Long-Term Trends and Its Best Functional Form Estimation of Yearly Maximum and Minimum Temperatures at Cotonou City by Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise Method

The understanding of the long-term trend in climatic variables is necessary for the climate change impacts studies and for modeling several processes in environmental engineering. However, for climatic variables, long-term trend is usually unknown whether there is a trend component and, if so, the functional form of this trend is also unknown. In this context, a conventional strategy consists to assume randomly the shape of the local trends in the time series. For example, the polynomial forms with random order are arbitrarily chosen as the shape of the trend without any previous justification. This study aims to 1) estimate the real long-term nonlinear trend and the changing rate of the yearly high temperature among the daily minimum (YHTaDMinT) and maximum temperatures (YHTaDMaxT) observed at Cotonou city, 2) find out for these real trend and trend increment, the best polynomial trend model among four trend models (linear, quadratic, third-order and fourth-order polynomial function). For both time series, the results show that YHTaDMinT and YHTaDMaxT time series are characterized by nonlinear and monotonically increasing trend. The trend increments present different phases in their nonmonotone variations. Among the four trend estimations models, the trend obtained by third-order and fourth-order polynomial functions ex-How hibits a close pattern with the real long-term nonlinear trend given by the Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (ICEEMDAN). But, the fourth-order polynomial function is optimal, therefore, it can be used as the functional form of trend. In the trend increment case, for the YHTaDMaxT time series, the fourth-order fit is sys-tematically the best among the four proposed trend models. Whereas for the YHTaDMinT time series, the third-order and fourth-order polynomial functions present the same performance. They can both be used as the functional form of trend increments. Overall, the fourth-order polynomial function presents a good performance in terms of trend and trend increments estimation.


Abstract
The understanding of the long-term trend in climatic variables is necessary for the climate change impacts studies and for modeling several processes in environmental engineering. However, for climatic variables, long-term trend is usually unknown whether there is a trend component and, if so, the functional form of this trend is also unknown. In this context, a conventional strategy consists to assume randomly the shape of the local trends in the time series. For example, the polynomial forms with random order are arbitrarily chosen as the shape of the trend without any previous justification. This study aims to 1) estimate the real long-term nonlinear trend and the changing rate of the yearly high temperature among the daily minimum (YHTaDMinT) and maximum temperatures (YHTaDMaxT) observed at Cotonou city, 2) find out for these real trend and trend increment, the best polynomial trend model among four trend models (linear, quadratic, third-order and fourthorder polynomial function). For both time series, the results show that YH-TaDMinT and YHTaDMaxT time series are characterized by nonlinear and monotonically increasing trend. The trend increments present different phases in their nonmonotone variations. Among the four trend estimations models, the trend obtained by third-order and fourth-order polynomial functions ex-

Introduction
To modelize several processes in environmental engineering, agriculture, climatology and hydrology, the understanding of the real nonlinear trend and trend model's estimation of climate variables as extrema temperatures are necessary [1]. Moreover, climate change impacts are evaluated through trend analysis of climate variables [2]. Climate change is marked with variation in temperature, which can directly influence the hydrological processes [3] [4] [5]. In Benin Republic, a sub-Saharan country affected by climate change effects, few studies are focused on the real nonlinear trend investigation and on the estimation of the functional form of this trend in the extrema temperatures during historical period. In this country, one of the major challenges for hydrologists and meteorologists is the estimation of the real functional form of historical climate variables trend. Indeed, if there is a trend in a real-time series of historical climatic variables, the functional form of this trend is usually unknown. In this context, certain authors assume randomly the form of the local trends as a polynomial function with arbitrarily chosen order (Abaje and Oladipo [6], Gadedjisso-Tossou et al. [7], Agbazo et al. [8]). Generally, linear regression technique is used to analyze the trends in the climatic variables time series. This situation can lead to erroneous conclusions.
Nowadays, to overcome these difficulties and to correctly represent the trend in climatic variables, with higher accuracy without ignorance of its nonstationarity and nonlinearity characteristics, some improved methods are developed in literature. Among these methods one can cite, the Empirical Mode Decomposition (EMD) developed by Huang et al. [9], and its variations, such as the Ensemble Empirical Mode Decomposition (EEMD) developed by Wu and Huang [10], Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) developed by Torres et al. [11] and Improve Complete Ensemble  [12]. The CEEMDAN is an important improvement of EEMD and ICEEMDAN is a further improvement of CEEMDAN [11] [12].
Based on ICEEMDAN method, the purpose of this article is, firstly to estimate the real long-term nonlinear trend and the changing rate of the yearly high temperature among the daily minimum (YHTaDMinT) and maximum temperatures (YHTaDMaxT) observed at Cotonou city, the most industrialized and developed city in Benin Republic, secondly, to find out the best (optimal) polynomial forms of these nonlinear trends and trend increments. The article is structured as follows: Section 2 describes data sets and the methods involved. The results are presented and interpreted in Section 3. Finally, a summary with concluding remarks forms and outlook for further research ends the study in Section 4.

Materials
Cotonou is located in the southern part of Benin Republic (West Africa), closer to the Atlantic Ocean. It is characterized by four seasons each year: a long rainy season (April to July) followed by a short dry season (August to September) and a short rainy season (October to November) followed by long dry season (December to March). In this study, the daily maximum temperature and minimum temperature data are recorded from 1953 to 2019 at Cotonou Airport's synoptic station and provided by "Météo-Benin". From these data, one generates the yearly high temperature among the daily minimum (YHTaDMinT) and the yearly high temperature among the daily maximum temperatures (YHTaD-MaxT) observed during each year. Figure 1 shows that YHTaDMinT and YH-TaDMaxT time series vary regarding to the year. The minimum values of YH-TaDMinT and YHTaDMaxT are respectively greater than 27˚C and 32˚C.

Methods
Improved Complete Ensemble Empirical Mode Decomposition with adaptive noise, namely ICEEMDAN, developed by Colominas et al. [9] is used for estimating nonlinear trends and trend increments in YHTaDMinT and YHTaDMaxT time series. The ICEEMDAN is an important improved version of the complete ensemble empirical mode decomposition with adaptive noise, CEEMDAN (developed by Torres et al. [11]). It is an expansion algorithm of Empirical Mode Decomposition (EMD), which is an adaptive (data-driven) method developed by Huang et al. [9] to analyze nonstationary time series stemming from nonlinear systems. According to Colominas et al. [12], ICEEMDAN's main steps can be described as follows: x n which is obtained after the specific white noise is added into the original signal ( ) where . represents the mean value,

( )
. M is the operation which produces the local average of the envelope that satisfies the IMF sifting threshold and 1 imf is the first IMF.

3) Construct the second signal
. This signal is decomposed by EMD and one obtains the second IMF component of ICEEMDAN: is the residual obtained after the first decomposition.

4) Construct the kth signal
. This signal is decomposed by EMD and obtains the kth IMF component of ICEEMDAN: r n is the further residual after the kth decomposition, and 1, 2, , k K =  . 5) Repeat step 4 for the next k until the residual cannot be decomposed any further, the residual is monotonous, or its amplitude is less than the indicated In order to find out the best (optimal) functional form of the real long-term nonlinear trend and its changing rate, different polynomials functions (linear, quadratic, third-order and fourth-order) are used to modelize this trend. Four main indicators are used to clearly reflect the error of each polynomial function's result and evaluate the fitting accuracy: the deterministic coefficient (DC), the mean squared error (MSE), the mean absolute error (MAE) and the weighted mean absolute percentage error (WMAPE). According to Chen et al. [13], Tao et al. [14], Xia and Bo [15], these parameters are calculated respectively as follows: where N is the series length; t D represents the real long-term nonlinear trend or trend increment's value at time t; t d is the predicted long-term nonlinear trend or trend increment's value at time t. o D is the mean value of the real long-term nonlinear trend or trend increment's value. Statistically, the tinier the values of the MSE, MAE and WMAPE are, the more precise the prediction accuracy of the model is. However, the closer the DC value is to 1, the better is the model's prediction, and when DC < 0.5, the prediction is unreliable.    Findings are in agreement with the global increase in temperatures indicated by IPCC [16], Trenberth et al. [17], Hartmann et al. [18], and Tang and Zhang [19]. They are also in agreement with the findings of (Trenberth et al. [17]), global temperatures are rising and will continue to do so throughout the 21st century.

Results and Discussion
It is clearly noted that the natural trend of YHTaDMinT and YHTaDMaxT time series cannot be modeling by linear trend model, but it seems that the quadratic, third-order, and fourth-order polynomials are the good candidates to fit these trends, however, their fitting accuracy cannot be evaluated visually. Thus, to find out the functional forms of these nonlinear trends, the fitting errors of each proposed trend model are calculated and shown in Table 1.  It can be observed that whatever the model, the time series (YHTaDMinT or YHTaDMaxT) considered the MSE, MAE, WMAPE values (Table 1)    Fourth-order 6.4 * 10 −11 (7.21 * 10 −10 ) 6.7 * 10 −6 (2.18 * 10 −5 ) 0.0156 (0.0749) 0.9997 (0.9895) trend model, but it seems that the quadratic, third-order, and fourth-order polynomials are good candidates for YHTaDMaxT time series. Thus, to find out the functional form of these trend increments, the fitting errors of each proposed trend increments model are calculated and shown in Table 2.

Summary and Conclusions
In this paper, the real long-term nonlinear trend and the changing rate of the yearly high temperature among the daily minimum (YHTaDMinT) and maxi- ture works will be focused on the investigation of the long-term trends in the climatic variables simulated by regional climate models in Benin republic and its best functional forms.

Data Availability
The data used in this study are supplied by the local service of Météo-Bénin in Cotonou. The data are not available online in any data base, so that we cannot provide a link to reach them. They are provided when researchers address requests to Météo-Bénin (http://www.asecna.aero).