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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2023, Vol. 40 ›› Issue (5): 763-778.doi: 10.3969/j.issn.1005-3085.2023.05.006

• • 上一篇    下一篇

含有复数解的灰色${\rm GM}(2,1)$模型及其应用

程毛林   

  1. 苏州科技大学数学科学学院，苏州 215009
  • 收稿日期:2022-02-18 接受日期:2022-07-08 出版日期:2023-10-15 发布日期:2023-12-15
  • 基金资助:
    国家自然科学基金(11401418).

Grey ${\rm GM}(2,1)$ Model with Complex Solution and Its Applications

CHENG Maolin   

  1. School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009
  • Received:2022-02-18 Accepted:2022-07-08 Online:2023-10-15 Published:2023-12-15
  • Supported by:
    The National Natural Science Foundation of China (11401418).

摘要：

在灰色系统预测中，${\rm GM}(2,1)$模型是一类重要模型，但有时它的白化方程的解为复数解，导致时间序列的预测值为复数，这时可以利用模与实际值的距离反映其误差大小。为了提高建模精度，提出拓展的灰色${\rm GM}(2,1)$模型，并给出时间响应方程为复数解的预测方法和参数估计方法。按照提出的模型和方法，针对中国人均国内生产总值和中国私人汽车拥有量分别建立了拓展的灰色${\rm GM}(2,1)$模型，并与常规方法和相关文献方法进行了比较，结果表明改进方法精度最高。该方法有助于灰色${\rm GM}(N,1)$模型的推广应用。

关键词: ${\rm GM}(2,1)$模型, 复数解, 参数估计, 时间响应方程, 模型精度

Abstract:

${\rm GM}(2,1)$ model is an important model in grey system prediction, but sometimes the solution of its whitening equation is complex, resulting in the predicted value of time series is complex, then we can use the distance between its mode and the actual value to reflect the size of its error. In order to improve the modeling accuracy, an extended grey ${\rm GM}(2,1)$ model is proposed in this paper, and the prediction method of complex time response equation and parameter estimation method are given. According to the model and method proposed in this paper, an extended grey ${\rm GM}(2,1)$ model is established for China's per capita GDP and China's private car ownership respectively, and compared with conventional methods and relevant literature methods. The results show that the improved method in this paper has the highest accuracy. The method presented in this paper is helpful to the generalization and application of grey ${\rm GM}(N,1)$ model.

Key words: ${\rm GM}(2,1)$ model, complex solutions, parameter estimation, time response equation, model accuracy

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