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?