Feedforward neural networks have been established as versatile tools for nonlinear black-box modeling, but in many data-mining tasks the choice of relevant inputs and network complexity still constitute major challenges. Statistical tests for detecting relations between inputs and outputs proposed in the literature are largely based on the theory for linear systems, and laborious retraining combined with the risk of getting stuck in local minima make the application of exhaustive search through all possible network configurations impossible but for toy problems. This paper proposes a systematic method to tackle the problem where an output shall be estimated on the basis of a (large) set of potential inputs. Feedforward neural networks of multilayer perceptron type are used in the three-stage approach: First, starting from sufficiently large networks, an efficient pruning method is applied to detect potential model candidates. Next, the best results of the pruning runs are extracted by forming a Pareto-frontier, with the contradictory objectives of minimizing network complexity and estimation error. The networks on this frontier are considered to contain promising hidden nodes with their specific connections to relevant input variables. These hidden nodes are therefore optimally combined by mixed-integer linear programming to form a final set of neural network models, from which the user can select a model of suitable complexity. The modeling method is applied on an illustrative test example as well as on a complex modeling problem in the metallurgical industry, i.e., prediction of the silicon content of hot metal produced in a blast furnace. It is demonstrated to find relevant inputs and to yield parsimonious sparsely connected neural models of the output.
Autoregressive models with exogenous inputs are useful tools for analyzing systems with unknown dynamics, but are limited by the assumption that the relations between inputs and output(s) are linear. For complex systems with nonlinear or abruptly changing dynamics it is possible to modify the technique by allowing for multiple local models and designing a strategy for switching between them. A method by which this can be realized is developed in the paper. The technique is applied on a complex problem in the metallurgical industry, i.e., the prediction of hot metal silicon content in the blast furnace. A set of local models is developed for different parts of a training set, using a statistical criterion for model selection. The resulting local models are then applied to predict future values of the silicon content. It is demonstrated that the method is capable to develop models, among which a proper choice can be made for prediction. The potential of multi-step predictions is also studied. Finally, some conclusions concerning the method and the results are drawn.
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