Comparison of genetic programming with neuro-fuzzy systems for predicting short-term water table depth fluctuations
Introduction
Physical-based numerical groundwater flow models are powerful tools for representing high spatial and temporal variations of aquifers. However, this capability renders the models data intensive, and to achieve acceptable simulations and prediction performance, the properties and conditions of the groundwater system must be accurately presented within the model's space and time domains (Coppola et al., 2003, Feng. et al., 2008). Because the properties and conditions of groundwater can never be ascertained with absolute accuracy, unavoidable discrepancies between the model and the real-world system reduce simulation accuracy hinders efforts to appropriately manage the groundwater resources (Coppola et al., 2005). Therefore, empirical models may be considered as alternative methods and can provide useful results without costly calibration time (Daliakopoulos et al., 2005, Box and Jenkins, 1976, Hipel and Mc Leod, 1994). However, these models have their own limitations, because they are data demanding models and they are not adequate when the dynamical behavior of the hydrological system changes in time (Bierkens, 1998).
In the recent past, the use of Artificial Intelligence techniques, such as Genetic Programming (GP), Adaptive Neuro-Fuzzy Inference System (ANFIS) and Artificial Neural Networks (ANNs) have become viable: Coulibaly et al. (2001) applied ANNs for modeling of monthly groundwater level fluctuations; Coppola et al. (2005) developed ANNs for accurately predicting potentiometric surface elevations; Daliakopoulos et al. (2005) applied ANN for forecasting groundwater level; Szidarovszky et al. (2007) introduced a hybrid ANNs-numerical model for groundwater problems; Coppola et al. (2007) applied a combination of ANN modeling with multi-objective optimization for a complicated real-world groundwater management problem in New Jersey; and Feng et al. (2008) applied ANNs to investigate the effects of human activities on regional groundwater levels; Yang et al. (2009) applied ANN for forecasting groundwater levels in Western Jilin Province, China.
The focus of the current paper is on the application of GP and ANFIS data driven models to forecast groundwater table depth time series. The methodology of GP was first proposed by Koza (1992), as a generalization of Genetic Algorithms (GA) (Goldberg, 1989). The fundamental difference between GP and GAs lie in the nature of individuals, where in GAs individuals are linear strings of fixed length (as chromosomes), while in GP individuals are nonlinear entities of different sizes and shapes (as parse trees). Major advantages of GP are that it can be applied to areas where (a) the interrelationships among the relevant variables are poorly understood (or where it is suspected that the current understanding may well be less than satisfactory), (b) finding the ultimate solution is hard, (c) conventional mathematical analysis does not, or cannot, provide analytical solutions, (d) an approximate solution is acceptable (or is the only result that is ever likely to be obtained), (e) small improvements in the performance are routinely measured (or easily measurable) and highly valued, and (f) there is a large amount of data, in computer readable form, that requires examination, classification, and integration (such as satellite observations) (Banzhaf et al., 1998). Also effective data driven neuro-fuzzy models have received more attention in the recent past. ANFIS was firstly introduced by Jang (1993), Jang and Sun (1995) and Jang et al. (1997), and later on widely applied in engineering problems. Jang (1993) introduced architecture and a learning procedure for the Fuzzy-Inference Systems (FIS) that uses a neural network learning algorithm for constructing a set of fuzzy if-then rules with appropriate membership functions (MFs) from the specified input–output pairs. This procedure is called an adaptive network-based-fuzzy inference system (ANFIS). There are largely two approaches for fuzzy inference systems, namely the approaches of Mamdani (Mamdani and Assilian, 1975) and Sugeno (Takagi and Sugeno, 1985). The differences between the two approaches arise from the consequent part. Mamdani's approach uses fuzzy membership functions, whereas Sugeno's approach uses linear or constant functions. The neuro-fuzzy model used in this study implements Sugeno's fuzzy approach (Takagi and Sugeno, 1985) to obtain the values for the output variable from those of input variables. For a given input–output data set, various Sugeno models may be developed by using different identification methods (i.e., grid partitioning, subtractive clustering and Gustafson–Kessel clustering methods). However, the recent researches demonstrated that the type of identification method does not affect the results rigorously (Vernieuwe et al., 2005). Therefore, the commonly used grid partitioning identification method was applied for constructing the neuro-fuzzy models in this paper. The grid partitioning method proposes independent partitions of each antecedent variable through defining the membership functions of all antecedent variables. A major problem with application of this method is that the construction of the membership functions of each variable is not dependent on each other, hence the relationship between the variables is omitted.
One of the strong points of using GP over other data driven techniques (e.g., ANFIS) is that it can produce explicit formulations (model expression) of the relationship that rules the physical phenomenon. Such expressions may be subject to some physical interpretations. Actually, the comprehensibility of GP models is also a way to reduce the risk of over-fitting to training data and improve generalization of resulting models. In this way, one may perform knowledge discovery using GP, finding some confirmation of well-known physical relationships and evolving interesting new formulae, as an upgrading of particular cases of study.
Review of all of the applications of GP and ANFIS in hydrology and water resources engineering is beyond the scope of this paper and only some limited studies are discussed here. Babovic et al. (2002) applied GP for modeling of risks in water supply. Aytek and Alp (2008) applied GP to rainfall-runoff modeling. Aytek and Kisi (2008) applied GP to suspended sediment transport streams. Ghorbani et al. (2010) applied GP to forecast averaged sea water level values. Kisi and Shiri (2010) applied GP and ANFIS techniques for predicting short-term and long term river flow.
Kisi (2005) estimated suspended sediment using neuro-fuzzy and neural network approaches. Kisi (2006) proposed a neuro-fuzzy computing technique for daily pan evaporation modeling. Partal and Kisi (2007) proposed a new wavelet-neuro-fuzzy conjunction model for precipitation forecast. Kisi (2009) applied evolutionary fuzzy models for river suspended sediment concentration estimation.
To the best knowledge of the authors, no study has been carried out to predict groundwater table fluctuations using GP and ANFIS. This provides an impetus for the current work. The aim of this study is the application and comparison of GP and ANFIS for forecasting short-term daily groundwater table depths. It is relevant to remarked that the models investigated here are normally applied within deterministic frameworks in professional practices, which has encouraged the practice of comparing the actual with predicted values. However, this is a black-and-white approach for selecting the merits of a method and does not necessarily measure the impact on the decision.
Section snippets
Used data
The data set used in this study was obtained from Illinois State Water Survey, U.S (www.isws.illinois.edu/data.asp). The time series of daily depth to water table records from two wells were used: Bondville (station no: 421832; FIPS code: 019; Latitude: 40°05′N; Longitude: 88°37′W; Altitude: 213 m) and Perry (station no: 421843; FIPS Code: 149; Latitude: 39°80′N; Longitude: 90°83′W; Altitude: 213 m). Groundwater levels are monitored continuously with Stevens Type-F paper chart recorders. The
Statistical measures and model implementations
Four statistical evaluation criteria were used to assess the model performance
- (1)
The coefficient of determination (R2); which ranges between 0 and 1, with higher values indicate the better performance of the model. Legates and McCabe (1999) argued that this indicator should not be applied as fitness measure alone. Therefore, it is appropriate to quantify the error in the same unit as for the variables, as discussed by Legates and McCabe (1999). One of these measures is
- (2)
the root mean square error (
Conclusions
The accuracy of GEP and ANFIS techniques in forecasting short-term (one-, two- and three-day ahead) ground water depth has been investigated in the present study. The inter-comparison of the results obtained using GEP and ANFIS indicated that the GEP models performed slightly better than the ANFIS models in forecasting ground water depths. It can be concluded that both the GEP and ANFIS models can be considered as promising tools for forecasting daily groundwater depths, based on previously
Acknowledgments
The data set used in this study was obtained from the website of Illinois State Water Survey, U.S. The authors are grateful to the staff of the Illinois State Water Survey, who were associated with data observation, processing and management of web site.
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