Methodology - Response of Flood Events to Land Use and Climate Change - page 90

3.12.4 Sequential Mann-Kendall Test

The sequential Mann-Kendall test Sneyers [ 127 ] is used to test an assumption
about the beginning of the development of a trend within a sample (x 1 , …, x n )of
random variable x based on a rank series of progressive and retrograde rows of the
sample. To see change of a trend with time, Sneyers [ 123 ] introduced sequential
values, u(t) and u 0 (t), from the progressive analysis of the Mann-Kendall test. The
Mann-Kendall test statistic u(t) is a value that indicates directions (or signs) and a
statistical magnitude of the trend in a series. When the value of u(t) is significant at
a 5 % significant level, it can be decided whether it is an increasing or a decreasing
trend, depending on whether u(t) [ 0 or u(t) \ 0. A 1 % level of significance was
also taken into consideration. Partial and short—period trends, and a change point
or beginning point of a trend in climatic series were investigated by using time—
series plot of the u(ti) and u 0 (ti) values. In order to obtain such a time- series plot,
sequential values of the statistics u(t) and u 0 (ti) were computed from the pro-
gressive analysis of the Mann-Kendall test. The following steps are applied to
calculate u(t) and u 0 (t):
First, original observations are replaced by their corresponding ranks y i , which
are arranged in ascending order. Then, for each term y i , the number of terms y i
preceding (i [ j) is calculated with (y i [ y j ). The values of x j annual mean time
series, (j = 1,…,n) are compared with x i ,(i = 1,…, j - 1). At each comparison,
the number of cases x j [ x i is counted and denoted by n j .
The test statistic t is then calculated by equation

t j ¼ X

j

n j ð 3 : 48 Þ

1

The distribution function of the test statistic t has a mean and a variance of the
test statistic respectively, given as [ 128 ]:

E ðÞ¼ n ð n 1 Þ
4

ð 3 : 49 Þ

Var t j ¼ jj 1

½ ð Þð 2j þ 5 Þ= 72 ð 3 : 50 Þ

The sequential values of the statistic u(t) are then computed as

U ðÞ¼ t j E ð t Þ p

Var ð t j Þ

ð 3 : 51 Þ

Finally, the values of u 0 (t) are similarly computed backward, starting from the
end of the series using the same equation but in the inverse series of data.
In two-sided trend test, the null hypothesis is accepted at a significance level if
U ð t j j U ð t Þ 1 ¼ a = 2 ; where U ð t Þ 1 ¼ a = 2 is the critical value of standard normal dis-
tribution with a probability exceeding a = 2 : Positive U(t) denotes a positive trend