59
where n is the quantity of data points, sign signifies the signum work, x j and x k
denotes the data points of time j and k,
!
1
0
0
0
1
0
if
if
if
x x
x x
x x
x x
j
k
j
k
j
k
j
k
sgn
(3.6)
The variance of S, VAR(S) is calculated as:
VAR S
n n
n
t t
t
g
i
i
i
i
¦
­
®
¯
½
¾
¿
1
18
1 2 5
1 2
5
1
(3.7)
where g denotes the tied groups number and t i indicates the extent i
th
ties number. A
set of sample data with similar value is called the tied groups.
The time series data Statistic (S) is identified with Kendall’s τ (tau) which given
as follows:
W
S
D
(3.8)
where
D
n n
t t
n n
g
i
i
i
¦
1
2
1
1
2
1
1
2
1
1
The estimation of (S) and VAR(S) is used to develop the standardized test measurement Z as follows when n > 10 (Gilbert 1987)
Z
S
S
S
S
S
S
S
!
{
,
,
,
1
0
0
0
1
0
VAR
if
if
VAR
if
(3.9)
The positive and negative value of Z specifies increasing and decreasing trends in
the time series. To test null hypothesis (H 0 ), of no trend against the alternative
hypothesis (H a ), of an upward or descending pattern at the α = 0.05 level of significance, H 0 is rejected if the estimation of Z is more than 1.96 which is obtained from
the standard ordinary distribution table (Pearson and Hartley 1966).
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