The necessary and sufficient condition for GM(1, 1) grey prediction model

The grey theory demonstrates the optimal and unique ability of performing fitting predictions using small data sets and limited information to allow fast, concise, accurate, and effective predictions and understand future trends. The GM(1, 1) model within the grey theory is used frequently by scholars as a prediction tool. To enhance the accuracy of GM(1, 1) prediction, numerous studies have provided satisfactory results on revised grey prediction models. However, because the sufficient and necessary condition of GM(1, 1) model that the grey development coefficient should not equal zero is neglected, singular phenomena occur when computers are used to calculate matrix values to obtain the grey development coefficient a. Additionally, if it is neglected that a should not equal zero and the result is substituted into the prediction equation, a meaningless predictive value is obtained. In this paper, we will demonstrate scientific and effective procedures to solve this singular phenomenon of GM(1, 1) prediction model with the practical cases and show its application in forecasting the moving path of the typhoon MORAKOT.