05/09/2009 Slide 1 of 19 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

05/09/2009 Slide 2 of 19 ♦ To develop data-based modelling techniques that can be used for environmental system analysis and forecasting ♦ As an example, to introduce a novel Fractional Power Autoregressive (FPAR) model for river flow modelling and forecasting Aim: Objective:

05/09/2009 Slide 3 of 19 The Thames River at Kingston — Some View Points

05/09/2009 Slide 4 of 19 The Thames River at Kingston Photos:

05/09/2009 Slide 5 of 19 Kinston Bridge over the River Thames Kingston Bridge over the River Thames at Kingston upon Thames, London. Photos:

05/09/2009 Slide 6 of 19 Kingston Upon Thames: Flood Events Resource: Environment Agency,

05/09/2009 Slide 7 of 19 Data Analysis and Modelling for 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 Slide 8 of 19 Data-Based Modelling and System Identification DataModel Applications Historically observed data e,g. river flow, rainfall-flow (rainfall-run- off), global temperature, and other environmental and space weather data Environmental and space weather data modelling and analysis, e.g. river flow forecasting Linear/nonlinear Parametric/nonparametric Time series (AR/NAR) Input-output models (ARX/NARX/NARMAX) Neural Networks, Wavelet models, etc. ■ The model considered here belongs to a class of nonlinear autoregressive (NAR) representations: Fractional Power AutoRegressive (FPAR) model

05/09/2009 Slide 9 of 19 Kingston Upon Thames— Historical Data Records

05/09/2009 Slide 10 of 19 River Flow of the Thames at Kingston [m 3 s -1 ] Resource: Environment Agency, Centre for Ecology and Hydrology, Wallingford, UK.

05/09/2009 Slide 11 of 19 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 Slide 12 of 19 The Fractional Autoregressive Model ■ The form of FPAR model are model parameters. 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. 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 Slide 13 of 19 FPAR Model for River Flow Forecasting: Thames at Kingston ■ The FPAR model can be estimated using existing methods, see references [1]-[8]. d =15. ■ The Data Training Data: Daily observations of the Thames river flow at Kingston, from 1 st 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 Slide 14 of 19 FPAR Model for River Flow Forecasting: One-day Ahead Prediction Root Mean Square Error (RMSE): m 3 s -1.

05/09/2009 Slide 15 of 19 FPAR Model for River Flow Forecasting: Two-day Ahead Prediction Root Mean Square Error (RMSE): m 3 s -1.

05/09/2009 Slide 16 of 19 FPAR Model for River Flow Forecasting: Five-day Ahead Prediction Root Mean Square Error (RMSE): m 3 s -1.

05/09/2009 Slide 17 of 19 FPAR Model for River Flow Forecasting: Ten-day Ahead Prediction Root Mean Square Error (RMSE): m 3 s -1.

05/09/2009 Slide 18 of 19 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 Slide 19 of 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, 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, 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, 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, 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, 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, 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, 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.