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Estimation of intensity–duration–frequency curves using max-stable processes

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Abstract

We present an approach to estimate intensity–duration–frequency (IDF) curves based on max-stable processes. The proposed method has been inspired by the seminal study of Nadarajah et al. (J R Stat Soc B 60(2):473–496, 1998), who used a multivariate extreme value distribution (MEVD) to estimate (IDF) curves from rainfall records. Max-stable processes are continuous extensions of MEVD (i.e. the marginal distributions of rainfall maxima at different durations are generalized extreme valued), which are more flexible, allow for extreme rainfall estimation at any arbitrary duration d (i.e. not just a discrete set, as is the case of MEVD), while preserving asymptotic dependencies. The latter characteristic of IDF estimates results from the combined effect of the statistical structure of rainfall (i.e. temporal dependencies), as well as the IDF construction process, which involves averaging of the original series to obtain rainfall maxima at different temporal resolutions. We apply the method to hourly precipitation data, and compare it to empirical estimates and the results produced by a semiparametric approach. Our findings indicate that max-stable processes fit well the statistical structure and inter-dependencies of annual rainfall maxima at different durations, produce slightly more conservative estimates relative to semiparametric methods, while allowing for extrapolations to durations and return periods beyond the range of the available data. The proposed statistical model is fully parametric and likelihood based, providing a theoretically consistent basis in solving the problem at hand.

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Acknowledgements

We thank two anonymous reviewers for their useful suggestions.

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Correspondence to Hristos Tyralis.

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Appendices

Appendix A: Pairwise likelihood

The bivariate density function of the distribution defined in Eq. (4) is given by (Padoan et al. 2010; Huser and Davison 2013):

$$f\left( {z\left( {d_{l} } \right),z\left( {d_{j} } \right)} \right): = f_{1} \left( {z\left( {d_{l} } \right),z\left( {d_{j} } \right)} \right) \times \left[ {f_{2} \left( {z\left( {d_{l} } \right),z\left( {d_{j} } \right)} \right) \times f_{3} \left( {z\left( {d_{l} } \right),z\left( {d_{j} } \right)} \right) + \left( {f_{4} \left( {z\left( {d_{l} } \right),z\left( {d_{j} } \right)} \right) + f_{5} \left( {z\left( {d_{l} } \right),z\left( {d_{j} } \right)} \right)} \right)} \right]$$
(A.1)

where

$$f_{1} \left( {z\left( {d_{l} } \right),z\left( {d_{j} } \right)} \right): = \exp \left( {{-}\varPhi \left( {w\left( {h_{l,j} } \right)} \right)/z\left( {d_{l} } \right) \, {-}\varPhi \left( {v\left( {h_{l,j} } \right)} \right)/z\left( {d_{j} } \right)} \right)$$
(A.2)
$$f_{2} \left( {z\left( {d_{l} } \right),z\left( {d_{j} } \right)} \right) \, : = \varPhi \left( {w\left( {h_{l,j} } \right)} \right)/z^{2} \left( {d_{l} } \right) \, + \varphi \left( {w\left( {h_{l,j} } \right)} \right)/\left( {a\left( {h_{l,j} } \right)z^{2} \left( {d_{l} } \right)} \right) \, {-}\varphi \left( {v\left( {h_{l,j} } \right)} \right)/\left( {a\left( {h_{l,j} } \right)z\left( {d_{l} } \right)z\left( {d_{j} } \right)} \right)$$
(A.3)
$$f_{3} \left( {z\left( {d_{l} } \right),z\left( {d_{j} } \right)} \right): = \varPhi \left( {v\left( {h_{l,j} } \right)} \right)/z^{2} \left( {d_{j} } \right) + \varphi \left( {v\left( {h_{l,j} } \right)} \right)/\left( {a\left( {h_{l,j} } \right)z^{2} \left( {d_{j} } \right)} \right){-}\varphi \left( {w\left( {h_{l,j} } \right)} \right)/\left( {a\left( {h_{l,j} } \right)z\left( {d_{l} } \right)z\left( {d_{j} } \right)} \right)$$
(A.4)
$$f_{4} \left( {z\left( {d_{l} } \right),z\left( {d_{j} } \right)} \right): = \left( {v\left( {h_{l,j} } \right)\varphi \left( {w\left( {h_{l,j} } \right)} \right)} \right)/\left( {a^{2} \left( {h_{l,j} } \right)z^{2} \left( {d_{l} } \right)z\left( {d_{j} } \right)} \right)$$
(A.5)
$$f_{5} \left( {z\left( {d_{l} } \right),z\left( {d_{j} } \right)} \right): = \left( {w\left( {h_{l,j} } \right)\varphi \left( {v\left( {h_{l,j} } \right)} \right)} \right)/\left( {a^{2} \left( {h_{l,j} } \right)z\left( {d_{l} } \right)z^{2} \left( {d_{j} } \right)} \right)$$
(A.6)

In eqs. (A.1)–(A.6), φ denotes the standard normal density function, a is given by Eq. (6), and w and v are defined as:

$$w\left( h \right): = a\left( h \right)/2 + \log \left( {z\left( {d_{j} } \right)/z\left( {d_{i} } \right)} \right)$$
(A.7)
$$v\left( h \right): = a\left( h \right){-}w\left( h \right)$$
(A.8)

Let i(d) ~ GEV(μ(d), σ(d), ξ(d)) be modelled by a Brown–Resnick process, and suppose that i(d) is observed at dj, j = 1, …, m, with n years of available observations for each j. Variables z(d) in the bivariate density (A.1) are unit Fréchet distributed, therefore, i(d) should be transformed using Eq. (8). Then the bivariate densities for i(d) are given by:

$$f_{{\underline{i}_{l} ,\underline{i}_{j} }} ,(i(d_{l} ), \, i(d_{j} )) = f_{{\underline{z}_{l} ,\underline{z}_{j} }} \left( {g^{ - 1} (i(d_{l} ), \, i(d_{j} ))} \right)\left| {J(i(d_{l} ),i(d_{j} ))} \right|$$
(A.9)

where g is defined by the bijection (i(dl), i(dj)) = g(z(dl), z(dj)), and J is the Jacobian:

$$\left| {J\left( {i\left( {d_{l} } \right),i\left( {d_{j} } \right)} \right)} \right| \, = \, \left( {1/\left( {\sigma \left( {d_{l} } \right)\sigma \left( {d_{j} } \right)} \right)} \right) \, \left( {1 \, + \xi \left( {i\left( {d_{i} } \right) \, {-}\mu \left( {d_{i} } \right)} \right)/\sigma \left( {d_{i} } \right)} \right)_{ + }^{1/\xi - 1} \times \, \left( {1 \, + \xi \left( {i\left( {d_{j} } \right) \, {-}\mu \left( {d_{j} } \right)} \right)/\sigma \left( {d_{j} } \right)} \right)_{ + }^{1/\xi - 1}$$
(A.10)

Define now:

$$L_{r,l,j} \left( {\varvec{\theta}|i_{r} \left( {d_{l} } \right),i_{r} \left( {d_{j} } \right)} \right): = \log \left( {f\left( {i_{r} \left( {d_{l} } \right),i_{r} \left( {d_{j} } \right)|\varvec{\theta}} \right)} \right),\quad r = 1, \ldots ,n\quad {\text{and}}\quad l,j = 1, \ldots ,m$$
(A.11)

then the log-pairwise likelihood is given by:

$$L\left( {\varvec{\theta}|i_{1} \left( {d_{1} } \right), \ldots ,i_{n} \left( {d_{m} } \right)} \right): = \sum_{r = 1, \ldots ,n} \sum_{l,j,l < j} L_{r,l,j} \left( {\varvec{\theta}|i_{r} \left( {d_{l} } \right),i_{r} \left( {d_{j} } \right)} \right)$$
(A.12)

When using the response surfaces defined in Eqs. (11), (14) and (15), the parameter vector \(\varvec{\theta}= \{ \rho ,\alpha ,\mu_{0} ,\sigma_{0} ,\xi ,c\}\) can be estimated by maximizing the log-pairwise likelihood in eq. (A.12). We note that the maximum pairwise likelihood estimator has similar asymptotic properties to the full likelihood estimator. Furthermore, the number m of the discrete durations dj, j = 1, …, m used to compute the pairwise likelihood does not affect the properties of the estimator (Padoan et al. 2010).

Appendix B: Extremal coefficient and F-madogram

To describe the dependence structure of simple max-stable processes, one usually uses the F-madogram vF(h), instead of the semivariogram, because the latter is not defined for processes with unit Fréchet marginals (see e.g. Ribatet 2013). The F-madogram is defined as (Cooley et al. 2006).

$$v_{F} \left( h \right): = \left( {1/2} \right)E\left[ {F_{d + h} (\underline{i}\left( {d + h} \right)) \, {-}F_{d} (\underline{i}\left( d \right))} \right],\quad d > 0,\quad h \ge 0$$
(B.1)

where Fd denotes the CDF of random variable i(d). In general, 0 ≤ vF(h) ≤ 1/6 ∀ h ≥ 0, with vF(h) = 1/6 corresponding to complete asymptotic independence.

The extremal coefficient θ(h) in Eq. (9) is linked to the F-madogram through (Schlather and Tawn 2003; Ribatet 2013):

$$\theta \left( h \right) = \left( {1 + 2v_{F} \left( h \right)} \right)/(1{-}2v_{F} \left( h \right)),\quad h \ge 0$$
(B.2)

Equation (B.2) implies that there is a one-to-one correspondence between vF(h) and θ(h), with θ(h) = 2 denoting asymptotic independence. The estimates of θ(h) in Fig. 6 (dots) have been obtained using Eq. (B.2), for vF(h) calculated empirically using the R function “fmadogram” as implemented by Ribatet and Singleton (2018). The latter is based on the estimator suggested by Ribatet (2013).

Appendix C: Used software

All analyses and visualizations were conducted in R Programming Language (R Core Team 2017), using the following packages: caTools (Tuszynski 2014), data.table (Dowle et al. 2017), devtools (Wickham and Chang 2018), extRemes (Gilleland 2016; Gilleland and Katz 2016), gdata (Warnes et al. 2017), ggplot2 (Wickham 2016), knitr (Xie 2014, 2015, 2018), lubridate (Grolemund and Wickham 2011), scales (Wickham 2017), SpatialExtremes (Ribatet 2013; Ribatet and Singleton 2018). The code to reproduce the method is available as supplemental information.

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Tyralis, H., Langousis, A. Estimation of intensity–duration–frequency curves using max-stable processes. Stoch Environ Res Risk Assess 33, 239–252 (2019). https://doi.org/10.1007/s00477-018-1577-2

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