Berardi, L.

Creaco, E.

Giustolisi, O.

Savic, D.

Sun, S.

I-Revues

NOVATECH 2016 
1 
Multi-objective evolutionary polynomial regression 
paradigm for the selection of relevant input variables 
in stormwater quality modelling 
Paradigme de la régression polynomiale évolutionnaire 
multi-objectif pour la sélection de variables d'entrée 
pertinentes de modèles de qualité des eaux pluviales 
 
E. Creaco1, L. Berardi2, Siao Sun3, O. Giustolisi4, D. Savic5 
 
1 Dept. of Civil Engineering and Architecture, University of Pavia, v. Ferrata 3, 
27100 Pavia, Italy. Email: creaco@unipv.it; previously at College of 
Engineering, Mathematics and Physical Sciences, University of Exeter, North 
Park Road, Exeter EX4 4QF, UK. Email: E.F.Creaco@exeter.ac.uk 
2 Dept. of Civil Engineering and Architecture, Technical University of Bari, v. E. 
Orabona 4, 70123 Bari, Italy. Email: luigi.berardi@poliba.it 
3 Institute of Geographical Sciences and Natural Resource Research, Chinese 
Academy of Sciences, Beijing, 100101, People’s Republic of China. Email: 
siao.sun05@163.com 
4 Dept. of Civil Engineering and Architecture, Technical University of Bari, v. E. 
Orabona 4, 70123 Bari, Italy. Email: o.giustolisi@poliba.it 
5 College of Engineering, Mathematics and Physical Sciences, University of 
Exeter, North Park Road, Exeter EX4 4QF, UK. Email: D.Savic@exeter.ac.uk 
 
RÉSUMÉ 
Cet article présente une procédure pour la sélection des variables d'entrée pertinents en utilisant le 
paradigme de la régression polynomiale évolutionnaire multi-objectif (EPR-MOGA). La procédure est 
basée sur l'examen des variables explicatives qui apparaissent à l'intérieur de l'ensemble des 
expressions symboliques de modèles de l'EPR-MOGA; de complexité et de qualité de calage avec la 
sortie cible croissantes. La stratégie permet également à la sélection d'être validée par un jugement 
technique. L'application de la procédure proposée sur la modélisation de la qualité des eaux pluviales 
sur deux bassins versants français montre qu'elle était en mesure de réduire considérablement le 
nombre de variables explicatives lors d'analyses successives. Enfin, les modèles EPR-MOGA obtenus 
après la sélection des variables d'entrée sont comparés avec ceux obtenus en utilisant la même 
technique sans bénéficier de sélection des variables d'entrée et à ceux obtenus dans des travaux 
précédents où d'autres techniques de modélisation des données ont été utilisées sur les mêmes 
données. La comparaison met en évidence l'efficacité à la fois de EPR-MOGA et de la procédure de 
sélection des variables d'entrée. 
ABSTRACT 
This paper presents a procedure for the selection of relevant input variables using the multi-objective 
evolutionary polynomial regression (EPR-MOGA) paradigm. The procedure is based on scrutinizing 
the explanatory variables that appear inside the set of EPR-MOGA symbolic model expressions of 
increasing complexity and goodness of fit to target output. The strategy also enables the selection to 
be validated by engineering judgement. The application of the proposed procedure on modelling storm 
water quality parameters in two French catchments shows that it was able to significantly reduce the 
number of explanatory variables for successive analyses. Finally, the EPR-MOGA models obtained 
after the input selection are compared with those obtained by using the same technique without 
benefitting from input selection and with those obtained in previous works where other data-modelling 
techniques were used on the same data. The comparison highlights the effectiveness of both EPR-
MOGA and the input selection procedure. 
KEYWORDS 
Data driven modelling, input selection, EPR, stormwater quality, pollutants 
SESSION 
2 
1 INTRODUCTION 
In the framework of data driven models, identifying the most relevant variables to describe a given 
phenomenon is of relevant interest from both the data collection and effective modelling viewpoints. 
On the one hand, understanding which information should be collected first can help in prioritizing 
what should be measured under budget constraints. On the other hand, the preliminary selection of 
inputs is essential for system identification through data-modelling. Irrelevant and redundant input 
variables used in data-modelling routines might result in poor modelling. As highlighted by Galelli et al. 
(2014), irrelevant input variables are uninformative about the underlying process and their eventual 
inclusion would cause noise and complexity to be added to the model. The inclusion of redundant, but 
relevant, input variables would instead increase the dimensionality of the model identification without 
providing any additional predictive benefit. One of the possible undesirable consequences of including 
irrelevant and redundant input variables is the construction of models that over-fit training data, while 
showing poor generalization capabilities on other similar contexts. Such a drawback is actually 
undesirable in engineering fields where the transferability of identified data-driven models from one 
case to another (new) is required. 
The issue of input variable selection in non-linear models of stormwater quality is addressed in this 
paper. In particular, the procedure for relevant input selection developed and described by Creaco et 
al. (2015) is applied in this work. This procedure is based on the use of the multi-objective evolutionary 
polynomial regression (EPR-MOGA), a nonlinear regression technique developed by Giustolisi and 
Savic (2009) which has been widely used in the field of Hydroinformatics  
The remainder of the paper is organized as follows. First, the main aspects of the methodology are 
described. Then, the applications of the methodology to the modelling of stormwater quality in two 
French sewer systems show how the input selection procedure works with a wide set of available 
data. Finally, EPR-MOGA is applied to construct stormwater quality models based on the selected 
input variables in the two sewer systems. The constructed models are then compared with those 
developed on the same data in a previous work (Sun and Bertrand-Krajewski, 2011). Further details 
about methodology and applications can be found in Creaco et al. (2016). 
2 METHODOLOGY 
The input selection procedure is based on the use of the nonlinear regression technique EPR-MOGA, 
which is able to yield data driven models taking on the following form: 
, (1) 
where Y and X are the output and input variables, respectively; aj is the j-th model parameter, 
ES(j,1,…,k) are the exponents of the inputs, and function f is a user-defined function that has 5 
possible settings: (1) no function, (2) natural logarithm, (3) exponential, (4) tangent hyperbolic and (5) 
secant hyperbolic; m and k represent the maximum number of polynomial terms and the number of 
input variables (details can be found in Giustolisi and Savic, 2009). In all the applications of the work, 
the first setting is used for function f, i.e., no function. 
The optimal values of parameters a and exponents ES are searched for inside EPR-MOGA in such a 
way as to obtain a trade-off between mutually contrasting objectives (Giustolisi and Savic, 2009). In 
particular, in the applications of the work, the objective functions calculated for the generic model 
proposed by EPR-MOGA were: 1 – coefficient of determination (CoD), calculated as a function of 
simulated and observed values for the output; 2 – number Nx of inputs in Eq. (1); 3 – number Na of 
monomials in Eq. (1). In fact, objective function 1 is representative of the goodness of fit of the model 
to data whereas objective functions 2 and 3 are representative of model complexity. 
As a result of the application of EPR-MOGA to a certain training dataset, a Pareto front of optimal 
models featuring various levels of goodness of fit and complexity is obtained. The goodness of fit of 
the models can then be assessed also in a testing dataset. 
The input selection procedure is based on counting the number of occurrences of the generic input in 
the optimal set of EPR-MOGA models. After fixing a certain occurrence threshold (equal to 0 in the 
applications of the work), the generic input is then considered relevant if it features a number of 
occurrences larger than the threshold. 
NOVATECH 2016 
3 
In the case of multiple available case studies, the total number of occurrences of the generic input is 
obtained as the sum of its numbers of occurrences in the various case studies. 
3 APPLICATIONS 
3.1 Case studies 
Two urban catchments, located in the West and East part of Lyon (France) respectively, namely the 
Ecully and Chassieu catchments (Métadier and Bertand-Krajewski, 2012), were considered for the 
applications. These catchments were two experimental sites in the OTHU project (Field Observatory 
for Urban Hydrology - www.othu.org). In particular, the Ecully catchment is a low density residential 
catchment, with a surface area of 245 ha and an imperviousness coefficient of 42%. The Chassieu 
catchment is an industrial catchment, with a surface area of 185 ha and an imperviousness coefficient 
of 75%. In the Ecully and Chassieu catchments, numerous variables were measured during 239 and 
263 rain events, respectively, in the period from 2004 to 2008. 
A typical stormwater quality indicator, i.e., the total amount (kg) of chemical oxygen demand (COD) 
recorded during a certain rain event, was chosen as target variable (model output). Other 56 variables, 
characterizing rainfall or runoff information, were considered potential explanatory variables (model 
inputs). These variables, whose symbols appear in the x axis in the graphs in Figure 1, are thoroughly 
described by Métadier and Bertand-Krajewski (2012). 
In the applications, the first 2/3 of the events were used for selecting the most relevant variables. 
Afterwards, the same set of events was used to develop and train the data driven models; the second 
1/3 of the events were used for testing the data driven models. 
3.2 Results 
The application of EPR-MOGA to the Ecully and Chassieu catchment yielded the results reported in 
the graphs in Figure 1, in terms of occurrences of each of the 56 input variables. The application of the 
input selection procedure enabled identification of 29 relevant inputs and, therefore, a substantial 
simplification of the problem. 
 
Figure 1 Modelling COD in the two catchments. Occurrences of the explanatory variables in EPR-MOGA models 
and selected variables highlighted with a rectangle. 
A subsequent application of EPR-MOGA to the set of relevant inputs is able to produce models with 
higher goodness of fit to data in both the training and testing phases, than those that are obtained by 
applying EPR-MOGA to the whole set of 56 inputs. EPR-MOGA models obtained after relevant input 
identification are also more robust, since they show smaller performance decay when passing from the 
training to testing phases. This is evident in the graphs in Figure 2, relative to the simulation of COD in 
the Ecully catchment. The CoD of the models obtained by applying EPR-MOGA to the relevant inputs 
were then reported in the graphs in Figure 3 and compared with the CoD values of the models 
SESSION 
4 
obtained by Sun and Bertrand-Krajewski (2011) using the genetic programming (GP) technique. 
Overall, the graphs show that various models with larger CoD values than the GP models in both the 
training and testing phases can be obtained thanks to the fitting performance of EPR and to the 
adoption of the input selection procedure. 
In the paper of Creaco et al. (2015), these results are described more accurately and other aspects, 
such as the better simplicity and interpretability of the EPR-MOGA models, are highlighted.  
 
Figure 2 Values of coefficient of determination CoD obtained by the EPR models in the Ecully catchment in the 
training (black dots) and testing (grey dots) phases for the representation of COD (Kg); results of a) application of 
EPR following input selection procedure; b) application of EPR alone. In the graphs EPR models are sorted 
according to ascending values of CoD. 
 
Figure 3 Values of coefficient of determination CoD obtained by the EPR models for the representation of COD 
(Kg) in the Ecully and Chassieu catchments in the training and testing phases. CoD values of the models 
obtained by Sun and Bertrand-Krajewski (2011) using the GP technique in the testing and training phases. 
 
ACKNOWLEDGEMENTS 
The data sets of the Chassieu and Ecully catchments have been collected and made available by 
OTHU – Field Observatory on Urban Hydrology (www.othu.org) with the financial support of the 
Greater Lyon and the Rhône-Méditerranée & Corse Water Agency. 
LIST OF REFERENCES 
Creaco, E., Berardi, L., Sun, S., Giustolisi, O. and Savic, D. (2016). Selection of relevant input variables in 
stormwater quality modelling by multi-objective evolutionary polynomial regression paradigm. Water 
Resources Research, doi: 10.1002/2015WR017971. 
Galelli, S., Humphrey, G.B., Maier, H.R., Castelletti, A., Dandy, G.C., and Gibbs, M.S. (2014). An evaluation 
framework for input variable selection algorithms for environmental data-driven models. Environmental 
Modelling & Software, 62, 33-51. 
Giustolisi, O., and Savic, D.A. (2009). Advances in data-driven analyses and modelling using EPR-MOGA. J. 
Hydroinformatics, 11, 225-236. 
Metadier, M., and Bertrand-Krajewski, J.-L. (2012). The use of long-term on-line turbidity measurements for the 
calculation of urban stormwater pollutant concentrations, loads, pollutographs and intra-event fluxes. Water 
Research, 46(20), 6836-6856. 
Sun, S., and Bertrand-Krajewski, J.-L. (2011). The calibration of urban storm water quality models using genetic 
programming (GP). In: Urban Water Management: Challenges and Oppurtunities. Exeter, UK, 5-7 September 
2011, Eds. D.A. Savic, Z. Kapelan, D. Butler, Centre for Water Systems, University of Exeter, pp. 663-668. 


Paradigme de la régression polynomiale évolutionnaire multi-objectif pour la sélection de variables d'entrée pertinentes de modèles de qualité des eaux pluviales

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Paradigme de la régression polynomiale évolutionnaire multi-objectif pour la sélection de variables d'entrée pertinentes de modèles de qualité des eaux pluviales

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