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Reliability of several statistical tests for detecting changes in extreme precipitation events

Reliability of several statistical tests for detecting changes in extreme precipitation events. Liu Kejing . Wang Wen Hohai University x13a13@yahoo.com.cn. 5 th International Symposium on IWRM 3 rd International Symposium on Methodology in Hydrology. Contents. Back ground Method

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Reliability of several statistical tests for detecting changes in extreme precipitation events

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  1. Reliability of several statistical tests for detecting changes in extreme precipitation events Liu Kejing . Wang Wen Hohai University x13a13@yahoo.com.cn 5th International Symposium on IWRM 3rd International Symposium on Methodology in Hydrology

  2. Contents Back ground Method Analysis Conclusion

  3. Back ground Precipitation is an important factor of climate and hydrological cycling, its change reflects climate change. The reliability of change tests is still not clear, to estimate their effects in different precipitation indices is also a problem.

  4. Method Precipitation series generation One order Markov chain model Calculate precipitation amount in Gamma distribution Scenarios & change designation 2 scenarios Secular change Step change

  5. Select precipitation extreme indices • 4 indices • Statistical tests to secular change: • Mann-Kendall (MK) test • Regional average MK (RAMK) test • Statistical tests to step change: • Kolmogrov-Smirnov (KS) test • Bayesian change point (BCP) test

  6. Markov chain model (Gabriel & Neumann, 1962)have 2 statements: Wet (rainfall depth>0.1mm); Dry (rainfall depth≤0.1mm) Precipitation occur at the day: r>P(W|W) else r>P(W|D) where r∈[0, 1] Rainfall generation

  7. Precipitation amounts in wet days follow a Gamma distribution Gamma probability density function

  8. Scenarios & change designation 2 scenarios Beijing Haikou parameters designed refers to Liao et al. (2004) Secular change: Change taking gradually in the whole series Step change: suddenly change occurs at some position of the series

  9. Parameters of the two scenarios

  10. Secular change & Step change

  11. The precipitation extreme indices

  12. Secular change test The method: MK test, a nonparametric statistical test (Mann, 1945; Kendall, 1975) RAMK test, regional average MK test (Douglas et al., 2000) Test results

  13. Results of MK test to secular change series

  14. “+” means the total of significant increase trend; “-” means the total of significant decrease trend. Our test significance level is 0.05 here.

  15. When the series length is 10a, β constant or increase with rate 0. 1%/a, no series show significant trend; Significance of increase trend increase with series length increase; In same length, R95p & Pr are more significant than RX1day & RX5day; The larger the increase slop of β, the more significant increase trends, and less decrease trends.

  16. Regional secular change test Why? For trend of a single station can not present trend of a whole region, we need to detect the trend of region and to estimate effect of the test How? Virtual region generation: a virtual “Region” include several “stations”, a station means a set of parameters Apply RAMK to virtual regions The result 16

  17. Region generation 2 “regions”: one generated from Scenario B (RB) another generated from Scenario H (RH) “Stations” generation assumption: stations in a region (not very large) have same distribution same trend in extreme indices similar parameters 17

  18. Any station’s parameters are random numbers in ranges of: • α’ ∈[α-0.03, α+0.03]; • β’ ∈[β-1.5, β+1.5]; • P(W|D)’ ∈[P(W|D)-0.05, P(W|D)+0.05]; • P(W|W)’ ∈[P(W|W)-0.05,P(W|W)+0.05] • α, β, P(W|D), P(W|W) follows the designed parameters of the two scenarios 18

  19. Results of RAMK test to RB 19

  20. Results of RAMK test to RH 20

  21. To 10a series, the effect of RAMK is similar with that of MK test to single stations, both are not significant; Its significance increase with series length increase, Pr is the most significant indice; To same length & slop, more “stations” makes the test more significant; Indices of RH have more significant increase trends and decrease trends than that of RB. 21

  22. Step change test The methods KS test, a nonparametric test for comparing two samples to determine whether they follow the same distribution; BCP test (Ebarry & Hartigan, 1993), based on the Bayesian statistics. The results

  23. KS test result

  24. “L1” refers length of the series before change ; “L2” refers that after change. “*” means total of significant in KS test in level 0.05

  25. Test significance depends on the length of series too; Change with increase magnitude 20% generates more significance in every indices; R95p & Pr show more significant than RX1day & RX5day in same series length & increase magnitude 25

  26. BCP test result

  27. “s” is the systematical deviation of the change point detected by BCP test and the designed position, negative means the detected ones precede to the designed ones. “#” is the SD of BCP test.

  28. Most of the results, their detected change point (by BCP test) are earlier than the designed position, thus most of the systematical deviation are negatives; The results are with large SDs; Effect of test depends on the change magnitude more than series length

  29. Conclusion About the tests to secular change Effect of MK test depends on series type series length slop of designed trend Effect of RAMK can differentiate regions under different climate; depends on series type series length slop of designed trend amount of stations in a region

  30. About the tests to step change • Effect of KS test depends on • series type • series length • the change magnitude • Effect of BCP test • tend to detect the points where in front of the actual ones • with large SD • To all case, Pr is the most significant indice

  31. Thank You

  32. Appendix All programs use in this study are based on the packages of R; Method of MK test: Let xi, i∈[1, n] represent n data points where xj represents the data point at time j, then the MK statistic S is given by

  33. where A high positive value of S indicates an increase trend, and a low negative value indicates a decrease trend; Calculate the variance of S, VAR(S), by:

  34. Compute a normalized test statistic Z as follows: Compute the probability density function as follows:

  35. Method of KS test: for goodness of fit usually involves examining a random sample from some unknown distribution in order to test the null hypothesis that the unknown distribution function is in fact a known, specified function. Let S(x)be the empirical distribution function based on the random sample xi, i∈[1, n]; let F*(x) be a completely specified hypothesized distribution function

  36. KS test statistic T is the greatest vertical distance between S(x)and F*(x), compute as follow: For testing H0: F(x) = F*(x) for all x ∈(- ∞, ∞) H1: F(x) ≠ F*(x) for at least one value of x

  37. Method of BCP test refers to : D.Barry & J.A.Hartigan (1993) A Bayesian analysis for change point problems. J. Am. Stat. Assoc., 88,309-319 C.Erdman & J.W.Emerson (2008) A fast Bayesian change point analysis for the segmentation of microarray data. Bioinformatics., 24,2143-2148

  38. RAMK test refers to : E.M.Douglas, R.M.Vogel, C.N.Kroll (2000) Trends in floods and flows in the United States: impact of spatial correlation. J. Hydrology., 240, 90-105 Basically, it is a average to statistics of MK test, S and VAR(S), its Z calculates same with MK test

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