A Statistical Evolutionary Analysis and Prediction of Carbon-dioxide Emission in Gujarat, Maharashtra and Madhya Pradesh of India

Purpose of the study: The change of climate due to global warming has become a burning problem throughout the world. India is a country where global warming causes a concern to the scientists. Unrestricted emission of different green house gases mainly carbon dioxide is responsible for this alarming situation. This paper develops a state-wise nonlinear emission model of carbon dioxide, an important green house gas. Methodology: Non linear least-square and regression analysis method used to explain the emission of the gas. Main Findings: The short term and long-term forecast of carbon dioxide emission trend in the states are presented. Applications of this study: Model is used in the states of Gujarat, Maharashtra and Madhya Pradesh of India. Novelty/Originality of this study: Future prediction of emission of CO 2 may be evaluated from our proposed model. This paper may be helpful for the future researchers for finding emission of CO 2 and other GHGs for the remaining states in India.


INTRODUCTION
Global warming has reached at an alarming stage throughout the world. As a result we see the abrupt change of climate and weather. It is not a local issue now, it is a global phenomenon and the scientists along with the environmentalists are seriously worried about the evil effects of the unnatural change of weather. Polar ice sheets and glaciers are melting very fast resulting the rise of sea level (Khan, Z. A. (2017)). All on a sudden we often observe very extra weather events like tsunami, typhoons, droughts, heat wave etc. (Khan, A. A. (2018)).
This unnatural and unpredictable diversity of weather are caused due to indiscriminate human activities. Uncontrolled use of pesticides in agricultural field, unplanned set up of industry, indiscriminate deforestation, abuses of natural resources generates huge amount of Green House Gases (GHG) which gives result the high rise of temperature of the Earth known as Global Warming (Basak, P. and Nandi, S. (2015)). Among the GHG, carbon dioxide (CO 2 ) is mainly responsible for the high rise of temperature of the Earth (Battle, M., et al. (2010), Ghoshal, T. and Bhattacharyya, R. (2008)). In augmentation of CO 2 concentration in atmosphere, India is one of the leading countries in the world. Since India is a developing country, so there occur industrials set up in a rapid and unplanned way. Moreover, there are vast agricultural lands in India and as a result emissions of GHGs are occurring for burning of fossil fuels and land use changes and other industrial processes (Murthy, N. S., Panda, M. and Parikh, J. (1997a), Murthy, N. S., Panda, M. and Parikh, J. (1997b)). Due to degradation of O 3 layer, the heat wave radiates back down to the Earth which gives rise of temperature (Dewitte, S. and Clerbaux, N. (2018)). Moreover, burning of fossil fuels and emission from transport vehicles along with the rapid industrialization releases CO 2 and other GHGs that increases the temperature of atmosphere and it is being able to absorb more radiation and would warm up rapidly (Khansis, A. A. and Nettleman, M. D. (2005), Rufael, Y. W. (2010)). The main harmful gas which is responsible for global warming is CO 2 and that is why emission of CO 2 is the burning problem throughout the country. Few states of India are badly affected mainly for agricultural, industrial and population growth and increase emission of transportation. While some states are comparatively less emitter and less affected. Hence, it is important to study about the nature of emission of CO 2 in different states in India (Kram, T., et al.(2000)). Several research works has done about CO 2 emission (Basak, P. and Nandi, S. (2014), Ghoshal, T. and Bhattacharyya, R. (2008), Mondal, K., Basak, P. and Sinha, S. (2019a). in India. Also the emission is studied for different states of India (Mondal, K., Basak, P. and Sinha, S. (2019b)). In this article, the pattern of emission in the states of Gujarat, Maharashtra and Madhya Pradesh are studied and we would like develop a mathematical model to find the trend of CO 2 emission.

Method of Least Square
The data of CO 2 emission for the 21 years  is utilized for the purpose of modeling. We formulated a third degree polynomial model for the analysis of CO 2 emission in the states of Gujarat, Maharashtra and Madhya Pradesh (Jin, R., et al.(2010)). For generating the model of CO 2 emission, we followed the works of (Basak, P. and Nandi, S. For given set of points (x i , y i ); (i=1, 2,…,n), the equations can be solved for , , and to find estimated It has been found that in all the cases, the value of the 2 nd order derivatives evolves to be positive at the points , , and . These satisfy the minimization criteria of . The equation (5) at a particular time is utilized for prediction of CO 2 .

Quality of estimates
Equation (4) may be used for obtaining the estimation of the CO 2 emission for short and medium terms of time in a state. It remains to examine the goodness of fit for this estimation. The matter depends on the quality of the developed analytical models using the raw data. The quality of the proposed analytical models is verified with the statistical criteria, namely the coefficient of determination, adjusted coefficient of determination and residual analysis.

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 R 2 is defined as, Where, Here, = Total sum of square (proportional to the sample variance), = the regression sum of squares or the explained sum of square, and = the sum of squares of residuals, also called the residual sum of square. and are observed and estimated values of CO 2 emission.

Adjusted Coefficient of Determination
The Adjusted Coefficient of Determination (Adj 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. It is, however, another advanced measure how good the model fit the actual data.

RESULTS / FINDINGS Gujarat
Using least square method, the CO 2 emission model for the state of Gujarat can be represent as, Where, x represents time in year.
A graphical display of the observed data and estimated data for Gujarat is given in the Figure 1.
Observed data Estimated data The values of R 2 (Adj R 2 ) are obtained and presented as: A residual analysis is calculated of the proposed model of emission of CO 2 for the state of Gujarat is given as: According to the Table 2, the residuals are very small compared to data and the so is standard error. The results of Table  2 suggest a good quality of model in Gujarat. The graphical visualization of the IROC is presented in Figure 2.

Maharashtra
The CO 2 emission model for the state of Maharashtra can be represented as, The observed data and model data can be represented graphically as below, Observed data Estimated data

Figure 3: Emission of CO 2 in Maharashtra
According to the Figure 3, observed status of CO 2 emission and the data evaluated by the model matches almost perfectly. The statistical evaluation is given below in Table 3. For verifying the goodness of fit of the model SS tot , SS reg , SS err and R 2 (Adj R 2 ) calculated. The value of R square is 0.99178 which is quite satisfactory and ensures that 99% of the total variation is extracted by the model. Residual analysis is computed and the result is presented in the following Table 4. It is observed that residuals and standard errors of the residuals are considerably small compared to observed emission. The result of statistical criteria and Residual analysis suggest that proposed model may be used as future prediction of emissions of CO 2 for the state of Maharashtra.

Prediction of emission in Maharashtra State
The IROC of CO 2 emission for the states of Maharashtra can be represent by the differential equation, .................. (7) The graphical display of expression (7) is given by,

Madhya Pradesh
The model for the CO 2 emission of Madhya Pradesh can be represent as, A graphical display of the actual data and model data for Madhya Pradesh is given by Figure 5.
Observed data Estimated data  Table 5. Finally, the residual analysis is computed on the proposed model of CO 2 emission for the state of Madhya Pradesh. From the Table 6 it is clear that the residuals and standard error of residuals are too small compared to the data. It endorses that a good model is identified. The expression (8) can expressed visually as ,

CONCLUSION
Regarding the prediction, for Gujarat state IROC graph is strictly decreasing, whereas for the states of Maharashtra and Madhya Pradesh it is increasing up to the year of 1990 and then it gradually decreases. Increasing trend of IROC indicates that uncontrolled emission of CO 2 from different sources. Certainly, it indicates that in Gujarat, the uncontrolled emission still persists and is somewhat controlled as a whole in the states of Madhya Pradesh and Maharashtra. In this article, a third degree polynomial has been developed as a CO 2 emission model utilizing the observed data of the emission of CO 2 of about 21 years and using the method of least square that characterize well the emission pattern for the states of Gujarat, Maharashtra and Madhya Pradesh. Besides, computing total emission of the gas for the subjected states, some statistical procedure applied to clarify the goodness of fit of the proposed model namely, coefficient of determination R 2 , R 2 (Adjusted) and residual analysis. The numerical value of the statistical procedure reflects the fact that we have chosen a good model. The value of R 2 for the states Gujarat and Madhya Pradesh is 0.9845 and 0.9868 respectively where that is for Maharashtra is 0.9903. These numerical values indicate that proposed model is fitted fairly well. Future prediction of emission of CO 2 may be evaluated from our proposed model. This paper may be helpful for the future researchers for finding emission of CO 2 and other GHGs for the remaining states in India.