# Mann-Kendall trend test This is a test for monotonic trend in a time series z[t] based on the Kendall rank correlation of z[t] and t. ## Details The test was suggested by Mann (1945) and has been extensively used with environmental time series (Hipel and McLeod, 2005). For autocorrelated time series, the block bootstrap may be used to obtain an improved signficance test. ```r MannKendall(x) ``` ## Arguments - `x`: a vector of data, often a time series ## Returns A list with class Kendall. - **tau**: Kendall's tau statistic - **sl**: two-sided p-value - **S**: Kendall Score - **D**: denominator, tau=S/D - **varS**: variance of S Generic function print.Kendall and summary.Kendall are provided to print the output. ## Note If you want to use the output from MannKendall, save the result as in res<-MannKendall(x,y) and then select from the list res the value(s) needed. ## References Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and Their Application. Cambridge University Press. Hipel, K.W. and McLeod, A.I., (2005). Time Series Modelling of Water Resources and Environmental Systems. Electronic reprint of our book orginally published in 1994. [http://www.stats.uwo.ca/faculty/aim/1994Book/](http://www.stats.uwo.ca/faculty/aim/1994Book/). Mann, H.B. (1945), Nonparametric tests against trend, Econometrica, 13, 245-259. ## Author(s) A.I. McLeod, aimcleod@uwo.ca ## See Also `Kendall`, `SeasonalMannKendall` `tsboot` ## Examples ```r # Annual precipitation entire Great Lakes # The time series plot with lowess smooth suggests an upward trend # The autocorrelation in this data does not appear significant. # The Mann-Kendall trend test confirms the upward trend. data(PrecipGL) plot(PrecipGL) lines(lowess(time(PrecipGL),PrecipGL),lwd=3, col=2) acf(PrecipGL) MannKendall(PrecipGL) # #Use block bootstrap library(boot) data(PrecipGL) MKtau<-function(z) MannKendall(z)$tau tsboot(PrecipGL, MKtau, R=500, l=5, sim="fixed") # # Deseasonalize a monthly time series and use the block bootstrap library(boot) data(manaus) z<-matrix(manaus, ncol=12, byrow=12) zm<-apply(z, MARGIN=2, FUN=mean) zs<-apply(z, MARGIN=2, FUN=sd) z2<-sweep(z, MARGIN=2, STATS=zm) #subtract monthly means z3<-sweep(z2, MARGIN=2, STATS=zs, FUN="/") #divide by monthly sd zds<-c(t(z3)) attributes(zds)<-attributes(manaus) plot(zds) #do Mann-Kendall trend test MannKendall(zds) #check robustness by applying block bootstrap MKtau<-function(z) MannKendall(z)$tau tsboot(zds, MKtau, R=500, l=12, sim="fixed") #the significance level has dropped! #it is significant now at about 2 sd ```

MannKendall function