Seasonal Autoregressive Integrated Moving Average Modelling and Forecasting of Monthly Rainfall in Selected African Stations

The global climate condition in recent years has had a significant impact on rainfall trends. Climate change is known to be driven by natural resource management, which aids in the forecasting and prediction of rainfall patterns [1]. According to the report, climate change has a significant ecological impact on hydrology, natural resource degradation, and human development in the environment [1]. As a result, it is critical to understand how rainfall patterns vary around the world. This is due to the global rainfall pattern, which has a significant impact on climate change [1]. Rainfall plays an important role in the development of any society, providing important information to agriculturalists in planning farming activities [2]. Rainfall reduction has been reported to affect agricultural product decrease as a result of global climate change [3], which includes African countries. According to study [1], information on the pattern of rainfall may have future consequences on the possibilities of prediction and forecasting, which have great implications on the agricultural product globally. According to research, the duration and frequency of rainfall intensity, that is, the amount of water that falls in any country are affected by the period and change in the weather of the environment [1]. The recurrence of rainfall flow volume shall affect the future level of water flow in any country, as a result of an increase in the volume of water downpours as a result of temperature changes [4]. Hyndman and Athanasopoulos [5] and Aweda et al. [6] have emphasized the need for government to put up actionable policies targeted towards mitigating the effects of rainfall on the environment. Researchers such as Hyndman and Athanasopoulos [5] and Aweda et al. [6] have reported that the main reason for forecasting is to be able to predict the future occurrence of any environment so that policymakers can make accurate decisions on rainfall and other climatic occurrences. The rainfall trend can be forecasted using the time series analysis method. As a result, for this study, rainfall prediction and forecasting will be based on the Seasonal Autoregressive Moving Average (SARIMA) as reported by Ramli et al. [7]. According to the report, there are various variations around an average length value, which is a modelling and prediction tool for the use of atmospheric data forecasting tools [8-10]. According to the study made by Heizer et al. [11], time series are various variable values observed at a specific time interval. According to the study made by Suhartono [12], future prediction is based on the forecasting of some parameters, which involved the use of some mathematical tools on historical data. However, using time series analysis [1], it is possible to predict the future trend of data for past events. Autoregressive Integrated Moving Average (ARIMA) model has been reported by Suhartono et al. [12-17] to solve various data issues. It is based on univariate stochastic prediction models [13]. It has been demonstrated that ARIMA time series have an accurate predictive method in comparison to other models [1]. As demonstrated in the forecasting of the periodic intensity of stream flow of the SERIMA model, a series of static analyses were created [14, 15]. Similarly Aweda et al. [6] and Ramli et al. [7] adopted a non-seasonal ARIMA model, ARIMA (p,d,q) while Subbaiah Naidu [16] and Chen et al. [17] employed Seasonal ARIMA (SARIMA) model. However, the SARIMA rainfall prediction model is known to be based on time series analysis which can also be used for the development of various data to follow the seasonal pattern mentioned by Liu et al. [18]. Thereby it has advantages over other models due to its ability to sense and house different datasets [1]. Various researchers such as Ramli et al. [1], Aweda et al. [2], Dindarloo et al. [19], El-Mallah and Elsharkawy [20] applied SARIMA model in their research due to its accuracy and predictive ability.

Short-term forecasting with the ARIMA model is good, but long-term forecasting of rainfall is better [21]. According to research, the SARIMA model and ANN can be used to aid education prediction in the study of rainfall and other parameters [1, 22, 23]. However, for this study, the author employs the ARIMA model to forecast rainfall across selected African stations. According to Afrifa-Yamoah et al. [23] and Sampson et al. [24], the SARIMA model forecasts monthly rainfall for the future and is used to predict what may occur in the future. According to Papalaskaris et al. [25] and Mohamed and Ibrahim [26], time series modelling using the ARIMA model on some meteorological data, hydrological data, and other data shows that some cities are prone to flooding due to the impact of rainfall around the world. As a result, the author intends to use the SARIMA model to forecast rainfall parameters in some selected African stations for this study. These stations were selected based on their climatic region and the activities that are likely to occur in the sub-region. However, the effect of climate change on rainfall has made a significant contribution to the development of Africa Station. As a result, there is a need to study and report on the future occurrence of rainfall in Africa using SARIMA models.

This study is very significant due to the fact that predicting the rainfall can help in decision making regarding agricultural production, weather planning as well as mitigating the risk that could result in excessive or inadequate rainfall. Proper understanding of the future rainfall pattern can inform agricultural decision making regarding what should be planted, when it should be planted and the possibility of having an alternative source of rainfall in order to guarantee food sufficiency. The issue of food insufficiency is a global problem, and Africa is blessed with arable lands suitable for agricultural production. It is believed that the result of this study will provide adequate information that will assist in making best use of rainfall while also providing possible mitigating measures to forestall any negative effect of rainfall on livelihood.

Table 2 presents the mean and coefficient of variation for rainfall in some of the cities in Africa and the results show that the peak of rainfall in Abuja is in August (536.11), in Cairo, it is in March (7.95) while in Nairobi (168.04), Pretoria (107.65) and Yaoundé (493.39) the peak of the rainfall is in April, January and October respectively. The dispersion around the mean was highest in December in Abuja (224.56%), March in Cairo (204.66%), February in Nairobi (114.54%), July in Pretoria (177.76%) and in January in Yaoundé (97.29%). The result also shows that Abuja which is the capital of Nigeria recorded the highest dispersion in rainfall compared with other major cities in Africa. This could be because Nigeria, in the west African region, is closer to the ocean than any other African country except South Africa. Furthermore, research has shown that Abuja has a high temperature, which could cause evaporation, condensation, and transpiration, followed by rainfall. However, Abuja is Nigeria's transition center between ocean breeze and Sahara breeze, giving it an advantage in terms of high rainfall. Furthermore, as reported in study [27], the higher the temperature, the more rainfall. The result also established that there was a higher rainfall in Yaoundé between March and June compared to other major cities in Africa while Abuja reported the highest rainfall in July and August than other African cities (Figure 2). In both Yaoundé and Abuja, a very significant downward trend in rainfall was observed towards the end of the year (September to December) as shown in Figure 2.

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Figure 2. Mean monthly rainfall in the selected countries in Africa

Table 2. Descriptive statistics for the monthly rainfall in some selected countries in Africa

Note: SD- standard deviation, COV- coefficient of variation

The result of the test of stationarity of the series was carried out using the Augmented Dickey-Fuller (ADF) test and the result obtained is presented in Table 3. The result reveals that rainfall in all selected African cities (p<0.05) was stationary at levels except for Yaoundé where stationarity was achieved after the first differencing (test statistic=-24.77805, p-value=0.0000, p<0.05). This implies that the order of differencing for Abuja, Cairo, Nairobi and Pretoria will be 0 while that of Yaoundé will be 1.

Different tentative models were identified based on the plots of the autocorrelation function (ACF) and partial autocorrelation (PACF) and the performances of these tentative models are presented in Table 3. The goodness of fits of each of these tentative models was examined using the Ljung box statistic and the result as presented in Table 4 reveals p-values greater than 0.05 (p>0.05) for each of the tentative models indicating that the error follows a white noise, implying that these models are all of the good fit (p>0.05).

Table 3. Augmented Dickey-Fuller (ADF) test for the rainfall series in the selected countries in Africa

**Significant at 1% (p<0.01)

Table 4. Performance evaluation for tentative models for forecasting rainfall in selected cities in Africa

Note: Bolded values are the least normalized Bayesian Information Criteria and highest adjusted R². NBIC- Normalized Bayesian Information Criteria.

Table 5. A summary estimate of the best model for each of the selected cities in Africa

*Significant at 5% (p<.05), **significant at 1% (p<0.01).

The performances of these models were examined using the adjusted R², normalized BIC and Root Mean Square Error (RMSE). In terms of fitness performance, ARIMA (1,0,0) (1,0,1)₁₂ (R²_adj.=0.769, NBIC=9.201) outperformed other models for Abuja, in Cairo, it was ARIMA (0,0,0) (1,0,1)₁₂ (R²_adj.=0.146, NBIC=3.453), in Nairobi, it was ARIMA (1,0,0) (1,0,1)₁₂ (R²_adj.=0.189, NBIC=8.501), in Pretoria, it was ARIMA (1,0,0) (1,0,1)₁₂ (R²_adj.=0.367, NBIC=7.886) while in Yaoundé, ARIMA (1,1,1) (1,0,1)₁₂ (R²_adj.=0.641, NBIC=9.349) outperformed other seasonal ARIMA models (Table 4). In terms of out-of-sample forecasting performance, ARIMA (1,0,0) (1,0,1)₁₂ (RMSE=70.7044), ARIMA (1,0,1) (1,0,1)₁₂ (RMSE=9.9279), ARIMA (0,0,1) (1,0,1)₁₂ (RMSE=92.8925), ARIMA (2,0,0) (1,0,1)₁₂ (RMSE=42.05462) and ARIMA (1,1,1) (1,0,1)₁₂ (RMSE=79.42084) were the best forecasting seasonal model for rainfall in Abuja, Cairo, Nairobi, Pretoria and Yaoundé respectively. All the fitted SARIMA models were found to be of good fit (Ljung box p-value >.05) and this is also substantiated by the ACF and PACF plots of the residuals as shown in Figures 3-7. The plots of the ACF and PACF of the residuals as presented in Figures 3-7 indicate that the models were of good fit as they were no significant spikes capable of undermining the goodness of fit of these models.

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Figure 3. Plot of the ACF and PACF of the residuals for ARIMA(0,0,1)(1,0,1)₁₂- Abuja

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Figure 4. Plot of the ACF and PACF of the residuals for ARIMA(1,0,1)(1,0,1)₁₂- Cairo

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Figure 5. Plot of ACF and PACF of the residuals for ARIMA(0,0,1)(1,0,1)₁₂- Nairobi

The parameter estimates of the best forecasting model are as presented in Table 5. The result shows that the seasonal terms were statistically significant in all models which justified the use of seasonal ARIMA models in modelling rainfall in these selected cities in Africa. This also underscored the significant role of the season in the rainfall pattern in the selected African cities. In Cairo, Pretoria and Yaoundé, the AR (1) terms were significant but was more positively significant in Cairo than other locations. This implies that the previous month’s rainfall has a positive influence on the present month’s rainfall in these locations. For Abuja and Nairobi, there were no AR (1) terms in the optimal model (Table 5).

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Figure 6. Plot of ACF and PACF of the residuals for ARIMA(2,0,0)(1,0,1)₁₂- Pretoria

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Figure 7. Plot of ACF and PACF of the residuals for ARIMA(2,0,0)(1,0,1)₁₂- Yaounde

The SARIMA models for forecasting rainfall in each of the selected African stations are presented below:

For Abuja:

$\begin{aligned} & \left(1-0.999 B^{12}\right) R_t=161.029 +\left(1+0.905 B^{12}\right)(1-0.020 B) \varepsilon_t\end{aligned}$     (11)

For Cairo:

$\begin{aligned} & \left(1-B^{12}\right)(1-0.534 B) R_t^* =0.5090+\left(1+0.975 B^{12}\right)(1+0.435 B) \varepsilon_t\end{aligned}$     (12)

where, $R_t^*=\ln \left(R_t\right)$.

For Nairobi:

$\begin{aligned} & \left(1-0.995 B^{12}\right) R_t=88.140 +\left(1+0.952 B^{12}\right)(1-0.161 B) \varepsilon_t\end{aligned}$  (13)

For Pretoria:

$\begin{aligned} & \left(1-0.999 B^{12}\right)\left(1-0.160 B-0.071 B^2\right) R_t =52.279+\left(1+0.976 B^{12}\right) \varepsilon_t\end{aligned}$    (14)

For Yaounde:

$\begin{aligned} & \left(1-0.995 B^{12}\right)\left(1-B^{12}\right)(1-0.235 B) R_t= -0.355+\left(1+0.882 B^{12}\right)(1+B) \varepsilon_t\end{aligned}$     (15)

The forecast of the rainfall in these selected cities is presented in Table 6 and Figure 8. Based on the one-year prediction of rainfall for 2023, the peak of the rainfall is predicted to happen in July in Abuja (531.54), January in Cairo (6.74), April in Nairobi (149.85), January in Pretoria (106.53) and October in Yaoundé (377.66). A downward trend in rainfall was also predicted towards the end of the year in both Abuja and Yaoundé (Figure 8).

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Figure 8. Forecast of the rainfall in some selected major cities in Africa (January 2023- December 2023)

Table 6. Summary of the forecast of rainfall from January 2023 to December 2023 in the selected cities in Africa

Figure 9 depicts the seasonal variation of rainfall across the African stations studied. It was discovered that all of the stations follow a similar pattern. The results show that Abuja (Figure 9(a)) has deep and evenly distributed rainfall, whereas Cairo (Figure 9(b)) has less rainfall. It was discovered that Abuja (Figure 9(a)) had the highest rainfall in 1980 with 950 mm, and the lowest rainfall was recorded in 2004 with 250 mm. The results also revealed that Cairo (Figure 9(b)) is in Northern Africa with less rainfall than Abuja (Figure 9(a)), with the highest rainfall in 2021 being 75 mm, which is far less than what was experienced in Abuja (Figure 9(a)) compared to the minimum value observed. However, Pretoria (Figure 9(d)) in South Africa receives more rainfall due to its proximity to the ocean, which causes stations in the south to receive more rainfall than stations in the north. However, when compared to Abuja (Figure 9(a)), Pretoria (Figure 9(d)) has less rainfall, with the highest recorded in 2009 at 340 mm and the lowest recorded in 2004 at 45 mm. This refutes the notion that the station in the south should receive more rainfall than the station in the north. The results revealed that Nairobi (Figure 9(c)) has a year-round rainfall pattern, indicating that the rainfall in the station is beneficial to the farmer's planting season. The highest value of rainfall observed in the station is 550 mm in 2020, and the lowest value observed in the station is 100 mm in 1980. Rainfall in Yaoundé (Figure 9(e)) decreases in a sinusoidal pattern throughout the study year. The highest amount of rainfall recorded in Yaoundé (Figure 9(e)) was 870 mm in 1985. However, the seasonal variation of rainfall results shows that Abuja (Figure 9(a)) has the highest rainfall of any African station. As a result of the high rainfall, Abuja (Figure 9(a)) will most likely experience flooding in the coming years.

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Figure 9. Seasonal variation of rainfall across the selected African station

This finding has shown that rainfall in most of the African stations considered were stationary at the level which is consistent with that of the study [2] on modelling and forecasting of selected meteorological parameters for the environmental awareness in Sub- Sahel Africa of which rainfall is one of the meteorological parameters considered. The finding also revealed that in most of the fitted models, the previous month’s rainfall was found to have a significant impact on the present month’s rainfall which is also corroborated [2]. Furthermore, the finding showed a significant effect of the seasonal terms showing evidence of seasonal factors in rainfall in the studied station as reported in other studies on rainfall forecasting [31, 32].

This study based on the forecast rainfall in the different African cities indicate show uniqueness in the peak of rainfall in the different cities (Figure 8). In Cairo, the peak of the rainfall was predicted to be in January and December while in Nairobi, it was predicted to be in April. For Pretoria, Yaoundé and Abuja, the peak of rainfall was predicted to occur in January, October and July respectively. This work contributes to existing literature by providing insights on the trend and future values of rainfall in five major cities that cut across the different regions of Africa. This is very different from other studies as it provides a more informative nature of rainfall that take care of the different regions of Africa: West Africa, North Africa, East Africa, South Africa and Central Africa. This work has provided an all-inclusive modelling of rainfall that reflect the peculiarities of the different sub-regions in Africa. This study has also provided insights on when rainfall is expected to be at the peak in each of these cities. This information, if properly harnessed, could help stimulate better agricultural production in Africa as agriculture is the backbone of Africa economy. Also, these predictions could help address some of the challenges associated with excessive rainfall such as flooding which is also one of the problems in Africa. This could help in formulating policies that would help avert the negative consequences of flooding, thereby reinforce the need for a more proactive measures that can help curb flooding in Africa.

The authors wish to recommend to the water management cooperation under the leadership of the federal republic of Nigeria as well as other stations considered to make available water storage tank so that agriculturists could take the period of low rainfall in their country for their plantation through the medium of irrigation so that there will be all round agricultural products in each of the countries. This is also applicable to other African countries, who should take note and do what is necessary for their countries. It is therefore recommended that more research centers should be built for each country in Africa in order to reduce the continent's food shortage.