Adaptive fuzzy modeling versus artificial neural networks
Introduction
The modeling of natural process has to cope with uncertainty in the parameters and the modeling methods. The crux of ecological modeling lies in our understanding: “depending upon the nature of the study, complexity can confound the analysis of an ecosystem model; the more interacting components a model has, the less straightforward it is to extract and separate causes and consequences; this is compounded when uncertainty about components obscures the accuracy of a simulation” (Ecological Model, 2006). This leads to the wish of the modelers to use expert's knowledge as model basis or to extract models from the data in a straightforward way (Ghielmi and Eccel, 2006). The focus of this paper is on these soft computing methods and their application to a simplified but realistic problem.
In this paper models will be discussed as function approximators. On the one hand there are some inputs (for example soil quality, nitrogen fertilizer, and cropping year) and on the other hand a single output (crop yield). There are a lot of possible methods for developing models from data sets, such as multiple regression, artificial neural networks, fuzzy methods, etc. Because of the lack in understanding ecological modeling the introduction of expert's knowledge in form of fuzzy modeling or artificial neural networks as universal function approximators can be one step towards creating models without an explicit mathematical equation. It can be shown that fuzzy modeling and neural networks are similar from a mathematical point of view. This offers the opportunity to introduce a training algorithm in fuzzy models and to compare these with artificial neural networks.
The complete investigation was based on the software “Spatial Analysis and Modeling Tool” (SAMT). This tool is available as free open source software (SAMT, 2007). Additionally were used free open source libraries like the gnu scientific library (GSL, 2007), the hierarchical data format (HDF, 2007) and for the graphical user interface the QT-library (QT, 2007).
To be useful for practical simulations, models should satisfy the following requirements.
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Accuracy: the error resulting between the simulated and measured values should be minimal.
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Generalization: the model should reduce the complexity of the real world using an approximation of the data based on fundamental knowledge.
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Portability: the model should be usable in different sites with slightly changed inputs (compared to the training data).
These requirements are contradictory (for example, a high accuracy can lead to low generalization) to some degree so a practical compromise between them must be found.
Section snippets
Description of the data sets
The handling of different artificial neural networks and fuzzy methods to develop simple models, their specifics (advantages and disadvantages) and their suitability for application in the field of ecology is shown using data sets for spatial grain yield (yield) estimation for winter rye and winter barley under practical field cropping conditions. An example from the literature using artificial neural networks in a similar application is given in Grzesiak et al. (2006); an introduction to the
Feed forward networks
An artificial neural network as function approximator is useful because it can approximate a desired behavior without the need to specify a particular function. This is a big advantage of artificial neural networks compared to multivariate statistics. The price for this is the difficult nonlinear optimization technique, that can lead to local optima or to the so called over training. Feed forward networks are the most established type of artificial neural networks. A good introduction is given
Radial basis function network (RBF)
A feed forward network used as a function approximator is a black box fed with inputs, which produces meaningful outputs. However, there are no methods for checking if the inputs are in the region where the network was trained. A further type of artificial neural network is the so called radial basis function network (Bishop, 1995), which uses a cluster algorithm in the first step (unsupervised training) and calculates the approximation in the second step (supervised training).
Fuzzy modeling
Fuzzy modeling is designed to use knowledge of an experienced expert directly as basis of modeling. This means that:
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fuzzy sets for the inputs of the model have to be created,
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the outputs have to be created, and
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a set of rules that combines the inputs with the outputs has to be created.
The rules are easy to understand and provide a basis for understanding of a fuzzy model and the discussion with other experts. We have learned that it is important to find the real expert and not a group of people
Comparison of feed forward network, radial basis function network, and a fuzzy system using an independent data set
The data sets B1 (winter barley) and B2 (winter rye) from the independent region are the basis for the comparison of different methods. The results shown here were produced with the trained neural network (NN) and the trained fuzzy system (FS) without any changes, and any retraining. The comparison shows how the models can cope with both these data sets.
Table 5 shows that the accuracy of the fuzzy model without training is very poor. At a first glance this is rather surprising, but it reflects
Conclusions and perspectives
It has been shown that the fuzzy system and neural networks have similar mathematical fundamentals, so that a training method used with neural networks could be applied to fuzzy models, and RBFs could be used as fuzzy models. An interesting approach to interpret Kohonen maps as linguistic variables is given in Predrycz and Card (1992). The combination of a fuzzy model with a training of the outputs has the advantage that the model can be understood by the expert and adapt to the measured
Acknowledgment
This contribution was supported by the German Federal Ministry of Consumer Protection, Food and Agriculture, and the Ministry of Agriculture, Environmental Protection and Regional Planning of the Federal State of Brandenburg (Germany). Thank you to Dr. W. Haberstock (Institute of Land Use Systems and Landscape Ecology) and to Dr. sc. G. Lutze (Institute of Landscape Systems Analysis) of the Leibniz-Center for Agricultural Landscape Research in Muencheberg (Germany) for the friendly
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