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Giorgio Corani, Giorgio Guariso
Dipartimento di Elettronica ed InformazionePolitecnico di Milano corani@elet.polimi.it
Fuzzy modelling of basin saturation state and neural networks for flood forecasting
iEMSs 2004
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
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
L’orizzonte previsionale raggiungibile dipende dal’area complessiva del bacino
Problematiche Idrologiche
Variabilità spaziale: piogge/permeabilità
Non linearità: imbibimento del terreno.
0.00
0.60
0 90Pioggia cumulata 5gg
Rai
nfal
l/R
unof
f
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)
...
Approccio Black Box Lineare (Arx)
n ingressi esogeni : pluviometri disponibili (es: 2)
stima parametrica MQ Problema: legame piogge-portate è non
lineare!
21
)2(22)1(111 ***ˆnbnbna
iktiiktiitit ububyay
AR X1 X2
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 S1S1=????
Soglia S2S2=????
Predittore 1 Predittore 2 Predittore 3
Dagli ARX alle Reti Neurali
Richiesta di modellizzazione non lineare
ARX vs reti neurali
Reti neurali usate in diversi lavori idrologici degli ultimi anni
Il cervello umano : reti neurali 100 miliardi di neuroni Ogni neurone collegato a migliaia di altri
neuroni Soglia di attivazione
(Marchese, 1987)
Reti neurali biologiche
Plasmabilità: le sinapsi variano nel tempo interagendo con segnali del mondo esterno
Modifiche nei collegamenti sinaptici: memorizzazione delle informazioni
Apprendimento
Reti Neurali Artificiali (ANN)
Idea di neurone artificiale: McCulloch (1943)
Simulazione delle strutture nervose cerebrali.
Scompone l’informazione in informazioni elementari contenute all’interno di ogni neurone artificiale
Algoritmi di apprendimento (1986)
Sono approssimatori universali
Modelli di neuroni artificiali
xt
xt-1
xt-2
...
w1,1
w1,r
b
1
input
neurone
= f(Wx+b)
xt-
xt --1
...
jkkjj bxwz
somma pesata degli ingressi (cfr. dendriti)
funzione logistica (cfr. assone)
Reti Neurali Artificiali (ANN)
x0
x1
x2
y
xr
...
f
w1,1
wn,r
input strato nascosto(n neuroni)
output
neurone d’uscita
Neural network modelling of rainfall-runoff process Data acquired from hydrometers and rain gauges (r1,..rn) 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 rj
delayed of kj hours
Input layer:
rj(t-kj)
rj(t-kj-1),…
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)
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
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)
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
Dyyt jjj 2ˆ)()(
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)
j
jj ktytkty )(ˆ)()(ˆ
jy
Olona case study Basin size: about 190 km2
Average flow: 2.5 m3/sec (100 m3/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
3-hours ahead prediction performances (testing set)
ModelEfficien
cy(R2)
Correlation
RMSEHigh flows error
FFNN .85 .93 .30 .294Fuzzy(=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
Simulation sample
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: what’s about modelling ozone peaks with temperature fuzzy classes?
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