forecast based on fractional opposite-direction accumulating nonlinear grey bernoulli markov model

: In this paper, a fractional opposite-direction accumulating nonlinear grey Bernoulli Markov model (FOANGBMKM) is established to forecast the annual GDP of Huizhou city from 2017 to 2021. The optimal fractional order number and nonlinear parameters of the model are determined by particle swarm optimization (PSO) algorithm. An experiment is provided to validate the high fitting accuracy of this model, and the effect of prediction is better than that of the other four competitive models such as autoregressive integrated moving average model (ARIMA), grey model (GM (1,1)), fractional accumulating nonlinear grey Bernoulli model (FANGBM (1,1)) and fractional opposite-direction accumulating nonlinear grey Bernoulli model (FOANGBM (1,1)), which proves the robustness of the opposite-direction accumulating nonlinear Bernoulli Markov model. This research will provide a scientific basis and technical references for the economic planning industries.

accumulating generation operator and the fractional opposite-direction inverse accumulating generation operator. Although many excellent works have been done in the above areas, the GM (1,1) model based on forward direction sequence accumulation cannot satisfy the priority principle of the new information, and there is no theory to prove that the opposite-direction sequence accumulation satisfies the new information principle for the GM (1,1) model with inverse accumulation sequence [6]. A novel Grey system model with fractional accumulation was proposed and the priority of new information can be better reflected as the fractional accumulation order number becomes smaller in the in-sample model [19].
The new information priority principle was embodied in the FAGO grey Bernoulli model [20]. Referring to the practice of Gao et al. [21], we list these representative complementary approaches to GDP in Table 1. Although a great deal of efforts are devoted to the grey prediction model, it is easy to generate random error among these models in fact. To capture the nonlinear trend in annual GDP data from Huizhou of China and obtain an appreciate prediction accuracy, this paper proposes a FOANGBMKM (1,1)), and the main contributions can be summarized as follows: 1) The FOANGBM (1,1) model is established based on the PSO algorithm. Under the condition of minimizing mean relative errors, we search for the optimal order and nonlinear parameters of the FOANGBM (1,1) model.
2) According to Markov transition probability matrix and state division, we construct concrete expressions for the estimated and predicted values of the FOANGBMKM (1,1) model. 3 Accumulating the original sequence by the fractional opposite-direction accumulation, the accumulation sequence is obtained as follows: Then, the whitening differential equation of the model FOANGBMKM (1,1) is and the grey differential equation is We obtain the following matrixes , Let , ) , ( T b a   By using the least square algorithm, we obtain where Eq (7) is obtained by the least squares formula.
Assume that ) ( ) ( ) 0 ( ) ( n u n u r  , and then the solution of Eq (4) is Applying the inverse accumulating on Eq (9), the result of simulation is as follows Next, we will calculate the fitting values (11) and we define the inverse accumulating operator with r order of the prediction sequence as follows: By applying the fractional opposite-direction inverse accumulating on Eq (12), we obtain the prediction values

Markov model
The Markov model is a general tool for data, statistics and analysis, which predicts the latest state according to the state transition probability of the previous time. Markov process has the characteristics of non-aftereffect property and good short-term prediction, which is suitable to be applied to analyze the fluctuation data. It has been widely applied in military, biology, meteorology and so on [32][33][34].

State partition
According to the Markov chain, the data sequence can be partitioned into multiply different states, which is denoted by 1 The state transition occurs only at countable moments such as 1 2 t , t , , t m  , and state partitions are where 1 2 , i i Q Q represents the lower and upper limits of relative error of state partition respectively, jdenotes the number of state partition.

State transition probability matrix
The transition probability of Markov chains from state i E to state j E after k steps is where i M denotes the total number for the occurrence of status i E , ij ( ) m k represents the number of state i E to state j E after k steps, m is the number of state partition.
The state transition probability matrix of one step is as following: Using the Chapman-Kolmogorov equation repeatedly, let V(0) be the initial vector for the original state i E of one variable, and we get the transition probability matrix after k steps and the state vector as the following, respectively. k P(k) (P(1)) , 

Determination of the value of prediction
Select j groups of data which are closest to the predicted data. According to the order of data groups from near to far, the number of the step t is determined as 1, 2, , j  . Then, a new matrix is constructed by choosing the row vectors of the t-step state transition matrix corresponding to each data, and the most probable state of prediction value is determined by the sum of the column vectors of the new matrix. The state partition can be determined after confirming the status. Choosing the midpoint of the state interval

The validation of model errors
In this part, the mean absolute percentage error (MAPE) and root mean square error (RMSE) are used to evaluate the model errors. With [35], we calculate statistics STD and 2 R , and their calculation formulas are listed as follows.
where u is the average of training data, and 1 1 n t t u u n    .

Determination the optimal order r and nonlinear parametersof the model
PSO algorithm is a swarm intelligence optimization algorithm in the field of computational intelligence excepted to the ant colony algorithm and the fish swarm algorithm, which is originated from the research on predation problems of birds and first proposed by Kennedy and Eberhart in 1995. The PSO algorithm has many advantages such as definite physical concept, good convergence, more stability, etc. In this section, we will use PSO algorithm to search for the optimal order r and nonlinear parameter  of the FOANGBM (1,1) model under the condition of minimizing mean relative errors, the mathematical expression of the PSO algorithm is

Estimation and prediction of the model FOANGBM (1,1)
Importing the statistical yearbook data of Huizhou into R, and combing the PSO algorithm with (2.2), we get the optimal order 0.01 r  and the nonlinear parameter 0.99  (FANGBM (1,1)) as the comparison model, the prediction results are listed in Table 2. As can be seen from Table 2, the prediction results of the FOANGBM (1,1) model are closer to the real values, and the relative error is smaller than that of other models. We also can obtain the smallest value of RMSE, STD and the MAPE when forecasting the test data and estimate the training data by using the FOANGBM (1,1) model. However, FOANGBM (1,1) model has a higher value of 2 R than that of the other models, such as FANGBM (1,1), GM (1,1) and ARIMA (in Table 3).

Huizhou GDP prediction
Constructing the new state transition matrix by using the several most recent groups of data, we get the state of 2017, which is listed in Table 4.  Table 3.
As can be seen from The results in Figure 1 show that the curve of FOANGBMKM (1,1) model is closer to the true values than that of FOANGBM (1,1) model.

Conclusions and future directions
In this paper, a novel FOANGBMKM is proposed to predict the annual GDP of Huizhou city. The suitable states are determined by using the transition matrix of Markov. PSO algorithm is used to search for the optimal order as well as the optimal nonlinear parameters of the accumulating generation operator. According to the results of prediction of the statistical yearbook data from 2005 to 2016, we calculate four statistics, MAPE, RMSE, STD and 2 R , for different models. According to the size of the values of these statistics, we find that the estimation accuracy of FOANGBM model is higher than that of GM, ARIMA and FANGBM models. At the same time, the fitting effect FOANGBMKM model is superior to FOANGBM model. Finally, the proposed model is applied to forecast the GDP of Huizhou city from 2017 to 2021. Compared to the opposite-direction accumulating linear Bernoulli model, the new model can more accurately and effectively to evaluate the development level of Huizhou GDP. The results show that the prediction effect of the proposed new model is better than that of the other four competitive models such as GM (1,1), ARIMA, FANGBM (1,1) and FOANGBM (1,1), which proves the greater accuracy and efficiency of the FOANGBMKM (1,1) model.
We will focus on the multi-variable GMs of electricity consumption that fully utilize potential factors. In addition, the other cutting-edge optimization algorithms are used to seek for the optimal parameters, such as ant lion optimization algorithm [36], grey wolf algorithm [37] and whale optimization algorithm [38]. Further, it is well known that the Optimal fractional accumulation GM with variable parameters is an efficient method to improve the prediction accuracy [39] and fractional time-varying grey traffic flow model based on viscoelastic fluid [40], which can be used for forecasting shorten prediction period, thus respectively establishing the Optimal fraction accumulation grey Markov model with variable parameters and fractional time-varying grey traffic flow Markov model will receive extensive attention in our next work.