A Gaussian process is an infinite-dimensional multi-variate Gaussian. Let y(t) be the complete observation (i.e., no missing function of time t), assuming y(t) follows the GP, then:

$\text{y}\left(\text{t}\right)~\text{GP}\left(\text{m},\text{K}\right)$ (1)

With mean m and a covariance K parametrized by the set of hyperparameters Θ.

For a single GPs where y(t) = (y(t1), y(t2), …, y(tN)), the covariance function considers only the covariance between any two points of time t and t', then the covariance K = k(t,t'), and N the number of samples. Many functions can be used to represent the covariance between two instants of time. The covariance functions play a crucial role in GPs. They are used to enforce prior knowledge about the data in GP regression by defining what constitutes as similarity between the data points.

The standard Gaussian Process (GP) formulation is typically applied to scalar functions, meaning it predicts only a single variable. When extending the standard GP formulation to multiple variables, Independent Gaussian Processes (IGPs) can be considered as an alternative. However, IGPs do not account for the correlation between outputs, which can reduce forecasting accuracy when dealing with real variables that are interdependent.

In contrast, the multi-task Gaussian process (MTGP) framework, as proposed by [9] , addresses the correlation between outputs. This approach incorporates the correlation between inputs and outputs by using the linear model of coregionalization (LMC) within the GP framework. The LMC represents outputs as linear combinations of Q independent random functions, u_q(x), which are modeled as Gaussian processes with zero mean and covariance k_q as their prior. Assuming M outputs represented by ${\left\{f_m\left(x\right)\right\}}_{m=1}^M$, $f_m$can be expressed as:

$f_m=\underset{q=1}{\overset{Q}{{\displaystyle \sum }}}{a}_{m,q}{u}_q\left(x\right)$ (2)

where $a_{m,q}$ are scalar coefficients and . The LMC assumes uncorrelated latent Gaussian process i.e. $cov\left(u_q\left(x\right),u_{q^{\prime }}\left(t\right)\right)=0$ for $q/\equiv q^{\prime }$. Then the covariance function of two outputs function f_m and f_m’ can be expressed as:

$cov\left(f_m\left(x\right),f_{m^{\prime }}\left(x\right)\right)=K\left(t,t^{\prime }\right)=\underset{q=1}{\overset{Q}{{\displaystyle \sum }}}{B}_q{k}_q\left(t,t^{\prime }\right)$ (3)

for Q = 1, we have the simplified case of LMC known as intrinsic corgeionalization matrix (ICM) then the covariance of M outputs is expressed as:

$K=B\otimes k\left(t,t^{\prime }\right)$ (4)

For the multi-task Gaussian process proposed by cite{multitask} the covariance function considered the correlation between inputs and outputs. The dataset used for training is the input represented by time and the daily rainfall data at M weather stations. The daily rainfall at M weather stations can be expressed as:

$y\left(t\right)={\left(y_1\left(t_1\right),\cdots ,y_1\left(t_N\right),\cdots ,y_2\left(t_1\right),\cdots ,y_2\left(t_N\right),\cdots ,y_M\left(t_1\right),\cdots ,y_M\left(t_N\right)\right)}^T$

where $y_{il}$ is the rainfall observation of $l^{th}$ station at time $t_i$ with $i=1\cdots N$and $l=1\cdots M$. The hyperparameters are obtained using dataset and the minimization of the likelihood, the predict mean and covariance can be expressed as Equations (5) and (6).

$m_{*}=K_{*}^T{K}^{-1}y.$ (5)

$C_{*}=K_{*,*}-K_{*}^T{K}^{-1}{K}_{*},$ (6)

The python framework for Gaussian process named GPy [15] and [16] is used for GP simulation.

We first organize the rainfall data by grouping the stations around synoptic stations. The synoptic stations, which have complete data over five years, serve as the central reference points for these groups. Each synoptic station (shown in red in Figure 1 forms a group with the stations located within a 100 km radius. This grouping ensures that nearby stations, which are likely to have correlated rainfall patterns, are considered together for data imputation.

For each group, the neighbouring stations are treated as dependent variables, and the missing data of each station are estimated based on the available data from its group members. The stations are grouped into 10 groups, with each group associated with a synoptic station. For example, the blue color in Figure 2 represents the neighbouring meteorological sensors at the Bobo synoptic station. Some synoptic stations, such as Bobo, Boromo, and Ouaga, have more neighbouring stations, while others, like Bogandé, have fewer.

To assess the performance of the MTGPs in imputing missing rainfall data, we compare its performance with several traditional statistical methods and machine learning techniques widely used for data imputation in the literature [1] [4] . The methods considered in this comparison include:

In order to evaluate the performance of different imputation methods listed in section above. Two performance indicators are used, namely the correlation coefficient and the root-mean-square error (RMSE).

The comparison is conducted for each method, and the one yielding the best results in terms of accuracy and reliability is selected. In the following subsections, we present the results of these comparisons and discuss the implications for improving the imputation of missing daily rainfall data.

To assess the performance of different imputation techniques, artificial missing data is introduced into the complete datasets of these stations. Missing data is simulated at levels of 20%, 40%, and 50% for stations located within a 100 km radius of the synoptic stations.

The performance of various imputation models is then evaluated and compared using the artificially generated missing data. The data is introduced on stations close to synoptic stations, defined as those within a 100 km radius.

In the following section, we identify the best Gaussian Process (GP) model for imputing daily rainfall data. Specifically, we compare the multitask GP model (MTGP) with well-known statistical and machine learning imputation methods such as MICE, KNN, LOCF, and mean imputation (ME). The comparison is made using artificially generated missing data at levels of 20%, 40%, and 50% for daily rainfall across all stations.

Tables 1-6 present performance indicators, including RMSE and the correlation coefficient, for each station in the Ouaga, Bobo, and Bogandé groups. As shown in these tables, the MTGPs model outperforms other imputation methods in each group. As the percentage of missing data increases, MTGPs continues to provide better results than traditional imputation techniques.

In the Bobo group, with 20% missing data, only two stations exhibit better performance than MTGPs. When the missing data percentage increases to 50%, only one station outperforms MTGPs. Across all stations in this group, MTGPs consistently outperforms MICE.

In the Ouaga group, MTGPs demonstrate superior performance compared to other methods. Among the 11 stations in the Ouaga group, MTGPs achieves the lowest RMSE and highest correlation coefficient ($rho$) in 8 stations for 20% missing data. When the missing data rises to 50%, MTGPs achieves the best results in 10 stations. For 20% missing data, KNN is the second-best method, while for 50%, MICE emerges as the second-best performer.

In the Bogandé group, for 20% missing data, MTGPs achieve the lowest RMSE and highest $\rho$ in 4 stations, KNN in 2 stations, and ME in 2 stations. However, for 50% missing data, MTGPs improve their performance, achieving the best results in 5 stations, while KNN outperforms in 3 stations.

This study explored the efficacy of Multi-Task Gaussian Processes (MTGPs) for imputing missing daily rainfall data, leveraging correlations between nearby meteorological stations. The results demonstrated that MTGPs outperform well-established statistical and machine learning methods, such as MICE, KNN, LOCF, and mean imputation (ME), across various missing data scenarios (20%, 40%, and 50%) in different climatic zones. The superior performance of MTGPs is particularly evident as the percentage of missing data increases, underscoring their robustness and adaptability in handling data sparsity.