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Giorgio Corani, Giorgio Guariso Dipartimento di Elettronica ed Informazione Politecnico di Milano corani@elet.polimi.it Fuzzy modelling of basin saturation state and neural networks for flood forecasting iEMSs 2004
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Outline Neural networks modelling of the rainfall-runoff relationship Basin saturation issues The proposed joint fuzzy-neural networks approach Results: flood forecasting on Olona case study Conclusions
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Requisiti Del Sistema Accuratezza previsionale.. anche nel caso in cui non siano disponibili i dati rilevati da tutte le stazioni (robustezza) Velocità computazionale Minimo orizzonte temporale utile per interventi: 3h Lorizzonte previsionale raggiungibile dipende dalarea complessiva del bacino
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Problematiche Idrologiche Variabilità spaziale: piogge/permeabilità Non linearità: imbibimento del terreno. Pioggia cumulata 5gg Rainfall/Runoff
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Schema Di Previsione y(t, t-1,..): termini autoregressivi (portate) u1,u2(t-,t - -1,..): termini esogeni (piogge) : tempo di corrivazione piogge portate (ritardo) PREVISORE y(t+1) y(t) y(t-1) y(t-m)... u2(t- ) u2(t- -1) u2(t- -m)... u1(t- ) u1(t- -1) u1(t- -m)...
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Approccio Black Box Lineare (Arx) n ingressi esogeni : pluviometri disponibili (es: 2) stima parametrica MQ Problema: legame piogge-portate è non lineare! ARX1X2
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Un diverso arx per ogni classe idrologica. Arx Con Soglie dominio di pioggia cumulata (mm) In corrispondenza delle soglie si ha un brutale cambio di modello Soglia S1 S1=???? Soglia S2 S2=???? Predittore 1Predittore 2Predittore 3
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Dagli ARX alle Reti Neurali Richiesta di modellizzazione non lineare ARX vs reti neurali Reti neurali usate in diversi lavori idrologici degli ultimi anni
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Il cervello umano : reti neurali 100 miliardi di neuroni Ogni neurone collegato a migliaia di altri neuroni Soglia di attivazione (Marchese, 1987)
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Reti neurali biologiche Plasmabilità: le sinapsi variano nel tempo interagendo con segnali del mondo esterno Modifiche nei collegamenti sinaptici: memorizzazione delle informazioni Apprendimento
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Reti Neurali Artificiali (ANN) Idea di neurone artificiale: McCulloch (1943) Simulazione delle strutture nervose cerebrali. Scompone linformazione in informazioni elementari contenute allinterno di ogni neurone artificiale Algoritmi di apprendimento (1986) Sono approssimatori universali
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Modelli di neuroni artificiali xtxt x t-1 x t-2... w 1,1 w 1,r b 1 input neurone = f(Wx+b) x t- x t - -1... somma pesata degli ingressi (cfr. dendriti) funzione logistica (cfr. assone)
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Reti Neurali Artificiali (ANN) x0x0 x1x1 x2x2 y xrxr... f w 1,1 w n,r input strato nascosto (n neuroni) output neurone duscita
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Neural network modelling of rainfall-runoff process Data acquired from hydrometers and rain gauges (r 1,..r n ) in the basin Forecast is issued after the arrival of the rainfall event Hidden layer: logistic Output layer: linear y(t+k) (direct predictor) y(t), y(t-1),… Autoregressive terms Exogenous terms: rain gauge r j delayed of k j hours Input layer: r j (t-kj) r j (t-k j -1),…
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Basin saturation issues The catchment response to rainfall impulses depends strongly on the saturation state of the basin An indirect measure at time (t) may be obtained by using the information R(,t), i.e. cumulated rainfall on the time window [t-,t] The proxy can be noisy (spatial interpolation from local rain measures, differences between saturation and precipitation)
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Coupling fuzzy logic and neural networks The rationale: each saturation class results in a different non-linear rainfall-runoff relationship The idea: to train a different, specialized neural network on each saturation class to issue the forecast by linearly combining the prediction of the different models the higher the membership related to a given saturation class, the higher the weight of the corresponding predictor on the forecast
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Fuzzyfication of cumulated rainfall R(,t) A set of centroids is identified on R(,t) We fuzzify the basin state at each time step of the dataset The basin state at time (t) is classified in a fuzzy way. For instance: 1 (t) : membership related to saturation class 1 (dry class) 2 (t) : membership related to class 2 (medium class) 3 (t) : membership related to class 3 (wet class) 1 (t) + 2 (t) =1 (constraint)
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Specialized predictors training We implemented a weighted least squares variant of the LM training algorithm: To prevent overfitting, we jointly use regularization and early stopping during the training The optimal architectures are selected via trial and error (20 estimates of each model) The model showing the lowest wls on the validation set is finally chosen
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Issuing the forecast As in Takagi Sugeno systems, we linearly combine the output of the specialized models: is the prediction of the j-th specialized model Switching between models is smooth and ruled by the state of the basin at time (t)
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Olona case study Basin size: about 190 km 2 Average flow: 2.5 m 3 /sec (100 m 3 /sec with a return period of 10 years) Forecast horizon of interest: 3 hours One hydrometer, three rain-gauges Dataset: about 1100 hourly steps of flood data
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3-hours ahead prediction performances (testing set) Model Efficiency (R 2 ) Correlation RMSE High flows error FFNN.85.93.30.294 Fuzzy ( =2 days).86.94.29.319 Fuzzy ( =5 days).88.95.27.284 The fuzzy framework with =5 days appears the most effective forecasting approach
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Simulation sample
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Conclusions The proposed approach uses specialized models and couples their output via fuzzy logic, in order to account for the basin saturation state The framework outperforms the classical FFNN rainfall-runoff approach The framework complexity does not involve significant computational overload nor additional measurement costs to issue the prediction Interesting extensions to other domains: whats about modelling ozone peaks with temperature fuzzy classes?
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