3.2.4. Validation

Station-measured data are the most direct observations of precipitation. In this study, 10-fold cross-validation was used to validate the precision of the OI outcomes and downscaling products. The observation stations were randomly divided into 10 groups. Nine groups (90%) were selected for the OI-based merging of the IMERG precipitation data with the observation data and GWR-based

downscaling. The remaining 10% constituted an independent dataset to validate the accuracy of the OI results and downscaling products. This was repeated 10 times to guarantee an independent validation for each station and to ensure the representativeness of the training samples and validation samples. Three statistical indicators, namely, the mean absolute error (MAE), root-mean-square error (RMSE), and correlation coefficient (CC), were utilized to validate the estimated values of the downscaled product against the observed values. The formulas for these indicators are as follows:

$$\text{MAE} = \frac{1}{n} \sum\_{i=1}^{n} |\mathbf{x}\_i - y\_i| \tag{7}$$

$$\text{RMSE} = \sqrt{\frac{1}{n} \sum\_{i=1}^{n} (x\_i - y\_i)^2} \tag{8}$$

$$\text{CC} = \frac{\sum\_{i=1}^{n} \left( x\_i - \overline{x} \right) \left( y\_i - \overline{y} \right)}{\sqrt{\sum\_{i=1}^{n} \left( x\_i - \overline{x} \right)^2 \sum\_{i=1}^{n} \left( y\_i - \overline{y} \right)^2}} \tag{9}$$

where *x* = <sup>1</sup> *n n i*=1 *xi*, *y* = <sup>1</sup> *n n i*=1 *yi*, *n* is the sample size, and *xi* and *yi* are the estimated values and station observations of precipitation, respectively.

In addition, a statistical analysis showed that July 2016 was the month with the most precipitation in the Tianshan Mountain area, while September 2017 was the month with the least precipitation. Therefore, in addition to the 10-fold cross-validation, the data from these two months were used to further validate the results of the proposed downscaling method in terms of their spatial distribution.
