mult.mk.test: Multivariate (Multisite) Mann-Kendall Test
In trend: Non-Parametric Trend Tests and Change-Point Detection

mult.mk.test

R Documentation

Multivariate (Multisite) Mann-Kendall Test

Description

Performs a Multivariate (Multisite) Mann-Kendall test.

Usage

mult.mk.test(x, alternative = c("two.sided", "greater", "less"))

Arguments

`x`	a time series object of class "ts"
`alternative`	the alternative hypothesis, defaults to `two.sided`

Details

The Mann-Kendall scores are first computed for each variate (side) seperately.

S = \sum_{k = 1}^{n-1} \sum_{j = k + 1}^n \mathrm{sgn}\left(x_j - x_k\right)

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

The variance - covariance matrix is computed according to Libiseller and Grimvall (2002).

\Gamma_{xy} = \frac{1}{3} \left[K + 4 \sum_{j=1}^n R_{jx} R_{jy} - n \left(n + 1 \right) \left(n + 1 \right) \right]

with

K = \sum_{1 \le i < j \le n} \mathrm{sgn} \left\{ \left( x_j - x_i \right) \left( y_j - y_i \right) \right\}

and

R_{jx} = \left\{ n + 1 + \sum_{i=1}^n \mathrm{sgn} \left( 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 \le 10:

z = \frac{\sum_{i=1}^d S_i}{\sqrt{\sum_{j=1}^d \sum_{i=1}^d \Gamma_{ij}}}

where

z denotes the quantile of the normal distribution S is the vector of Mann-Kendall scores for each variate (site) 1 \le i \le d and \Gamma denotes symmetric variance - covariance matrix.

Value

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

Note

Ties are not corrected. Current Version is for complete observations only.

References

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, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/env.507")}.