含时变时滞函数的GM(1,1|τ_i)模型及其应用

针对带有时滞效应的小样本数据序列的预测建模问题,现有模型通常假设时滞期为固定值,忽略了时滞值动态变化对模型效果的影响.为了克服这一局限性,本文考虑系统时滞的动态变化效应,将GM(1,1|τ,r)模型的静态时滞参数推广为时变时滞函数,设计出非整数时滞取值区间对应的时变时滞参数表达式.提出以灰关联理论为基础的时变时滞函数的参数优化方法,推导出GM(1,1|τ_i)模型参数估计值以及预测序列的时间响应式.该方法不仅提高了模型对所分析序列的拟合度,还可充分利用时滞参数函数的数学性质,进一步研究时滞因素对系统发展趋势的影响.最后,将GM(1,1|τ_i)模型应用于福建省全省沿海港口货物吞吐量预测,并将建模预测结果与经典的GM(1,1)模型和GM(1,1,τ)模型进行比较.结果表明当原始序列具有时滞效应时,GM(1,1|τ_i)模型具有更高的建模精度,能够反映出更为复杂的系统时滞变化情况,扩展了含时滞参数灰色预测模型的适用范围.

GM(1,1|τi) prediction model with time-varying delay function and its application

As to the problem of time series prediction for small sample data with time delays, the dynamic changes of system time delays should be considered and expressed in the process of modeling. This paper extends the GM(1,1|τ_i) model to a more applicable GM(1,1|τ_i) model, which contains a time-varying delay function to describe the possible time-varying delays in series. An efficient algorithm for the model parameter estimation is given, together with the time response formula of GM(1,1|τ_i) model. Parameters of the time-varying delay function used in the model algorithm are optimized by the gray correlation degree theory. The method designed in this paper improves the fitting degree of the GM(1,1|τ_i) model to the analyzed sequence. It also helps to analyze the development trend of system based on the mathematical properties of time-varying delay functions. Finally, the model is applied to forecast the cargo throughput of coastal ports in Fujian province, and the results are compared with those based on GM(1,1) and GM(1,1,τ). Results show that the GM(1,1|τ_i) model has higher modeling precision when the raw data contains complex time-varying delays and this will enlarge the class of existing grey series prediction models with time delays.