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Statistical Modelling of Extreme Rainfall Indices using Multivariate Extreme Value Distributions

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Abstract

Multivariate extreme value models are used to investigate the combined behaviour of several weather variables. To investigate joint dependence of extreme rainfall events, a multivariate conditional modelling approach was considered to analyse the behaviour of joint extremes of rainfall events at selected weather stations in South Africa. Moreover, 1-day to 5-day indices of rainfall events were constructed to investigate the frequencies and intensities of rainfall events for selected weather stations. Then, the conditional multivariate modelling was fitted to investigate dependence between series of extreme rainfall events. The conditional multivariate modelling has provided all forms of dependence, using Laplace marginal transformations, for which all weather stations are not equally extreme. Bootstrap sampling was also employed to account for models uncertainty in computing the prediction standard errors and compared with the prediction obtained from the conditional modelling that was fitted to extreme data. The results obtained from predictions reflected both the marginal and the dependence features, as well as the extremal dependence structure described consistently for indices of rainfall events between weather stations. The modelling framework and results of this study contribute towards understanding the salient features on the extremal dependence of rainfall extremes which are associated with, e.g. flash floods and landslides. This knowledge has practical applications in disaster risk preparedness by communities.

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Acknowledgements

The authors are very indebted to the University of South Africa for the financial support. The authors also would like to thank the South African Weather Service for providing the data.

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Authors and Affiliations

  1. Department of statistics, University of South Africa, Floor 6, Florida Park, Roodepoort, Private Bag X6, Florida, 1710, South Africa

    Tadele Akeba Diriba & Legesse Kassa Debusho

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  1. Tadele Akeba Diriba
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  2. Legesse Kassa Debusho
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Contributions

TAD and LKD conceptualised the research. TAD obtained the data and cleaned it. TAD and LKD analysed the data, and TAD drafted the original manuscript. TAD and LKD reviewed the manuscript. TAD and LKD read and approved the final manuscript. TAD and LKD approved the final manuscript and agreed to submit it for publication.

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Correspondence to Tadele Akeba Diriba.

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Appendices

A Appendix

Details on the bootstrapping procedure are given as follows:

  1. (1)

    Convert the recorded extreme data to have Laplace margins following the marginal transformation techniques.

  2. (2)

    Calculate the nonparametric bootstrap sample by sampling with replacement from the new dataset obtained in step 1.

  3. (3)

    Convert the marginal values of the bootstrap sample obtained in step 2, provided the marginal distributions are all Laplace and preserving the connections between the ranked points in each marginal component. Particularly, for each \(i=1, \ldots , d\), replace the ordered sample of the component \(Y_i\) margin with the ordered sample of the component from the standard Laplace marginally transformed distribution.

  4. (4)

    Transform the resulting sample obtained in step 3 back to the original margins by using the Laplace marginal transformation technique that was estimated from the first recorded extreme data. Note that the new extreme data that are produced following this approach have univariate marginal distributions for which the exceedances are simulated from the fitted GPD model. Also, the dependence structures are entirely consistent with the extreme data as determined by the dependence between components of the variables.

B Appendix

Fig. 5
figure 5

Profile log-likelihood surface for indices of rainfall events for Cape Point, Durban, George, Port Elizabeth and Ulundi from top to bottom, respectively. (\(\alpha _{R_2|R_1},\beta _{R_2|R_1}\)), (\(\alpha _{R_3|R_1},\beta _{R_3|R_1}\)), (\(\alpha _{R_4|R_1},\beta _{R_4|R_1}\)), (\(\alpha _{R_5|R_1},\beta _{R_5|R_1}\)); \(\mathbf{a }- {\alpha }\), \(b- {\beta }\) left to right row, respectively

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Diriba, T.A., Debusho, L.K. Statistical Modelling of Extreme Rainfall Indices using Multivariate Extreme Value Distributions. Environ Model Assess 26, 543–563 (2021). https://doi.org/10.1007/s10666-021-09766-6

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