2022 Tianfu Cup National Undergraduate Mathematical Contest in Modeling
Question C Environmental Protection and Green Economy
Reproduction of the original title:
"Energy saving and emission reduction" and the development of green economy are a major national strategic plan. "Green water and green mountains are mountains of gold and silver". Environmental protection is of great significance to the development of the national economy. With the acceleration of the industrialization process, environmental protection issues have become urgent. The appendix provides some economic and environmental data for reference, including GDP, carbon dioxide emissions, energy use, agricultural land and forest area, etc. In recent years, with the destruction of vegetation on a global scale, the forest area has gradually decreased, while industrial development has increased carbon emissions. Compared with global trends, China's vegetation area is gradually increasing.
Now based on these data, your team needs to solve the following problems:
(1) Please predict the domestic forest area in 2025-2035 based on the data given in the attachment.
(2) By analyzing the relevant data and finding the required data by yourself, construct a mathematical model to analyze the impact of the policy of returning farmland to forests on the national industrial level and national economy.
(3) The appendix gives the data of many countries in the world. Please build a model based on the national data given in the appendix to evaluate the economic level, industrial level and forest vegetation area of each country in the world. What do you find?
(4) If your team wants to submit a report to the United Nations, please write a report to describe your findings and propose some specific measures.
Overview of the overall solution process (abstract)
"In the fragrance of rice flowers, it is said that there is a good year, and one can hear the sound of frogs." With just a few lines of poems, a scene of natural and harmonious coexistence comes to mind. In the process of human historical development, people have increasingly realized that the rapid economic and social development must not be at the cost of environmental damage and waste of resources. In this context, this paper discusses the following three issues: (1) Predicting China's forest area in 2025-2035. (2) Build a mathematical model to analyze the impact of the policy of returning farmland to forests on the national industrial level and national economy (3) Evaluate the economic level, industrial level and forest vegetation area of various countries in the world.
For question 1, we performed an outlier test on the data in the attachment through the box plot and found that no outliers appeared. On this basis, based on the time series model, the data stationarity test was first carried out through the time series diagram, autocorrelation diagram and Daniel test, and the results showed that the attached data was non-stationary. Then according to this result, the first and second differences are used to construct a stationary sequence and its stationarity is checked, and the second difference shows that it is stable. According to the characteristics of the autocorrelation diagram and partial correlation diagram of the data, the parameters were selected and the ARIMA (2, 2.1) model was established by SPSS24.0, and the stability of the model was tested based on the model residual diagram, and the results showed that the model was stable. On this basis, we use SPSS24.0 to predict and estimate the results, and compare some of the data prediction results with the actual data. The results show that the data simulation results and the attached known actual data results have a high degree of fitting. Finally, based on the estimated values obtained, we estimate the 2025-2035 China's forest area (square kilometers) prediction with a 95% confidence interval to take into account the invisible errors caused by other factors to make the estimated results more accurate and credible.
For the second problem, first use the Raida criterion algorithm to eliminate abnormal data, and then perform statistical analysis on the eliminated data. Subsequently, the employment benefits and potential benefits are used as indicators to measure the impact of returning farmland to forests on the national industrial level, economic benefits and biodiversity are used as indicators to measure the impact of returning farmland to forests on the national economy, and on this basis, relevant mathematical expressions are constructed . Next, discuss and determine the parameters in the constructed mathematical expression. The average value of the collected data over the years is substituted into the expression, and on this basis, the standard value of the impact of returning farmland to forestry on the national industrial level and national economy is obtained. Then substitute the data over the years into the expression, compare the obtained value with the standard value, and then obtain the converted results of the impact of returning farmland to forestry on the national industrial level and national economy in different years. The analysis of the results shows that: (1) The implementation of the policy of returning farmland to forestry has an overall upward trend in the employment level of the secondary and tertiary industries. (2) The potential value impact of the conversion of farmland to forest policy on the national industrial level shows a certain periodic fluctuation law in the steady growth. (3) The implementation of the policy of returning farmland to forestry has injected more power and vitality into the development of the national industrial level. (4) The implementation of the policy of returning farmland to forests has greatly improved economic benefits. (5) The marginal effect of forest area growth and biodiversity growth is diminishing. (6) Since the implementation of the policy of returning farmland to forests in 2003, the national economic development trend has become more and more favorable.
For question three, after weighing, select four indicators - carbon dioxide emissions, GDP, industrial added value, and forest area, and conduct principal component analysis on the average value of the data from 1990 to 2020 to obtain comprehensive evaluation results. Then, draw a graph of the comprehensive evaluation results to visually display the comprehensive evaluation results. Finally, the analysis of the results shows that: (1) The countries ranked first and second in the comprehensive evaluation results are China and the United States respectively. (2) The comprehensive evaluation results of low- and middle-income countries are generally lower than those of middle- and high-income countries. This may be because low-income countries urgently need to solve basic social problems such as food, clothing and medical care, and their GDP has not yet reached the ideal level of development. (3) Most European countries have relatively high comprehensive evaluation results. Sensitivity analysis was performed on the relationship between forest area and biodiversity in question 2. When the value of β changes by 0.3 (equivalent to about 5.7%), the result changes by less than 3%, which shows that the model is relatively stable.
Model assumptions:
1. Assume that the data given in the attachment is completely stationary after second-order difference processing
2. Assume that the data given in the title is true and reliable
3. Assume that there are no major accidents in the predicted future time period
4. Assume that the collected data is true and reliable
problem analysis:
(1) Analysis of Question 1 Question
1 belongs to the forecasting mathematical problem. As far as the characteristics of this kind of problem are concerned, it is required to predict the future value and the possible interval of the predicted value on the basis of observing the change law of historical data. The important premise assumption is that things The past can continue to the future under normal development conditions, which is in line with the characteristics of time series analysis. The data provided in the attachment is a numerical sequence formed at each time point, and the title requires predicting the forest area in 2025-2035, that is, the forest area (square kilometers) in each year from 2025 to 2035. Based on the above reasons, we will first create a time series and build a time series model. Then, compare the existing predicted value with the actual value, and on this basis, predict the forest area (square kilometers) in each year from 2025 to 2035.
(2) Analysis of Question 2
This question requires the construction of a mathematical model to analyze the impact of the policy of returning farmland to forestry on the national industrial level and national economy. The main point of this question is the selection of evaluation indicators and the selection of parameter values. Considering that there may be outliers in the data found by yourself, it is first necessary to test and eliminate outliers on the original data, and then analyze on the newly obtained data. Employment benefits and potential benefits are used to measure the impact of returning farmland to forestry on the national industrial level. Use economic benefits and biodiversity to measure the impact of returning farmland to forests on the national economy. By constructing a mathematical expression, the collected data is substituted into the expression, and then the equivalent score is calculated to analyze the impact of returning farmland to forestry on the national industrial level and national economy.
(3) Analysis of Question 3
This question belongs to the evaluation category, and principal component analysis can be performed on some data in the appendix to obtain comprehensive evaluation results. The results obtained are then analyzed. Four indicators can be selected: carbon dioxide emissions, GDP, industrial added value, and forest area, and the average value of the data of each indicator from 1990 to 2020 can be used for principal component analysis. The method of drawing makes the comprehensive evaluation result more intuitive.
Model establishment and solution Overall paper thumbnail
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Part of the program code: (code and documentation not free)
function [MaxA,darth]=Maxchara(A)
%用于计算矩阵的最大特征值及特征向量
[X,Y]=eig(A);
[Max,index]=max(max(Y));
MaxA=Max;
thA=X(:,index);
darth=(1/sum(thA))*thA;
end
A=[1 2 3 7;1/2 1 2 4;1/3 1/2 1 2;1/7 1/4 1/2 1];
B1=[1 2 3;1/2 1 2;1/3 1/2 1];
B2=[1 1 3;1 1 3;1/3 1/3 1];
B3=[1 1/3 1;3 1 1/3;1 3 1];
B4=[1 1/4 1/3;4 1 2;3 1/2 1];
RI3=0.58;
RI4=0.90;
disp('Matriax A');
[MaxA,darthA]=Maxchara(A);
fprintf('矩阵A的最大特征值为 %d\n',MaxA);
disp('矩阵A的最大特征向量为：');
disp(darthA);
CIb1=(MaxA-4)/(4-1);
fprintf('矩阵A的一致性指标为 %f\n',CIb1);
CRa=CIb1/RI4;
fprintf('矩阵A的随机一致性比率为 %f\n',CRa);
%Matrix B1
disp('Matriax B1');
[MaxB1,darthB1]=Maxchara(B1);
fprintf(' 矩阵B1的最大特征值为 %d\n',MaxB1);
disp('矩阵B1的最大特征向量为：');
disp(darthB1);
CIb1=(MaxB1-3)/(3-1);
fprintf('矩阵B1的一致性指标为 %f\n',CIb1);
CRb1=CIb1/RI3;
fprintf('矩阵B1的随机一致性比率为 %f\n',CRb1);
%Matrix B2
disp('Matriax B2');
[MaxB2,darthB2]=Maxchara(B2);
fprintf('矩阵B2的最大特征值为 %d\n',MaxB2);
disp('矩阵B2的最大特征向量为：');
disp(darthB2);
CIb2=(MaxB2-3)/(3-1);
fprintf('矩阵B2的一致性指标为 %f\n',CIb2);
CRb2=CIb2/RI3;
fprintf('矩阵B2的随机一致性比率为 %f\n',CRb2);
%Matrix B3
disp('Matriax B3 ');
[MaxB3,darthB3]=Maxchara(B3);
fprintf(' 矩阵B3的最大特征值为 %d\n',MaxB3);
disp('矩阵B3的最大特征向量为：');
disp(darthB3);
CIb3=(MaxB3-3)/(3-1);
fprintf('矩阵B3的一致性指标为 %f\n',CIb3);
CRb3=CIb3/RI3;
fprintf('矩阵B3的随机一致性比率为 %f\n',CRb3);
%Matrix B4
disp('Matriax B4');
[MaxB4,darthB4]=Maxchara(B4);
fprintf('矩阵B4的最大特征值为 %d\n',MaxB4);
disp('矩阵B4的最大特征向量为：');
disp(darthB4);
CIb4=(MaxB4-3)/(3-1);
fprintf('矩阵B4的一致性指标为 %f\n',CIb4);
CRb4=CIb4/RI3;
fprintf('矩阵B4的随机一致性比率为 %f\n',CRb4);
temp=zeros(1,3);
for i=1:3
temp(i)=darthA(1)*darthB1(i)+ darthA(2)*darthB2(i)+ darthA(3)*darthB3(i)+ darthA(4)*darthB4(i);
fprintf('措施A%d的总层次排序为%f\n',i,temp(i));
end
CIZ=darthA(1)*CIb1+darthA(2)*CIb2+darthA(3)*CIb3+darthA(4)*CIb4;
RIZ=darthA(1)*RI3+darthA(2)*RI3+darthA(3)*RI3+darthA(4)*RI3;
CRZ=CIZ/RIZ;
fprintf('措施层的总排序的随机一致性比率 %f\n',CRZ);