smk.test: Seasonal Mann-Kendall Trend Test

In trend: Non-Parametric Trend Tests and Change-Point Detection

Seasonal Mann-Kendall Trend Test

Description

Performs a Seasonal Mann-Kendall Trend Test (Hirsch-Slack Test)

Usage

smk.test(x, alternative = c("two.sided", "greater", "less"), continuity = TRUE)

Arguments

x	a time series object with class ts comprising >= 2 seasons; NA values are not allowed
alternative	the alternative hypothesis, defaults to two.sided
continuity	logical, indicates, whether a continuity correction should be done; defaults to TRUE

Details

The Mann-Kendall statistic for the $g$-th season is calculated as:

S_g = \sum_{i = 1}^{n-1} \sum_{j = i + 1}^n \mathrm{sgn}\left(x_{jg} - x_{ig}\right), \qquad (1 \le g \le m)

with \mathrm{sgn} the signum function (see sign).

The mean of S_g is \mu_g = 0. The variance including the correction term for ties is

\sigma_g^2 = \left\{n \left(n-1\right)\left(2n+5\right) - \sum_{j=1}^p t_{jg}\left(t_{jg} - 1\right)\left(2t_{jg}+5\right) \right\} / 18 ~~ (1 \le g \le m)

The seasonal Mann-Kendall statistic for the entire series is calculated according to

\begin{array}{ll} \hat{S} = \sum_{g = 1}^m S_g & \hat{\sigma}_g^2 = \sum_{g = 1}^m \sigma_g^2 \end{array}

The statistic S_g is approximately normally distributed, with

z_g = S_g / \sigma_g

If continuity = TRUE then a continuity correction will be employed:

z = \mathrm{sgn}(S_g) ~ \left(|S_g| - 1\right) / \sigma_g

Value

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

References

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, \Sexpr[results=rd]{tools:::Rd_expr_doi("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.

Examples

res <- smk.test(nottem)
## print method
res
## summary method
summary(res)

trend documentation built on Oct. 10, 2023, 9:06 a.m.