## 1 Introduction  
The Grey prediction is an effective tool for understanding an uncertain environment with limited information [1]. These models consists of the basic model GM(1,1), the GM(2,1), Verhulst and direct grey model DGM(2,1), which are known as the common prediction models in Grey system theory [2]. However, some defects have been found in the performance of the grey forecasting model and can still be improved [3]. Among these, the DGM(2,1) is considered when the sampling data occurs the fluctuating phenomenon [4], but the accuracy of this model is not high and is difficult to apply directly to the actual estimation [5]. As a result, there has been much attention paid in recent years to studies on improving the precision of DGM(2,1). For example, [PERSON] [5] successfully utilized [PERSON]'s mean value theorem to optimize the existing DGM(2,1) model. Literature in [6] also proposed high accuracy for the optimized DGM(2,1) model, which has the white exponential superposition. Moreover, along with the enhancement of the performance of DGM(2,1), a new 2-order grey derivative expression deduct based on the solution structure of white differential equation of DGM(2,1) was also introduced by [7]. In the above discussions, the improved precision of DGM(2,1) mainly focused on modifying the whitening differential equation or optimizing the parameter of the grey model. On the other hand, in order to increase the accuracy of the forecasting model, some combinations of forecasting approaches have been considered under special circumstances. Several integrated effective models have been introduced in recent research, including the novel dynamic grey model which integrates GM(1,1); the exponential smoothing technique and the Fourier series [8]; the modified grey Markov model [9]; and the grey fuzzy predictive model [10] or Genetic algorithms-based forecasting model [11], _etc._ Empirical testing or simulation has proven that these combined models dramatically improve the precision of GM(1,1). In addition, the Fourier Markov Grey forecasting model, or FMGM(1,1), which has been integrated into the GM(1,1), the Fourier series, and the Markov chain, is a combined grey model offering highly accurate predictions. This model has successfully forecast the typhoon rainfall system, the steel price system [3] and the turning time of the stock market [12]. The findings from these studies confirmed that the FMGM(1,1) model accurately predicted in unstable environments, with randomly fluctuating data, and limited data samples. Although the combined models based on the GM(1,1) have been proposed in many previous works, the literature on improving the precision of DGM(2,1) through integration with other approaches is still limited in current studies. There are few combined DGM(2,1) forecasting models that have been introduced in recent studies. For instance, the study of [13] combined the Taylor approximation method and the DGM(2,1) model to predict the number of foreign students in Taiwan, and the results proved that the combined model has a higher precision than the original DGM(2,1). In the same manner as the FMGM(1,1) model, a combination of the DGM(2,1), the Fourier series and the Markov chain has also been developed, and has been successfully applied to forecast trends in Taiwan's electronic paper industry [14]; the results indicated that the FMDGM(2,1) model gave a better predicted precision than the FMGM(1,1) and the FM-Verhulst models. However, the evaluation of the forecast performance of the single DGM(2,1) model and its modified models has not been conducted in previous literature. Furthermore, its applications are quite uncommon in the past research. For this reason, this work aims to evaluate the forecast performance of the DGM(2,1) model and its modified models in order to determine the range of application of the DGM(2,1) models in practical applications and academic research.