pbmk {modifiedmk}

Bootstrapped Mann-Kendall Trend Test with Optional Bias Corrected Prewhitening

Description

The empirical distribution of the Mann-Kendall test statistic is calculated by bootstrapped resampling. The Hamed (2009) bias correction prewhitening technique can optionally be applied as the default for prewhitening before the bootstrapped Mann-Kendall test is applied (Lacombe et al., 2012).

Usage

pbmk(x, nsim=1000, pw="Hamed")

Arguments

x

- Time series data vector

nsim

- Number of bootstrapped simulations

pw

- Optional bias corrected prewhitening suggested by Hamed (2009)

Details

Bootstrapped samples are calculated by resampling one value at a time from the time series with replacement. The p-value (p_s) of the resampled data is estimated by (Yue and Pilon, 2004):

p_s = m_s/M

The Mann-Kendall test statistics (S) is calculated for each resampled dataset. The resultant vector of resampled S statistics is then sorted in ascending ordering, where p_s is the rank corresponding the largest bootstrapped value of S being less than the test statistic value calculated from the actual data. M is the total number of bootstrapped resamples. The default value of M is 1000, however, Yue and Pilon (2004) suggest values between 1000 and 2000. If the user does not choose to apply prewhitening, this argument 'pw' can be set to NULL.

Value

Z Value - Mann-Kendall Z statistic from original data

Sen's Slope - Sen's slope from the original data

S - Mann-Kendall S statistic

Kendall's Tau - Mann-Kendall's Tau

BCP Z Value - Bias corrected prewhitened Z value

BCP Sen's Slope - Bias corrected prewhitened Sen's slope

BCP S - Bias corrected prewhitened S

BCP Kendall's Tau - Bias corrected prewhitened Kendall's Tau

Bootstrapped P-Value - Mann-Kendall bootstrapped p-value

References

Hamed, K. H. (2009). Enhancing the effectiveness of prewhitening in trend analysis of hydrologic data. Journal of Hydrology, 368: 143-155.

Kendall, M. (1975). Rank Correlation Methods. Griffin, London, 202 pp.

Kundzewicz, Z. W. and Robson, A. J. (2004). Change detection in hydrological records - a review of the methodology. Hydrological Sciences Journal, 49(1): 7-19.

Lancombe, G., McCartney, M., and Forkuor, G. (2012). Drying climate in Ghana over the period 1960-2005: evidence from the resampling-based Mann-Kendall test at local and regional levels. Hydrological Sciences Journal, 57(8): 1594-1609.

Mann, H. B. (1945). Nonparametric Tests Against Trend. Econometrica, 13(3): 245-259.

van Giersbergen, N. P. A. (2005). On the effect of deterministic terms on the bias in stable AR models. Economic Letters, 89: 75-82.

Yue, S. and Pilon, P. (2004). A comparison of the power of the t test, Mann-Kendall and bootstrap tests for trend detection, Hydrological Sciences Journal, 49(1): 21-37.

Examples

x<-c(Nile[1:10])
pbmk(x)