2.4 The Modified Mann-Kendall (MKK) test
Hamed and Rao (1998) pioneered the MKK test was used which take in accordance all auto-correlation coefficients in time-series datasets. In this test, modified variance (Var(S)) is used for calculating statistics of the most common MK test (Islam et al. 2020a; Praveen et al. 2020). The following Eqs. (3-5) was used to calculate the modified coefficient of autocorrelation datasets:
\(\text{Var}\left(S\right){}=Var\left(S\right)\times\left(\frac{n}{n{}}\right)\)…………………………………………………………..(3)
\(\left(\frac{n}{n{}}\right)=1+\left(\frac{2}{n(n-1)(n-2)}\right)\times\sum_{k=1}^{n-1}{(n-k)(n-k-1)(n-k-2)}^{r}\text{.k}\)…………. (4)
\(r_{k\ =}\frac{\left(\frac{1}{n-k}\right)\sum_{i=1}^{n-k}{(x_{i}-x)(x_{i+k}-x)}}{\left(\frac{1}{n}\right)\sum_{i=1}^{n}{\ ({x_{i}-x)}^{2}\text{\ \ \ }}}\)…………………………………………………………(5)
  Where, Var(S) is calculated via the Eq. (3), nn* represents the modified coefficient of autocorrelated data,\(\ r_{\text{k\ }}\) represents the autocorrelation coefficient of kth, and x represents the mean of time series. The significance of autocorrelation coefficient of k that 95 % confidence level can be calculated by the following Eq.(6):
\(\left(\frac{-1-1.96\sqrt{n=k-1}}{n-k}\right)\leq r_{\text{k\ }}(95\%)\leq\left(\frac{-1-1.96\sqrt{n=k-1}}{n-k}\right)\)………………………………. (6)
If the attained \(r_{\text{k\ }}\) follows the above situation, 95 % confidence level data can be obtained. Else, the data are dependent and the impact of autocorrelation coefficient with different lags should be eradicated to estimate the trend of time series. Lastly, a trend in time series data can obtained if Z is less or greater than the Z of standard normal distribution at 95 % confidence level.