Consider the series of rain { x t , t ∈ T } ⊂ ℝ n , T ⊂ ℤ as a realization of the possible set value of the monthly rainfall height { X t , t ∈ T } , here taken over 732 months. The SARIMA (Sesonal Autoregressive Integrated Moving Average) model take into account the effect of seasonality and non-stationarity.

ϕ ( B ) Φ ( B S ) Δ d Δ S D X t = θ ( B ) Θ ( B ) ε t (1)

B , Δ are the “backward shift” and “Differences” operator such as B X t = X t − 1 , Δ X t = X t − X t − 1 , Δ S = 1 − B S , ( ε t ) t ∈ ℤ is a low white noise B B f ( 0, σ 2 ) corresponding to unpredictable phenomena causing a disturbance of the series and including an independent residual component ( η t ) t ∈ ℤ [12] . ϕ ( B ) , Φ ( B ) , θ ( B ) et Θ ( B ) being polynomials defining the lags order p of the autoregressive part, P with seasonality, q the moving average part and Q with seasonality. The exponents d, D are the orders of seasonal differentiation and S, the period (or seasonality) of the process. Any time series data can be decomposed into trend, saisonic and residual components.

X t = C T + S t + ε t (2)

A process is said to be stationary in the strict sense when the law of joint probability is invariant for any translation of time [3] . Several tests make it possible to detect it,

· Augmented Dickey-Fuller (ADF) test

{ H 0 : ϕ = 1 ⇔ unityrootexistence : theseriesisnotstationary   ( d ≥ 1 ) H 1 : ϕ ≠ 1 ⇔ noexistenceofrootsunity   ( ϕ < 1 )

· Phillips-Perron (pp)/Kwiostowski & al. (KPSS) test

{ H 0 : d = 1 ( theseriesisnotstationary ) H 1 : d = 0 ( theseriesisstationary )

A non-stationary ( X t ) t ∈ ℤ process follows a model SARIMA(p,d,q)(P,D,Q)[S] orders p , d , q , P , D , Q , S ∈ ℕ , if the process ( Δ d Δ S D X t ) t ∈ ℤ is SARMA(p,q)(P,Q)[S] and becomes a stationary process.

The partial autocorrelation function η ( . ) takes into account the strength of the bond between X t et X t + h by removing links induced by intermediate values X t − ( h − 1 ) , ⋯ , X t − 1 . We note by X t * = ∑ k = 1 h − 1     α k X t − k :

η ( h ) = C o r r ( X t , X t − h | X t − 1 , ⋯ , X t − h + 1 ) = C o r r ( X t − X t * , X t − h − X t − h * ) (3)

The ordinary least squares estimator of σ for K = p + q + P + Q constraints:

σ ^ m c o 2 = 1 n − K S ( β ^ ) (4)

The model parameters β ^ = ( ϕ ^ , Φ ^ , θ ^ , Θ ^ ) ∈ M 1 × p × M 1 × q × M 1 × P × M 1 × Q , are estimated by minimizing the log mean square error and the sum of residual squares (7).

β ^ = arg min β ∈ ℝ 4 ( 1 n ∑ t = 1 n log ( R t ( β ) ) + log ( 1 n S ( β ) ) ) (5)

To test the significance of the parameters, the hypothesis H₀ is assumed that each parameter ϕ i , Φ i ′ , θ j ou Θ j ′ is not zero, for i = 1, ⋯ , p , i ′ = 1, ⋯ , P , j = 1, ⋯ , q , and j ′ = 1, ⋯ , Q and test statistic:

T j = β ^ k σ ^ β j →       n → ∞       L t n − K (6)

H₀ is accepted at threshold α ∈ [ 0 ; 1 ] if | T j o b s | > q 1 − α / 2 t n − K , where T j o b s is the observed value.

Consider X t | X t − 1 , ⋯ , X t − p ~ N ( X ^ t ( β 0 ) ; σ ^ 0 2 ( β 0 ) ) and ε ^ t = X t − X ^ t ( β 0 ) the forecast error where X ^ t is the value of the estimated rainfall height at time t with the parameters β 0 = ( ϕ 0 , Φ 0 , θ 0 , Θ 0 ) . We note by R t ( β 0 ) , the mean square error or forecast error and S ( β ) , the sum of the residual squares. The best model is the one that returns the low value of the AIC, BIC information criteria and standard criteria such as MAE, RMSE or MAPE errors and having the most significant parameters.

( R t ( β 0 ) , S ( β 0 ) ) = ( 1 σ 0 ∑ t = 1 n ( X t − X ^ t ( β 0 ) ) 2 , ∑ t = 1 n ( X t − X ^ t ( β 0 ) ) 2 R t ( β 0 ) ) (7)

Information criteria and standards:

AIC = log ( σ ^ ε 2 ) + 2 K n     ( AkaikeInformationCriterion ) (8)

SIC   ou   BIC   ou   SBC = log ( σ ^ ε 2 ) + K log ( n ) n     ( SwartchtzBayesianCriterion ) (9)

( ME , MAE , MAPE , RMSE ) = ( 1 n ∑ t = 1 n     ε ^ t , 1 n ∑ t = 1 n | ε ^ t | , 100 n ∑ t = 1 n | ε ^ t x t | , 1 n ∑ t = 1 n     ε ^ t 2 ) (10)

Lower the value of these criteria, better the model. To study the nullity of the mean of the residualss, we use the classic tests of student (or conformity). Ljung-Box tests can identify processes of white noise. The test statistic being:

Q = K ∑ k = 1 n − k [ ρ ^ ( k ) ] 2 →       n → ∞       L χ K 2 (11)

where ρ ^ ( k ) = ∑ t = 1 n − k     ε ^ t ε ^ t − k ∑ t = 1 n     ε ^ t 2 . H₀ is accepted at threshold α ∈ [ 0 ; 1 ] if | Q o b s | < q 1 − α χ K 2 where Q o b s is the observed value.

The forecast made at time t for the t + k date (k = forecast horizon) is defined by:

X ^ t + k = E [ X t + k | I t ] (12)

where I t = σ ( X 1 , ⋯ , X t , ε 1 , ⋯ , ε t ) is a filtration. This predictor represents the best prediction ( X t ) t ∈ ℤ conditionally at the time of the information available at date t, I t .

The study focuses on monthly rainfall data over six decades of the three regions that include Lake Guiers. They are collected from the National Agency of Civil Aviation and Meteorology (ANACIM), Dakar (Senegal).

We have a series of data on monthly rainfall in Saint Louis, Podor and Louga, three regions in which Lake Guiers is located (16˚15'N; 15˚48'W), from 1960 to 2020. This zone located in the Sahel, on a strip of 10 to 25 km wide, is characterized by an extreme fragility of ecological balance (Kane A., 2005) [13] due to the alternation of a long dry season (from November to June) and a short wet season (from July to October) [14] .

“Figure 1” shows the monthly precipitation in each region and shows that there is no trend. The discontinuous lines in blue represent the monthly averages.

The red zones (Figure 2) showing changes in the precipitation index correspond

to dry periods in the region. In [15] , successive environmental transformations naturally linked to more than two decades of drought (1970-1997).

The simple randomness of a chronological sequence is due to the fact that all the observed values come from the same population by random and independent draws [16] . These tests aim at the constancy of the dispersion of the series, that is to say, they study whether the variability of the series is uniform over time and assume that “the series is random” [17] . The parametric test of Student makes it possible to see if the samples share the same mean value while the non-parametric Mann-Whitney [18] one makes it possible to evaluate if the ranks of the values are 2 to 2 homogeneously distributed in the 3 distributions.

With the Student tests, the p-values are <0.05 except for Podor and Saint Louis, where the hypothesis of proximity of rainfall averages is rejected. This may be due to high rainfall variability and its spatial distribution of measuring stations near the desert. As for the p-value obetunes with the Mann-Whitney test are >0.05, we accept the hypothese according to which, the distributions are significantly homogeneous (Table 1).

Persistence or autocorrelation, it represents the short-term dependence of observations. This violates the assumption of independence of observations ( x t ) t or errors ( ε t ) t . Thus, its presence may lead to the rejection of the stationarity hypothesis, that is to say, it may indicate the existence of a trend that, while in reality, there is no [3] . Problems induced by the presence of persistence or seasonality have a related effect.

The autocorrelation function is used to determine the lags order of the terms (Figure 3). In the following, we will consider only the 6-year moving average series of the monthly series. This value, centered of order k at time t being given

by: m m k , t = m m k ( x t ) = 1 k ∑ i = − k k     x t − i . We get 5 moving series of sizes 660 each.

The choice of this period is to remove missing data from the dry season that lasts 9 months each year. This is worth a 3-year loss of data on both sides of the tails.

These diagrams (Figure 4) show the presence of autoregressive terms with lags of order 8.

The trend represents the general upward or downward trend over a fairly long period of time.

In “Figure 5”, the 6-year moving average rainfall graphs match the trend components from the decomposition of the raw series, with the maiximal and

minimum rainfall explained respectively to (40.46; 25.49; 34.05)% et (19.49; 47.15; 33.36)% of monthly rainfall respectively on Louga, Podor and Saint Louis.

The nonparametric Mann-Kendall test shows the presence of a trend and determines whether the series includes a seasonal component. The values of the phi3 test statistics are significantly below the critical value 6.25 with the Dickey-Fuller test (Table 2), so the assumption that each series has a unity root and no trend is accepted.

Moreover, the results from the Mann-Kendall test reveal the presence of both trend and seasonality in all series except that of Louga. This can be due to a high spatio-ternporel variability that characterizes this area which has undergone flows of trade winds and monsoons according to the position of the FIT (Intertropical Front) and the note of a decrease in rainfall according to a South-North and West-East gradient [14] .

Three parametric tests (Dickey-Fuller [12] , Phillips-Perronet [12] and Kwiatkowski & al. [19] ) were used for trend detection.

The p-values in the first two tests are >0.05, so the assumption of non-stationarity H₀ is accepted. This is confirmed by the third test (KPSS), which, for a p-value < 0.05, accepts H₀ as a stationarity hypothesis (Table 3).

The choice of significance level of 5% is arbitrary, but appears to be a reasonable limit for the type of data under study [11] . Thus, given the results of the three tests, it can be concluded that the moving average series are not stationary.

The data can be graphically assimilated as sequences whose evolution is normal, even if the normality test of D’Agostino seems to reject this hypothesis (Figure 6).

With the R software, the commende auto-arima returns the best ARIMA model according to the value of the information criteria.

The values obtained from σ² successively for the monthly rain series (Louga, Podor, Saint Louis, rain min, rain max) are 1121; 914; 1144; 413; 1294 as well as for moving average series: 0.0905; 0.0602; 0.0879; 0.0339; 0.1016.

“Table 4” gives the list of the best models obtained with the lowest values of AIC, BIC and criteria of eurreur with the pourtcentage of significant parameters in each model in the 4^th column.

Notes: df = degree of freedom; TLP = Truncation lag parameter.

The hypothesis of nullity of the residualss is accepted by the Student test as well as that of the portmenteau test (Box-Pierce or Box-Ljung), because they follow a white noise, since the p-values are >0.05 (Table 5).

Note: signif. par. = number of significant parameters.

Notes: df = degree of freedom, conf. int. = confidence interval.

Gaussity tests appear to reject the H₀ hypothesis because of the high Kurtosis (thick-tailed distribution), as shown in this example of residuals density from the ARIMA (8, 1, 0) (6, 1, 1) [12] the moving average series of the Saint Louis station.

The Quantile-Quantile plot allows to compare two distributions that are considered similar. In abscissa, we have the quantiles of the theoretical distribution of the normorle law adjusted to the mean and slope of the theoretical sample compared to those of the residues from the selected models. Here, the five graphs of the residuals show that most of the points are the theoretical normal line except at the ends which move further and further away, due to heavy tail distributions ( κ > 3 “Table 5”) as shown in “Figure 7”.

The following table allows the comparison of the results of the estimated values on the test bases.

“Table 6” gives an overview of forecast errors when the model applies on 87% and 76% of the initial bases. And by comparison with the test base respectively 2011-2017 and 2005-2017, gets fairly low errors except the parameters in the Louga series model.

The monthly moving average of the three stations from 1963 to 2017 is represented by the horizontal cyan solid line of ordinates 22.58 on each of the graphs (Figure 8).

A slightly oscillatory downward trend in the previewing results of the Louga series and maximum rainfall from 2023 to 2032 and a constant wave after a short period of climb in Podor and Saint Louis during this same period. As for the minimum rainfall series, it remains moderately stable (Table 7).