Abstract
In this paper we show how the grey box methodology can be applied to find models that can describe the flow prediction uncertainty in a sewer system where rain data are used as input, and flow measurements are used for calibration and updating model states. Grey box models are composed of a drift term and a diffusion term, respectively accounting for the deterministic and stochastic part of the models. Furthermore, a distinction is made between the process noise and the observation noise. We compare five different model candidates’ predictive performances that solely differ with respect to the diffusion term description up to a 4 h prediction horizon by adopting the prediction performance measures; reliability, sharpness and skill score to pinpoint the preferred model. The prediction performance of a model is reliable if the observed coverage of the prediction intervals corresponds to the nominal coverage of the prediction intervals, i.e. the bias between these coverages should ideally be zero. The sharpness is a measure of the distance between the lower and upper prediction limits, and skill score criterion makes it possible to pinpoint the preferred model by taking into account both reliability and sharpness. In this paper, we illustrate the power of the introduced grey box methodology and the probabilistic performance measures in an urban drainage context.
Similar content being viewed by others
Notes
Continuous-Time Stochastic Modelling - http://www.imm.dtu.dk/ctsm
References
Baadsgaard M, Nielsen JN, Spliid H, Madsen H, Preisel M (1997) Estimation in stochastic differential equations with a state dependent diffusion term. SYSID’97—11th IFAC symposium of system identification, IFAC
Barbera PL, Lanza LG, Stagi L (2002) Tipping bucket mechanical errors and their influence on rainfall statistics and extremes. Water Sci Technol 45(2):1–9
Bertrand-Krajewski JL, Bardin JP, Mourad M, Béranger Y (2003) Accounting for sensor calibration, data validation, measurement and sampling uncertainties in monitoring urban drainage systems. Water Sci Technol 47(2):95–102
Breinholt A, Thordarson FÖ, Møller JK, Mikkelsen PS, Grum M, Madsen H (2011) Grey box modelling of flow in sewer systems with state dependent diffusion. Environmetrics 22(8):946–961
Gneiting T, Raftery AE (2007) Strictly proper scoring rules, prediction and estimation. J Am Stat Assoc 102(477):359–378
Gneiting T, Balabdaoui F, Raftery AE (2007) Probabilistic forecasts, calibration and sharpness. J R Stat Soc B 69(2):243–268
Giraldo JM, Leirens S, Díaz-Grenados MA, Rodríguez JP (2010) Nonlinear optimization for improving the operation of sewer systems: the Bogotá Case Study. International Environmental Modelling and Software Society (iEMSs). 2010 International Congress on Environmental Modelling and Software Modelling for Environments Sake, Fifth Biennial Meeting, Ottawa, Canada
Iacus SM (2008) Simulation and Inference for stochastic differential equations—with R examples. Springer series of Statistics
Jacobsen JL, Madsen H, Harremoës P (1997) A stochastic model for two-station hydraulics exhibiting transient impact. Water Sci Technol 36(5):19–26
Jazwinski AH (2007) Stochastic processes and filtering theory. Dover Publications, Mineola, NY
Jonsdottir H, Jacobsen, JL, Madsen H (2001) A grey-box model describing the hydraulics in a creek. Environmetrics 12:347–356
Jonsdottir H, Madsen H, Palsson OP (2006) Parameter estimation in stochastic rainfall-runoff models. J Hydrol 326(1–4):379–393
Kloeden P, Platen E (1999) Numerical solutions of stochastic differential equations. Springer
Kristensen NR, Madsen H (2003) Continuous time stochastic modeling—CTSM 2.3—mathematics guide. Technical Unversity of Denmark
Kristensen NR, Madsen H, Jørgensen SB (2004a) Parameter estimation in stochastic grey-box models. Automatica 40:225–237
Kristensen NR, Madsen H, Jørgensen SB (2004b) A method for systematic improvement of stochastic grey-box models. Comput Chem Eng 28(8):1431–1449
Limpert E, Stahel WA, Abbt M (2001) Log-normal distributions across the sciences: keys and clues. BioScience 51(5):341–352
Madsen H (2008) Time series analysis. Chapman & Hall/CRC
Mannina G, Freni G, Viviani G, Saegrov S, Hafskjold L (2006) Integrated urban water modelling with uncertainty analysis. Water Sci Technol—WST 54(6–7):379–386
Møller JK, Nielsen HA, Madsen H (2008) Time-adaptive quantile regression. Comput Stat Data Anal 52:1292–1303
Møller JK, Madsen H, Carstensen J (2011) Parameter estimation in a simple stochastic differential equation for phytoplankton modelling. Ecol Model 222:1793–1799
Molini A, Lanza LG, Barbera Pl (2005) The impact of tipping-bucket raingauge measurement errors on design rainfall for urban-scale applications. Hydrol Process 19:1073–1088
Nielsen HA, Madsen H (2006) Modelling the heat consumption in district heating systems using a grey-box approach. Energy Build 38(1):63–71
Ocampo-Martinez C, Puig V (2010) Piece-wise linear functions-based model predictive control of large-scale sewage systems. IET Control Theory Appl 4(9):1581–1593
Øksendal B (2003) Stochastic differential equations—an introduction with applications, 6th edn. Springer
Pedersen L, Jensen NE, Christensen LE, Nielsen HA, Madsen H (2010) Quantification of the spatial variability of rainfall based on a dense network of rain gauges. Atmospheric Res 95(4):441–454
Pinson P, Nielsen HA, Møller JK, Madsen H (2007) Non-parametric probabilistic forecasts of wind power: required properties and evaluation. Wind Energy 10(6):497–516
Puig V, Cembrano G, Romera J, Quevedo J, Aznar B, Ramoń G, Cabot J (2009) Predictive optimal control of sewer networks using CORAL tool: application to Riera Blanca catchment in Barcelona. Water Sci Technol 60(4):869–878
Shedekar VS, King KW, Brown LC, Fausey NR, Heckel M, Harmel DR (2009) Measurement errors in tipping bucket rain gauges under different rainfall Intensities and their implication to hydrologic models. Conf. paper, ASABE Annual International Meeting, June 21–24, pp 1–9
Tornøe CW, Jacobsen J, Pedersen O, Hansen T, Madsen H (2006) Grey-box Modelling of pharmacokinetic/pharmacodynamic systems. J Pharmacokinet Pharmacodyn 31(5):401–417
Vaes G, Willems P, Berlamont J (2005) Areal rainfall correction coefficients for small urban catchments. Atmospheric Res 77(1–4):48–59
Vestergaard M (1998) Nonlinear filtering in stochastic volatility models. Master’s thesis, Technical University of Denmark. Department of Mathematical Modelling, Lyngby, Denmark
Willems P (2001) Stochastic description of the rainfall input errors in lumped hydrological models. Stoch Environ Res Risk Assess 15:132–152
Willems P (2010) Parsimonious model for combined sewer overflow pollution. J Environ Eng 136(3):316–325
Acknowledgements
We appreciate the help and support of flow meter data from Spildevandcenter Avedøre I/S. The research was funded by a PhD fellowship, including DTU Informatics and DTU Environment, and by the Danish Strategic Research Council (Sustainable Energy and Environment Programme).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Thordarson, F.Ö., Breinholt, A., Møller, J.K. et al. Evaluation of probabilistic flow predictions in sewer systems using grey box models and a skill score criterion. Stoch Environ Res Risk Assess 26, 1151–1162 (2012). https://doi.org/10.1007/s00477-012-0563-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00477-012-0563-3