We used a CART simulation environment to develop the decision trees (www.salford-systems.com/products-cart.html). We selected the minimum cost tree regardless of tree size. Figures 12 and 13 illustrate the variation of error with reference to the number of terminal nodes for Datasets A and B. For Data A, the developed tree has 122 terminal Figure 11. Neural network training using SCGA
Table 5. Training and test performance of neural networks versus decision trees
Data A Data B
Training Testing Training Testing RMSE
CART 0.00239 0.00319 0.00227 0.00314 Neural
Network
0.00105 0.00095 0.00041 0.00062
Figure 12. Dataset A - Variation of relative error versus the number of terminal nodes
Figure 13. Dataset B - Variation of relative error versus the number of terminal nodes
Figure 14. Dataset A - Developed decision tree with 122 nodes
Figure 15. Dataset B - Developed decision tree with 128 nodes
nodes as shown in Figure 14, while for Data B, the rest of the tree had 128 terminal nodes as depicted in Figure 15. Training and test performance are summarized in Table 5.
Figure 16 compares the performance of the different intelligent paradigms used in developing the TACDSS (for clarity, we have chosen only 20% of the test results for Dataset B).
DISCUSSION
The focus of this research is to create accurate and highly interpretable (using rules or tree structures) decision support systems for a tactical air combat environment problem.
Experimental results using two different datasets revealed the importance of fuzzy inference engines to construct accurate decision support systems. As expected, by providing more training data (90% of the randomly-chosen master data set), the models were able to learn and generalise more accurately. The Takagi-Sugeno fuzzy inference system has the lowest RMSE on both test datasets. Since learning involves a complicated Figure 16. Test results illustrating the efficiency of the different intelligent paradigms used in developing the TACDSS
procedure, the training process of the Takagi-Sugeno fuzzy inference system took longer compared to the Mamdani-Assilian fuzzy inference method; hence, there is a compromise between performance and computational complexity (training time). Our experiments using different membership function shapes also reveal that the Gaussian membership function is the “optimum” shape for constructing accurate decision support systems.
Neural networks can no longer be considered as ‘black boxes’. Recent research (Setiono, 2000; Setiono, Leow, & Zurada, 2002) has revealed that it is possible to extract rules from trained neural networks. In our experiments, we used a neural network trained using the scaled conjugate gradient algorithm. Results depicted in Figure 5 also reveal with the trained neural network could not learn and generalise accurately compared with the Takagi-Sugeno fuzzy inference system. The proposed neural network outperformed both the Mamdani-Assilian fuzzy inference system and CART.
Two important features of the developed classification and regression tree are its easy interpretability and low complexity. Due to its one-pass training approach, the CART algorithm also has the lowest computational load. For Dataset A, the best results were achieved using 122 terminal nodes (relative error = 0.00014). As shown in Figure 12, when the number of terminal nodes was reduced to 14, the relative error increased to 0.016.
For Dataset B, the best results could be achieved using 128 terminal nodes (relative error
= 0.00010). As shown in Figure 13, when the terminal nodes were reduced to 14, the relative error increased to 0.011.
CONCLUSION
In this chapter, we have presented different soft computing and machine learning paradigms for developing a tactical air combat decision support system. The techniques explored were a Takagi-Sugeno fuzzy inference system trained by using neural network learning techniques, a Mamdani-Assilian fuzzy inference system trained by using evolutionary algorithms and neural network learning, a feed-forward neural network trained by using the scaled conjugate gradient algorithm, and classification and adaptive regression trees.
The empirical results clearly demonstrate that all these techniques are reliable and could be used for constructing more complicated decision support systems. Experiments on the two independent data sets also reveal that the techniques are not biased on the data itself. Compared to neural networks and regression trees, the Takagi-Sugeno fuzzy inference system has the lowest RMSE, and the Mamdani-Assilian fuzzy inference system has the highest RMSE. In terms of computational complexity, perhaps regression trees are best since they use a one-pass learning approach when compared to the many learning iterations required by all other considered techniques. An important advantage of the considered models is fast learning, easy interpretability (if-then rules for fuzzy inference systems, m-of-n rules from a trained neural network (Setiono, 2000) and decision trees), efficient storage and retrieval capacities, and so on. It may also be concluded that fusing different intelligent systems, knowing their strengths and weak-ness could help to mitigate the limitations and take advantage of the opportunities to produce more efficient decision support systems than those built with stand-alone systems.
Our future work will be directed towards optimisation of the different intelligent paradigms (Abraham, 2002), which we have already used, and also to develop new adaptive reinforcement learning systems that can update the knowledge from data, especially when no expert knowledge is available.
ACKNOWLEDGMENTS
The authors would like to thank Professor John Fulcher for the editorial comments which helped to improve the clarity of this chapter.
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