Statistical Analysis of Trends in the Red River Over a 45 Year Period

The N ′
individual slope estimates, Q, are computed for each time period:
where
Q = x′ i − x i
i ′ − i ,
x ′ i and x i are the data values at time i ′
and i (in days), respectively, i ′ > 1 and;
N ′
is the number of observations in the pth group
Sen’s slope estimator is then calculated by choosing the middle-ranked slope as
follows;
⎧
⎪⎨ Q [N ′ =n(n−1)/2] if N ′ is odd
(
)
1 ⎪⎩ Q ′
2
N
+ Q ′ N =n(n+2)
if N ′
is even;
2
2
where n is the number of time periods;this value is multiplied by 365 to give the
yearly slope value.
The seasonal Mann-Kendall test is an extension of the Mann-Kendall test for
trend that removes seasonal cycles. To compute the seasonal Mann-Kendall statistic,
S i , for each season there must be a minimum sample size of four data points in
each season.
S i =
n∑
i −1
∑n i
k=1 l=k+1
sgn(x il − x ik )
where S i is the statistic for the ith season and
⎧
−1, if x < 0;
⎪⎨
sgn(x) = 0, if x = 0;
⎪⎩
1, if x > 0.
With use of the normal approximation (i.e.: greater than 41 observations) the
Mann-Kendall test statistic, S is calculated. When there are no tied values, the
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