1. Introduction

2. Materials and Methods

2.3. Multi-Model Ensemble Techniques

The simplest approach employed in the multi-model ensemble is the arithmetic mean technique. It involves calculating the average of simulation results obtained from various models. This approach offers convenience because it eliminates the need for comparisons between the models and the observations. However, it is important to note that different models often possess varying simulation capabilities. One model may excel in simulating a specific variable or aspect, while another may perform less effectively. In contrast, the weighted mean method takes into account these varying model capabilities. It accomplishes this by assigning distinct weighting coefficients to each model, which are determined based on comparisons with observational data. Consequently, this method relies on observational data for its implementation. The ensemble method utilizing multivariate linear regression enjoys extensive use in ensemble research (Crawford, et al., 2018). Furthermore, there is a growing trend of incorporating nonlinear regression and artificial neural networks into ensemble modelling. According to standard procedure, 80% of the data are used for the training (1950-1994; observation period) and the remaining 20% (1995-2005; evaluation period) are used to verify the performance of the outputs of the constructed neural network. The output from the CRU dataset served as the dependent variable while GCMs served as the independent variables.

2.3.1. Arithmetic Mean (AM) Ensemble

The use of arithmetic means of GCMs projections in climate ensemble is a fundamental technique that strengthens the robustness and reliability of climate predictions (Crawford, et al., 2018). However, this method has no parameters to optimise its performance and it is employed in this study as a comparative baseline (Venkatarama et al. 2016). While climate ensemble driven by the arithmetic mean method continues to be a cornerstone of climate science alternative methods, such as weighted ensembling, may be applied to assign more significance to models with superior performance. Equation 1 below shows the formula for climate ensembling using the AM method

$$\overline{x}=\frac{{\sum }_{i=1}^n{x}_i}{n}$$

(1)

where:

̅x represents the arithmetic mean (average).

xi represents individual values in a dataset.

n represents the total number of GCMs.

2.3.2. Multiple Linear Regression (MLR) Ensemble

Linear regression can be employed to assign weights to GCMs based on their historical performance. It assumes that the relationship between the observed and GCMs is approximately linear, and aims to estimate and correct this bias. This means that models that have demonstrated better predictive ability in the past are given more influence in the ensemble. MLR models provides the coefficients that can be used to correct the observed data. In linear regression, homoscedasticity is a crucial assumption that demands consistent error variance across all independent variable values. Violating this assumption, termed heteroscedasticity, can harm the model’s reliability and clarity. The presence of homoscedasticity can be influenced by factors like the scale, climate variable, and location. For example, when comparing GCM outputs to long-term climate trends over large regions, the assumption of constant variance is more likely to be met. The presence or absence of homoscedasticity can differ based on the specific climate variable and the geographic location under consideration. The MLR models with independent variables (GCMs) and dependent variable (CRU) were expressed in the form in Equation (2):

y = β₀ + β₁x₁ + β₂x₂ + . . . + β _kx _k ,

(2)

where:

y is the dependent variable,

x_i (i = 1, . . ., k) is the independent variable,

β₀ is the intercept of y, and

β₁ (i = 1, . . ., k) is the regression coefficient,

2.3.3. Artificial Neural Network (ANN) Ensemble

The neural network ensemble on the hand similar to the human brain in two ways: information is acquired by the network, and synaptic weights, which are the strengths of interneuron connections, are stored. In this sense, a multi-layered feedforward neural network, trained using a supervised-learning approach and operated in the MATLAB numerical computing environment was used for this study. The multi-layered feedforward neural networks consisting of three layers; the input, hidden, and output layers as shown in Figure 2. was deployed. Learning and output were calculated through backpropagation algorithms amongst the different layers, interconnected by means of synaptic weights. The input variable x = (x1, …,xn) is introduced into the neural network through the input layer. This layer is interconnected with the hidden layer h = (h1 ….,hn) through weighted edges, and subsequently, it is connected to the output layer. The weight of the edge linking the input layer to the hidden layer is represented as wkj, while wk denotes the weight connecting the hidden layer to the output layer. The output produced by the kth hidden neuron can be expressed in Equation (3) as follows:

$$h_k=\mathrm{ᶲ}\left({\omega }_{k0}^{\left(1\right)}+{\sum }_{j=1}^p{\omega }_{kj}^{\left(1\right)}{x}_j\right)$$

(3)

where:

ᶲ is the activation function.

Figure 2. diagram of a multilayer feed-forward neural network.

Figure 2. diagram of a multilayer feed-forward neural network.

The hyperbolic tangent sigmoid function (tansig) and linear transfer function (purelin) were used for the activation function of the network while the conjugate gradient algorithm is used to change the synaptic weights and learning rate in optimization of the outputs. The search and conjugate directions are carried out simultaneously in the conjugate gradient training algorithm and it has been noted to produce fast convergence. In the conjugate gradient training algorithm, the search is performed along with conjugate directions. They generally produce faster convergence than gradient descent directions. The hyperbolic tangent function is used for the activation function and the conjugate gradient method is used as the optimization algorithm. In the multilayer perceptron of network, the minimal relative variation of the learning error is set at 0.001, and the minimum variation of the learning error was fixed at 0.0001. The calculations are repeated until the error is as small as possible. The neurons in the input layer serves as a buffer. The number of neurons in the hidden layer was adjusted to prevent overfitting, set at twice the number of independent variables, considering the optimal stimulation condition and the number of independent variables (Jang, 2017). Input selection for ANN models patterned after the method of Jang, (2021) were generated for individual study areas using the top three models (“ANN3”) or the top five models (“ANN2”) based on their ranked correlation during the historical period, as detailed in Table 2. The “ANN1” configuration, on the other hand, includes all seven of the GCMs studied.

2.4. Statistical Metrics for Model Evaluations

2.4.1. Taylor Diagram

The Taylor diagram provides a graphical summary of the statistical characteristics of the climate models and the relative skill which GCMs simulate the spatial pattern of annual mean of climate variable. The Taylor diagram is used in this study to rank the performance of the seven GCMs in simulating the CRU dataset (observed). Three important metrics, the correlation coefficient (Cor.), standard deviation (σ), and centred root mean square error (cRMSE) are displayed on the diagram (Taylor, 2005). The centred root-mean-square is difference between the GCMs and the CRU and represented as the radial distance to the point on the x-axis identified as “observed.” Also, the σ measures the average dispersion of the outputs of the datasets away from its mean, represented on the Taylor diagram as the distance from the origin. Finally, the Correlation coefficient measures the strength and direction of the linear association between the GCMs and the observed. The description of the correlation is then made using the delineation by (Handoko et al., 2019) as displayed on Table 3. Equations (3)–(6) below, displays the formula for the estimation of the correlation coefficient, the centred root men square error and the standard deviation between two Models a and b as displayed on the Taylor diagram.

$$Cor.=\frac{\frac{1}{n}{\sum }_{i=1}^n(a-\overline{a})(b-\overline{b})}{{\sigma }_a{\sigma }_b}$$

(3)

$${\sigma }_a=\sqrt{\frac{1}{n}{\sum }_{i=1}^n(a-\overline{a}{)}^2}$$

(4)

$${\sigma }_b=\sqrt{\frac{1}{n}{\sum }_{i=1}^n(b-\overline{b}{)}^2}$$

(5)

$$cRMSE=\sqrt{\frac{1}{n}{\sum }_{i=1}^n\left[(b-\overline{b}{)}^2-(a-\overline{a}{)}^2\right]}$$

(6)

where:

σ_a and σ_b are the standard deviations of Model “A” and Model “B” respectively

Table 3. ANN Ensemble Design.

Table 3. ANN Ensemble Design.

S/N	Station Name	Case	Variable	ANN Ensemble Inputs
1	Ilorin	ANN 3	Precipitation	CNRM-CM5, MPI-ESM-LR, GFDL-ESM2M
		ANN 2		CNRM-CM5, MPI-ESM-LR, GFDL-ESM2M , EC-EARTH, MIROC5
		ANN 3	Tmax	CNRM-CM5, CanESM2, MPI-ESM-LR
		ANN 2		CNRM-CM5, CanESM2, MPI-ESM-LR, MIROC5, NorESM1-M
		ANN 3	Tmin	CanESM2, NorESM1-M, GFDL-ESM2M
		ANN 2		CanESM2, NorESM1-M, GFDL-ESM2M, EC-EARTH, MPI-ESM-LR
		ANN 1	Precipitation/ Tmax/Tmin	CanESM2, CNRM-CM5, EC-EARTH, MIROC5, MPI-ESM-LR, NorESM1-M, GFDL-ESM2M

2.4.2. Mean Absolute Error (MAE), Root Mean Squared Error (RMSE) and Nash–Sutcliffe Efficiency (NSE)

The mean absolute error (MAE), root mean square error (RMSE), and Nash-Sutcliffe efficiency (NSE) were used to evaluate the accuracy of the ANN ensembles developed. The RMSE compared to the MAE is designed to penalise large prediction errors in the dataset as each error influences the overall error in proportion to its square rather than its direct magnitude. The accuracy assessment of the ensembles is done by comparing the values generated from the ensemble networks and the observed in the validation dataset. When MAE and RMSE are smaller, the estimated value indicates higher accuracy and shows how big of an error we can expect from the forecast on average.

The Nash-Sutcliffe model efficiency coefficient (NSE) is commonly used to assess the predictive power of hydrological discharge models. Nash-Sutcliffe efficiency ranges from negative infinity to 1 with an efficiency of 1 indicating a perfect match between the ensembles. whereas an efficiency less than 0.0 occurs when the observed mean is a better predictor than the model.

$$\mathbf{M}\mathbf{A}\mathbf{E}=\frac{\sum _{\mathit{i}=1}^{\mathit{n}}\left|{\mathit{y}}_{\mathit{i}}-{\mathit{x}}_{\mathit{i}}\right|}{\mathit{n}}$$

(7)

$$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}=\sqrt{\frac{\sum _{\mathit{i}=1}^{\mathit{n}}({\mathit{y}}_{\mathit{i}}-{\mathit{x}}_{\mathit{i}}{)}^2}{\mathit{n}}}$$

(8)

$$\mathbf{N}\mathbf{S}\mathbf{E}=1-\frac{{\sum }_{\mathit{i}=1}^{\mathit{n}}({\mathit{y}}_{\mathit{i}}-{\mathit{x}}_{\mathit{i}}{)}^2}{\sum _{\mathit{i}=1}^{\mathit{n}}({\mathit{y}}_{\mathit{i}}-\overline{\mathit{y}}{)}^2}$$

(9)

where:

y_i=prediction

x_i=true value

n=total number of data points

3. Results

3.4. ANN Ensemble Accuracy Assessment

The accuracy of the ANN ensembles developed for each variable in each of the analysis station were assessed. Performance of the ensemble methods were compared for an evaluation period; 1995-2005. Three error metrics are used to assess the performance of the ensembles, the MAE, RMSE and NSE metrics. This metrics were selected to project the multidimensional nature of assessing climate and hydrological modelling. It has been established that the use of a single error metric in model performance evaluation will only highlight a limited aspect of the error characteristic (Willmott & Matsuura, 2005). Accuracy assessment of GCMs/ensembles varied across the analysis stations based on the error metric and are presented in Figure 10, Figure 11 and Figure 12.

Figure 10. Rainfall GCMs/ ANN Ensembles accuracy assessment using: (a) MAE, (b) MSE, (c) NSE.

Figure 10. Rainfall GCMs/ ANN Ensembles accuracy assessment using: (a) MAE, (b) MSE, (c) NSE.

Figure 11. Tmax GCMs/ ANN Ensembles accuracy assessment using: (a) MAE, (b) MSE, (c) NSE.

Figure 11. Tmax GCMs/ ANN Ensembles accuracy assessment using: (a) MAE, (b) MSE, (c) NSE.

Figure 12. Tmin GCMs/ ANN Ensembles accuracy assessment using: (a) MAE, (b) MSE, (c) NSE.

Figure 12. Tmin GCMs/ ANN Ensembles accuracy assessment using: (a) MAE, (b) MSE, (c) NSE.

Figure 13. Comparison of Ensemble methods for (a) Rainfall (b)Tmax, (c) Tmin.

Figure 13. Comparison of Ensemble methods for (a) Rainfall (b)Tmax, (c) Tmin.

To assess ensemble agreement with observations quantitatively, accuracy evaluation was structured to address general statistical aspects of model performance that do not pertain to any specific application. Similarly, in order to present evaluation results in line with the natural temporal resolution of the observational datasets, without any aggregation or alteration, we limited our analysis to the monthly scale. The graphical representation of the best performing ANN ensemble in each station against the least accurate GCM are displayed in the Supplementary Materials section. An illustration of climate model enhancement for a specific application is evident in the comparison of monthly precipitation between less accurate GCMs and an optimized ensemble for the ALT, as presented in the Supplementary Figures. The temporal trend of the outputs provides strong evidence of reduced overestimation of rainfall during the rain seasons by GCMs. This “wet bias”, noted by Gnituo et al. (2019), brought about by the exaggerated depiction of the West African Monsoon convergence zone, especially from June to September were corrected (see Supplementary Figure S4).

The ANN ensembles developed all outperformed the individual GCMs outputs for all stations vis-à-vis the three-error metrics with varying differences, except for rainfall at Birnin-Kebbi station. Selecting the ideal model requires discrete consideration based on the intension for use. By way of illustration, accuracy assessment of precipitation models using MAE at the Birnin-Kebbi station ranked the MIROC5 model highest. However, it is important to note that the proportional scale of the error is not always clear with the MAE, a major disadvantage of the metric (Willmott and Matsuura, 2005). Demonstrably, the RMSE which best shows the impacts of large errors in the forecast ranks the MIROC5 fourth while the NSE ranks it sixth. Therefore, the ANN1 ensemble which is jointly ranked best by NSE and RMSE may be more suitable if considerations of weather extremities will be executed.

The performance of ensembles varied across different factors, including the specific methods used, variables examined, the choice of error metrics, and geographical locations. In general, ensembles significantly improved predictive performance, particularly as measured by the NSE. These improvements were notable, with a 34% enhancement for rainfall, a substantial 255% improvement for maximum temperature (tmax), and an impressive 660% boost for minimum temperature (tmin). Furthermore, the ensembles contributed to a reduction in errors, as assessed by the RMSE. This reduction was quantified at 26.51% for rainfall, 77.09% for tmax, and 64.58% for tmin (refer to the Supplementary Table S6 for details). The performance differences between ensembles for rainfall and temperature were primarily linked to variations in the complexity and complementarity of the individual models associated with each variable.

For instance, when simulating rainfall, it was observed that the ensemble accuracy was higher when excluding the NorESM1-M and GFDL-ESM2M models from the ensemble design. Moreover, in specific locations like the Maiduguri station (SAS), the ANN3 ensemble, which included the CNRM-CM5, MIROC5, and MPI-ESM-LR models, displayed a notably low average absolute error of 2.17mm per month, a substantial improvement compared to the large error margin of 27.47mm per month observed in the individual CanESM2 model. Conversely, the RMSE and NSE metrics ranked the ANN2 ensemble as the best performer in simulating precipitation at the station. Similarly, at the ALT observation station, individual GCMs exhibited substantial errors, with the MIROC5 model showing an average error of 140mm per month during the evaluation period. These deficiencies were markedly rectified using the MME approach, which reduced the error to a minimal 1.4mm per month for the ANN3 ensemble, accompanied by a notably high predictive power of 93%.

4. Conclusions

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