2.4. Multi-linear regression analysis
The MLRA (multi-linear regression analysis), a statistical technique used to model the relationships between a dependent variable and one or more independent variables was used to characterize, trends, annual and semi-annual amplitudes in rainfall. It uses a least square approach and has been widely applied in hydrology and climate science to explain the possible relationships between key variables (see, e.g., Ndehedehe et al., 2016a; Rieser et al., 2010). The focus here is to separate the harmonic components (mean annual and semi-annual amplitudes) of rainfall over different 30-year climatological periods (1901- 2014) and then compare whether they are statistically different or have different means using analysis of variance (ANOVA). ANOVA is a prominent statistical method that is employed to assesses if the means of two or more groups (in our case, trends, mean annual and semi-annual rainfall patterns from each climatological period) are significantly different from each other. In the MLRA technique, the trends and harmonic components (mean annual and semi-annual amplitudes) for each climatological period (1901- 1930; 1931- 1960; 1961- 1990; 1991- 2010) were compartmentalised from the precipitation time series (P ) through parameterizations as (e.g., Ndehedehe and Ferreira, 2020; Rieser et al., 2010),
P (l, k, t ) =β0 + β1t + β2sin (2πt ) + β3cos (2πt ) + β4sin (4πt ) + β5cos (4πt ) + ε(t ); ( 2)
where (l; k ) are the grid locations, t is the time component, β0 is the constant offset, β1is the linear trend, β2 and β3 account for the annual signal, β4 and β5represent the semi-annual signal, while ε(t ) is the random error term. The amplitudes of rainfall (i.e., mean annual and semi-annual) over the region are then estimated as,
\(Annual\ =\left[\left(\ \beta 2\right)^{2}+\ \ \left(\ \beta 3\right)^{2}\ \right]^{\frac{1}{2}}\)and \(\text{\ Semi\ }Annual=\)\(\left[\left(\ \beta 4\right)^{2}+\ \ \left(\ \beta 5\right)^{2}\ \right]^{\frac{1}{2}}\)(3)