Integrating solar energy into the electricity grid is an important but challenging task. Forecasting errors can not only break the supply demand balance but also cause additional costs. Therefore, accurately and effectively forecast the global horizontal irradiance is the key feature to the photovoltaic installation. In this paper, sparse quadratic radial basis function neural network (QRBF) is established. Through mining the association rules, Eclat algorithm is applied to determine relevant meteorological variables to forecast the global horizontal irradiance. QRBF is reformulated as a linear-in-the-parameters problem and a novel approach called square root progressive quantile variable selection procedure (SRPQVSP) is proposed to reduce the complexity of model structure. Furthermore, cuckoo search (CS) algorithm is utilized to optimize the parameters in the model so as to boost forecasting accuracy. Finally, the developed model is verified at four sites of Qinghai province in China with different features of terrain, latitude and other meteorological sources. The experimental results reveal that the developed models composing of selected variables deliver superior performances over other existing approaches.