Analysis of rainfall data of some West African countries using wavelet transform and nonlinear time series techniques

ABSTRACT The chaotic behaviour of monthly rainfall data of Benin, Cote d’Ivoire, Cameroon, Ghana, Niger, Nigeria, Senegal and Togo between January 1901 and December 2015 were investigated using wavelet transformation analysis and time series techniques. Wavelet power spectrum was used to split the time series into different scales. Power concentrations between 8 and 16 months were observed for the selected locations. The embedding dimension, delay and largest Lyapunov exponent (LE) were calculated. We observed positive LE ranging from 0.13 to 0.36, indicating the rainfall was chaotic. Ghana had the highest values of LE, while the lowest LE was observed at Niger..


Introduction
Rainfall analysis has gained a better understanding in recent times. It is an essential factor in considering the measure of water obtainable to meet the different demands of agriculture, industry and other human activities. Analysis of rainfall distribution over time and space is essential for the benefit of a countrywide economy. Climatic occurrences and relations between the atmosphere, ocean and ground surface over different time scales have influence on rainfall (Wallace and Hobbs 2006). Various applications of rainfall data are improved by comprehension of the real distribution of rainfall over a particular region, rather than relying on simple statistics.
Several reports on rainfall have engaged different methods, such as temporal analysis of rainfall (Turkes 1996, Serrano et al. 1999, De Luis et al. 2000, Estrela et al. 2000, Agnese et al. 2002 and analysis of precipitation series (Brunetti et al. 2002(Brunetti et al. , 2004. Matteo et al. (2019) analysed the rainfall trends and extreme precipitation in the middle Adriatic side of the Marche Region with 128 rain gauges from 1921 to 2017. It was observed that the growth of extreme precipitation events is significant in the southern part of the Marche Region, categorized mostly by a Mediterranean-type climate which helps extreme events as related to the central northern part. Sekela and Manfred (2019) investigated the trends in annual and seasonal rainfall time series in the Wami River Basin during the period 1983-2017 and its influence on the water supply services in rural areas, using simple regression, the Mann-Kendal Test and Sen's Slope Estimator. The water points were found to be significantly affected by seasonal changes, in terms of both availability and quality of water between rural water services and seasons. Zeineddine and Ovidiu (2020) examined 27 Sahelian climatic stations in three countries: Burkina Faso, Mauritania and Senegal. The chronological graphic method of information processing (MGCTI) of the Bertin Matrix and continuous wavelet transform (CWT) were used to determine the modes of this variability and patterns of rainfall. The results showed rain resumption perceived in the latest years over the Sahelian region and a convincing connection with the surface temperature of the Atlantic Ocean. Katzenberger et al. (2020) used 32 CMIP-6 models to investigate the Indian summer monsoon's reactions to climate change. They were compared with the models' simulations in the past to 285 WFDE5 reanalysis data. It was noted that 16 out of 32 models are able to capture the monsoon rainfall twice the standard deviation in the period 1985-2015. They established that all models showed intensification in mean summer monsoon rainfall. Recent studies using global coupled models established that the Indian monsoon rainfall intensified due to climate change in the 21st century (Varghese et al. 2020).
In recent years, wavelet transform has become an important mathematical device that offers a time to frequency demonstration of an analysed signal in the time series area (Percival and Walden 2000). Lafreniere and Sharp (2003) established that wavelet analysis is an efficient way to describe runoff, temperature, precipitation, their interrelationships and inter-annual variability. The wavelet power spectrum was used to examine the rainfall and runoff stations in the Piranhas-Açu River basin, located in the semiarid north-eastern part of Brazil. A total of 12 rainfall and runoff time series were evaluated and their respective global wavelet spectra, alongside with the band scale-average time series, were considered suitable for determining the hydrological zones within a region (Celtos and Sandra 2006). A wavelet power spectrum approach was used to investigate the wind speed data in the framework to examine the characteristics of the wind potential and the choice of suitable locations that might be the subject of a wind farm construction (Avdakovic et al. 2011). Falayi et al. (2017 investigated the impact of temperature and wind speed over Covenant University, Nigeria, using a wavelet spectrum approach to analyse the temperature and wind speed monthly series in a sequence of monthly scales from January to December 2013. Adepitan and Falayi (2020) applied wavelet transformation to investigate the variation of temperature and rainfall time series for a period of 115 years in Nigeria. Power concentration between 8 and 16-month bands was noticed for both temperature and rainfall. It was suggested that the climate of Nigeria changed with variability of the climatic parameter.
The chaos concept is widely used in numerous applied fields such as space weather, solar physics, meteorology, hydrology, psychology, exchange rates, economics and traffic flow (Rodriguez-Iturbe et al. 1989, Lorenz 1993, Ayers 1997, Chen 1998, Sivakumar 2000, Nair et al. 2001, Elshorbagy et al. 2002, Letellier et al. 2005, Das and Das 2007, Unnikrishnan 2008, Rabiu et al. 2015, Suresh and Selvaraj 2017. This theory is used to recognize deterministic nonlinear mechanisms (Zhongda 2020a). Chaotic systems can be used to recognize deterministic components which are merged with other stochastic components in the data , Ogunjo et al. 2021. Adewole et al. (2020) investigated the chaotic time series analysis of meteorological parameters in some selected stations in Nigeria. It was concluded that the meteorological parameters used were good tools for modelling weather-predicting systems with different dynamic variables for the selected locations. Zhongda (2021Zhongda ( , 2020b investigated the chaotic dynamic behavior of wind power time series at different time scales. The results of the study have a certain theoretical value and practical importance for grasping the oscillation law of wind power and improving the estimation accuracy of wind power.
This study evaluates the variation of rainfall over Niger, Nigeria, Senegal, Cameroon, Togo, Benin, Ghana and Cote d'Ivoire from 1901 to 2015 using the wavelet power spectrum and nonlinear time series. Rainfall is a vital meteorological input for modelling of agricultural systems and water resources planning. The analysis of rainfall variability in West Africa is useful in providing additional theoretical studies for a weather-predicting system that could aid productivity in a region heavily dependent on agriculture. The paper is arranged as follows. Section 2 gives details on the datasets and methodology for the rainfall distributions. Section 3 presents wavelet transforms to investigate the rainfall difference over selected stations in West Africa. In Section 4, we apply chaotic time series to examine the rainfall variation over selected stations. Discussion of results is provided in Section 5 and conclusions are presented in Section 6.

Methodology
We considered monthly mean rainfall data for 115 years  of some stations in West African countries. The data are collected from the World Bank Data Group (http:// sdwebx.worldbank.org/climateportal/index.cmf?page=downloadscaled_data_down load&menu=historical). The dataset was designed to contain monthly variations for each country. The time series of rainfall were acquired from observations. Both wavelet analysis and the chaos concept were used to examine the monthly rainfall variation. For nonstationary signals, wavelet analysis measures in both the time and frequency domains. Wavelet analyses decompose one-dimensional time series into two-dimensional timefrequency representations at the same time. This depicts the amplitude of phase signals within time series and the variation of amplitude with time. The non-linear system framework can be described by chaotic theory. Chaotic theory can make known whether an inconsistent time series is truly deterministic. It also points out if long-term prediction for the system is realistic. Table 1 displays the geographic latitudes and longitudes.

Monthly rainfall variations
Rainfall is an atmospheric phenomenon restraining solar radiation at the ground surface. Figure 1 shows contour plots of monthly spatial distributions of rainfall for eight stations in West Africa (Benin, Cote d'Ivoire, Cameroon, Ghana, Niger, Nigeria, Senegal and Togo) between 1901 and 2015. Figure 1 exhibits the strength of rainfall recorded between the months of June and September (wet season). Low values of rainfall are noticed during the dry season (January, February, March, April, October, November and December). During the wet period relative humidity is more enhanced compared to the dry period. In Figure 1, less monthly rainfall was recorded in Niger, which is attributable to a strong influence of latitude.

Wavelet power spectrum (WPS)
The Morlet wavelet transformation is regarded as a more suitable mathematical technique, introduced in 1984, to break down time series into a time-frequency space and to determine the prevailing modes of variability (Roddam andBhattacharyya 2010, Falayi et al. 2018). The Morlet function is expressed in Equation 1 as: where t indicates time. The WPS transformation is given in Equation 2 as  where the dilation factor is denoted as a, b is a symbol of the location factor, the conjugate of the wavelet function is ψ � , and the time series is represented as f ðtÞ. The wavelet coefficient of the time index n and scale a is defined in Equation 3 where the data length of the time series corresponds to N, while the time interval is dt.
The GWS is expressed in Equation 4 Figure S2

Nonlinear dynamical techniques to examine chaotic theory of rainfall
Nonlinear low-dimensional chaotic behaviour can be investigated using analysis with non-linear time series techniques. Dynamical attributes that link existing and prospective states of the system are the Lyapunov exponent (LE), false nearest neighbour (FNN) and average mutual information (AMI). The quantifier that identifies the nonlinear chaotic behaviour in a dynamic system is the Lyapunov exponent. The evaluation of this quantity from the time series data of a dynamical method engages phase space reconstruction. For time series data, phase space reconstruction with embedding dimension (m) and time delay (τ) is necessary to acquire the dynamic attributes of a structure and is expressed in Equation 5: where Y n represent vectors in phase space. The AMI is a techniques aimed at deciding an appropriate embedding parameter, the delay for nonlinear systems analysis (Fraser and Swinney 1986). The phase space construction is derived from the first minimum of AMI (Shaw 1981), and the first minimum of the AMI is picked as the delay time. The advantage of AMI over autocorrelation is that it accounts for the nonlinear structure in the fundamental dynamics and quantifying the dependence measures between two states (Oludehinwa et al. 2018). Equation 6 is used to calculate the AMI: where j is an integer number and symbols P q and P r represent the probabilities that the variables assume a value in the qth and rth bins, respectively. The p qr(τ) is the mutual probability that x n is in bin q and x( n+ τ) is in bin r. Figure 2 illustrates the AMI with time delay. The AMI depicts a noticeable minimum; this is considered to be the best possible way to acquire the time delay.
The FNN technique can be used to establish the accurate minimal embedding dimension to describe nonlinear time series (Kennel et al. 1992). The fraction of false neighbours is generated from FNN which has a good connection with the distances among samples reconstructed in embedding dimensional spaces. When the fraction is minimal for an embedding dimension, the reconstruction in embedding dimensions is enhanced. The FNN technique supports the geometric concept of phase space reconstruction using time series. Figure 3 depicts decreases in false nearest neighbours while the embedding dimensions are enhanced. The FNN declines significantly after the initial values of m. For this reason the lowest value of m equivalent to the lowest number of FNN can be considered as the primary information for the embedding dimension.

Lyapunov exponent (LE)
The Lyapunov exponent is a vital quantifier to describe the chaotic behaviour of nonlinear time series. The Lyapunov exponent is the exponential rate of growth among the neighbouring trajectories, a vital quantifier of chaotic dynamics. The divergence of trajectory or increase in a dimension shows a positive LE, which signifies evidence of chaos, while the convergence of trajectory or reduction in the dynamic system corresponds to a negative LE. The Lyapunov exponents for the selected stations were calculated between 1901 and 2015 to examine the annual trend of rainfall variation using Equation 7 (Wolf et al. 1985).  The annual variation of the Lyapunov exponent of the rainfall values between 1901 and 2015 is presented in Figure 4. It was noted that the annual values of the Lyapunov exponent for rainfall show latitudinal dependence.

Discussion of the results
The resulte of the inter-annual rainfall investigation for 115 years (Figure 1) depict the monthly average of rainfall in Benin, Cote d'Ivoire, Cameroon, Nigeria, Niger, Ghana, Senegal and Togo stations. The largest amount of rainfall observed at these stations occurs during the wet season, excluding November, December, January, February, March and April. The maximum monthly average rainfall recorded in Benin, Cote d'Ivoire, Cameroon, Ghana, Nigeria and Togo stations ranges between 250 mm and 200 mm during the wet season, while Niger and Senegal received low rainfall in the period of June to August. A significant decrease in rainfall variability was noticed from   Odekunle (2006). The Atlantic Ocean and moist air masses of the westerly influx moisture might have an effect on rainfall during the dry and wet seasons. In the intertropical conversion zone (ITCZ) stations, rainfall is higher during the wet season in association with variations in solar radiation and the equivalent convergence pattern. Figure S2(a-d) depicts the rainfall variation for Niger, Nigeria, Togo and Ghana between 1901 and 2015, while Figure S3(a-d), Figure S4(a-d) and Figure S5(a-d) display contour plots of the wavelet power spectrum, the monthly global wavelet power spectrum and monthly scale-average time series between 1901 and 2015 respectively. We used a straightforward mathematical technique on the wavelet power spectrum (WPS) to investigate the rainfall variation in these West African regions between 1901 and 2015. The relative power at a certain scale and a certain time is generated from WPS analysis. Figure S3(a-d) shows the power (absolute value squared) of the WPS for the monthly rainfalls presented in Figure S2(a-d). The Morlet wavelet was employed due to the precise frequency information when related to other mother wavelets. Figure S3(ab) displays the definite fluctuation of the specific wavelet and the concentration of power can simply be recognized in both frequency and time domain during the entire time series . Power concentrations between 8 and 16 months were noticed in Figure S3(a-d) across the selected West African stations. The WPS colour bar varies from blue, which denotes a low power coefficient, to the red, which indicates a high power coefficient. Considerable regions are denoted by red, orange and yellow, which demonstrate the mathematical intensity of the variables ( Figure S3(a-d)). When the power concentration is high it indicates wet years and the power diminishes considerably in dry years. The global wavelet spectrum (GWS) is used to investigate the prevailing phase of the signal of the rainfall data for different stations between 1901 and 2015 ( Figure S4(a-d)). The patterns of the time series variation are established by an integration of power across the time spectrum, denoted by the dashed lines ( Figure S4(a-d). The GWS offers an impartial and reliable evaluation of the factual power spectrum of the time series. Figure S4(a-d) exhibits one significant peak with confidence level of 95% denoted as dotted lines. The 5% significance level is the same as the 95% confidence interval; it denotes series of confidence in the statistical value of a parameter of a given population deduced from the test of a sample against a certain background level: 5% of the WPS would be above this level (Torrence andCompo 1998, Santos andMorais 2013). Figure S5(a-d) shows the scaled average power wavelet, which is used to study the modulation of a time series and frequency within the same time series. The scale-average wavelet power for rainfall at the Niger, Nigeria, Togo and Ghana stations shows high and low variance periods ( Figure S5(a-d)).
We employed three techniques to assess the dimensional time series using the calculation of the AMI function, FNN and LE as illustrated in Figures 2 and 3 and Table 2 respectively. The initial minimums of both AMI and FNN functions are representative of both the delay and the embedding dimension (Figures 2 and 3) Lali 1996, Marwan et al. 2006). Figure 2 gives an idea about the mutual information estimation of the rainfall time series. Figure Table 2. We noticed positive Lyapunov exponents for rainfall time series in Benin, Cote d'Ivoire, Cameroon, Ghana, Nigeria, Niger, Senegal and Togo, which were 0.22, 0.25, 0.22, 0.36, 0.22, 0.13, 0.17 and 0.26 respectively. This is a strong indicator of chaotic behaviour in the parameters. We observed that the highest value of Lyapunov exponents is in Ghana, while Niger and Senegal exhibit low values of the Lyapunov exponent ( Figure 4). Variations in the Lyapunov exponent show a strong impact of the latitude, meaning that rainfall reduces with high latitude at which the solar radiation is highest. This variation in the Lyapunov exponent might be due to the ITCZ location, which may be persistent in the northern locations due to the enhanced heating from the larger land mass in the northern part of the region as a result of sensitivity to solar heating. The ITCZ trails the movement of the sun, which results in seasonal distribution of rainfall during the wet and dry season in the tropical regions. The time series analysis of rainfall data in the southern part of Nigeria displays chaotic dynamics due to El Nino/Southern Oscillation, the Intertropical Discontinuity and the coastally induced rainfall (Fuwape et al. 2017). Rainfall reduces slowly as it shifts northwardly due to the increase in distance of cold winds from the Atlantic Ocean, which might have an effect on the Lyapunov exponent values. The nonlinear dynamics revealed that ITCZ affects the stability and instability conditions of rainfall due to change in the degrees of freedom generated through an external stochastic driver and may alter the dynamics of the system. Finally, we perceived that the nonlinear time series analysis unveiled through the nonlinear dynamics technique confirms that monitoring the impact of climate change on the rainfall through chaos theory can assist remarkably in identifying, forecasting and estimating the state of the climate change.

Conclusion
The study examined the variability of rainfall time series between 1901 and 2015 in Benin, Cote d'Ivoire, Cameroon, Ghana, Nigeria, Niger, Senegal and Togo using both wavelet spectrum analysis and nonlinear dynamic time series techniques. The WPS demonstrated an enhanced concentration of power in the 8-16 months band for all the stations examined. Wavelet transforms exposed significant characteristics of the rainfall time series. It is interesting to note that the occurrences of strong oscillations in rainfall at the West Africa stations are in the same phase. Different methods such as AMI, FNN and LE were used to examine chaotic behaviours of rainfall time series in Benin, Cote d'Ivoire, Cameroon, Ghana, Nigeria, Niger, Senegal and Togo. The results depicted that chaotic characteristics are clearly present in the rainfall data with positive Lyapunov exponents for all the selected stations. The prevalent Lyapunov exponent notifies us of the highest time span of a precise prediction. The Lyapunov exponent variation demonstrated a strong effect of latitude on rainfall distribution. This signifies that rainfall diminishes with high latitude while the solar radiation is at a peak. The ITCZ and intertropical discontinuity (ITD) might have an influence on chaotic dynamics noticed in the rainfall time series data of selected locations in West Africa. These results can further be worked upon to model climate-change-based research for predictability of future behaviour of rainfall in the selected region.