Xiaoling HAO，Ruixia SUO

School of Management，Xi'an University of Science and Technology，Xi'an 710054，China

1 Introduction

The total power of agriculturalmachinery refers to the sum of all mechanical power used for farming，forestry，animal husbandry，fishery production and transport，which is themain indicator to reflect the overall level of developmentof agriculturalmechanization in a region［1］.The forecasting of total power of agriculturalmachinery is to extrapolate the future trends of power variables by establishing a stable relationship between power variables and time variables.Currently，the forecasting methods concerning total power of agriculturalmachinery include linear regression model，moving average，exponential smoothing，least squares method，Compartz curve，and artificial neural network forecasting model［2］.According to the actual situation of annual agriculturalmachinery in Heilongjiang Province，this paper performs the predictive analysis of total power of agricultural machinery in Heilongjiang Province，to provide a reference for the future development of agriculturalmechanization strategy in Heilongjiang Province.Since the combination forecasting theory was first developed by Bates and Granger in the1960s，the research and application of combination forecastingmethod have been developed quickly.The basic idea of combination forecasting is to use an appropriate method to integrate the calculation results of various singlemodels so as to improve the forecasting accuracy and increase the forecasting reliability.By processing the historical data of total power of agriculturalmachinery and analyzing the time series graphs in Heilongjiang Province，this paper employs the exponentialmodel［3］，GM（1，1）model［4］and BP neural networkmodel and uses coefficient of variation［5］，quadratic programming［6］and Shapley value method［7-8］to build the combination forecasting model，respectively，in order to carry out the combination forecasting of total power of agricultural machinery in Heilongjiang Province.The precision is also compared to obtain the practicalmethod to forecast total power of agriculturalmachinery.

2 Single forecasting model

2.1 Exponential curvemodelTable 1 shows that the historical statistical data about total power of agricultural machinery in Heilongjiang Province continue to increase over time，so the exponential curve regression analysis is used for forecasting.Using SPSS statistical software，the forecastingmodel is established.

Table 1 Total power of agriculturalmachinery in Heilongjiang Province during 1980-2007

where Y（t）is the total power of agriculturalmachinery；t is the time variable，taking values of1-11 for 1997-2007 respectively.The analysis of variance shows that the tail probability of significance test of exponential curvemodel is less than 0.0001，the coefficient of determination is R2＝0.947，and F＝161.866＞F0.01.Themodel is significant，and the fitting accuracy is high.The average relative error is 4.976%，and the fitting results are shown in Table 2.

2.2 GM（1，1）modelThe grey series GM（1，1）model forecasting is realistic and dynamic，and if the original series X（0）＝｛X（0）（1），X（0）（2），…，X（0）（n）｝，we select the continuous data of different length from X（0）series as the sub-series.For the sub-series，GM（1，1）model is established.We determine any sub-series as follows：

We perform the accumulated generating operation of sub-series as follows：

We build the cumulativematrix B and the constant term vector Ym：

We build the GM（1，1）model as follows：

We perform the derivation and restoration of X（1）＾：

Using the statistics concerning total power of agriculturalmachinery in Heilongjiang Province during 1997-2007，and the above GM（1，1）model，we build the grey forecastingmodel of total power of agriculturalmachinery in Heilongjiang Province：

At the same time，the posterior ratio c＝0.3231＜0.35，and small error probability p＝1.000＞0.95.Thus it can be found that the established GM（1，1）model has better fitting accuracy，and the average relative error is 5.2692%，so it can be used to forecast and analyze the total power of agriculturalmachinery in Heilongjiang Province.The fitted value and relative error ofmodel are shown in Table 2.

2.3 BP neural network modelBP neural network is widely used in the field of forecasting，because the three-layer neural network that contains a hidden layer can approximate to any nonlinear function，so this paper uses the three-layer BP network model to predict.According to the historical data about the total power of agriculturalmechanization in Heilongjiang Province during1980-2007，we take the data of total power of agriculturalmachinery in the previous four years as input data（namely select four nodes in the input layer），and the data of total power of agriculturalmachinery in the fifth year as the output data，to construct the input and outputsamples.The network input is［Xi，Xi+1，Xi+2，Xi+3］，，and the output is X4，i＝1，2，…，n.The number of nodes in the hidden layer of network is determined using themethod of trialand error，and finally，the number of nodes in the hidden layer is taken 9.Then the network can quickly converge to the required accuracy.Thus，the topological structure of the selected network is4-10-1.We take the historical data of total power of agriculturalmachinery in Heilongjiang Province during 1980-1996 for network training，and take the data during1997-2007 asnetwork calibration data.When the network transfer function is Sigmoid function，the error is E＝0.0001；the initial learning rate is0.5；themomentum term is 0.85；themaximum number of training is 5000.The network accuracy meets the requirements.The fitting results of sample data are shown in Table 2.The fitted average relative error is3.4788%.

Table 2 The forecasting results and error ofmodels

3 Combination forecasting model

It isassumed thatwe use N different forecastingmodels to forecast the same problem，then the combination forecastingmodel consisting of N different forecastingmodels is as follows：

where ytis the forecasting value of combinationmodel I；y1is the forecasting value of exponentialmodel；y2is the forecasting value of GM（1，1）model；y3is the forecasting value of BP neural network.

The fitting results of total power of agriculturalmachinery in Heilongjiang Province during 1998-2007 using this combination model are shown in Table 3.

3.2 Determ ining the combinationmodel based on the quadratic programm ing methodAccording to the minimum quadratic sum of error of each single forecasting model，we establish themathematicalmodel as follows：

where w is the quadratic sum of error；eiis the combination forecasting error at time t；ytis the observed value.

By substituting the forecasting value ofeach single forecasting model into formula（12），we can get the following combination forecastingmodel：

Formula（13）is consistent with the mathematicalmodel of quadratic programming，sowe can call quadprog function ofMATLAB to calculate the minimum value.Enter the following command in Matlab window：

We can calculate the combination coefficient of forecasting modelas：w＝（0.021 7 0 0.978 3）.The combination forecastingmodel II is established as follows：

where y11is the forecasting value of combinationmodel II，and the fitting results of total power of agricultural machinery in Heilongjiang Province during 1998-2007 using this combination model are shown in Table 3.

3.3 Determ ining the combination model based on Shapley valuemethodIn game theory，the Shapley value is a solution concept in cooperative game theory.To each cooperative game it assigns a unique distribution（among the players）of a total surplus generated by the coalition of all players.The Shapley value is characterized by a collection of desirable properties.Assuming the average relative forecasting error of forecasting method i is Ei，there is：

The weight distribution formula of Shapley valuemethod is as follows：

According to formula（15）to（17），we can calculate the combination coefficient based on Shapley value：w＝（0.3004

0.2764 0.4232）.The combination forecasting model III is established as follows：

where yIIIis the forecasting value of combination model III.

The fitting results of total power of agriculturalmachinery in Heilongjiang Province during 1998-2007 using this combination model are shown in Table 3.

From Table 2 and Table 3，it can be found that each combination forecastingmodel is superior to the single forecastingmodel.The fitting results of combination model show that the fitting precision of the combinationmodelbased on Shapley value isgreater than that of the combinationmodel based on dispersion coefficient or quadratic programming.Therefore，this paper uses combinationmodel IIIbased on Shapley value to forecast the total power of agriculturalmachinery in Heilongjiang Province during 2008-2015.The forecasting results are shown in Table 4.

where w（｜ui｜）is the weighting factor，representing themarginal contribution that i should make；ui-｛i｝means the removal of model i from the combination；u stands for all sub-sets containing i；｜ui｜is the number of forecastingmodel in the combination.

By formula（16），the weight of each forecastingmethod in the combination forecasting is calculated as follows：

Table 3 The fitted values of each combination model

Table 4 The forecasting value of total power of agriculturalmachinery in Heilongjiang Province during 2008-2015

4 Conclusions

Using exponentialmodel，GM（1，1）model and BP neural network model，this paper forecasts the total power of agricultural machinery in Heilongjiang Province，and the forecasting prediction of the three models is 4.9761%，5.2692%and 3.4788%，respectively.Using dispersion coefficient，quadratic programming and Shapley value，this paper establishes the combination model for the combination forecasting of total power of agriculturalmachinery in Heilongjiang Province.The fitting precision of Shapley valuemethod is3.16%，lower than thatof each single forecasting model and also lower than that of combination forecastingmodels based on coefficient of variation and quadratic programming（3.32%and 3.42%）.Therefore，during the forecasting of agriculturalmechanization，it isnecessary to establish the combination forecastingmodel to improve forecasting precision in order to provide a reference for the developmentofagriculturalmechanization.

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