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针对经典Verhulst模型背景值建模机理的不严密和初始值设定的不科学性,该文给出了灰导数改进模型及模型参数的最优估计式。采用原始数据一次累加与其拟合值的残差平方和最小作为约束准则,推导出虚拟初始值的计算公式,建立了无须设定初始值约束的优化模型。以南水北调工程沉降监测实例,比较了在3种背景值构造方法和两种初始值约束条件下的预测精度。结果表明,该文提出的初始值优化模型与灰导数法构造背景值,所得残差的平方和最小,从而验证了优化模型的可行性,为沉降监测中长期预报建模提供了合理的解决方案。
In view of the unscientific nature of the modeling mechanism of the classical Verhulst model and the unscientific setting of the initial value, the optimal estimation formula of the gray derivative and the model parameters are given. The formula of virtual initial value is deduced by using the sum of the squares of residuals of the original data and its fitting value as the constraint criterion. The optimal initial value constraint model is established. Taking the example of settlement monitoring of South-to-North Water Diversion Project, the prediction accuracy under three kinds of background value construction methods and two kinds of initial value constraints was compared. The results show that the proposed initial value optimization model and the gray derivative method construct the background value, and the sum of squares of the residuals obtained is the smallest, which verifies the feasibility of the optimization model and provides a reasonable solution for medium and long term forecasting modeling of settlement monitoring .