Proposal of reliability index in search for reliable solutions of reverse calculation based on fuzzy neural network modeling

The artificial neural network (ANN) or fuzzy neural network (FNN) is one of the superior modeling methods that can be used to estimate the relationship between input and output with excellent accuracy. When an FNN model is constructed, reverse calculation is often carried out using this model. However, solutions estimated by reverse calculation may not necessarily have low estimation errors. In this paper, we propose a reverse calculation method that can be used to search input values with high reliability. For estimating the reliability of an arbitrary point in the input, many equations were tested in our study using the distance between the point of interest and the nearest learning point, and the estimation error of the nearest learning point. As a result, we selected the following equation to calculate the reliability index value (RI). RI = 1/n ∑ n i=1 (log e_i/I_i) where I, e, and n represent the distance between the point of interest and the nearest learning point, the estimation error of each nearest learning point, and the number of input variables, respectively. The larger the RI value is, the smaller the estimation error for the point of interest is. The reverse calculations using the genetic algorithm (GA) combined with the RI (referred to as RIGA henceforth), and the conventional GA, were conducted in the study using the FNN learned by the learning data with three or six inputs and one output. The relative error for the target value of GA searching was 1.9% to 32%, whereas that of the RIGA was only 0.1% to 2.6%.