Handling ill-posedness and overparameterization of rational function model using Bi-objective particle swarm optimization

Document Type : Original Article


Department of Geomatics Engineering, Faculty of Civil Engineering and Transportation, University of Isfahan, Isfahan, Iran.


The existence of both ill-posedness and overparameterization phenomena in the rational function model (RFM), makes it difficult to determine rational polynomial coefficients (RPCs). In this regard, Meta-heuristic algorithms have been widely used. Despite the extensive efforts in this field, it is still challenging to find optimum structures of RFM due to the above-mentioned phenomena. The existing meta-heuristic methods focus on overparameterization and try to remove some unnecessary RPCs using binary particles. Although solving overparameterization can automatically address the ill-posedness phenomenon, meta-heuristics do not achieve desired results by solely focusing on overparameterization. Therefore, it seems necessary to consider both ill-posedness and overparameterization phenomena to achieve an optimum structure of the RFM. Accordingly, in this study, a bi-objective particle swarm optimization (PSO) algorithm, namely BOPSO-RFM, is proposed to determine the optimum RFM structure. This method has two objective functions that should be minimized: 1) the Root Mean Square Error (RMSE) over some of the ground control points (GCPs), and 2) the maximum Pearson correlation coefficient between the columns of the design matrix, each of which corresponding to one of RPCs. While binary meta-heuristic algorithms mostly address the overparameterization phenomenon by considering binary particles and calculating the RMSE over some GCPs, the added objective function tries to address ill-posedness. Experiments conducted on three high-resolution datasets show that the proposed method has led to average improvements of 95% and 29% in terms of accuracy and RMSE values and 99% and 76% improvements in terms of stability, over well-known PSORFO and the state-of-the-art PSO-KFCV method, respectively. Moreover, the analysis of the final design matrix obtained from the final RFM structure revealed that the average of condition numbers corresponding to the BOPSO-RFM results had been 1.14e+9 and 7.39e+4 times lower than those of PSORFO and PSO-KFCV.


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