Academic Journal of Science and Technology ISSN: 2771-3032 | Vol. 11, No. 1, 2024 13 Employ Mathematical Modeling to Summarize, Analyze, And Predict the Relationship Between Carbon Dioxide and Temperature, Location, And Other Factors. Yuheng Xie1, Yanxi He2, Yufei Ye3 and Chenfei Liu4 1Beijing National Day School, Beijing 100039, China 2Alcanta International School. Guangzhou 511458, China 3Nanjing Xuanwu Foreign Language School, Nanjing 213012, China 4Nanwai Xianlin Foreign Language School Xianlin Branche, Nanjing 210023, China Abstract: Fluctuations in the amount of greenhouse gases (GHGs) are affected by a variety of factors, both natural and artificial. Natural factors, such as volcanic eruptions and changes in the intensity of solar radiation, can cause fluctuations in greenhouse gases. However, in recent years, human activities have surpassed natural factors and have become a major driver of climate change. During the industrial revolution, the widespread use of fossil fuels, large-scale deforestation and emissions from factories caused irreversible damage to ecosystems. Rising global temperatures are destroying natural ecosystems around the world and are a potential threat to human survival. In this paper, we systematically examine and analyze the main sources of greenhouse gas emissions, explore the factors contributing to air pollution, and evaluate the specific impacts of these changes on global warming. We analyze the relationship between global warming and different factors (economic development, clean energy, and CO2 emissions) by using both one-way linear regression models and multiple linear regression models. A quadratic linear regression model shows that solar energy is the most promising renewable energy source. Meanwhile, using the global GDP as a criterion, we have shown that economic growth contributes to CO2 emissions. However, due to the governments having valued economic development more, the upward growth trend of the economic development curve is intensified, while the downward growth trend of the global average temperature is slowed down. Finally, we developed a multivariate linear regression model to evaluate whether using new energy sources could help mitigate global warming. Based on the previous analysis, we focus on the factor of CO2 emission. By using the linear regression model, we obtained the projected values of CO2 emissions, clean energy usage and global GDP in 2050, and the CO2 emissions in 2050 are 36.59439 billion tons, which meet the goal of carbon neutrality. Keywords: Carbon dioxide, greenhouse gases, clean energy use, temperature, modeling. 1. Introduction 1.1. Background Over the past century, there have been significant changes in the climate, mainly in the form of global warming. Since 1950s, global greenhouse gas (GHG) concentrations have continued to rise rapidly, and global average surface temperatures have maintained an overall upward trend. Meanwhile, a number of climatic problems have occurred, such as rising sea levels, melting of the highland snowpack, retreating of Arctic Sea ice, and rising sea surface temperatures. By the beginning of the twenty-first century, there was little dispute that greenhouse gas emissions were causing global warming. On February 2, 2007, the U.N. Intergovernmental Panel on Climate Change (IPCC) released a summary of its Fourth Global Assessment of Climate Change, which consolidated six years of research by scientists around the world[1]. The report concludes that global warming has become an "indisputable" fact. According to the report, the global average temperature from now to 2100 is likely to rise by 1.8ºC to 4ºC, while sea level could rise by 18 to 59 centimeters. The IPCC notes that the global average surface temperature has risen by 0.74°C over the past 100 years, and the past 50 years have been the warmest on record[2]. According to the World Meteorological Organization (WMO), the global average temperature in 2006 was 0.42°C higher than the 1961-1990 average, it was the sixth warmest year on record. The IPCC committee concluded that the sharp rise in global temperatures coincided with a period of large-scale emissions of greenhouse gases, and that human activities are the main cause of global warming. To help people better understand climate change, we need to develop a simplified model that describes the relationship between carbon dioxide concentrations and changes in the Earth's temperature. With this model, we can predict future climate changes and CO2 emission, and explain how they relate to global warming. This will help to demonstrate the trend of global climate change and its impact on the environment, as well as to clarify the causes of global warming. By alerting people to potential risks, we can motivate decision makers to adopt policies and regulations to address global climate issues. 1.2. Problem description Not only do we need to perform a linear regression of location factors such as latitude and longitude, but we also need to predict the average annual temperature for the year 2050 based on the correlation between time and temperature. Finally, we will create a report that examines the relationship between global temperature and factors that could be the causes, and determines how important a role has CO2 played. 14 2. Notation and Hypothesis Analysis 2.1. Notation Description Table 1. The Description of Notations Notations Meaning T Global average annual temperature/°C G Global annual GDP C Global annual CO2 emissions/ton P Global population/0.1 billion people E Global clean energy use a1 GDP weighting a2 Annual CO2 emissions weighting a3 Global population weight a4 Clean Energy Use Weight l1 Latitude l2 Longitude b1 Latitude Weights b2 Longitude Weights Y Time b3 Time weights W Weight Matrix b4 Error term 2.2. Assumptions 1. It is assumed that the increase in CO2 emissions due to the use of fossil fuels will not stop abruptly; 2. A long-term assessment is made, i.e. there are enough data sources; 3. Human use of clean energy increases gradually according to current policies and trends; 4. Any fuel usage by 2050 is consistent with the current trend in fuel use levels; 5. The location factor is divided into latitude and longitude, the temperature longitude is used as the independent variable. 3. Model Analysis and Establishment 3.1. Explore the relationship between temperature change and time To explore the relationship between location, time and temperature, we use the regression model to analyze the global temperature change and time between 1958-2021. Firstly, for the quadratic equation, the parameters are set as: y = p1x 2 + p2x + p3 (1) According to the data obtained from the Heywhale Community[3], the quadratic regression analysis shows that the parameters are p1=0.0001.59375997, p2=-0.617179359, and p3=597.357428. The covariance matrix of the fitted parameters was used for evaluation. The values on the diagonal of the covariance matrix represent the magnitude of the error for each parameter, and the parameter error for each coefficient is a1=0.0000399855504, a2=0.159063862, and a3=158.179765. It can be seen that the predictions of a1 and a2 are more accurate than parameter a3. For the linear equation, we set the parameter as: y = q1x + q2 b (2) Similarly, according to the linear regression, the q1 and q2 coefficients are 0.016818356374807917 and - 33.100599718381844, which shows that the slope is less than 0.1, indicating that the effect of time on the temperature is relatively small. 3.2. Argumentation of the relationship between temperature and time through time series 3.2.1. Augmented Dickey-Fuller Test The test procedure of ADF test is the same as that of Dickey-Fuller test, but applies to the model ∆y = a + βt + a1∆y1 + ⋯ + a(n + 1)∆y(n − 1) (3) In statistics, the Augmented Dickey Fuller Test (ADF) tests the original hypothesis that there is a unit root in the time series sample[5]. The alternative hypothesis varies depending on the version of the test used, but is usually smoothness or trend smoothness. ADF can test larger and more complex sets of time series models. 3.2.2. Code Implementation and Modeling The next step is to determine if the time series is white noise. Since it is daily data over several years, we can assume that the seasonal component repeats itself every 365 days, thus we set the period as 365. Then we should verify that the residuals are white noise (which is the same requirement as for smoothness), and we have the following figure: 15 Figure 1. Seasonal decomposition for a time series To verify that the residuals are white noise, we can check for smoothness using the Augmented Dickey-Fuller Test, get: p-value = 0.0. The p-value obtained is close to 0.0, which is below our significance level (0.05). Therefore, we can reject the null hypothesis (H₀: the time series is non-stationary), and our time series can be considered as white noise. 3.3. Relationship between temperature and longitude and latitude by linear regression 3.3.1. Data sources With data obtained from the Heywhale Community, the autocorrelation coefficients of longitude and latitude are set to β1 and β2. Since the time series has already been proved to be white noise, it is not taken into consideration. Table 2. (excerpt) 1950/11/1 7.327 1.07 Kabul Afghanistan 34.56N 70.05E 1950/12/1 1.849 0.881 Kabul Afghanistan 34.56N 70.05E 1951/1/1 1.403 0.292 Kabul Afghanistan 34.56N 70.05E 1951/2/1 2.903 0.504 Kabul Afghanistan 34.56N 70.05E 1951/3/1 7.948 0.607 Kabul Afghanistan 34.56N 70.05E 1951/4/1 12.697 0.588 Kabul Afghanistan 34.56N 70.05E 1951/5/1 19.484 0.253 Kabul Afghanistan 34.56N 70.05E 1951/6/1 23.355 0.484 Kabul Afghanistan 34.56N 70.05E 1951/7/1 25.22 0.443 Kabul Afghanistan 34.56N 70.05E 1951/8/1 24.432 0.249 Kabul Afghanistan 34.56N 70.05E 1951/9/1 22.042 0.361 Kabul Afghanistan 34.56N 70.05E 1951/10/1 17.926 0.784 Kabul Afghanistan 34.56N 70.05E 1951/11/1 10.02 0.318 Kabul Afghanistan 34.56N 70.05E 1951/12/1 5.549 0.391 Kabul Afghanistan 34.56N 70.05E 1952/1/1 2.766 0.412 Kabul Afghanistan 34.56N 70.05E 1952/2/1 5.294 0.472 Kabul Afghanistan 34.56N 70.05E 1952/3/1 8.049 0.511 Kabul Afghanistan 34.56N 70.05E 3.3.2. The steps of Modeling After cleaning the data, the latitude and longitude of each region and the average annual temperature are obtained as in table.3 16 Table 3. Average temperatures by region Table 4. Least squares fitting results Using multiple linear regression, the relationship between temperature and latitude/longitude was set as T = l1 b1 + l2 b2 + b4 (4) The results were obtained by using least squares fitting as shown in table 4. The coefficient in the result is the value of b1, b2. And the final correlation is obtained: T = (−0.1402) ∗ l1 + (0.0156) ∗ l2 + 22.4135 (5) And we get the R-squared of 0.8. The R-squared measures the fitness of the equation, the larger the R-squared, the better fitness. In general, greater than 0.8 means that the equation fits the sample points very well and meets the requirements of the question. Thus, the relationship between temperature and location and time is obtained. 3.4. Prediction model based on previous years' global average temperature data and CO2 concentration data 3.4.1. Data sources In order to test the effect of the above factors on CO2 emissions, we use a regression model for graphical analysis. By analyzing the historical data[4], we plan to examine the relationship between the following variables and CO2 emissions, and whether carbon neutrality can be achieved in the future by adjusting the use of clean energy. 17 Table 5. Global clean energy use, 2012-2021 2 Year Total installed capacity (MW) Geothermal energy (MW) Biomass (MW) Solar (MW) Wind energy (MW) Hydropower (MW) 2012 1443923 10479 77198 104312 266918 1090111 2013 1566148 10717 84545 141169 299840 1137100 2014 1698319 11157 90605 180216 349307 1175994 2015 1851651 11812 96351 228054 416170 1210655 2016 2013932 12137 105245 300199 466866 1245903 2017 2184709 12697 110966 395276 514215 1270876 2018 2357495 13169 118256 488820 563466 1293619 2019 2541688 13738 123940 591059 621270 1311765 2020 2807265 14073 133025 716788 731763 1335114 2021 3063926 15644 143371 849473 824874 1360054 Table 6. Global CO2 emissions, global GDP and global mean surface temperature, 2012-2021 Year CO2 emission/billion tons GDP/billion dollars Baseline global mean surface temperature change 8°C (°C） 2012 34.94 75277.89 0.89617 2013 35.23 75011.41 1.02535 2014 35.47 76265.41 1.12538 2015 35.76 77439.7 1.20485 2016 36.11 79500.78 1.23379 2017 36.53 81088.95 1.2988 2018 36.82 86095.66 1.32017 2019 37.01 87325.54 1.35691 2020 37.42 90960.9 1.38187 2021 38.25 96487.67 1.41853 3.4.2. Analysis of CO2 and Temperature After testing, we believe that the quadratic regression model is more accurate for graphing. Figure 2. Total global CO2 emissions/billion tons Figure 3. Global average annual temperature/°C By comparing the recent trends of global CO2 emissions and global average annual temperature, we find that the temperature increases with the increasing of CO2 emissions. It is obvious that CO2 emission is one of the important factors causing temperature increase and global warming. Reducing CO2 emissions can effectively mitigate the greenhouse effect. 3.4.3. Analysis of Clean Energy Use In order to study the development potential of various green energy sources, we collected data on the global use and development of five kinds of energy sources, as well as data on the total global use of renewable energy, and set up a quadratic linear regression model to analyze which energy source is most efficient in achieving carbon neutrality. Figure 4. Global geothermal energy use/MW 18 Figure 5. Global biomass energy use/MW Figure 6. Global solar energy use/MW Figure 7. G lobal wind energy use/MW Figure 8. Global hydro energy use/MW Figure 9. Total installed energy/MW (total renewable energy use) The figures show that global renewable energy use is growing significantly, which is a major factor in the slowing down of CO2 emissions and the warming trend. The most used renewable energy is water, at 136,054 MW. Solar energy has seen the most significant growth in use and is still rising rapidly, making it the most promising renewable energy source for the future. 3.5. Other Factors that Affect Global Warming Figure 10. 2012-2021 Global GDP Figure 11. 2012-2021 Global Average Temperature 19 Figure 12. 1970-2021 Total Global Population Figure 13. 2012-2021 Global Average Annual Temperature Figure 10 shows the global GDP over the last 10 years, and we can see that economic development has been rising rapidly and is gradually accelerating. Figure 12 shows how the global population has increased steadily over the last 50 years. The global average annual temperature has also been on the rise for the last 10 years, with a slowing down growth trend. 3.6. Multiple Linear Regression Modeling 3.6.1. Modeling Based on the above analysis, we find that there is a certain relationship between global warming and factors including economic development, CO2 emission and population. We can establish a multiple linear regression model to evaluate whether controlling these factors is effective in reducing global warming. We set the global average annual temperature as the dependent variable (T), and the independent variables are the global annual GDP (G), global annual CO2 emission (C), and global population (P). Based on the graphical data, the equation can be derived: T = a1 ∗ G + a2 ∗ C + a3 ∗ P + e (6) where e is the error term and a1,a2,a3 are the regression coefficients. 3.6.2. Model Calculation We consider the effects of global GDP, global population and CO2 emissions on global temperature. Firstly, we standardize the data. Then, by programming the linear regression, the equation is obtained as: T = 0.0303 ∗ G + 0.0250 ∗ P + 0.1445 ∗ C − 6.1874 (7) The coefficients of G, P, C are positive, which means that G, P, C is positively correlated with T. The coefficient of C is 0.1445, which is larger than the coefficients of G and P, indicating that CO2 emissions have the greatest impact on global temperature. Therefore, our next step will focus on CO2. 3.6.3. CO2 and clean energy, GDP modeling Based on the above results, we can analyze the relationship between the three by setting up a one-way linear regression model and a multiple linear regression model. [6] We set CO2 emission (C) as the dependent variable, and the independent variables include: global clean energy use (E) and global GDP (G). G is positively correlated with C and E is negatively correlated with C. The following equation can be obtained: C = b1G + b2E + f (8) By normalizing the data, we model and program the equation as: C = 0.1281 ∗ G − 0.2824 ∗ E + 38.9363 (9) For every $1 billion increase in total annual GDP (G), global CO2 emissions are projected to increase by 0.1281 billion tons. For every 1 MW increase in clean energy use (E) by the global population, global CO2 emissions are expected to decrease by 0.2824 billion tons. From this analysis, we can conclude that in the future, clean energy use will reach carbon neutrality as the trend of clean energy use continues to grow. At that time, CO2 emissions will peak at about 3,89363 billion tons. 3.6.4. Clean Energy, GDP Annual Forecast Modeling In order to investigate whether carbon neutrality can be achieved by 2050, a one-way linear regression model can be developed to calculate the projected values of clean energy use and total GDP in 2050. Assume that the relationship between global clean energy use (E) and year (y) is given by: E = c1 ∗ y + d (10) The equation can be obtained by programming: E = 1.91956 ∗ y - 3848.76689 (11) Substituting y = 2050 into this equation yields E2050 = 86.3311 Assume that total global GDP (G) is related to the year as: G = c2 ∗ y + g (12) 20 The equation can be obtained by programming: G = 2.63256y - 5224.71060 (13) Substituting y = 2050 into this equation gives G2050 = 172.03740 3.6.5. Projecting Carbon Emissions in 2050 Next, we substitute G2050, E2050: C = 0.1281 ∗ G - 0.2824 ∗ E + 38.9363 Yield C2050 = 36.59439. Since 36.59439 < 38.9363, the carbon emissions in 2050 have already decreased, and carbon neutrality is expected to be reached by 2050. 4. Summary and Discussion 4.1. Summary Through optimization and improvement of the model, we can take more factors into account that affect global warming, such as energy efficiency, to further predict the time to achieve carbon neutrality. Therefore, we should continue to promote environmental protection measures and increase the use of clean energy to help protect the environment. 4.2. Advantages and Shortages of the Model 4.2.1. Advantages 1. In Model 1, we have analyzed the influence of time on temperature through different methods. 2. Studying the time series, we found that the effect of time on temperature is relatively small, thus eliminate the effect of time and leaves only two independent variables, which is more accurate. 3. The relationship between location and temperature is demonstrated by linear regression. 4. Having considered multiple influencing factors and using multiple sets of data. 5. Both GDP and population are considered. 4.2.2. Shortages 1. The model only uses linear fitting, quadratic function fitting, linear regression, etc., which is relatively simple, and there are few comparisons between different models. 2. The model will have bias, and the generalizability of some models is still not good. 3. There will be some errors in the weighting analysis. References [1] IPCC climate report 2022 summary: The key findings. (2023, June 19). Climate Consulting. https://climate.selectra.com/en/news/ipcc-report-2022 [2] Carbon dioxide now more than 50% higher than pre-industrial levels. (2022, June 3). National Oceanic and Atmospheric Administration. https://www.noaa.gov/news-release/carbon- dioxide-now-more-than-50-higher-than-pre-industrial-levels [3] - heywhale.com. (n.d.). https://www.heywhale.com/home/dataset [4] Data Overview - Berkeley Earth. (2023, November 10). Berkeley Earth. https://berkeleyearth.org/data/ [5] Wikipedia contributors. (2024, February 22). Dickey–Fuller test. Wikipedia. https://en.wikipedia.org/wiki/Dickey%E2%80%93Fuller_test [6] Best, V. a. P. B. C. (2013b, September 22). A fit to global temperature data. Clive Best. https://clivebest.com/blog/?p=2353