The main factors affecting Taiwan’s economic growth rate via dynamic grey relational analysis

Ever since the grey system theory was proposed about 40 years ago, its characteristics such as small samples, few data, and uncertainty have been used for study in the literature with increasingly wider scope. Recent studies on grey relation analysis have included static data analyses, and most of them have adopted initial values with only a relational order. Under the same study conditions, if different data preprocessing methods are used, then the relational order will be ranked differently. This study took Taiwan as the object to explore seven economic indices (birth rate (%), Taiwan’s total population (thousand people), unemployment rate (%), income per capita (USD), weighted average interest rate on deposits (%), Consumer Price Index (CPI), and national income (NI)) and how they affect the economic growth rate. The traditional static grey relational analysis treated the collected data with taking consideration of time effect which is irrational under some circumstance. An innovative dynamic grey relational analysis was carried out by shifting the raw data due to the time leading or lagging effect which is a mean to improve the capability of traditional grey relational analysis. The differences in analyses between static grey relational analysis and dynamic grey relational analysis via different data preprocessing methods were further discussed, finding that different data preprocessing methods generated a new set of relational orders through the latter. Finally, the prosperity index was used to identify the effects of all factors on economic growth (leading, synchronization, and lagging indices).


Factors affecting economic growth rate
According to IHS Markit's latest economic forecast for February 2020, the COVID-19 outbreak has dramatically reduced global demands and impacted supply chains, tourism, transportation, and international trade. The global economy has been greatly affected with its growth rate slowing down from the original 2.5% to 0.7%. Economic growth rate forecasts are of great importance to the future development of all countries in the world, and seeking out those factors affecting the changes in the economic environment is a topic of great concern by many global scholars and national governments [1].
when the system contains uncertainty, which may arise either due to insufficient data or uncertain relationships among the parameters associated with the system or its processes. As the application of grey relational analysis is getting popular in variety of academic realms, Javanmardi [21] conducted a systematic review of grey systems theory-based methods and applications in sustainability studies.
Based on the literature review, grey relational analysis is a powerful mathematic tool to analyze the relationship among data sets. The method will be inapplicable if the time effect is unneglectable due to data nature. For example, the unemployment rate will not affect the economic rate simultaneously. It is usually deemed as lagging index. Therefore, the shift of raw data when the grey relational analysis is performed is required. Also, the difficulty of analysis is greatly increased and the computer program is needed to execute tedious calculation. In this study, the strategy of shifting raw data is called dynamic grey relational analysis. The innovative method will extend the grey rational analysis to even more application and contribute to the knowledge of grey systems theory (GST) to make GST more sound. This paper is organized as follows. Research method was used to introduce the mathematical tool adopted in this study. Empirical Results and Analysis showed the calculation process and presented numerical outcome. And the final section is conclusion to conclude the important findings of this research.

Research method
The literature related to grey relation has employed static analysis in data studies, without considering leading or lagging time. A dynamic method (to compare the effects of sequential data movement) can be used to generate a time order, as well as the leading, synchronization, and lagging modes of a prosperity index. The present study realizes that more changes and combinations of original data could be generated by this movement method. In addition to that, there are various data preprocessing methods. Therefore, professional databases are needed to store larger data sizes. However, the current tools in the market do not meet the needs of this study. It is thus necessary to develop a customized system to process the tools needed. An innovative dynamic grey relational analysis is therefore used to explore the main factors affecting the economic growth rate.

Data collection and software and hardware equipment
Based on the above data, birth rate (%), Taiwan's total population (thousand people), unemployment rate (%), income per capita (USD), weighted average interest rate on deposits (%), Consumer Price Index, NI (million dollars), and economic growth rate are the factors discussed in this paper. Statistical data from 2006 to 2017 published on the statistical information network of Taiwan are used as the original data for numerical study. Microsoft asp.net c# and SQL2008 database are used as the software, and three personal computers are used as the hardware.

Research process
A local GRA is adopted in this paper, and the explanations are as follows.
Step 1 Establish a reference sequence and a comparative sequence: x 0 ðkÞ ¼ x 0 ð1Þ; x 0 ð2Þ; . . .x 0 ðnÞ ð1Þ where k = 1,2,3,� � �,n, i = 1,2,3,� � �,m. x 0 (k) is a reference sequence and x i (k) is a comparative sequence Step 2 Standardize the original data: Step 3 Obtain the grey relational coefficient: The equation is : gðx 0 ðkÞ; where k = 1,2,3,� � �,n, i = 1,2,3,� � �,m, x 0 is the reference sequence, and x i is a specific comparative sequence. γ is a grey relational coefficient Δ oi = kx 0 (k)−x i (k)k: the absolute value of the k th difference between x 0 and x i . 8kkx 0 ðkÞ À x j ðkÞk z: distinguishing coefficient; z2 [0,1]; in general, the distinguishing coefficient is 0.5 and can be adjusted as needed. The adjusted value will only change the relative value and will not affect the order of grey relational grade [18].
Step 4 Obtain the grey relational grade: Obtain the mean value of the grey relational coefficient The closer the value of grey relational grade is to unity, the higher is the relational grade of the reference sequence; otherwise, it is lower.
Step 5 Obtain the grey relational order: The grey relational grade indicates the relational grade between each sequence and the standard sequence. The order in which the relational grade are between all comparative sequences and the standard sequence is ranked by their values, and it is called the grey relational order.
When γ(x 0 ,x i )>γ(x 0 ,x j ), it means that the relational grade between x i and x 0 is greater than that between x j and x 0 ; that is, x i is more similar to x 0 .
Step 6 Computer system development: In this system, Microsoft asp.net c# is used to develop the front-end web program, and the SQL2008 database is used in the back-end for access to the front-end web data. The system is divided into 3 stages for operation.
Phase 1 Import the data of Table 1 into SQL database and develop the first front-end web program by using Step 1. All comparative sequences are based on the data from 2009 to 2014. By randomly moving 0 to 3 grids, either to the left or to the right, six pieces of data can be continuously acquired to regenerate a comparative sequence. After movement, there are seven comparative sequences. Moreover, the original data from 2009 to 2014 are regularly used in the reference sequence. Therefore, a new set of original data is formed, and the original data generated by the above sequence are unrepeatable. In total, of 2,097,153 original data combinations were generated.
Phase 2 Develop 6 standardized front-end web programs by using Step 2, and each program transforms 2,097,153 combinations in the database into standard data.
Phase 3 Develop the 6 th front-end web program by using Step 3 to Step 5, and each program generates the final total relational grade for the standard data in the database.

Empirical results and analysis
According to the final execution results of system development in Phase 3, the closer the characteristic index of grey relational grade is to unity, the higher is the relational grade between the comparative sequence and the reference sequence; otherwise, it is lower. However, there are 2,097,153 groups of relational grade in each standardization. In this study, the relational grade of all groups are added up to obtain the group with the maximum total value of relational grade for subsequent analysis.

The ranking of grey relational grade
The maximum total values of six standardized dynamic and static relational grades are taken from the database, and all the relational orders are organized and ranked from the largest to the smallest, as shown in Table 2. The seven factors of all relational orders in Table 2 are ranked from the largest to the smallest and divided into (relation ranking 1, 2, 3, 4, 5, 6, 7) 7 groups, as shown in Table 3. The frequencies of appearance of all factors are added up by using the 7 ranking groups in Table 3. The results are shown in Tables 4 and 5.
The factors with the highest frequencies in all rankings are selected from Table 4. The relational orders are re-ranked from large to small as follows: CPI > (NI (million dollars), unemployment rate (%) > Taiwan's total population (thousand people) > income per capita (USD) > birth rate > unemployment rate (%) > weighted average interest rate on deposits (%), unemployment rate (%). The unemployment rate appeared twice in succession, respectively ranking 2 and 6. Thus, the new ranking is simplified to: CPI > (NI (million dollars), unemployment rate (%) > Taiwan's total population (thousand people) > income per capita (USD) > birth rate > weighted average interest rate on deposits (%).
The factors with the highest total frequencies in all rankings are selected from Table 5. The relational orders are re-ranked from large to small as follows: weighted average interest rate on deposits (%) > (Taiwan's total population (thousand people), CPI) > (birth rate, unemployment rate (%)) > (birth rate, Taiwan's total population (thousand people)) > NI (million dollars) > income per capita (USD) > (Taiwan's total population (thousand people), CPI). Taiwan's total population appears three times in succession, respectively ranking 2, 4, and 7, and birth rate appears twice in succession, respectively ranking 3 and 4, and so the new relational order is simplified to: weighted average interest rate on deposits (%) > (Taiwan's total population (thousand people), CPI) > (birth rate, unemployment rate (%)) > Taiwan's total population (thousand people) > NI (million dollars) > income per capita (USD) > CPI, Taiwan's total population (thousand people).

Distribution of prosperity indices of all factors
The static and dynamic grey relations are integrated and compared, as shown in Table 6. The relational grades of initial value x6 (CPI) and interval value x7 (NI (million dollars)) decline, and the relational grade of interval value x1 (birth rate) remains unchanged, whereas other relational grades increase. The data from 2009 to 2014 are synchronous-namely, the  Table 7.
After that, the data are moved to a maximum of three grids, with each grid representing one year. The indices are subdivided as shown in Table 8. Finally, the original data are retrieved from the database according to the grey relational grade, to create Table 9. The prosperity indices of all factors of six data preprocessing methods are organized as shown in Table 10, by using Table 9 for the coordinate codes of original data and Table 7. As seen, except for the interval value of x1 (birth rate) being synchronous, the other standardizations are more relevant if they are more into the lead. Aside from the mean value and the maximum range being in the lag, other values of x2 (Taiwan's total population (thousand people)) are more relevant if they are more in the lead. Aside from the maximum range being in the lead, other values of x5 (weighted average interest rate on deposits (%)) are more relevant if they are more in the lag. Aside from the multiple being synchronous and the interval value being in the lead, other values of x6 (CPI) are more relevant if they are more in the lag. Aside from the maximum range and the interval value being in the lead, other values of x7 (NI (million dollars)) are more relevant if they are more in the lag. The analysis on the prosperity indices of other factors is shown in Table 10.
The prosperity indices of the static grey relation in Table 10 are changed to those of the dynamic grey relation, and the data movement directions are organized into Table 11. As seen, the standardized prosperity indices closer to the left are more in the lead except that the prosperity indices of the interval value of birth rate are synchronous. The standardized prosperity indices closer to the right are more in the lead, except that the prosperity indices of the maximum range of weighted average interest rate on deposits (%) and NI (million dollars) are closer to the left. The standardized prosperity indices closer to the right are more in the lead, except that the prosperity indices of the multiple of CPI were synchronous and the prosperity indices of the interval value are closer to the left. Table 11 shows the analysis on the prosperity indices of other factors.

Conclusions
In traditional studies on the economic growth rate by grey relation, the original data were collected and preprocessed by a selected method. The data used are static to finally generate a set of relational order, but different data preprocessing methods result in different rankings. Under inconsistent conditions, in this study, the grey relation was divided into dynamic and static analyses by using an innovative method, and the characteristic of moving original data dynamically was used, in order to finally integrate various data preprocessing methods and generate a new set of rankings. The time order of the original data produced by various factors was formed in a dynamic manner, to generate new data corresponding to the prosperity indices (leading, synchronization, and lagging). The final results can be used to solve the inconsistency arising from the use of the different data pre-processing methods.
There is no past literature talking about dynamic grey relation. It was thus found in this study that the ranking of static grey relation changes after dynamic grey relation, and that the relational grade increased or decreased. If the grade of decline in the relational grade can be found-i.e., deleting the affecting factors-then the accuracy of the study on affecting factors can be improved. The integrated achievements of this study are noted below. 1. In the process of moving the original data from dynamic grey relation, the original data should not be less than that of the static grey relation, and the number of grids of original data movement should be greater than the number of grids of original data movement of this study.
2. In this study, 6 data preprocessing methods were used for dynamic grey relational analysis, and the results showed that, except for the relational grades of initial value x6 (consumer price index) and interval value x7 (NI (million dollars)) in the static grey relation declining after dynamic grey relation, the relational grades of other factors were higher than those in the original static grey relation. In addition, any factor originally ranking first in the static grey relation might become last after dynamic grey relational analysis.
3. The prosperity indices were used to identify which factors are the leading, synchronization, and lagging indices affecting economic growth, so as to verify whether the static state (synchronization) has a leading or lagging relation.