Regression Analysis

A polynomial regression model is established at all six upscaled resolutions (i.e., also called the low-resolution scales) using geospatial predictors to predict each individual precipitation variable. The equation deriving the relationship between geospatial predictors and precipitation variables at each low-resolution scale is as follows in Equation (4):

$$P\_{D\\_GPM\_{LR}} = p\_1 \cdot x\_{DEM\_{LR}}^{\beta} + p\_2 \cdot x\_{DEM\_{LR}}^{2} + p\_3 \cdot x\_{DEM\_{LR}} + p\_4 \tag{4}$$

where *PD*.*GPMLR* is the predicted precipitation for each GPM variable at each low-resolution scale, and *p*1, *p*2, *p*<sup>3</sup> *and p*<sup>4</sup> are polynomial coefficients, *xi DEMLR* is the low-resolution geospatial variable(s).

In addition, a linear regression model is established to evaluate the relationship between the EDBF-based weighted precipitation and geospatial predictor, i.e., the latitude, which is as follows in Equation (5):

$$P\_{\rm D.WTP\_{Rx}} = p\_1 \text{.x}\_{\rm latitude} + p\_2 \tag{5}$$

where *PD*.*WTPRes* is the predicted weighted precipitation at investigated resolution scale(s), e.g., the low-resolution (*PD*.*WTPLRes*) or the high-resolution (*PD*.*WTPHRes*), *p*<sup>1</sup> *and p*<sup>2</sup> are linear coefficients, and *xlatitude* is the geospatial predictor.

#### Calculation of *r* Values

To execute EDBF algorithm for predicting the weighted precipitation from the multitemporal precipitation variables, the generation of initial weight vector for each contributing precipitation variable is a key process. In this regard, the *r* value is needed to formulate the initial weight vector for each contributing variable. The equation deriving *r* values at each low-resolution scale is given by Equation (6):

$$r = \frac{\text{Cov}\{Pr\_{i\prime}, Pr\_j\}}{\left(\sigma\_{Pr\_i} \times \sigma\_{Pr\_j}\right)}\tag{6}$$

where *Cov*-*Pri*, *Prj* is the covariance and σ*Pri* , σ*Prj* is the standard deviation of predictors (*i*, *j*) at *i*-th and *j*-th pixels, respectively. The equation deriving the covariance, the standard deviation and the mean for each investigated predictor is given by Equations (7)–(9), respectively:

$$Cov\left(Pr\_{i\prime}, Pr\_{j}\right) = \frac{\sum\_{ij=1}^{n} (Pr\_{i} - \mu\_{i}) \left(Pr\_{j} - \mu\_{j}\right)}{N} \tag{7}$$

$$\sigma = \left(\sum\_{i=1}^{n} (Pr\_i - \mu)^2 \Big| \mathcal{N}\right)^{\frac{1}{2}} \tag{8}$$

$$\mu = \frac{\sum\_{i=1}^{n} Pr\_i}{N} \tag{9}$$

where *Pri*, *Prj* are the two investigated predictors; μ*i*, μ*<sup>j</sup>* are the mean of investigated predictors, respectively, and *N* is the total number of observations.
