Multivariate (Multisite) Mann-Kendall Test
Performs a Multivariate (Multisite) Mann-Kendall test.
mult.mk.test(x, alternative = c("two.sided", "greater", "less"))x |
a time series object of class "ts" |
alternative |
the alternative hypothesis, defaults to |
The Mann-Kendall scores are first computed for each variate (side) seperately.
S = ∑_{k = 1}^{n-1} ∑_{j = k + 1}^n sgn(x[j] - x[k])
with sgn the signum function (see sign).
The variance - covariance matrix is computed according to Libiseller and Grimvall (2002).
Γ_{xy} = \frac{1}{3} ≤ft[K + 4 ∑_{j=1}^n R_{jx} R_{jy} - n ≤ft(n + 1 \right) ≤ft(n + 1 \right) \right]
with
K = ∑_{1 ≤ i < j ≤ n} \mathrm{sgn} ≤ft\{ ≤ft( x_j - x_i \right) ≤ft( y_j - y_i \right) \right\}
and
R_{jx} = ≤ft\{ n + 1 + ∑_{i=1}^n \mathrm{sgn} ≤ft( x_j - x_i \right) \right\} / 2
Finally, the corrected z-statistics for the entire series is calculated as follows, whereas a continuity correction is employed for n <= 10:
z = sum(S) / sum(Γ)
where
z denotes the quantile of the normal distribution S is the vector of Mann-Kendall scores for each variate (site) 1 <= i <= d and Γ denotes symmetric variance - covariance matrix.
An object with class "htest"
data.name |
character string that denotes the input data |
p.value |
the p-value for the entire series |
statistic |
the z quantile of the standard normal distribution for the entire series |
null.value |
the null hypothesis |
estimates |
the estimates S and varS for the entire series |
alternative |
the alternative hypothesis |
method |
character string that denotes the test |
cov |
the variance - covariance matrix |
Ties are not corrected. Current Version is for complete observations only.
Hipel, K.W. and McLeod, A.I. (1994), Time Series Modelling of Water Resources and Environmental Systems. New York: Elsevier Science.
Lettenmeier, D.P. (1988), Multivariate nonparametric tests for trend in water quality. Water Resources Bulletin 24, 505–512.
Libiseller, C. and Grimvall, A. (2002), Performance of partial Mann-Kendall tests for trend detection in the presence of covariates. Environmetrics 13, 71–84, http://dx.doi.org/10.1002/env.507.
data(hcb) mult.mk.test(hcb)
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