【作者】 李娟；

【导师】 李晔；

【作者基本信息】 河南农业大学 ， 管理科学与工程， 2020， 硕士

【摘要】 灰色预测模型是一种处理带有不确定性预测问题的重要方法。传统灰色预测模型的建模对象为实数序列,为了使模型能够对其他类型的数据建模从而可以应用于更多实际问题的解决,本文研究了面向灰数信息的预测模型构建问题。主要从建模对象为区间灰数和三参数区间灰数两方面展开,具体内容包括以下两点。(1)区间灰数预测模型研究。针对现有预测模型的建模对象大多以实数为主,研究了如何将其拓展到区间灰数。采用灰色属性法,从核和认知程度两个维度实现区间灰数灰信息的提取,保证了灰数的完整性和独立性。对白化后的序列分别构建灰色Verhulst预测模型,然后根据核和认知程度的计算表达式,通过逆推的方法得到区间灰数上、下界的模拟值,从而提出基于核和认知程度的区间灰数Verhulst预测模型。此外,考虑到区间灰数在白化的过程中未涉及灰数取值可能性的大小,研究了白化权函数已知的区间灰数建模问题。从几何角度出发,将白化权函数映射到二维坐标平面上,计算所围图形的面积和重心;同时得到标准化的区间灰数,提取其白部和灰部信息。从几何和灰信息分解两个方面实现灰数的白化,得到考虑白化权函数的区间灰数预测模型。针对模型在求解时由差分方程向微分方程跳跃导致误差的问题,构建新信息优先的无偏区间灰数预测模型。以一次累加序列与其模拟值的误差平方和最小为准则,得到关于模型参数的线性方程组,并基于Cramer法则进行求解。同时,根据新信息优先原理,选取x(1)(n)为初值条件,采用递推迭代法给出时间响应函数,得到预测模型。(2)三参数区间灰数预测模型研究。定义了三参数区间灰数的上、下信息域以体现“重心”点在取值区间的位置。从核和上、下信息域三个方面挖掘蕴含的灰信息,建立基于核和双信息域的三参数区间灰数预测模型。为了进一步分析三参数区间灰数在取值区间不同位置的取值可能性对模型构建的影响,研究了具有简单线性分布特征的三参数区间灰数可能度函数,根据“数形结合”的思想计算可能度函数与坐标轴所围图形的面积和几何中心。另一方面,核反映了灰数的发展趋势。因此通过核和面积、几何中心实现三参数区间灰数的白化过程,得到基于可能度函数的三参数区间灰数预测模型。更多还原

【Abstract】 As one of the most significant components of grey system theory,grey prediction model is an important method to solve the prediction problem with uncertainty.The modeling object of traditional grey prediction model is real number sequence.In order to expand its application scope to deal with more actual problems,this paper studies how to construct prediction model which is appropriate for grey number sequence.(1)Research on interval grey number prediction model.The grey attribute method is adopted to extract grey information of interval grey number from two dimensions of kernel and cognition degree.This method ensures the integrity and independence of grey number.Then the Verhulst prediction models are established for the whitened sequences respectively.According to the calculation formulas of the kernel and cognition degree,the inversion process of the upper and lower limits of the interval grey number is realized.Then the Verhulst prediction model of interval grey number based on kernel and cognition degree is proposed.On the other hand,considering that the whitening process of interval grey number does not involve the possibility of taking value of grey number,the interval grey number prediction model with known whitening weight function is established.The whitening method of interval grey number with known whitening weight function is analyzed.From the perspective of geometry,the whitening weight function is mapped to a two-dimensional coordinate plane,and the area and center of gravity of the surrounding graph are calculated.At the same time,the interval grey number is standardized,and the white part and grey part information are extracted.Then the whitening of grey number is realized from the two aspects of geometry and grey information decomposition.Aiming at the problem of error caused by jumping from difference equation to differential equation when solving interval grey number prediction model,an unbiased interval grey number prediction model with new information priority is constructed.Based on the principle of minimizing the sum of squares of errors between the accumulative sequence and its simulation value,the linear equations about the model parameters are obtained and solved based on Cramer’s rule.At the same time,according to the new information priority principle,selects x(1)(n)as the initial value.The expression of the time response function is given by the recursive iteration method,and the interval grey number prediction model is established.(2)Research on three-parameter interval grey number prediction model.Since the interval grey number does not involve the possibility of taking value of grey number,the "center of gravity" point with the largest possibility of taking value is considered in the interval grey number,and the prediction model of three-parameter interval grey number is studied.In order to reflect the position of the "center of gravity" point of three-parameter interval grey number in the value interval,the definition of the upper and lower information domains of three-parameter interval grey number is given.Then the grey information is mined from the aspects of the kernel,the upper and lower information domains.And the three-parameter interval grey number prediction model based on kernel and double information domains is established.In order to further analyze the influence of the possibility of taking value of three-parameter interval grey number at different positions in value interval on the prediction model,the possibility degree function of three-parameter interval grey number with simple linear distribution characteristics is studied.According to the idea of "combination of numbers and shapes",the area and geometric center of the graph surrounded by the coordinate axis and the possibility function are calculated.On the other hand,kernel reflects the development trend of grey number.Therefore,the whitening process of three-parameter interval grey number is realized through the kernel,the area and the geometric center.Then the three-parameter interval grey number prediction model based on the possibility degree function is obtained.更多还原