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Hua-Liang Wei, Stephen A. Billings Department of Automatic Control & Systems Engineering

Practical Linear and Nonlinear Modelling of Environmental Data: A Case Study for River Flow Forecasting. Hua-Liang Wei, Stephen A. Billings Department of Automatic Control & Systems Engineering The University of Sheffield Sheffield, Mappin Street, S1 3JD

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Hua-Liang Wei, Stephen A. Billings Department of Automatic Control & Systems Engineering

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  1. Practical Linear and Nonlinear Modelling of Environmental Data: A Case Study for River Flow Forecasting Hua-Liang Wei, Stephen A. Billings Department of Automatic Control & Systems Engineering The University of Sheffield Sheffield, Mappin Street, S1 3JD w.hualiang@sheffield.ac.uk , s.billings@sheffield.ac.uk 05/09/2009

  2. Aim: ♦ To develop data-based modelling techniques that can be used for environmental system analysis and forecasting Objective: ♦ As an example, to introduce a novel Fractional Power Autoregressive (FPAR) model for river flow modelling and forecasting 05/09/2009

  3. The Thames River at Kingston — Some View Points 05/09/2009

  4. The Thames River at Kingston Photos: http://commons.wikimedia.org/wiki/File:River_Thames_at_Kingston.JPG 05/09/2009

  5. Kinston Bridge over the River Thames Kingston Bridge over the River Thames at Kingston upon Thames, London. Photos:http://www.britannica.com/EBchecked/topic/318762/Kingston-upon-Thames 05/09/2009

  6. Kingston Upon Thames: Flood Events 05/09/2009 Resource:Environment Agency,http://www.environment-agency.gov.uk/static/documents

  7. Data Analysis and Modellingfor River Flow Forecasting ■Forecasting of river flow activities is helpful in planning and utilising local and national water resource systems, as well as avoiding disastrous floods. ■Data-based modelling, aimed at building mathematical models based on limited observational data, provides a powerful tool for river flow data modelling and analysis. ■The basic idea behind the data-based modelling approach is that: the process under study is treated to be a black-box where the inherent dynamics/mechanisms are unknown. 05/09/2009

  8. Applications Data Model Historically observed data e,g. river flow, rainfall-flow (rainfall-run- off), global temperature, and other environmental and space weather data Linear/nonlinear Parametric/nonparametric Time series (AR/NAR) Input-output models (ARX/NARX/NARMAX) Neural Networks, Wavelet models, etc. Environmental and space weather data modelling and analysis, e.g. river flow forecasting Data-Based Modelling and System Identification ■The model considered here belongs to a class of nonlinear autoregressive (NAR) representations: •Fractional Power AutoRegressive (FPAR) model 05/09/2009 Slide 8 of 19

  9. Kingston Upon Thames— Historical Data Records 05/09/2009

  10. River Flow of the Thames at Kingston [m3s-1] 05/09/2009 Resource:Environment Agency,Centre for Ecology and Hydrology, Wallingford, UK.

  11. River Flow Forecasting Learning a model from existing data (e.g. observations of the period from 1987 to 2000) The resultant model will be used to forecast future behaviour 05/09/2009

  12. The Fractional Autoregressive Model ■The form of FPAR model •k is the sampling index (the number of days for river flow observations) • s is an index to indicate that the model is for s-day ahead forecasting. • are model parameters. • e(k) is the modeling error. • are the fractional power numbers. • Traditional AR model is a special case of the FPAR model. 05/09/2009

  13. FPAR Model for River Flow Forecasting: Thames at Kingston ■The FPAR model • d =15. • can be estimated using existing methods, see references [1]-[8]. ■The Data • Training Data: Daily observations of the Thames river flow at Kingston, from 1st January 1987 to 31th December 2000, a total of 5114 observations • Testing Data: Daily observations from 1ts January 2001 to 31th December 2006, a total of 2191 samples. 05/09/2009

  14. FPAR Model for River Flow Forecasting: One-day Ahead Prediction Root Mean Square Error (RMSE): 12.69 m3s-1. 05/09/2009

  15. FPAR Model for River Flow Forecasting: Two-day Ahead Prediction Root Mean Square Error (RMSE): 13.96 m3s-1. 05/09/2009

  16. FPAR Model for River Flow Forecasting: Five-day Ahead Prediction Root Mean Square Error (RMSE): 18.43 m3s-1. 05/09/2009

  17. FPAR Model for River Flow Forecasting: Ten-day Ahead Prediction Root Mean Square Error (RMSE): 24.75 m3s-1. 05/09/2009

  18. Conclusions ♦Short-term (e.g. 1- and 2-day ahead) and medium-term ( e.g. 5-day ahead) forecasts of river flow are available by means of system identification techniques. ♦Indeed, the proposed FPAR model produces reliable short- and medium-term forecasts for the river flow in the Thames at Kingston. ♦The FPAR model can produce satisfactory results for medium-term predictions of river flow data. ♦Data based modelling, coupled with physical insights about the system, will produce more reliable results for medium-and long-term predictions. 05/09/2009

  19. Key References 1. S. A. Billings and H.L. Wei, ‘Sparse model identification using a forward orthogonal regression algorithm aided by mutual information’, IEEE Transactions on Neural Networks, Vol 18, 306–310, 2007. 2.S. A. Billings and H. L. Wei, ‘An adaptive search algorithm for model subset selection and nonlinear system identification’, International Journal of Control, Vol 81, 714–724, 2008. 3. H.L. Wei, S. A. Billings, and J. Liu, ‘Term and variable selection for nonlinear system identification’, International Journal of Control, Vol 77, 86–110, 2004. 4. H.L. Wei, S.A. Billings, M.A. Balikhin, ‘Prediction of the Dst index using multiresolution wavelet models’ Journal of Geophysical Research, Vol. 109, A07212, 2004. 5. H.L. Wei and S. A. Billings, ‘Long term prediction of noninear time series using multiresolution models’, International Journal of Control, Vol 79, 569–580, 2006. 6. H.L. Wei and S. A. Billings, ‘An efficient nonlinear cardinal B-spline model for high tide forecasts at the Venice Lagoon’, Nonlinear Processes in Geophysics, Vol 13, 577–584, 2006. 7. H.L. Wei, D. Zhu, S.A. Billings, M.A. Balikhin, ‘Forecasting the geomagnetic activity of the Dst index using multiscale radial basis function networks’, Advances in Space Research, Vol. 40, pp.1863–1870, 2007. 8. H.L. Wei and S. A. Billings, ‘Model structure selection using an integrated forward orthogonal search algorithm assisted by squared correlation and mutual information’, International Journal of Modelling, Identification and Control, Vol 3, 341–356, 2008. 05/09/2009

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