New sampling scheme for neural network-based metamodelling with application to air pollutant estimation

摘要

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 approach 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 proposed 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 X {x_1~(1), x_1~(2), … , x_1~(I); …; x_j~(1), x_j~(2), … x_j~(i)|i=1, …, m, j=1, … n} and Y = {y~(1), y~(2), …, y~(i) i =1, 2, …, m}, where m is the maximum number of the data points. Each data point has its own unique weight obtained from the distance factors between point pi and a common reference point c, by using the Euclidean distance measure (i.e. d_i (p~i, c)). 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 S = {X; Y}. The solution can be simplified further if we set a common reference centre at the coordinate origin by firstly normalising the design space to [s] = [-1, 1]~(n+1). A list of distance weight values, D {d_1, d_2, … d_|i=1, 2, …, m}, is then sorted and clustered by using an available clustering algorithm. In this work, the k-means algorithm based on the Voronoi iteration 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 accuracy 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 sampled at different sample size N, and then mapped using RBFNN metamodel.