| Summary: | 博士 === 義守大學 === 工業管理學系 === 101 === The GM(1,1) model within the grey theory is used frequently by scholars as a prediction tool which with the optimal and unique ability of performing fitting predictions using small data sets to allow fast, concise, accurate, and effective predictions and understand future trends. Because the appropriate and necessary GM(1,1) requirement that the grey development coefficient a cannot equal zero, the sufficient and necessary condition of GM(1,1) model is typically ignored. However, singular phenomena, the grey development coefficient a is equal or close to zero, such as 5.551×10-16, frequently occur when sampled specific raw data pattern then after computers are used to calculate matrix values, because of the floating-point error calculation. This results in an indeterminate form 0．(－∞) in the GM(1,1) grey prediction equation and extremely significant errors in the grey prediction values.
Generally, the most common four raw data case is discussed and the method of symbolic operation by software package is used to examine the second and fourth values are identical that the type of raw data pattern will cause singular phenomena, where the grey development coefficient a equals to zero, leading to erroneous grey prediction. This finding suggests that this specific data distribution should be the focus to avoid erroneous predictions. The L'Hôpital's rule being recommend that must be used to calculate the next grey predictive value, which is the b value that presents as x ̂^((0) ) (k+1)=b. Therefore, we propose adding the sufficient and necessary condition of a≠0 into the original of GM(1,1) prediction equations.
In order not to lead to the erroneous prediction result, the discriminant for judgment is eager to be developed. In this research, the method of symbolic operations is adopted to seek for the general discriminant. The result shows when when odd number of raw data obtained, the discriminant is ∑_(n=0)^(k-2)▒〖[k-2(n+1)] 〖X^((0))〗_(n+2)=0〗, for example. And the result is successfully applied to practical examples.
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