \varvec{e}_{n} } \right)^{2} } \right]/\left( {\varvec{e}_{n}^{\text{T}} \varvec{R}_{n}^{ - 1} \varvec{e}_{n} } \right) \hfill \\ \end{aligned} $$
(35)

From (6) and (7) we obtain the following:

$$ \pi \left(\varvec{\theta}\right)f\left( {\varvec{x}_{n} |\varvec{\theta}} \right) \propto \sigma^{{ - \left( {n + 2} \right)}} \left| {\varvec{R}_{n} } \right|^{ - 1/ 2} { \exp }\left[ {\left( { - 1/ 2\sigma^{ 2} } \right) \, \left( {\varvec{x}_{n} - \mu \varvec{e}_{n} } \right)^{\text{T}} \varvec{R }_{n}^{ - 1} \left( {\varvec{x}_{n} - \mu \varvec{e}_{n} } \right)} \right] $$
(36)

From (34), (35) and (36) we obtain (8). After integration of (36) we obtain (37) which proves (9):

$$ \pi \left( {\sigma^{ 2} |\varvec{\varphi },\varvec{x}_{n} } \right) \propto \left( {\sigma^{ 2} } \right)^{{ - \left( {n + 1} \right)/ 2}} \left| {\varvec{R}_{n} } \right|^{ - 1/ 2} { \exp }\left[ {\left( { - 1/ 2\sigma^{ 2} } \right)\left[ {\varvec{e}_{n}^{\text{T}} \varvec{R}_{n}^{ - 1} \varvec{e}_{n} \varvec{x}_{n}^{\text{T}} \varvec{R}_{n}^{ - 1} \varvec{x}_{n} - \left( {\varvec{x}_{n}^{\text{T}} \varvec{R}_{n}^{ - 1} \varvec{e}_{n} } \right)^{ 2} } \right]/\left( {\varvec{e}_{n}^{\text{T}} \varvec{R}_{n}^{ - 1} \varvec{e}_{n} } \right)} \right] $$
(37)

After integration of (36) we obtain (38), which proves (10) after integration:

$$ \pi \left( {\varvec{\varphi }|\varvec{x}_{n} } \right) \propto \iint \sigma^{{ - \left( {n + 2} \right)}} \left| {\varvec{R}_{n} } \right|^{ - 1/ 2} { \exp }\left[ {\left( { - 1/ 2\sigma^{ 2} } \right) \, \left( {x_{n} - \mu \varvec{e}_{n} } \right)^{\text{T}} \varvec{R}_{n}^{ - 1} \left( {x_{n} - \mu \varvec{e}_{n} } \right)} \right]{\text{ d}}\mu {\text{d}}\sigma^{ 2} $$
(38)

See also Falconer and Fernadez (2007) for some results.Now for the case where truncation is applied we obtain from (7) and (11):

$$ \pi \left(\varvec{\theta}\right)f\left( {\varvec{x}_{n} |\varvec{\theta}} \right) \propto \sigma^{{ - \left( {n + 2} \right)}} \left| {\varvec{R}_{n} } \right|^{ - 1/ 2} { \exp }\left[ {\left( { - 1/ 2\sigma^{ 2} } \right) \, \left( {\varvec{x}_{n} - \mu \varvec{e}_{n} } \right)^{\text{T}} \varvec{R}_{n}^{ - 1} \left( {\varvec{x}_{n} - \mu \varvec{e}_{n} } \right)} \right]{\text{ I}}_{{ [ {{a,b]}}^{{n}} }} \left( {x_{ 1} , \ldots ,x_{n} } \right) $$
(39)

Conditional on μ ∈ [a, b], a, b ∈ R∪{− ∞,∞} the derivation of (12), (13) and (14) from (39) is then trivial.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tyralis, H., Koutsoyiannis, D. A Bayesian statistical model for deriving the predictive distribution of hydroclimatic variables. Clim Dyn 42, 2867–2883 (2014). https://doi.org/10.1007/s00382-013-1804-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00382-013-1804-y

Keywords

Search

Navigation