Seasonal Mann-Kendall Trend Test
Performs a Seasonal Mann-Kendall Trend Test (Hirsch-Slack Test)
smk.test(x, alternative = c("two.sided", "greater", "less"), continuity = TRUE)x |
a time series object with class |
alternative |
the alternative hypothesis, defaults to |
continuity |
logical, indicates, whether a continuity correction
should be done; defaults to |
The Mann-Kendall statistic for the $g$-th season is calculated as:
S_g = ∑_{i = 1}^{n-1} ∑_{j = i + 1}^n \mathrm{sgn}≤ft(x_{jg} - x_{ig}\right), \qquad (1 ≤ g ≤ m)
with sgn the signum function (see sign).
The mean of S_g is μ_g = 0. The variance including the correction term for ties is
σ_g^2 = ≤ft\{n ≤ft(n-1\right)≤ft(2n+5\right) - ∑_{j=1}^p t_{jg}≤ft(t_{jg} - 1\right)≤ft(2t_{jg}+5\right) \right\} / 18 ~~ (1 ≤ g ≤ m)
The seasonal Mann-Kendall statistic for the entire series is calculated according to
\begin{array}{ll} \hat{S} = ∑_{g = 1}^m S_g & \hat{σ}_g^2 = ∑_{g = 1}^m σ_g^2 \end{array}
The statistic S_g is approximately normally distributed, with
z_g = S_g / σ_g
If continuity = TRUE then a continuity correction will be employed:
z = \mathrm{sgn}(S_g) ~ ≤ft(|S_g| - 1\right) / σ_g
An object with class "htest" and "smktest"
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 |
Sg |
numeric vector that contains S scores for each season |
varSg |
numeric vector that contains varS for each season |
pvalg |
numeric vector that contains p-values for each season |
Zg |
numeric vector that contains z-quantiles for each season |
Hipel, K.W. and McLeod, A.I. (1994), Time Series Modelling of Water Resources and Environmental Systems. New York: Elsevier Science.
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.
R. Hirsch, J. Slack, R. Smith (1982), Techniques of Trend Analysis for Monthly Water Quality Data, Water Resources Research 18, 107–121.
res <- smk.test(nottem) ## print method res ## summary method summary(res)
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