A Descriptive Time Series Analysis Applied to the Fit of Carbon-Dioxide (CO2)

The study examined the use of population spectrum in determining the nature (deterministic and stochastic) of trend and seasonal component of given time series. It also adopts the use of coefficient of variation approach in the choice of appropriate model in descriptive time series technique. Illustrations were carried out using average monthly atmospheric Carbon dioxide (C02) from 2000-2017 with 2018 used for forecast. Spectrum analysis showed that the descriptive technique of time series is more appropriate for analysis of the study data. The coefficient of variation revealed that the multiplicative model was appropriate for the CO2 data while the forecast and the actual values showed no significant mean difference at 5% level of significance.


Introduction
Atmospheric Carbon-dioxide (CO2)is one of the greenhouse gasesand it is known for its heat absorption and sunlight warming. In addition to this, CO2 in particular dissolves into the ocean thereby producing carbon acid and lowering the ocean's pHvalue [1]. It has been reported that higher concentration of CO2 increases the growth of cereal crops but reduces the nutritional value of key staple crops [2]. Its occurrence has been traced to some socio-economic characteristics, income, car ownership [2], poor rail accessibility and urban density [3], human activity and natural system [4]. The impact has been projected to be greater in the southern Asia and African region [1,5] and may likely result to inability to keep global average temperature increase below 2 o c [6]. With these in view, it suffices to consider the fundamental processes that are believed to generate the subject matter and then build a detailed model that can be used for control and also make predictions. Since the study data is a time series data, such model process will be a stochastic process indexed with time. To build such model, we adopt a time series technique. Commonly used time series techniques include; descriptive technique, probability models technique and spectral density technique [7,8]. Any appropriate technique to be adopted depends on the composition of the time series data. If a descriptive technique is to be applied, the time series is assumed to comprise of four components namely;trend, seasonal variation, cyclical variation and irregular component [9].Apart from comprising all or some of these components, the trend and seasonal component has to be deterministic in nature. When the nature of trend and seasonal component has been ascertained, then an appropriate model can be chosen among the competing model of descriptive time series.
Time Series decomposition involves the estimation, isolations and removal of various components of time series using descriptive techniques [10,11]. To choose among competing models for decomposition, a graphical approach [12] which involves the use of a time series plot has been recommended.If the seasonal variation stays roughly the same size regardless of the mean level, then additive model is more appropriate, but if it increases in size in direct proportion to the mean level, then the multiplicative model is considered more appropriate [12].Another graphical approach to the choice of model is the use of Buys-Ballot table [10]in which the relationship between the seasonal means and seasonal standard deviations provides guide to thedesired model.
All the above have some degree of subjectivity and thus may not be dependable for selection of appropriate model, hence an objective approach is required. One objective approach is to use both additive and multiplicative model and choose any that has minimum mean square of its residual [10]. However, this is time consuming and often,it has been observed that the mean square error and other accuracy measures may be misleading [13].Therefore, this study is aimed at recommending an objective approach to ascertain the nature of trend and seasonal component of a given time series as well as determining the appropriate choice among competing models in the application of descriptive technique of time series analysis.

Methodology
Descriptive techniques assume that a series is the combination of four major components which could be represented thus: where, t X is theoriginal value of the series at time t, t T is the Trend component, t S is the Seasonal variation, t C is the Cyclical component, and, t e the Irregular component.

Determination of the Nature of Trend and Seasonal Components of a Given Series
For a given series t X with its assumed various components, the trend component could be deterministicand stochastic. To ascertain the nature of trend and seasonal component, we will adopt spectral analysis approach. The method will be to evaluate the estimate of the population spectrum of the first differenced series at zero frequency.
Let T t X t ∈ , be an observed time series at time t, let is covariance stationary with constant mean and summable autocovariance given by k r , then the auto-covariance generating function is given by: where z denotes a complex scalar which could be represented as . Therefore, the population spectral is given by where ω is a real scalar.
Putting z =1 in equation (2), we have The population spectrum in equation (3) evaluated at z =1 and zero frequency becomes From equation (5), when a series is difference stationary, the population spectrum of the first difference at zero frequency is positive.
On the other hand, if the process is trend stationary,it has been shown that the auto-covariance generating function is given by

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Therefore, with z = 1 in equation (6) ( ) 0 Hence, the population spectrum of the first difference of a trend stationaryseries at zero frequency is zero.

Choice of Appropriate Model
The representation of the model in equation (1) depends on its various functional forms. They include: Additive Model - Mixed or Pseudo Additive Model - Mixed or Pseudo Additive model is mostly applied when the series comprises of small values. The two competing model are the additive and multiplicative model.
To choose the appropriate model for t X , we adopt a coefficient of variation approach [14] outlines below: (i) Calculate the seasonal difference of the series t X given by and its corresponding seasonal quotient using (iii) Apply the following decision rule: , Multiplicative model is appropriate.

Time Series Decomposition
In fitting the trend, the least squares method is considered appropriate, the seasonal component are isolated using ratio-to-trend method while the cyclical component, where present, are estimated using moving average approach.

Results and Discussion
To illustrate the above methodology, we use a monthly atmospheric CO2 data from 2000 to 2017 as our study data while 2018 will be use to examine the predictive performance.

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The series in figure 1 shows an evidence of upward trend with regular periodic movement at almost equal interval indicating presence of seasonal variation.

Determination of the Nature of the Trend and Seasonal Component
The application of spectrum analysis in the determination of the nature of trend and seasonal component The spectrum represented in Figure 2 is the periodogram of the first differenced series of the study data, the value of the spectrum at zero frequency is zero. This based on equations 3-7 indicates that the trend and seasonal component in the study data is deterministic. Hence, the descriptive technique of time series is more appropriate for analysis of the study data.
, it suffices that the multiplicative model is more appropriate.Hence, further analysis shall be based on the multiplicative model.

Estimation and Isolation of Trend Component
Using the least square method to fit the trend on the original series of C02, we have the result as presented inTable 2. The trend analysis shows common similarities on the accuracy measures of the likely competing trend lines (Linear, quadratic and exponential model). Hence, based on theory of parsimony the linear trend is chosen to fit the trend line. The fitted trend on the original series is displayed in figure 3 while the trend equation is shown in Table 1.

Seasonal Index Estimation and Isolation
The result of application ratio-to-trend on the de-trended series for estimation of seasonal index is as displayed in table 4.

Removal of Cyclical Variation and Diagnosis of Residual
The pattern of the detrended-deseasonalized series suggests that there is a presence of cyclical variation in the series.
To extract the cyclical component, the detrended-deseasonalized is fitted to a moving average and the result of the fit is as shown in Figure 4. Examining the residual for normality and serial correlation, the result is as displayed in figure5 and 6. The result on figure 5 is on normality test, which show Anderson-Darling (AD) of 0.48 with p-value greater than the level of significance. This suggests that the residual is normally distributed. This is also evident with the histogram having almost a bell shape of the normality curve. Figure 6 is the autocorrelation function of the residual. The result showed that none of the residual is significantly different from zero, indicating that the residual is a white noise. This implies that the model was appropriately fitted.

Conclusion
In this study, the average monthly atmospheric Carbon-dioxide CO2 from 2000 to 2018 was analyzed using descriptive time series technique. The population spectrum of the first difference of the series showed that the nature of the trend and seasonal component inherent in the CO2 data is deterministic which prompted the use of descriptive technique. Also, the coefficient of variation approach applied for choice of model revealed that the multiplicative model was appropriate for the CO2 data. Finally, the forecasted values and the actual values showed no significant mean difference at 5% level of significance.