2.3. Evolutionary patterns of rainfall and discharge
Three different multivariate methods were employed to analyse the spatial and temporal patterns of rainfall and discharge. First, the principal component analysis (PCA, e.g., Jolliffe, 2002) was used to decompose time series of GPCC-based precipitation into spatial and temporal patterns. PCA is an important tool to assess the spatio-temporal variability of continuous climate data such as rainfall due to its simplicity and capability to isolate both inter-annual signals, and long-term periodic variations (Ndehedehe et al., 2016a). The method reduces the dimensions of multivariate data by creating new variables that are linear functions of the original variables. Significant modes of variability for each climatological period (1901- 2014) were assessed based on the statistical rotation of PCA. To understand the spatio-temporal variability of rainfall during each climatological period with a 30-year window, rainfall grids for each window were decomposed using this technique. This window allows a
reasonable conclusion of climate change or climate variability impacts on rainfall and river discharge,
especially whether the latter has changed significantly in response to the former. The singular spectrum analysis (SSA, Ghil et al., 2002) was employed to decompose monthly river discharge through a singular value decomposition (SVD) of the lagged covariance matrices. The reconstructed discharge time series (i.e., matrix of reconstructed principal components of discharge) obtained from the SSA scheme were compared with the leading temporal patterns of PCA-derived rainfall using correlation and regression analyses (next sections) for the common period. Second, to assess the relationship between local rainfall and discharge, rainfall was further localised over the Congo basin using the independent component analysis (ICA, e.g., Cardoso, 1999; Cardoso and Souloumiac, 1993). The ICA method employed here is based on the JADE (Joint Approximate Diagonalisation of Eigen matrices) algorithm, which exploits the fourth order cumulants of the data matrix and is fully detailed in Cardoso and Souloumiac (1993). Several applications of PCA and ICA methods in droughts and hydro-climatic studies and localization of groundwater signals have been documented (e.g., Agutu et al., 2017, 2019; Montazerolghaem et al., 2016; Ndehedehe et al., 2016b, 2017; Sanogo et al., 2015). The temporal variations of rainfall associated with the localised spatial patterns were correlated with
discharge to determine the influence of local rainfall on discharge. Using the ICA method, rainfall grids were localised as:
\begin{equation} \mathbf{X}_{\text{GPCC}}\left(x,\ y,\ t\right)=\ \mathbf{\text{TS}};\ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(1\right)\nonumber \\ \end{equation}
where (x, y ), and t are grid locations and time steps (months), respectively. T is the temporal patterns and is unit-less since it has been normalised using its standard deviation while the spatial patterns S associated with T have been scaled using the standard deviation of its temporal patterns.T and S are interpreted together and integrated to form what is traditionally called the ICA mode of variability (e.g., Ndehedehe et al., 2017).