INTERNATIONAL JOURNAL OF ENVIRONMENTAL SCIENCES Volume 4, No 5, 2014 Copyright by the authors - Licensee IPA- Under Creative Commons license 3.0 Research article ISSN 0976 4402 A statistical analysis and prediction of carbon dioxide emission in some eastern and northern states of India Pijush Basak 1 and Sumit Nandi 2 1 Department of Mathematics, Narula Institute of Technology, Kolkata-700109, India 2 Department of Chemistry, Narula Institute of Technology, Kolkata-700109, India pijushbasak@rediffmail.com, sumitnandi5@gmail.com doi: 10.6088/ijes.2014040404534 ABSTRACT Environmental and climatic change due to global warming is a serious concern for scientists and researchers for the sake of existence of life on earth. One of the important causes for global warming is the uncontrolled emission of carbon dioxide from various sources which is responsible for this unwanted situation. The emission is enhanced in temporal and spatial scales since few decades. In India, an uneven variation in emission of the gas is observed in Northern and Eastern states for last thirty years and it is fear-provoking. In this paper, an attempt has been made to construct state-wise carbon-dioxide emission model for few northern and eastern states in India. A statistical procedure, namely, Least Square method is applied for the development of the model. Our models may be experimentally utilized for forecast of carbon dioxide emission in future. Keywords: Carbon dioxide, Least Square method, Polynomial, Coefficient of determination, analysis, Regression sum of square, sum of square, Null hypothesis. 1. Introduction During the last few decades, there has been a serious change in climate of the world due to global warming. Climate change due to global warming attracts tremendous attention to scientists, researchers and academicians as this change results in alarming situation for the existence of life on earth. Increased emission of different green house gases are responsible for this unwanted situation. Amongst them, an uncontrolled emission of carbon dioxide (CO2) from different sources is supposed to be the most important one (Battle et al, 2010). In India, the amount of emission of CO2 for the last fifty years from fossil fuels is tremendous. It may be mentioned here that India is the fourth largest CO2 emitter in the world (Ghoshal and Bhattacharyya, 2008). Carbon dioxide is emitted from different sources like solid fuels, liquid fuels, gaseous fuels, cement industry and gas flaring. Source wise, India is significantly different from global averages. The major global sources of CO2 are liquid fuels whereas solid fuels come second in importance. However, in India, solid fuel is much more responsible than the liquid fuels. For the world, solid: liquid ratio is 33:44 whereas in India, it is 68:24. Emission of CO2 in different states in India for last thirty years is much fear-provoking. In Indian scenario, CO2 emission from different sources mainly depends on industrialization, urbanization, economic growth. As an example, Uttar Pradesh leads as far as mean CO2 emission between 1980 and 2000 in India which is closely followed by central states like Madhya Pradesh. Consumption pattern of the solid and liquid fuels in India reveals that Received on March 2013 Published on April 2014 956
during 1980 to 2000, consumption of petroleum products was high for the western and southern states like Maharastra and Tamil Nadu, but coal consumption was high in northern and central states such as Uttar Pradesh and Madhya Pradesh. For the state level CO2 emission from cement manufacturing industry, southern state Andhra Pradesh atop the list, followed by western state like Rajasthan. The total CO2 emission for India in 2000 (from fossil fuel and cement industry) became 334431000 metric tons (MT) of carbon of which contribution of CO2 from cement manufacturing is 3.11 per cent which is in conformity with the percentage of CO2 emission from cement industry (3 per cent) at the global level (Marland et al, 1999). Emission of carbon dioxide for the eastern state of Assam is not high earlier but for the last ten years or so, emission rate is increasing slowly. Kram et. al. (2000) studied global and regional greenhouse gas (GHG) emissions scenarios and introduced key sources and parameters in emissions. Similarity research and results obtained by Fenhann (2000) to study on industrial non-energy resources for GHG. Ritter et al. (2002) recommended methodologies for consistent estimation of GHG emissions from oil and gas industry facilities. Singh et al. (2008) elaborated the trends of energy consumption and consequent emissions of GHG from the road transport sector in India. Efforts are made to study the nature of change CO2 emission and its constituents. Analysis of emission of CO2 by mathematical modeling has been performed by several researchers worldwide but in Indian context, it is perhaps neglected. Basak and Nandi (2014) applied a model of differential equation for the emission dynamics of CO2 in Indian perspective. Tokos and Xu (2009) made a study on the modeling of CO2 emission with a system of differential equations for six attribute variables for the continental United States from 1950 to 2005. Nandi and Basak (2014) also made a study on emission of carbon dioxide from main four attributes in India through differential equations. Parikh et al (2009) described CO2 emission structure of Indian economy based on fuel type, sector wise, final demand and expenditure classes. Goraeu (1990) recommended a differential equation for presentation of CO2 emission in the atmosphere. Based on this scenario, in this research article, we have formulated mathematical model based on emission of CO2 in some eastern and northern states in India using least square method. The states considered here are Delhi, Uttar Pradesh, Bihar in northern India and West Bengal and Assam in eastern India utilizing the data set of emission of CO2 for twenty one years (1980-2000). Emission data are collected from different literatures (Ghoshal and Bhattacharyya, 2008; Marland et al, 1999). 2. Material and methods 2.1 Method of Least Square The Method of Least Squares is a procedure, requiring some calculus and linear algebra to determine what the best fit line is to the data. The literature survey proposes [Tokos, 2009; Goraeu, 1990] that emission of CO2 in a region is well represented by a third degree polynomial. Without any loss of generosity, denoting emission of CO2 as Y, the concerned polynomial can be written as Y= a + b.x + c.x 2 + d.x 3 (1) where x stands for year and a, b, c and d are constants. Given data (x1, y1), (x2, y2) (xn,yn), the error associated the polynomial of best fit are 957
The goal is to estimate values of a, b, c and d that minimize the error. The corresponding normal equations are (2) (3) For given set of points ( ); (i=1, 2 n), the equations (3) can be solved to obtain the estimated a, b, c and d (noted as, that minimizes the error equation (2). It has been found that in all the cases, the values of the 2 nd order derivatives viz.,, come out to be positive at the points a, b, c and d respectively indicating that, of E. are the solutions of the normal equations and provide minimum The third degree fitted polynomial of CO2 emission is estimated as Y=, + x +.x 2 +.x 3 (4) 2.2 Quality of estimates Thus, one can utilize either equation (4) on the above to obtain the estimate of the CO2 emission for short and medium terms of time in a state. The related question is how good are these estimates? The answer depends on the quality of the developed analytical models using the raw data. The quality of the proposed analytical models are verified with the statistical criteria, namely the coefficient of determination R 2 (adjusted R 2 ) and residual analysis. 2.3 Coefficient of determination The coefficient of determination R 2 is defined as the proportion of the total response variation that is explained by the model. It provides an overall measure of how well the model fits. The general definition of the coefficient of determination is where Here, SStot = Total sum of square (proportional to the sample variance); SSreg = the regression sum of squares or the explained sum of square and SSerr = the sum of squares of residuals, also called the residual sum of square. yi and fi are observed and estimated values of CO2 emission. 958
2.4 Adjusted Coefficient of determination The adjusted R 2 is defined as where p is the total number of regressors in the model (not counting the constant term) and n is the sample size. 2.5 Test of mean of residual We provide careful attentions to the residual (error) term that enters the model. For hypothesis testing of small sample, we have to assume that for the model (4) stochastic residual term E follows the t distribution. The test statistic is t=(x m)/(s/ n) where x, m, s and n are residual mean, expected residual mean (=0), standard deviation and sample size respectively. Under the null hypothesis H0= Mean of the residual is zero; the t statistic is tested with (n-1) degrees of freedom at 5% level of significance. If t > tabulated t at (n-1) degrees of freedom, H0 is rejected. 3. Results and conclusion 3.1 West Bengal For the model, Least Square is applied to the data set of West Bengal and the estimates are computed as = -259707.875, = 52.9310341, = -0.145273298 and = 9.46642831E-005 The model for the emission of CO2 for the state of West Bengal can be represented as Y = -259707.875 + 52.9310341* X - 0.145273298 * X 2 + 9.46642831E-005* X 3 where X represents time in year. A graphical display of the actual data and the model for West Bengal is given by figure. 1. Figure 1: Emission of CO2 in West Bengal In the figure, the change of CO2 emission is compared with the model data and it is evident from the figure that CO2 emission model matches well with the actual status of CO2 emission. One can utilize the above model equation on the above graph to obtain the estimate of emission of CO2 in West Bengal for short or medium terms of time. However, it is now very much justifiable to quantify the goodness of the estimates. The answer depends on the quality of developed analytical models using the raw data. Intermediately, the regression sum of square (SSREG), in other words the variation explained by the model is computed. The 959
residual sum of squares (SSERR) is the variation that is left unexplained is calculated. The total sum of square (SSTOT) is proportional to the sample variance and equals the sum of SSERR and SSREG. As explained earlier, the coefficient of determination R 2 is defined as the proportion of the total response that is explained by the model is worked out. It provides an overall measure of how well the model fits and Adjusted R 2 will adjust for degree of freedom of the model that works better when we have a lot of parameters. The coefficient of determination R 2 (R 2 adjusted) reflect the fact that we have identified a good model. The details of statistical analysis are given in Table 1. R Square Table 1: Statistical evaluation criteria R square adjusted FVU (Frac. Var. Unexplained) 0.871701777 0.849060893 0.128298223 SSTOT SSREG SSERR 315834048 276639392 40520944 Furthermore, the residual analysis is performed on the proposed model of emission of CO2 of West Bengal State is given in Table 2 below. As seen from the table, the residuals are extremely small compared to data and so is the standard error. Under the assumption of H0: mean of the residuals as zero, for small sample (21 years), the t-statistic is computed insignificant at 5% level indicating good fit of the model. These results attest to the good quality of the proposed model for emission of CO2 in West Bengal. Table 2: Analysis The predicted total CO2 emissions in West Bengal for 2015 and 2020 are 31587.83 and 34701.29 ( 000 MT carbon) respectively. 1980 11499.33 10387.41 1111.91 1991 17909.15 16939.13 970.01 1981 12501.49 10978.86 1522.62 1992 17197.92 17539.79-341.86 1982 12753.50 11571.08 1182.41 1993 18439.53 18141.27 298.25 1983 12313.33 12164.22 149.10 1994 20435.90 18743.58 1692.31 1984 11663.23 12758.14-1094.91 1995 21432.41 19346.76 2085.65 1985 12848.32 13352.93-504.61 1996 20984.91 19950.76 1034.15 1986 12742.41 13948.55-1206.13 1997 20984.91 19950.76 1034.15 1987 13307.96 14544.99-1237.02 1998 17753.24 21161.34-3408.10 1988 14190.19 15142.25-952.06 1999 23457.05 21767.86 1689.18 1989 14268.26 15740.39-1472.12 2000 23363.71 22375.26 988.44 1990 15175.41 16339.00-1163.94 Standard Deviation of residual 1423.39 Mean of residual 0.018 Standard Error of residual 37.72 960
3.2 Assam Estimating the value of,, and from the data set of Assam, the model for the emission of CO2 for the state of Assam can be represented as Y = -23651.6563 + 4.4630022* X -3.41706909E-005* X 2 + 2.02274555E-006* X 3 A graphical display of the actual data and the model data for Assam is given by figure. 2. Figure 2: Emission of CO2 in Assam The change of CO2 emission is compared with the model data and it is evident from the figure that CO2 emission model match fairly well with the actual status of CO2 emission. The SSREG, SSERR and SSTOT are computed and presented in Table 3. The magnitude of R 2 reflects the fact that we have identified a good model. The details are given in Table 3. R Square Table 3: Statistical evaluation criteria R square adjusted FVU (Frac. Var. Unexplained) 0.561999977 0.538823557 0.438000023 SSTOT SSREG SSERR 1413390.75 794325.563 619216.25 Furthermore, the residual analysis is performed on the proposed model of emission of CO2 of Assam State is given in Table 4 below. The residuals as well as the standard errors are small compared to data. T-statistic is insignificant at 5% level of significance. These results endorse good quality of the proposed model for emission of CO2 in Assam. Table 4 Analysis 1980 589.84 752.46-162.62 1991 1329.32 1063.21 266.11 1981 682.14 780.59-98.45 1992 1245.81 1091.60 154.20 1982 657.02 808.75-151.73 1993 1093.22 1120.02-26.79 1983 622.90 836.93-214.02 1994 1134.67 1148.46-13.79 1984 668.75 865.13-196.38 1995 1298.34 1176.93 121.41 961
1985 856.71 893.35-36.64 1996 1277.20 1205.42 71.78 1986 956.76 921.60 35.15 1997 1078.78 1233.94-155.16 1987 1043.89 949.88 94.01 1998 923.27 1262.47-339.20 1988 1252.71 978.17 274.54 1999 1205.09 1291.04-85.94 1989 1257.10 1006.49 250.61 2000 1097.00 1319.62-222.62 1990 1470.40 1034.84 435.55 1991 1329.32 1063.21 266.11 Standard Deviation of residual 199.28 Mean of residual -0.0010 Standard Error of residual 14.11 The model may be utilized for the prediction. The predicted total CO2 emissions in Assam for the years 2015 and 2020 are 1751.34 and 1896.47 ( 000 MT carbon) respectively. 3.3 Bihar The model for the emission of CO2 for the state of Bihar can be represented as Y = -115580.508 + 27.4314423* X - 0.0591151789* X 2 + 4.00160643E-005* X 3 A graphical display of the actual data and the model for Bihar is given by figure. 3. Figure 3: Emission of CO2 in Bihar In the figure, the actual change of CO2 emission is compared with the model data and it is evident from the figure that CO2 emission model match very well with the actual status of CO2 emission except at a few years at the end period of data. The above graph may be utilized to obtain the estimate of emission of CO2 in Bihar for short or medium terms of time. As before, the SSERR, SSREG and SSTOT are computed. As explained earlier, the coefficient of determination R 2, the proportion of the total response that is explained by the model is worked out. The significance test of R 2 suggests the goodness of the model. The details are given in Table 5. Table 5 Statistical evaluation criteria R Square R square adjusted FVU (Frac. Var. unexplained) 0.6400 0.5600 0.3500 SSTOT SSREG SSERR 153891376 99026744 55122984 962
Furthermore, the residual analysis is performed on the proposed model of emission of CO2 of Bihar State is given in Table 6. As seen from the table, the residuals are extremely small compared to data and so is the standard error. The t-statistic under the assumption of mean residual as zero is insignificant at 5% level significance. These results attest to the good quality of the proposed model for emission of CO2 in Bihar. Table 6: Analysis The mean of the error under the assumption of population mean of the residual as 0 is with significant limit at 5% level of significance. Thus, it can be stated that the model fits well with the observed data. The total CO2 emissions in Bihar in 2015 and 2020 are estimated at 27058.48375 and 28446.15796 ( 000 MT carbon) respectively. 3.4 Uttar Pradesh The model for the emission of CO2 for the state of Uttar Pradesh can be represented as Y = -814178.875 + 119.478363* X - 0.41999507* X 2 + 0.000287490955* X 3 A graphical display of the actual data and the model for Uttar Pradesh is given by figure. 4. 1980 16785.76 17598.98-813.21 1991 20383.77 20524.32-140.54 1981 18632.26 17863.13 769.13 1992 22025.58 20792.42 1233.16 1982 19639.46 18127.62 1511.83 1993 20414.13 21060.88-646.75 1983 18779.63 18392.51 387.11 1994 20313.46 21329.69-1016.23 1984 16956.35 18657.72-1701.36 1995 21261.19 21598.87-337.67 1985 18854.35 18923.31-68.95 1996 22867.63 21868.40 999.22 1986 18429.89 19189.25-759.36 1997 18291.14 22138.31-3847.17 1987 18757.25 19455.55-698.30 1998 29905.86 22408.57 7497.29 1988 20135.11 19722.20 412.91 1999 23184.21 22679.18 505.02 1989 20552.89 19989.22 563.66 2000 19011.97 22950.17-3938.20 1990 20345.26 20256.59 88.66 Standard Deviation of residual 2225.15 Mean of residual 0.011 Standard Error of residual 47.17 Figure 4: Emission of CO2 in Uttar Pradesh In the figure, the change of CO2 emission as extracted from model data and the actual status of CO2 emission indicates almost a perfect matching. One can utilize the model data to obtain the estimate of emission of CO2 in Uttar Pradesh for short or medium terms of time. It is 963
justifiable to quantify the goodness of the estimates. As a routine analysis, SSTOT, SSERR and SSREG are computed. As explained earlier, the coefficient of determination R 2, defined as the proportion of the total response that is explained by the model is worked out. R 2 is an astonishing result of 0.99 meaning that 99% of the total variation is explained by the model. Such a value of R 2 reflects the fact that we have identified an almost perfect model. The details are given in Table 7. Table 7: Statistical evaluation criteria R Square R square adjusted FVU (Frac. Var. Unexplained) 0.993178368 0.991974533 0.00682163239 SSTOT SSREG SSERR 2.67932954E+009 2.67365734E+009 18277480 The residual analysis is performed on the proposed model of emission of CO2 of Uttar Pradesh and is presented in Table 8 below. As seen from the table, the residuals are extremely small compared to data and so is the standard error. Under the assumption of H0: mean of the residual as 0, the observed mean of the residual is found insignificant at 5% level of significance. These results attest to the good quality of the proposed model for emission of CO2 in Uttar Pradesh. Table 8: Analysis 1980 9343.07 7457.12 1885.94 1991 26963.81 27826.15-862.34 1981 10338.96 9295.99 1042.96 1992 29599.18 29693.46-94.27 1982 10824.98 11137.25-312.27 1993 31449.67 31563.32-113.64 1983 12455.88 12981.36-525.47 1994 31961.24 33435.72-1474.48 1984 14090.16 14827.87-737.70 1995 35823.35 35310.82 512.53 1985 16437.53 16677.06-239.52 1996 38313.28 37188.46 1124.82 1986 17317.67 18528.81-1211.13 1997 40284.55 39068.79 1215.76 1987 20510.24 20383.10 127.13 1998 40046.37 40951.67-905.29 1988 23014.17 22239.93 774.24 1999 44122.21 42837.09 1285.11 1989 24423.24 24099.46 323.77 2000 44268.30 44725.21-456.91 1990 24603.24 25961.53-1358.29 Standard Deviation of residual 955.96 Mean of residual 0.045 Standard Error of residual 30.91 Thus, it can be stated that the model fits well with the observed data. The predicted total CO2 emissions in Uttar Pradesh for 2015 and 2020 are 63360.63837 and 83037.2813 ( 000 MT carbon) respectively. 3.5 Delhi Estimating the value of,, and by using Least Square method, the model for the emission of CO2 for the state of Delhi can be represented as 964
Y = -73296.875 + 7.5317297* X -0.0164620783* X 2 + 1.62818451E-005* X 3 A graphical display of the actual data and model data for Delhi is given by figure. 5. Figure 5: Emission of CO2 in Delhi One can utilize the above equation on the above graph to obtain the estimate of emission of CO2 in Delhi for short or medium terms of time. In the usual way, SSTOT, SSERR and SSREG are computed. As explained earlier, the coefficient of determination R 2 defined as the proportion of the total response and provides an overall measure of how well the model fits that is explained by the model is worked out. The values of R 2 (R 2 adjusted) reflect the fact that we have identified a good model. The details are given in Table 9. Table 9: Statistical evaluation criteria R Square R square adjusted FVU (Frac. Var. Unexplained) 0.633252919 0.568532825 0.366747081 SSTOT SSREG SSERR 22223178 14126435 8150286 Furthermore, the residual analysis is performed on the proposed model of emission of CO2 of Delhi State is given in Table 10 below. As seen from the table, the residuals are extremely small compared to data and so is the standard error. Moreover, the t-statistic of Null hypothesis of zero mean residual is found insignificant. These results attest to the good quality of the proposed model for emission of CO2 in Delhi. Table 10: Analysis 1980 2791.54 3464.08-672.54 1991 5500.18 4946.00 554.18 1981 3113.23 3598.00-484.76 1992 5696.10 5081.69 614.41 1982 3673.13 3732.07-58.93 1993 6101.74 5217.54 884.19 1983 3657.64 3866.31-208.66 1994 6288.66 5353.55 935.10 1984 3890.39 4000.71-110.31 1995 5691.08 5489.73 201.35 1985 3772.78 4135.27-362.49 1996 5625.16 5626.06-0.89 965
1986 4559.56 4269.99 289.56 1997 6143.62 5762.56 381.05 1987 4696.66 4404.87 291.78 1998 4018.42 5899.22-1880.79 1988 4881.27 4539.91 341.35 1999 5129.22 6036.04-906.82 1989 4727.43 4675.11 52.32 2000 6033.83 6173.03-139.19 Standard Deviation of residual 638.36 Mean of error 0.0008 Standard Error of residual 25.26 The total CO2 emission in the state of Delhi for 2015 and 2020 based on the model can be predicted as 8247.34684 and 8946.964997 ( 000 respectively. The emission of CO2 is presently a major concern in different parts of the country. Some parts of country are severely affected due to mainly deforestation, growing of industry, growing of population and uncontrolled emission. However, the pattern of growth is not uniform; it is time and area dependent. This pattern changes from less industrial to rapidly industry oriented and less populated area to densely populated part of the country. It is therefore essential to have study of emission of CO2 in different parts of the country. In the present study, we have developed a polynomial that characterize the behavior of total emission of CO2 consisting of different attributable variables namely, fossil fuels, cement industry and gas flaring for each of the five states in northern and eastern regions of India making use of the actual data from 1980 to 2000. In addition to have given the analytical form for emission in each state, we have utilized different statistical procedures, namely R 2 (R 2 adjusted) and residual analysis to evaluate the quality of the proposed cubic polynomial model using Least Square method. All these statistical procedures attest the good to quality of the proposed systems (cubic polynomial model). The outcomes are region specific. The model fits extremely well for the states Uttar Pradesh and West Bengal with coefficient of determination of, 0.99 and 0.87 respectively. For other states namely, Assam, Delhi and Bihar a reasonably well coefficient of variation, namely 0.56, 0.63 and 0.64 respectively are extracted. Those states need further improvement of model. Regarding the nature, it is needless to mention that the emission pattern shows increasing tendency towards years all the states considered with the state Bihar showing fluctuation at the end years. Finally, we have used these models to predict regarding CO2 emissions mainly in 2015 and 2020. A theoretical basis for the future researchers of CO2 emission in different regions in India may be obtained from the study and the model may be utilized for functional planning and strategic applications for the reduction of appalling global warming in near future. 4. References 1. Basak P. and Nandi S, (2014), an analytical study of emission dynamics of carbon dioxide in India, IOSR Journal of Applied Chemistry, 1, pp 16-21. 2. Battle M., Bender M. L., Tans P. P., White J. W., Ellis J. T., Conway T. and Francey R. J., (2010), Global carbon sinks and their variability inferred from atmospheric oxygen and d 13 C, Science, New Series, 287 (5462), pp 2467-2470. 966
3. Fenhann J., (2000), Industrial Non-Energy, Non-CO2 Greenhouse Gas Emissions, Technological Forecasting and Social Change, 63, pp 313 334. 4. Ghoshal T. and Bhattacharyya R., (2008), State level carbon dioxide emissions of India, 1980-2000, Contemporary Issues and Ideas in Social Sciences. 5. Goreau T. J., (1990), Balancing Atmospheric Carbon dioxide, Ambio Magazine, 19 (5), pp230-236. 6. Kram T., Morita T., Riahi K., Roehrl R. A., Rooijen S.V., Sankovski A. and Vries B. D., (2000), Global and Regional Greenhouse Gas Emissions Scenarios, Technological Forecasting and Social Change, 63, pp 335 371. 7. Marland G., Andres R. J., Boden T. A., Hohnston C. and Brenkert, A., (1999), Global, regional, and national CO2 emission estimates from fossil fuel burning, cement production, and gas flaring: 1751-1996, Carbon Dioxide Information Analysis Center, Oak Ridge, Tenn., USA. 8. Nandi S. and Basak P., (2014), Emission of carbon dioxide from different attributes in India: A mathematical study, Journal of Applied Chemistry, 1, pp 06-10. 9. Parikh J., Panda M., Ganesh K. A. and Singh V., (2009), CO2 emissions structure of Indian economy, Energy, 34 (8), pp1024-1031. 10.Ritter, K., Lev-On, M. and Shires, T., (2002), Development of a Consistent Methodology for Estimating Greenhouse Gas Emissions from oil and Gas Industry Operations, Paper Presented at the 11th Emissions Inventory Conference of the U.S. Environmental Protection Agency, April 2002, Atlanta, GA, Canada. 11. Singh A., Gangopadhyay S., Nanda P.K., Bhattacharya S., Sharma C. and Bhan C., (2008), Trends of Greenhouse Gas Emissions from the Road Transport Sector in India, Science of the Total Environment, 390, pp124 131. 12. Tokos C.P. and Xu Y. (2009), Modeling carbon dioxide emissions with a system of differential equations, Non linear Analysis: Theory, Methods and Applications, 71(12), pp1182-1197. 967