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.