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).
pbmk(x, nsim=1000, pw="Hamed")- Time series data vector
- Number of bootstrapped simulations
- Optional bias corrected prewhitening suggested by Hamed (2009)
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
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.
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.
# NOT RUN {
x<-c(Nile[1:10])
pbmk(x)
# }
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