Frontiers in Computing and Intelligent Systems 
ISSN: 2832-6024 | Vol. 3, No. 1, 2023 

 

138 

Combination Forecasting Model of R&D Intensity in 
Anhui Province Based on IGOWA Operator 

Weilun Xu, Yumei Wang 

School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233000, China 

 

Abstract: Taking the R&D intensity data of Anhui Province from 2006 to 2020 as the sample, the grey prediction model, 

ARIMA model and Holt Winters non seasonal model were selected to fit and predict the R&D intensity of Anhui Province, and 

then the IGOWA operator variable weight coefficient combination prediction model based on the minimum criterion of the sum 

of squares of errors was constructed, and the combination prediction model corresponding to four special values of the operator 

parameters was taken, Establish an error evaluation index system to illustrate the effectiveness of the model, and analyze the 

sensitivity of the parameters. Through model evaluation, it can be found that the prediction effect of the combined prediction 

model in the sample period is significantly better than the three single prediction models, greatly improving the prediction 

accuracy. Finally, the combined forecasting model is used to predict the R&D intensity of Anhui Province from 2021-2025. The 

prediction results show that in 2021-2025, the R&D intensity of Anhui Province will continue to rise, and the growth rate is 

increasing year by year. The scientific research and innovation ability will continue to develop steadily. 

Keywords: R&D intensity; IGOWA operator; Combination forecasting; Sum of squares of errors; Technological innovation. 

 

1. Introduction 

In today's world tide, the position of science and 

technology is increasingly prominent, and scientific and 

technological innovation is gradually becoming a key factor 

affecting the comprehensive national strength, which 

highlights the primary position of scientific and technological 

innovation, and also shows that in order to achieve rapid and 

stable development in the current society, we must adhere to 

the principle of independent research and development, and 

drive innovation. Since the 21st century, China's scientific 

research enthusiasm has continued to rise, and many scientific 

research achievements have emerged in combination with the 

corresponding scientific and technological development 

strategy and system construction. General Secretary pointed 

out in the report of the 20th National Congress of the 

Communist Party of China that innovation is the core driving 

force and plays a very important role in high-quality 

development. China has successfully entered the ranks of 

innovative countries. The "14th Five Year Plan" also clearly 

indicates five development directions, indicating that China is 

moving from the "quantitative expansion era" to the "quality 

expansion era".In order to enter the "quality expansion era" 

faster and better, it is necessary to promote efficiency through 

scientific and technological innovation and development. In 

2020, the R&D expenditure of Anhui Province will reach 

88.32 billion yuan, and the R&D intensity will reach 2.28%, 

0.25 percentage points higher than that of the previous year, 

ranking 10th in the country and 2nd in the central region, 

successfully realizing the leap forward development from "a 

big province of science and education" to "a source of 

scientific and technological innovation". 

The research of foreign scholars on R&D intensity is 

mainly based on the relationship between scientific and 

technological investment and the contribution rate of 

economic growth. Paul M. Romer et al. [1], Gene M. 

Grossman et al. [2], Paul S. Segerstrom et al. [3], all the 

studies had found that increasing investment in science and 

technology and R&D can boost the contribution rate of 

economic growth, and can better encourage enterprises to 

invest resources in R&D activities. At present, domestic 

scholars' research on R&D intensity mainly starts from the 

combination of R&D funds and GDP. Xuanwen Fang et al.[4] 

used the vector autoregression model and the grey prediction 

model GM(1,1), by selecting the data of China's R&D 

expenditure and GDP output value from 1990 to 2010, this 

paper forecasted the R&D expenditure at constant prices and 

current prices during the "12th Five Year Plan" period, and 

determined an equilibrium relationship with an elasticity of 

0.6 in economic growth; Liyu Zhao et al.[5] selected the 

cointegration test of China's GDP and financial investment in 

science and technology from 1989 to 2007 showed that there 

was a long-term equilibrium relationship between economic 

growth and the total internal expenditure of R&D funds; Shi 

Chen et al.[6] believed that China's current R&D investment 

intensity was at the middle level of the world, but it 

maintained a good development trend. The government 

should increase relevant investment to further enhance 

China's R&D intensity; Jianwen He [7] compared China's 

R&D intensity data from 2006 to 2013 and the data of five 

developed countries verifies that China's R&D intensity is 

growing fast at present, and will not enter the slow growth 

stage immediately like most developed countries. 

Scientific prediction requires integrating multiple 

prediction methods according to the history and reality of 

social and economic phenomena to minimize the loss of 

information and improve the prediction accuracy. Therefore, 

the concept of combined forecasting was first introduced by 

Bates and Granger [8]. It was proposed in 1969. Both 

theoretical and practical studies show that [9] in the case of 

different single prediction models and different data sources, 

the results derived from the combined prediction model may 

be better than any independent prediction value. The 

combined prediction model can reduce the systematic error of 

prediction, significantly improve the prediction effect, and 

reveal the development and change law of objective things. 

To sum up, many scholars have analyzed and predicted the 

R&D intensity, but most of them still stay in the stage of using 



 

139 

single prediction model to predict the data. Although these 

single prediction models can predict the time series data, they 

contain limited factors that can be considered and have a 

certain randomness, which may not make full use of all the 

effective information reflected in the sample data to provide 

prediction. Therefore, this paper selects the research and 

development intensity data of Anhui Province from 2006 to 

2020, first analyzes and forecasts the data through the gray 

prediction model, ARIMA time series model, Holt-Winters 

non-seasonal prediction model, and then forms a combination 

prediction method with the help of the IGOWA (generalized 

induction weighted average) operator and the optimal 

criterion of minimizing the sum of squares of prediction 

errors [10]. The variable weight coefficient combination 

forecasting model based on IGOWA operator is constructed 

to make short-term prediction on the R&D intensity of Anhui 

Province in the next five years, in order to improve the 

prediction accuracy and increase the stability of the model. 

And according to the data predicted by the more stable model, 

it is compared with the actual data of research and 

development intensity of Anhui Province in 2021 to verify the 

effectiveness and statistical significance of the model.  

2. Model Construction 

2.1. Operator 

2.1.1. OWA Operator 

Set up 𝑓𝜔  by n-ary function: 𝑅𝑛 → 𝑅 . There is a 

nonnegative weight vector 𝜔 = (𝜔1, 𝜔2, ⋯ ,𝜔𝑛)  and meets 

∑ 𝜔𝑖
𝑛
𝑖=1 = 1,𝜔𝑖 ∈ [0,1]. 

𝑓𝜔(𝑎1, 𝑎2, ⋯ 𝑎𝑛) = ∑ 𝜔𝑖𝑏𝑖
𝑛
𝑖=1          (1) 

then 𝑓𝜔  is called n-dimension ordered weighted average 

operator, also known as OWA operator. 𝑏𝑖 is the i-th largest 

number among 𝑎1, 𝑎2, ⋯ , 𝑎𝑛, which indicates that the weight 

coefficient 𝜔𝑖  of the ordered weighted average operator is 

independent of 𝑎𝑖, but only related to the first i position after 

ordering n numbers in descending order. 

2.1.2. IOWA Operator 

Record two-dimensional array < 𝑣1, 𝑎1 >,< 𝑣2, 𝑎2 >
,⋯ ,< 𝑣𝑛 , 𝑎𝑛 >. There is a nonnegative weight vector 𝜔 =
(𝜔1, 𝜔2, ⋯ , 𝜔𝑛) and meets ∑ 𝜔𝑖

𝑛
𝑖=1 = 1,𝜔𝑖 ∈ [0,1]. Then 

𝐼𝑂𝑊𝐴𝜔(< 𝑣1, 𝑎1 >,< 𝑣2, 𝑎2 >,⋯ ,< 𝑣𝑛, 𝑎𝑛 >) =
∑ 𝜔𝑖𝑎𝑣−𝑖𝑛𝑑𝑒𝑥(𝑖)
𝑛
𝑖=1           (2) 

is called n-Dimension induced ordered weighted average 

operator, also known as IOWA operator. 𝑣𝑖 is denoted as the 

induced value of the number 𝑎𝑖 , and 𝑣 − 𝑖𝑛𝑑𝑒𝑥(𝑖)  is the 

subscript of the number 𝑣1, 𝑣2, ⋯ , 𝑣𝑛  which is the greatest i 

in descending order [11]. This shows that the weighted 

averaging operator arranges n numbers according to their 

corresponding induced values in descending order, and then 

the weighted averaging cofficient 𝜔𝑖 has nothing to do with 

numerical value of 𝑎𝑖 , but only relates to the i-th position 

after sorting according to the induced values. 

2.1.3. IGOWA Operator 

Set up 𝑓𝜔 y  n-ar  function: 𝑅𝑛 → 𝑅   Record two-
dimensional arra < 𝑣1, 𝑎1 >,< 𝑣2, 𝑎2 >,⋯ ,< 𝑣𝑛, 𝑎𝑛 >   
There is a nonnegative weight vector 𝜔 =
(𝜔1, 𝜔2, ⋯ , 𝜔𝑛) 

and meets ∑ 𝜔𝑖
𝑛
𝑖=1 = 1,𝜔𝑖 ∈ [0,1]  

𝑓𝜔(< 𝑣1, 𝑎1 >,< 𝑣2, 𝑎 >2 , ⋯ , < 𝑣𝑛 , 𝑎𝑛 >) =

(∑ 𝜔𝑖(𝑎𝑣−𝑖𝑛𝑑𝑒𝑥(𝑖))
𝑛
𝑖=1

𝜆
)

1

𝜆
.

 

           (3) 

𝑓𝜔  is called generalized induced ordered weighted 

average operator and also called IGOWA operator where 
parameters 𝜆 ∈ (−∞, 0) ∪ (0,∞)   𝑣𝑖   

is denoted as the 
induced value of the number 𝑎𝑖 , and 𝑣 − 𝑖𝑛𝑑𝑒𝑥(𝑖)  is the 

subscript of the number 𝑣1, 𝑣2, ⋯ , 𝑣𝑛  which is the greatest i 

in descending order [11]. We can take any value within the 

effective value range of the parameter 𝜆 , so as to build 

different information aggregation operators. The following 

are several common operators when 𝜆  takes special 

values[12]： 

When 𝜆 = 1 , 𝑓𝜔(< 𝑣1, 𝑎1 >,< 𝑣2, 𝑎 >2 ,⋯ , < 𝑣𝑛 , 𝑎𝑛 >
) = ∑ 𝜔𝑖𝑎𝑣−𝑖𝑛𝑑𝑒𝑥(𝑖)

𝑛
𝑖=1 . The IGOWA operator at this time is 

the IOWA (induced ordered weighted average) operator; 

When 𝜆 = −1 , 𝑓𝜔(< 𝑣1, 𝑎1 >,< 𝑣2, 𝑎 >2 , ⋯ , <

𝑣𝑛 , 𝑎𝑛 >) =
1

∑ (
𝜔𝑖

𝑎𝑣−𝑖𝑛𝑑𝑒𝑥(𝑖)
)𝑛

𝑖=1
 . The IGOWA operator at this 

time is the IOWHA (induced ordered weighted harmonic 

average) operator; 

When 𝜆 → 0 , 𝑓𝜔(< 𝑣1, 𝑎1 >,< 𝑣2, 𝑎 >2 , ⋯ , < 𝑣𝑛 , 𝑎𝑛 >
) = ∏ 𝑎𝑣−𝑖𝑛𝑑𝑒𝑥(𝑖)

𝜔𝑖𝑛
𝑖=1 . The IGOWA operator at this time is 

the IOWGA (Induced Ordered Weighted Geometric Average) 

operator; 

When 𝜆 = 0.5 , 𝑓𝜔(< 𝑣1, 𝑎1 >,< 𝑣2, 𝑎 >2 , ⋯ , <
𝑣𝑛 , 𝑎𝑛 >) = (∑ 𝜔𝑖√𝑥𝑣−𝑖𝑛𝑑𝑒𝑥(𝑖)

𝑛
𝑖=1 )2 . The IGOWA operator 

at this time is the IOWSA (Induced Ordered Weighted Square 

Average) operator. 

2.2. IGOWA Operator Combination Prediction 

Model Based on the Criterion of Minimum 

Sum of Squares of Errors 

Assume that the actual observation value of a required 

prediction phenomenon is {𝑥𝑡 , 𝑡 = 1,2,⋯ ,𝑁} , n kinds of 

single prediction methods to predict this phenomenon, and 

𝑥𝑖𝑡  is the model prediction value of the first i kind of single 

prediction method in the t period. There are nonnegative 

weighting coefficients 𝜔 = (𝜔1, 𝜔2, ⋯ , 𝜔𝑛)  and meet 

∑ 𝜔𝑖
𝑛
𝑖=1 = 1,𝜔𝑖 ∈ [0,1] , then 𝑊 = (𝜔1, 𝜔2, ⋯ , 𝜔𝑛)

𝑇 

constitutes the weight vector of all single forecasting methods 

in the combined forecasting model. The prediction precision 

is constructed by the absolute value of the prediction relative 

error of each single prediction at each time point to 

characterize the induced variable of the prediction value at 

each time point. Note 

𝑣𝑖𝑡 = {
1 − |

𝑥𝑡−𝑥𝑖𝑡

𝑥𝑡
|， |

𝑥𝑡−𝑥𝑖𝑡

𝑥𝑡
| < 1

0, |
𝑥𝑡−𝑥𝑖𝑡

𝑥𝑡
| ≥ 1

  

    (4) 

as the prediction accuracy of the i-th single prediction in 

the t period, then using 𝑣𝑖𝑡   and 𝑥𝑖𝑡   can form a two-

dimensional array (< 𝑣1𝑡 , 𝑥1𝑡 >,< 𝑣2𝑡 , 𝑥2𝑡 >,⋯ ,<
𝑣𝑛𝑡 , 𝑥𝑛𝑡 >). Take it into the IGOWA operator for calculation, 

and the optimized combined predictive value is 𝑥𝑡
∧
=

(∑ 𝜔𝑖(𝑥𝑣−𝑖𝑛𝑑𝑒𝑥(𝑖𝑡))
𝜆𝑛

𝑖=1 )
1

𝜆. According to this, it can be seen 

that the 𝜆  power prediction error and induced prediction 

error in the t period are 𝑒𝑡
(𝜆) = 𝑥𝑡

𝜆 − 𝑥𝑡
∧ 𝜆

 and 𝑒𝑣−𝑖𝑛𝑑𝑒𝑥(𝑖𝑡) =

𝑥𝑡
𝜆 − 𝑥𝑣−𝑖𝑛𝑑𝑒𝑥(𝑖𝑡)

𝜆. In this paper, the four special values of 𝜆 

listed in 2.1.3. are used for analysis and discussion, so the 

specific form of induced prediction error is as follows 



 

140 

.

5.0,

0,lnln

1,
11

1,

)(

)(

)(

)(

)(














=−

→−

−=−

=−

=

−

−

−

−

−









itindexvt

itindexvt

itindexvt

itindexvt

itindexv

xx

xx

xx

xx

e     (5) 

From the induced prediction error, the n-order induced 

ordered weighted average prediction error matrix can be 

further denoted as      𝐸 = (𝐸𝑖𝑗)𝑛×𝑛 = (𝐸𝑗𝑖)𝑛×𝑛 =

(∑ 𝑒𝑣−𝑖𝑛𝑑𝑒𝑥(𝑖𝑡)𝑒𝑣−𝑖𝑛𝑑𝑒𝑥(𝑗𝑡)
𝑁
𝑡=1 )

𝑛×𝑛
; 𝑖 , 𝑗 = 1,2,⋯ , 𝑛. It can be 

seen that the induced ordered weighted average prediction 

error matrix is a symmetric n-order invertible matrix. 

Therefore the sum of squares of the total forecast errors in N 

period  is 𝑄2 = ∑ (∑ 𝜔𝑖𝑒𝑣−𝑖𝑛𝑑𝑒𝑥(𝑖𝑡)
𝑛
𝑖=1 )𝑁

𝑡=1

2
=

∑ ∑ 𝜔𝑖𝜔𝑗(∑ 𝑒𝑣−𝑖𝑛𝑑𝑒𝑥(𝑖𝑡)𝑒𝑣−𝑖𝑛𝑑𝑒𝑥(𝑗𝑡)
𝑁
𝑡=1 )𝑛

𝑗=1
𝑛
𝑖=1 =

∑ ∑ 𝜔𝑖𝜔𝑗𝐸𝑖𝑗
𝑛
𝑗=1

𝑛
𝑖=1 , where 𝑖, 𝑗 = 1,2,⋯ , 𝑛. 

In conclusion, the IGOWA operator combination prediction 

model based on the optimal criterion of minimizing the sum 

of squares of prediction errors can be constructed as 

𝑚𝑖𝑛𝑄2 =𝑊𝑇𝐸𝑊 

𝑠. 𝑡. {
∑ 𝜔𝑖 = 1 ;𝑛
𝑖=1

𝜔𝑖 ≥ 0, 𝑖 = 1,2,⋯ , 𝑛.

 

      (6) 

3. Empirical Analysis 

3.1. Data Source and Descriptive Statistics 

In recent years, R&D intensity has gradually become an 

important indicator reflecting the scientific and technological 

innovation strength and core competitiveness of a country or 

region. This article uses Jun Liu and others for reference [13]. 

The proposed R&D intensity measurement standard uses the 

proportion of R&D expenditure in GDP to measure the R&D 

intensity of a region. The research and development intensity 

index data of Anhui Province from 2006 to 2020 were 

collected by searching China Statistical Yearbook, China 

Science and Technology Statistical Yearbook and other data, 

and the trend line chart was drawn as shown in Figure 1. It 

can be seen from Figure 1 that the R&D intensity of Anhui 

Province has an obvious growth trend. 

 
Figure 1. Trend chart of research and development intensity of Anhui Province from 2006 to 2020

3.2. Single Prediction Model 

3.2.1. Grey Prediction Model GM (1,1) 

In the early 1980s, Professor Julong Deng proposed an 

effective prediction method for small sample data or data with 

low integrity — grey system theory [14], which fully 

excavates the essence of data through differential equations, 

has a very significant effect. Before establishing the grey 

prediction model GM (1,1), it is necessary to carry out rank 

comparison test on the time series data. The purpose of this 

step is to verify whether the time series data has suitable 

regularity, so as to judge whether a satisfactory model can be 

obtained. According to the actual value of research and 

development intensity of Anhui Province from 2006 to 2020 

as the original data of the model, the time series data 𝑋 =
(𝑥1, 𝑥2, ⋯ , 𝑥15)  is defined. According to the calculation 

formula of grade ratio 

𝜙(𝑡) =
𝑥𝑡−1

𝑥𝑡
, 𝑡 = 2,3,⋯ ,15,         (7) 

calculate all the grade ratios, if all the grade ratios are 

located in the interval (𝑒−
2

𝑛+1, 𝑒
2

𝑛+1)  (𝑛 = 15 ), the data is 

suitable for model construction. The calculation results of 

grade ratios are shown in Table 1. It can be seen from Table 1 

that the grade ratios of the original data do not all fall within 

the allowable coverage of the interval, so the original data is 

shifted by three units to obtain a new sequence 𝑍 =
(𝑧1, 𝑧2, ⋯ , 𝑧15) = (𝑥1 + 3, 𝑥2 + 3,⋯ , 𝑥15 + 3) . We can get 

that all the grade ratios of the translation-transformed series 

are located in the interval (0.882,1.133), indicating that the 

translation-transformated series are suitable for building the 

grey forecasting models. 
The time series data 𝑍 = (𝑧1, 𝑧2, ⋯ , 𝑧15)  is accumulated 

once to generate a  sequence 𝑌 = (𝑦1, 𝑦2 , ⋯ , 𝑦15) , an 

0

0.5

1

1.5

2

2.5

2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020

R
&

D
 I

N
T

E
N

SI
T

Y
(%

)

YEAR



 

141 

accumulation matrix 

























+−

+−

+−

=

1)(
2

1

1)(
2

1

1)(
2

1

1514

32

21

yy

yy

yy

B


  and s 

constant vector 𝐿 = (𝑧2, 𝑧3, 𝑧4, ⋯ , 𝑧15)
𝑇  are constructed. 

Use OLS method to calculate parameters 

.)( 1 LBBB TT −


=







=




         (8) 

By calculating the coefficient 𝛼 = −0.019 , 𝜇 = 4.029 

with SPSSPRO software, the prediction formula of 

accumulating sequence 𝑌  is 𝑦𝑡+1
∧

= 216.013𝑒0.019𝑡 −
212.053. According to which the model prediction value and 

prediction accuracy of grey prediction model GM (1,1) are 

shown in Table 4. It can be seen from Table 4 that the posterior 

difference ratio 𝐶  of the model 0.082, and the average 

relative error is 6.347%, which means that the model has high 

precision and good fitting effect, and is suitable for 

forecasting by using this model. 

Table 1. Grade ratio test results 

Year Original value Grade ratio Serial value after translation conversion Grade ratio after translation conversion 

2006 0.96 - 3.96 - 

2007 0.99 0.970 3.99 0.992 

2008 1.12 0.884 4.12 0.968 

2009 1.35 0.830 4.35 0.947 

2010 1.32 1.023 4.32 1.007 

2011 1.40 0.943 4.40 0.982 

2012 1.64 0.854 4.64 0.948 

2013 1.83 0.896 4.83 0.961 

2014 1.89 0.968 4.89 0.988 

2015 1.96 0.964 4.96 0.986 

2016 1.97 0.995 4.97 0.998 

2017 2.09 0.943 5.09 0.976 

2018 1.91 1.094 4.91 1.037 

2019 2.03 0.941 5.03 0.976 

2020 2.28 0.890 5.28 0.953 

3.2.2. ARIMA Prediction Model 

ARIMA model, also called autoregressive moving average 

model, is widely used to analyze various types of time series 

data and to model and predict. The ADF test results of the 

model are obtained through SPSS software processing, as 

shown in Table 2. When the difference is 0 order, the 

significance 𝑃  value is 0.832, which does not show 

significance horizontally, so the original hypothesis that the 

series is unstable time series cannot be rejected. When the 

difference is 1 order, the significance 𝑃  value is 0.012, 

showing significance at the 5% significance level, so the 

original hypothesis is rejected, and the series is a stable time 

series. When the difference is 2 order, the significance 𝑃 

value is 0.000, showing significance at the significance level 

of 1%, so the original hypothesis is rejected, and the series is 

a stable time series. Then judge whether the differential 

sequence is stable by looking at the differential sequence 

diagram. At the same time, the autocorrelation and partial 

autocorrelation analysis are carried out on the time series, and 

estimate the model's P and Q values according to the 

truncation and trailing of ACF and PACF diagrams. The 

ARIMA (1,1,2) prediction model is finally established by 

combining the model comparison of AIC information criteria. 

ARIMA model requires that the model has pure 

randomness, that is, the residual error of the model is white 

noise. The residual error of the model can be tested by 

calculating the 𝑃 value of the 𝑄 statistic of the model, and 

the 𝑃 value of the statistic 𝑄6 obtained from the calculation 

result is equal to 0.845 and greater than 0.1, so the original 

hypothesis cannot be rejected, which means that the residual 

sequence of ARIMA (1,1,2) model does not have 

autocorrelation and meets the requirements of parameter 

diagnosis. Therefore, the ARIMA model established meets 

the requirements, and can be used for fitting and prediction. 

The specific parameter information is shown in Table 3. 

According to the data in Table 3, the prediction formula of 

ARIMA (1,1,2) model built in this paper is 

𝑥𝑡 = 0.065 + 0.299𝑥𝑡−1 − 0.596𝜀𝑡−1 − 0.397𝜀𝑡−2.  (9) 

The R&D intensity data of Anhui Province from 2006 to 

2020 can be brought into formula (9) to obtain the prediction 

values and prediction accuracy of ARIMA (1,1,2) model in 

each period. The results are shown in Table 4. 

Holt-Winters Non-Seasonal Exponential Smoothing 

Prediction Model 

Holt-Winters non-seasonal exponential smoothing model 

is one of the exponential smoothing prediction models, which 

can be used to predict the type of R&D intensity data collected 

in this paper — data with no obvious seasonal change but with 

obvious temporal change trend. The prediction formula is 

𝑦𝑡+𝑘
∧

= 𝑎𝑡 + 𝑏𝑡𝑘, 𝑘 > 0 , where 𝑦𝑡
∧

  is the parameters 𝑎𝑡 , 𝑏𝑡 
after three times from the original data sequence 𝑦𝑡 , which is 

calculated by the following formula 

{
𝑎𝑡 = 𝛼𝑦𝑡 + (1 − 𝛼)(𝑎𝑡−1 + 𝑏𝑡−1) ;
𝑏𝑡 = 𝛽(𝑎𝑡 − 𝑎𝑡−1) + (1 − 𝛽)𝑏𝑡−1.

      (10) 

In the formula, 𝛼, 𝛽 ∈ (0,1)  is called damping factor, 

which can be estimated by EViews software based on the 



 

142 

principle of minimizing the sum of squares of errors and the 

minimum RMSE of root mean square error value [15][15]. In 

this way, the intercept 𝑎𝑡 and slope 𝑏𝑡 of the model can be 

calculated and brought into the prediction formula. Later, the 

predicetion results of R&D intensity data in Anhui Province 

are shown in Table 4. 

Table 2. ADF Inspection Table 

Variable 
Difference 

order 
T P AIC 

Critical value 

0.01 0.05 0.1 

Research and 

development 

strength 

0 0.754 0.832  -11.419    -4.012 -3.104 -2.691 

1 -3.369 0.012**  -7.084    -4.069 -3.127 -2.702 

2 -4.554 0.000***  -2.997    -4.138 -3.155 -2.714 

Note:  * * *, * * and * represent the significance levels of 1%, 5% and 10% respectively. 

Table 3. ARIMA (1,1,2) model parameters 

Term Symbol  Coefficient Standard error Z-value P-value 95% CI 

Constant term 𝑐 0.065 0.089 0.727 0.467 -0.110 ~ 0.240 

AR parameters 𝛼1 0.299 1.116 0.268 0.788 -1.887 ~ 2.486 

MA parameters 
𝛽1     -0.596 23.374 -0.025 0.98 -46.407 ~ 45.216 

𝛽2     -0.397 8.16 -0.049 0.961 -16.391 ~ 15.596 

AIC value: -14.225 

BIC value: -11.030 

Table 4. Predicted value and forecast precision of each single forecast model 

Year 
Actual value of 

R&D intensity (%) 

Grey prediction model ARIMA model Holt-Winters model 

Predicted 

value (%) 

Prediction 

accuracy 

Predicted 

value (%) 

Prediction 

accuracy 

Predicted 

value (%) 

Prediction 

accuracy 

2006 0.9600 0.9600 1.0000 0.9800 0.9792 0.9600 1.0000 

2007 0.9900 1.1430 0.8455 1.0500 0.9394 0.9570 0.9667 

2008 1.1200 1.2220 0.9089 1.0900 0.9732 0.9890 0.8830 

2009 1.3500 1.3030 0.9652 1.2300 0.9111 1.1540 0.8548 

2010 1.3200 1.3850 0.9508 1.4200 0.9242 1.4610 0.8932 

2011 1.4000 1.4690 0.9507 1.3800 0.9857 1.4670 0.9521 

2012 1.6400 1.5540 0.9476 1.5200 0.9268 1.5160 0.9244 

2013 1.8300 1.6410 0.8967 1.7100 0.9344 1.7750 0.9699 

2014 1.8900 1.7300 0.9153 1.8500 0.9788 2.0200 0.9312 

2015 1.9600 1.8200 0.9286 1.9100 0.9745 2.0860 0.9357 

2016 1.9700 1.9120 0.9706 2.0100 0.9797 2.1140 0.9269 

2017 2.0900 2.0060 0.9598 2.0400 0.9761 2.0720 0.9914 

2018 1.9100 2.1020 0.8995 2.1800 0.8586 2.1590 0.8696 

2019 2.0300 2.1990 0.9167 2.0400 0.9951 1.9290 0.9502 

2020 2.2800 2.2990 0.9917 2.2400 0.9825 1.9950 0.8750 

3.3. Combined Forecasting Model 

3.3.1. IGOWA Combined Forecasting Model for 𝜆 = 1 

According to the model building method in 2.2., the third-

order induced ordered weighted average prediction error 

matrix for calculating 𝜆 = 1 is: 

𝐸 = [
0.0626 0.0558 0.1057
0.0558 0.1841 0.0946
0.1057 0.0946 0.4108

]. 

Then, the IGOWA operator combination prediction model 

based on the optimal criterion of minimizing the sum of 

squares of prediction errors can be constructed as: 
𝑚𝑖𝑛𝑄2 =𝑊𝑇𝐸𝑊 = 0.0626𝜔1

2 + 0.1116𝜔1𝜔2 + 0.2114𝜔1𝜔3

+ 0.1841𝜔2
2 + 0.1892𝜔2𝜔3

+ 0.4108𝜔3
2𝑠. 𝑡. {

∑𝜔𝑖 = 1 ;

3

𝑖=1

𝜔𝑖 ≥ 0, 𝑖 = 1,2,3.

 

Using LINGO 18.0 software to solve, the weight 

coefficient obtained is: 

𝜔1 = 0.9497，𝜔2 = 0.0503，𝜔3 = 0. 
Then the corresponding IGOWA operator combination 

prediction model is 

𝑥𝑡
∧
= 0.9497𝑥𝑣−𝑖𝑛𝑑𝑒𝑥(1𝑡) + 0.0503𝑥𝑣−𝑖𝑛𝑑𝑒𝑥(2𝑡).   (11) 

The predicted value and prediction precision of the 

combined prediction model calculated according to equation 

(11) are shown in Table 5. During the sample period, the sum 

of squares of prediction errors of the model is 𝑄2 = 0.06226. 

3.3.2. IGOWA combined forecasting model for 𝜆 = −1 

According to the model building method in 2.2., the third-

order induced ordered weighted average prediction error 

matrix for calculating 𝜆 = −1 is: 

𝐸 = [
0.0081 0.0038 0.0110
0.0038 0.0287 0.0181
0.0110 0.0181 0.0766

]. 



 

143 

Then, the IGOWA operator combination prediction model 

based on the optimal criterion of minimizing the sum of 

squares of prediction errors can be constructed as: 

𝑚𝑖𝑛𝑄2 =𝑊𝑇𝐸𝑊
= 0.0081𝜔1

2 + 0.0076𝜔1𝜔2 + 0.022𝜔1𝜔3 + 0.0287𝜔2
2

+ 0.0362𝜔2𝜔3

+ 0.0766𝜔3
2𝑠. 𝑡. {

∑𝜔𝑖 = 1 ;

3

𝑖=1

𝜔𝑖 ≥ 0, 𝑖 = 1,2,3.

 

Using LINGO 18.0 software to solve, the weight 

coefficient obtained is: 

𝜔1 = 0.9852，𝜔2 = 0.0148，𝜔3 = 0. 
Then the corresponding IGOWA operator combination 

prediction model is 

𝑥𝑡
∧
=

1
0.9852

𝑥𝑣−𝑖𝑛𝑑𝑒𝑥(1𝑡)
+

0.0148

𝑥𝑣−𝑖𝑛𝑑𝑒𝑥(2𝑡)

.        (12) 

The predicted value and prediction precision of the 

combined prediction model calculated according to equation 

(12) are shown in Table 5. During the sample period, the sum 

of squares of the reciprocal prediction errors of the model is 

𝑄2 = 0.00744. 

3.3.3. IGOWA Combined Forecasting Model for 𝜆 → 0 

According to the model building method in 2.2., the third-

order induced ordered weighted average prediction error 

matrix for calculating 𝜆 → 0 is: 

𝐸 = [
0.0204 0.0159 0.0333
0.0159 0.0653 0.0381
0.0333 0.0381 0.1535

]. 

Then, the IGOWA operator combination prediction model 

based on the optimal criterion of minimizing the sum of 

squares of prediction errors can be constructed as: 

𝑚𝑖𝑛𝑄2 =𝑊𝑇𝐸𝑊
= 0.0204𝜔1

2 + 0.0318𝜔1𝜔2 + 0.0666𝜔1𝜔3 + 0.0653𝜔2
2

+ 0.0762𝜔2𝜔3

+ 0.1535𝜔3
2𝑠. 𝑡. {

∑𝜔𝑖 = 1 ;

3

𝑖=1

𝜔𝑖 ≥ 0, 𝑖 = 1,2,3.

 

Using LINGO 18.0 software to solve, the weight 

coefficient obtained is: 

𝜔1 = 0.9164，𝜔2 = 0.0836，𝜔3 = 0. 
Then the corresponding IGOWA operator combination 

prediction model is 

𝑥𝑡
∧
= 𝑥𝑣−𝑖𝑛𝑑𝑒𝑥(1𝑡)

0.9164 𝑥𝑣−𝑖𝑛𝑑𝑒𝑥(2𝑡)
0.0836 .         (13) 

The predicted value and prediction precision of the 

combined prediction model calculated according to equation 

(13) are shown in Table 5. During the sample period, the sum 

of squares of the prediction logarithmic error of the model is 

𝑄2 = 0.02002. 

3.3.4. IGOWA Combined Forecasting Model for 𝜆 = 0.5 

Table 5. Predicted value and Prediction Precision of IGOWA operator combination prediction model 

Year 

Actual 

value of 

R&D 

intensity 

(%) 

λ=1 λ=-1 λ→0 λ=0.5 

Predicted 

value (%) 

Prediction 

accuracy 

Predicted 

value (%) 

Prediction 

accuracy 

Predicted 

value (%) 

Prediction 

accuracy 

Predicted 

value (%) 

Prediction 

accuracy 

2006 0.9600 0.9600 1.0000 0.9600 1.0000 0.9600 1.0000 0.9600 1.0000 

2007 0.9900 0.9617 0.9714 0.9583 0.9679 0.9644 0.9742 0.9627 0.9724 

2008 1.1200 1.0966 0.9791 1.0917 0.9748 1.1005 0.9826 1.0981 0.9804 

2009 1.3500 1.2993 0.9625 1.3019 0.9643 1.2967 0.9605 1.2984 0.9617 

2010 1.3200 1.3868 0.9494 1.3855 0.9504 1.3879 0.9486 1.3872 0.9491 

2011 1.4000 1.3844 0.9888 1.3812 0.9866 1.3871 0.9908 1.3854 0.9896 

2012 1.6400 1.5523 0.9465 1.5535 0.9472 1.5511 0.9458 1.5519 0.9463 

2013 1.8300 1.7717 0.9682 1.7740 0.9694 1.7695 0.9669 1.7709 0.9677 

2014 1.8900 1.8586 0.9834 1.8523 0.9801 1.8636 0.9861 1.8605 0.9844 

2015 1.9600 1.9189 0.9790 1.9124 0.9757 1.9241 0.9817 1.9208 0.9800 

2016 1.9700 2.0051 0.9822 2.0085 0.9805 2.0016 0.9840 2.0038 0.9829 

2017 2.0900 2.0704 0.9906 2.0715 0.9912 2.0693 0.9901 2.0700 0.9904 

2018 1.9100 2.1049 0.8980 2.1028 0.8990 2.1067 0.8970 2.1056 0.8976 

2019 2.0300 2.0344 0.9978 2.0383 0.9959 2.0305 0.9998 2.0329 0.9986 

2020 2.2800 2.2960 0.9930 2.2981 0.9921 2.2940 0.9939 2.2953 0.9933 

According to the model building method in 2.2., the third-

order induced ordered weighted average prediction error 

matrix for calculating is: 

𝐸 = [
0.0088 0.0075 0.0147
0.0075 0.0268 0.0147
0.0147 0.0147 0.0607

]. 

Then, the IGOWA operator combination prediction model 

based on the optimal criterion of minimizing the sum of 

squares of prediction errors can be constructed as: 

𝑚𝑖𝑛𝑄2 =𝑊𝑇𝐸𝑊
= 0.0088𝜔1

2 + 0.0015𝜔1𝜔2 + 0.0294𝜔1𝜔3 + 0.0268𝜔2
2

+ 0.0294𝜔2𝜔3

+ 0.0607𝜔3
2𝑠. 𝑡. {

∑𝜔𝑖 = 1 ;

3

𝑖=1

𝜔𝑖 ≥ 0, 𝑖 = 1,2,3.

 

Using LINGO 18.0 software to solve, the weight 

coefficient obtained is: 

𝜔1 = 0.9372，𝜔2 = 0.0628，𝜔3 = 0. 
Then the corresponding IGOWA operator combination 



 

144 

prediction model is 

𝑥𝑡
∧
= (0.9372√𝑥𝑣−𝑖𝑛𝑑𝑒𝑥(1𝑡) + 0.0628√𝑥𝑣−𝑖𝑛𝑑𝑒𝑥(2𝑡))

2. (14) 

The predicted value and prediction precision of the 

combined prediction model calculated according to equation 

(14) are shown in Table 5. During the sample period, the sum 

of squares of prediction errors of the model is 𝑄2 = 0.00872. 

3.4. Evaluation of Model Effectiveness 

In order to reflect the prediction effectiveness of different 

models, this paper selects the following five error indicators 

to build the effectiveness evaluation index system of IGOWA 

operator combination prediction model based on the principle 

of minimum sum of squares of errors [16]: 

Sum of squares error: 

𝑆𝑆𝐸 = ∑ (𝑥𝑡 − 𝑥𝑡
∧
)𝑁

𝑡=1

2

;    (15) 

Mean square error: 

𝑀𝑆𝐸 =
1

𝑁
√∑ (𝑥𝑡 − 𝑥𝑡

∧
)2𝑁

𝑡=1 ;      (16) 

Square absolute error: 

𝑀𝐴𝐸 =
1

𝑁
∑ |𝑥𝑡 − 𝑥𝑡

∧
|𝑁

𝑡=1 ;        (17) 

Square absolute percentage error: 

 𝑀𝐴𝑃𝐸 =
1

𝑁
∑ |

𝑥𝑡−𝑥𝑡
∧

𝑥𝑡
|𝑁

𝑡=1 ;            (18) 

Mean square percentage error: 

𝑀𝑆𝑃𝐸 =
1

𝑁
√∑ (

𝑥𝑡−𝑥𝑡
∧

𝑥𝑡
)2𝑁

𝑡=1 .        (19) 

According to the effectiveness evaluation index system, 

each error index under three kinds of single prediction models 

and four kinds of special values of combined prediction 

models is calculated and normalized [17]. The smaller the 

error index value is, the better the model can predict and fit 

the data in the sample period. The larger the error index value 

is, the worse the effect of the model on data prediction and 

fitting in the sample period is. The average prediction 

accuracy of various prediction methods in the sample period 

is supplemented, and the results are shown in Table 6. 

Table 6. Prediction effect evaluation indicators of each model 

Evaluating indicator 
Grey prediction 

model 

ARIMA 

model 

Holt-Winters 

model 

Combined forecasting model 

λ=1 λ=-1 λ→0 λ=0.5 

SSE 0.2095 0.1413 0.3067 0.0623 0.0625 0.0625 0.0623 

SSE normalization 0.6831 0.4607 1.0000 0.2031 0.2038 0.2038 0.2031 

MSE 0.0305 0.0251 0.0369 0.0166 0.0167 0.0167 0.0166 

MSE normalization 0.8266 0.6802 1.0000 0.4499 0.4526 0.4526 0.4499 

MAE 0.1022 0.0727 0.1200 0.0449 0.0464 0.0436 0.0444 

MAE normalization 0.8517 0.6058 1.0000 0.3742 0.3867 0.3633 0.3700 

MAPE 0.0635 0.0454 0.0717 0.0273 0.0283 0.0265 0.0270 

MAPE 

normalization 
0.8865 0.6332 1.0000 0.3808 0.3947 0.3696 0.3766 

MSPE 0.0193 0.0150 0.0217 0.0096 0.0097 0.0096 0.0096 

MSPE normalization 0.8894 0.6912 1.0000 0.4424 0.4470 0.4424 0.4424 

Average precision 0.9365 0.9546 0.9283 0.9727 0.9717 0.9735 0.9730 

From the data in Table 6, among the three single item 

prediction models, the prediction error of the time series 

model ARIMA (1,1,2) is relatively minimum, which indicates 

that the model has a good effect on single item prediction of 

data in the sample period. The Holt-Winters non-seasonal 

exponential smoothing prediction model has the largest 

prediction error index value, which indicates that the model 

performs poorly in single prediction fitting of data in the 

sample period. Then observe the prediction performance of 

the combined prediction model. No matter what parameters 

IGOWA operator takes or what error indicators are used, the 

prediction error index value of IGOWA operator is far less 

than the prediction error index value of various single 

prediction models. The average precision of the combined 

forecasting model is also higher than the average forecasting 

precision of each single forecast, which is higher than 97.17%. 

The high prediction accuracy indicates that the IGOWA 

operator combination prediction model has better prediction 

effect, more accurate fitting, and the results are closer to the 

true value, which has excellent prediction performance and is 

significantly superior to other single prediction models. 

Therefore, it is completely reasonable and effective to use the 

IGOWA operator combination prediction model based on the 

minimum sum of squares of errors to estimate and predict the 

sample data. 

3.5. Model Sensitivity Analysis 

When the parameter 𝜆 of IGOWA operator takes different 

values, the optimal weight coefficient of the combined 

forecasting model will also change. In order to show more 

intuitively the influence of parameter 𝜆 variation on weight 

coefficient 𝜔  and five prediction error evaluation indexes, 

the sensitivity of parameter 𝜆 is carried out [18]. The results 

are shown in Figure 2 and Figure 3 respectively. 

It can be seen from Figure 2 that when 𝜆 gradually falls 

within the interval [−1,0) , the weight coefficient 𝜔1 

decreases and 𝜔2 increases. When 𝜆 is gradually within the 

interval (0,1] , the weight coefficient 𝜔1  rises slightly and 

𝜔2decreases slightly. However, in the parameter interval, the 

weight changes little. 𝜔1  is always within (0.9,1) , 𝜔2  is 

always within (0,0.1), 𝜔3 is always 0. This shows that the 

single prediction contribution degree ranks first in the 

prediction accuracy of each period when the combined 

prediction model is used in the sample period of this paper. 

It can be seen from Figure 3 that, in the parameter interval, 

the MAE and MAPE prediction error indexes of 𝜆  change 

obviously. The variation of 𝜆  has little contribution to the 



 

145 

variation of three prediction error indicators of SSE, MSE and 

MSPE. Among the five index values, the MSPE value of the 

mean square percentage error is the smallest, and all the error 

index values are below the 0.0625 horizontal line, which 

indicates that the prediction effect of the combined prediction 

model in this paper is very good, with high sensitivity, and 

meets the expectations. 

 

Figure 1. Influence of λ variation on optimal weight coefficient 

 

Figure 2. Influence of λ variation on five prediction error indexes 

3.6. Prediction Results and Analysis 

Based on the above analysis and test, it can be found that 

the forecasting effect of the combined forecasting model in 

the data sample period is significantly better than that of the 

grey forecasting model GM (1,1), time series model ARIMA 

(1,1,2), Holt-Winters non-seasonal exponential smoothing 

model. Therefore, it is naturally considered to use the IGOWA 

operator combination forecasting model based on the 

minimum sum of squares of error criterion to forecast the 

research and development intensity data of Anhui Province 

from 2021-2025, which will play a basic analysis and 

exploration role in the investment and planning of Anhui 

Province in scientific research and innovation in the next few 

years. In order to use the combination forecasting model to 

predict the R&D intensity data of the next five periods, we 

need to use the prediction accuracy of each single forecast in 

the same period to calculate. Since we cannot know the actual 

value of the R&D intensity of Anhui Province in the next 

several periods, we cannot know the prediction accuracy of 

each single forecast model in the next five periods, so we 

cannot determine how to assign the weight coefficient. 

Referring to the prediction precision processing method of 

prediction period proposed by Hongjun Yuan and Guiyuan 

Yang [19], based on the coherence principle of prediction, if 

we want to predict the future k periods in the prediction 

interval [𝑁 + 1,𝑁 + 2,⋯ ] , we can use the fitting average 

accuracy 
1

𝑘
∑ 𝑣𝑖𝑡
𝑁
𝑡=𝑁−𝑘+1  of the latest k periods of the i single 

prediction model to reflect the prediction accuracy of 𝑁 + 𝑘 

periods in the prediction interval. Therefore, according to the 

prediction values of the next five periods of the three single 

prediction models in this paper, the prediction values of 

IGOWA operator combination prediction model with 

different values of parameter 𝜆 can be calculated respectively, 

as shown in Table 7. 

It can be seen from Table 7 that no matter what parameter 

value is taken, the R&D intensity of Anhui Province will grow 

steadily from 2021-2025, and the scientific research and 

innovation ability will continue to improve. According to the 

2022 Anhui Statistical Yearbook provided by the Anhui 

Provincial Bureau of Statistics, the actual value of Anhui's 

R&D intensity in 2021 is 2.34%, which is similar to the result 

0

0.2

0.4

0.6

0.8

1

1.2

- 1 0 0 . 5 1

PARAMETER VALUE

ω1 ω2 ω3

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

- 1 0 0 . 5 1

PARAMETER VALUE

SSE MSE MAE MAPE MSPE



 

146 

predicted by the combination forecasting model, which 

proves the effectiveness of the model, and the model has 

statistical and forecasting significance. 

Table 7. Prediction Results of Anhui R&D Intensity (Unit: %) from 2021-2025 

Year 
Single forecast model Combined forecasting model 

Grey prediction model ARIMA model Holt Winters model λ=1 λ=-1 λ→0 λ=0.5 

2021 2.4000 2.4020 2.3100 2.4019 2.4020 2.4018 2.4019 

2022 2.5030 2.4870 2.4060 2.4878 2.4872 2.4883 2.4880 

2023 2.6080 2.5770 2.5070 2.5786 2.5775 2.5796 2.5789 

2024 2.7150 2.6690 2.6120 2.6713 2.6697 2.6728 2.6719 

2025 2.8240 2.7620 2.7220 2.7651 2.7629 2.7671 2.7659 

4. Conclusion 

In this paper, the grey prediction model GM (1,1), time 

series ARIMA model and Holt Winters non-seasonal 

exponential smoothing prediction model are used to build and 

test the model, so that the three single prediction models are 

used to predict the R&D intensity data of Anhui Province 

from 2006 to 2020 and calculate the corresponding prediction 

accuracy. Then, the IGOWA operator combination prediction 

model based on the optimal criterion of the minimum sum of 

squares of errors is constructed. The prediction accuracy of 

three single prediction models is taken as the induced value 

in the combination prediction model with different 𝜆 values 

is established. Then the model error evaluation index system 

is constructed to test the effectiveness of the model and the 

sensitivity of parameter 𝜆 . By comparison, it can be found 

that the combined forecasting model optimizes the 

forecasting process and is more accurate and effective than 

each single forecasting method. Finally, the verified IGOWA 

operator combination prediction model is used to predict the 

R&D intensity of Anhui Province from 2021-2025 based on 

the data of the sample period. The research results show that 

the prediction results are in line with the actual situation, and 

in the next five years, the R&D intensity of Anhui Province 

will continue to rise, and the growth rate will increase year by 

year, and the scientific research and innovation ability will 

continue to develop steadily. In order to guarantee the 

investment in scientific research and innovation in Anhui 

Province, drive the synchronous development of R&D and 

society, and improve the quality of life of the people, the 

government needs to strengthen the dominant position of 

enterprise R&D investment, stabilize the R&D investment of 

colleges and universities, scientific research institutes, play 

the role of financial funds to guide, improve the level of 

incentive services, and gradually improve the social security 

system and relevant regulations. In turn, it will drive the 

sustained high-quality development of the province's R&D 

and innovation level, and lay a solid foundation for social 

transformation and upgrading. 

Acknowledgements 

The research is supported by the Natural Science 

Foundation of Anhui University of Finance and Economics 

(Item number: acszjyzd2022008). 

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