Methodology - Response of Flood Events to Land Use and Climate Change - page 88

3.12.3 Detection of Trend: Mann-Kendall Test

The Mann-Kendall test was applied by many researchers and investigators to
estimate the trend in climatological and hydrological parameters from the past to
present [ 104 , 107 , 114 - 117 ]. This method as a nonparametric test identifies trends
in time series data [ 118 ]. The Mann-Kendall method is widely used in environ-
mental and natural science as a simple, robust method which can cope with
missing values and values below a detection limit [ 110 , 119 ]. The first original
proposals of this test derived by Mann [ 105 ] and Kendall [ 106 ], the covariance
between the Mann-Kendall statistics, were suggested by Dietz and Killen [ 120 ].
Then, the test was developed in order to include seasonality by Hirsch and Slack
[ 103 ], multiple monitoring sites were proposed by Lettenmaier [ 121 ] and
covariates representing natural fluctuations were considered by Libiseller and
Grimvall [ 119 ].
Khambhammettu [ 122 ] realized that one of the advantages of this test is that the
data does not need to conform to any particular distribution. Moreover, data as
nondetects can be included as a common value that is smaller than the smallest
measured value in the data set. Also, the data values are evaluated as an ordered
time series. Each data value is compared to all subsequent data values. As men-
tioned above, the nonparametric Mann-Kendall rank correlation test [ 123 ] was
applied to detect any possible trend in precipitation, discharge, and temperature
series, and to test whether or not such trends are statistically significant [ 124 ]. This
test, usually known as Kendall's s statistics, has been used in hydrology and
climatology to test randomness against trends of hydrologic time series. As it is a
rank-based procedure, it is robust to the influence of extremes and a good test for
skewed data. Kendall [ 106 ] reported a normal approximation test that could be
used for data sets with more than 10 values, provided there are not many tied
values within the data set. The procedure is as follows:
For the nonparametric Mann-Kendall method, for any sample of n variables,
x 1 ,…, x n , the null hypothesis (H 0 ) indicates that the sample is independent and
distributed randomly. The alternative hypothesis of a two-sided test (H 1 ) also
indicates that x i and x j are not identical for all k & j B n with i = j.
The Mann-Kendall test is based on test statistic S, with a zero mean and
computed variance from Eqs. ( 3.45 ) and ( 3.46 )[ 125 ] as:

S ¼ X

X

n i 1

sign ð x i x j Þ ð 3 : 43 Þ

i ¼ 2 j ¼ 1

8
<

þ 1 f ð x i x j Þ S [ 0
0if ð x i x j Þ S ¼ 0
1 f ð x i x j Þ S\0

sgn ð x j x jk Þ¼ ð 3 : 44 Þ

: