New Sampling Scheme for Neural Network-Based Meta-Modelling with Application to Air Pollutant Estimation

Purpose A new method for the design of experiments (DOE) or sampling technique is proposed, using a distance weight function and the k-means theory. The radial basis function neural network metamodelling approach1 is used to evaluate the performance of the proposed DOE by using an n-degree of test function, applied to the complex nonlinear problem of spatial distribution of air pollutants. A comparison study is included to analyse the performance of the pro-posed technique against available methods such as the n-level full fractional design method and the Latin Hypercube Design method. Method For one design objective and n number of input design variables, a set of input-output training dataset are where m is the maximum number of the data points. Each data point has its own unique weight obtained from the distance factors be-tween point pi and a common reference point c, by using the Euclidean distance measure (i.e. ),(cpdii). The weights represent the distinct patterns between each data point. A neighbour can be clustered as a group where the data point is taken as a candidate. To generalise the solution, the pairs of the input and output data points are combined to become the design space, given as {}YXS;=.The solution can be simplified further if we set a common reference centre at the coordinate origin by firstly normalising the design space to []11,1ˆ+−=nS. A list of distance weight values, {}midddDi,...,2,1|,...,21==, is then sorted and clustered by using an available clustering algorithm. In this work, the k-means algorithm based on the Voronoi iteration2 is used due to its fast computation especially in the 1-dimensional case. Here, the initial points are replicated randomly, to expectedly result in a global minimum solution. The maximum number of k corresponds to the number data points that will be sampled. Results & Discussion To initially validate the accu-racy of the scheme, a known test function called as “Hock–Schittkowski Problem 100” is used in which this nonlinear problem involving of 7 variables, 1 objective, and 4 constraints. A prepared dataset which generated randomly, are sam-pled at different sample size N, and then mapped using RBFNN metamodel.