3.3 Evaluation of Interpolation Methods
We are estimating rainfall at ungagged locations with different models
as discussed above. There are as many as twenty models which are
utilised in this work to assess the rainfall at ungagged locations.
Therefore, it is necessary to know the ability of these models to
predict rainfall accurately. But there is a challenge to find out which
one of these models is giving most reliable result. For this we have
done cross validation of the estimated value with the observed value.
Cross validation is carried out following Vicento-Serrano et al (2003)
and Muller et al (2004). Borges et al (2016) and Xie et al (2011) stated
that cross validation should be performed by removing one of the
stations and then applying one of the interpolation techniques to
measure the predicted value at the removed station taking into account
the other observing stations in the search radius. This method is
repeated for the remaining stations and the predicted value is noted
down after measuring the estimated rainfall at all the stations
comparisons is carried out to see how efficiently the model is
predicting. The comparison was done following the performance measures
discussed by Luo et al (2008), Tabio and Salas (1985) and Li and Heap
(2011). Cross validation is performed using seven performance measures
for all the twenty methods. The seven measures are as follows
Mean bias error
MBE =\(\frac{1}{N}\sum_{j=1}^{N}{[F\left(r_{j}\right)-f(r_{j})]}\)8
Mean absolute error
MAE =\(\frac{1}{N}\sum_{j=1}^{N}\left|F\left(r_{j}\right)-f(r_{j})\right|\)9
Mean squared error
MSE
=\(\frac{1}{N}\sum_{j=1}^{N}{[F\left(r_{j}\right)-f(r_{j})]}^{2}\)10
Root mean square error
RMSE =\(\sqrt{\frac{1}{N}\sum_{j=1}^{N}{[F(r_{j})-f(r_{j})]}^{2}}\)11
Model efficiency
ME = 1 -\(\frac{\sum_{j=1}^{N}{[F\left(r_{j}\right)\ -\ f\left(r_{j}\right)]}^{2}}{\sum_{j=1}^{N}{[f\left(r_{j}\right)-\ \overline{f(r)}]}^{2}}\)12
Coefficient of determination
R2 =\(\left\{\frac{\left[\sum_{j=1}^{N}{F\left(r_{j}\right)-\overline{f(r)}}\right]\left[\sum_{j=1}^{N}{f\left(r_{j}\right)-\ \overline{f(r)}}\right]}{\sqrt{\sum_{j=1}^{N}{[F\left(r_{j}\right)\ -\overline{f(r)}\ ]}^{2}}\sqrt{\sum_{j=1}^{N}{[f\left(r_{j}\right)\ -\overline{f(r)}\ ]}^{2}}}\right\}^{2}\)13
Where N is the number of observed data points
f(rj) is the observed value at station
F(rj) is the value estimated at station