Copula Candidates for T₁

As shown in Figure 11.3 and the discussions in the previous chapters, we need to first compute the marginal distributions nonparametrically (e.g., Weibull plotting position formula Equation (3.103), or kernel density) or parametrically with fitted marginal distributions. Here we will use the Weibull plotting position formula to compute the marginals, as shown in Table 11.3. Before we choose the copula candidate, we assess the tail dependence of (Q, V) and (V, D) such that we can make a better judgment to choose the candidate. Figure 11.4 shows the scatter plot using the empirical marginals of each of (Q, V) and of each of (V, D). Compared to the left tail (i.e., the lower tail) dependence, we are usually more interested in the right tail (i.e., the upper tail) dependence for these extreme events. Based on the tail dependence concept discussed in Chapter 3, we will first introduce how to evaluate the empirical tail dependence coefficient in what follows.

Figure 11.4 Scatter plots for the marginal of (Q, V) and of (V, D).

The tail dependence may be evaluated either graphically (Abberger, 2005) or numerically (Frahm et al., 2005; Schmidt and Stradtmuller, 2006). Here the nonparametric estimation is discussed in detail. Following Frahm et al. (2005), the nonparametric estimation is based on the empirical copula (i.e., Equation (3.64)) without any assumption on either parametric copula or marginals. In general, there are three types of nonparametric estimation (i.e., log-estimator [LOG], secant of the copula’s diagonal [SEC], and CFG; Poulin et al., 2007) for the upper-tail dependence (λ̂U) that can be expressed as follows:

λ̂ULOG=2−logCmn−knn−knlogn−kn,0<k<n(11.2)

λ̂USEC=2−1−Cmn−knn−kn1−n−kn,0<k<n(11.3)

λUCFG=2−2exp1n∑i=1nloglog1Uilog1Vi/log1maxUiVi(11.4)

In Equations (11.2)–(11.4), n is the sample size; U_i, V_i are the marginal variables; and k is the chosen threshold of the LOG and SEC methods.

The LOG method was proposed by Coles et al. (1999). The SEC method first appeared in Joe (1997). The threshold k can be estimated using the heuristic plateau-finding algorithm proposed by Frahm et al. (2005), which can be formulated as follows:

1. 
   

   1. Smooth using the box kernel with bandwidth b ∈ ℕb∈ℕ (usually each moving average window should maintain 1% data) to compute the average of (2b + 1)2b+1 successive points from λ̂1,…,λ̂n (i.e., mapping k↦λ̂k,k=1,2,…,n) to obtain λ¯1,…,λ¯n−2b.

   
   
2. 
   

   2. Set plateau length m=⌊n−2b⌋ and define a vector: pk=λ¯k…λ¯k+m−1,k=1,…,n−2b−m+1.

   
   
3. 
   

   3. Set the stopping criteria using the standard deviation of λ¯1,…,λ¯n−2b. The threshold k can then be estimated from the first plateau p_kpk that satisfies the condition:

   
   
   ∑i=k+1k+m−1λ¯i−λ¯k≤2σ(11.5)
   
   
   
   

   If k is un-identified, λ̂U is set as 0; otherwise, move on to step 4.

   
   
4. 
   

   4. Estimate the upper-tail dependence coefficient for threshold k as follows:

   
   
   λ̂Uk=1m∑i=1mλ¯k+i−1(11.6)
   
   

The CFG method (i.e., Equation (11.4)) first appeared in Capéraà et al. (2007) that does not require the estimation of a threshold. However, there exists a strong underlying assumption: the empirical copula may be approximated by the extreme value (EV) copula (e.g., the Gumbel–Hougaard copula as an example). It is worth noting that the lower-tail dependence is the same as the upper-tail dependence of the survival copula.

The empirical upper-/lower-tail dependence coefficient is computed, as listed in Table 11.4. To illustrate the procedure, the empirical upper-tail dependence coefficient is further explained using Q and V with the LOG method. From the sample data listed in Table 11.1, the sample size is n = 33n=33. Applying Equation (11.2), we compute λ̂k for k = 1, 2, …, 32,k=1,2,…,32, as listed in Table 11.5. With the initial λ̂ks estimated for the LOG method, we can now move on to evaluate the tail dependence. With the sample size of 33, we set the bandwidth b = 0. With b = 0b=0, we have λ̂k=λ¯k, and the standard deviation of vector λ¯s is 0.2114. The plateau length m = 5 yields the vector with size of 27 by 5 for the non-NaN values that are listed in Table 11.6. Finally, applying Equation (11.5), we obtain the first p vector that satisfies the condition that index k = 3k=3 that results in the following:

∑i=4:7λ¯i−λ¯3=0.3155<20.2114=0.4229.

We obtain the upper tail dependence as λULOG≈0.29.

Table 11.5. λ̂k computed using Equation (11.2).

k	CmCm	λ̂k	k	CmCm	λ̂k
1	0.9697	1.0000	17	0.3636	0.6026
2	0.9091	0.4755	18	0.3333	0.6066
3	0.8485	0.2761	19	0.3030	0.6076
4	0.7879	0.1549	20	0.2727	0.6053
5	0.7576	0.3102	21	0.2121	0.4672
6	0.7273	0.4131	22	0.1818	0.4483
7	0.6667	0.2993	23	0.1818	0.5721
8	0.6061	0.1963	24	0.1515	0.5476
9	0.5758	0.2664	25	0.1515	0.6683
10	0.5455	0.3210	26	0.1212	0.6391
11	0.4848	0.2146	27	0.1212	0.7622
12	0.4545	0.2556	28	0.0909	0.7293
13	0.4242	0.2878	29	0.0606	0.6715
14	0.3939	0.3126	30	0.0606	0.8309
15	0.3939	0.4631	31	0.0303	0.7527
16	0.3939	0.5956	32	0

Table 11.6. Vector p with the plateau length m = 5.

k	λ¯k	λ¯k+1	λ¯k+2	λ¯k+3	λ¯k+4
1	1.0000	0.4755	0.2761	0.1549	0.3102
2	0.4755	0.2761	0.1549	0.3102	0.4131
3	0.2761	0.1549	0.3102	0.4131	0.2993
4	0.1549	0.3102	0.4131	0.2993	0.1963
5	0.3102	0.4131	0.2993	0.1963	0.2664
6	0.4131	0.2993	0.1963	0.2664	0.3210
7	0.2993	0.1963	0.2664	0.3210	0.2146
8	0.1963	0.2664	0.3210	0.2146	0.2556
9	0.2664	0.3210	0.2146	0.2556	0.2878
10	0.3210	0.2146	0.2556	0.2878	0.3126
11	0.2146	0.2556	0.2878	0.3126	0.4631
12	0.2556	0.2878	0.3126	0.4631	0.5956
13	0.2878	0.3126	0.4631	0.5956	0.6026
14	0.3126	0.4631	0.5956	0.6026	0.6066
15	0.4631	0.5956	0.6026	0.6066	0.6076
16	0.5956	0.6026	0.6066	0.6076	0.6053
17	0.6026	0.6066	0.6076	0.6053	0.4672
18	0.6066	0.6076	0.6053	0.4672	0.4483
19	0.6076	0.6053	0.4672	0.4483	0.5721
20	0.6053	0.4672	0.4483	0.5721	0.5476
21	0.4672	0.4483	0.5721	0.5476	0.6683
22	0.4483	0.5721	0.5476	0.6683	0.6391
23	0.5721	0.5476	0.6683	0.6391	0.7622
24	0.5476	0.6683	0.6391	0.7622	0.7293
25	0.6683	0.6391	0.7622	0.7293	0.6715
26	0.6391	0.7622	0.7293	0.6715	0.8309
27	0.7622	0.7293	0.6715	0.8309	0.7527

From the tail dependence coefficients evaluated and listed in Table 11.4, it is seen that there exist both upper and lower tail dependences for the bivariate (Q, V) and (V, D) flood variables. To this end, we will have the following choices to investigate the dependence:

1. 
   

   i. Use a mixed copula to model the bivariate flood variables.

   
   
2. 
   

   ii. Use two-parameter copulas (Joe, 1997) to model the bivariate flood variables.

   
   
3. 
   

   iii. Use copulas with upper-tail dependence to model the bivariate flood variables.

In theory, (a) all three approaches should be able to capture the overall dependence structure; (b) compared with approaches ii and iii, approach i may better capture both upper and tail dependences; (c) among the three approaches, parameter estimation for approach i is most complex; and (d) if we are only concerned with the upper-tail dependence, we may prefer approach iii. In what follows, we will discuss the copula candidates for all three approaches.

Approach ii: Two-Parameter Copulas for Bivariate Variables

As discussed in Joe (1997), the two-parameter Archimedean copulas may be capable of capturing both the overall dependence and the tail dependence. Following Joe (1997), we will briefly introduce BB1 BB4, and BB7 copulas.

BB1 Copula

Cuvθ1θ2=1+u−θ1−1θ2+v−θ1−1θ21θ2−1θ1=ϕ−1ϕθ1,θ2u+ϕθ1,θ2v;θ1>0,θ2≥1(11.9)

Its generating function and tail dependence function can be written as follows:

ϕθ1,θ2t=t−θ1−1θ2,λU=2−21θ2,λL=2−1(θ1θ2(11.9a)

The BB1 copula can only be applied to model the positive dependence and may be considered as a two-parameter Archimedean copula. It possesses both upper- and lower-tail dependences. The limiting copulas are Gumbel–Hougaard copula (θ₁→0θ1→0) and Clayton copula (θ₂ = 1θ2=1). With the combination of the Gumbel–Hougaard and Clayton copulas, the BB1 copula is able to capture both upper- and lower-tail dependences in which the upper-tail dependence is independent of parameter θ₁θ1.

BB4 Copula

Cuvθ1θ2=u−θ1+v−θ1−1−u−θ1−1−θ2+v−θ1−1−θ2−1θ2−1θ1(11.10)

where  θ₁ ≥ 0, θ₂>0.θ1≥0,θ2>0. Its tail dependence functions can be written as follows:

λU=2−1θ2,λL=2−2−1θ2−1θ1(11.10a)

Unlike the BB1 copula, the BB4 copula is not a two-parameter Archimedean copula. Its limiting copulas are the Clayton copula when θ₂→0θ2→0 and the Galambos copula when θ₁→0θ1→0. The Glambos copula belongs to an extreme value copula given as follows:

Cuvδ=uvexp−logu−δ+−logv−δ−1δ,δ>0(11.10b)

As seen from Equation (11.10a), the upper-tail dependence of the BB4 copula is independent of parameter θ₁.θ1.

BB7 Copula

Cuvθ1θ2=1−1−1−1−uθ1−θ2+1−1−vθ1−θ2−1−1θ21θ1(11.11)

where θ₁ ≥ 1, θ₂>0θ1≥1,θ2>0.

BB7 is the same as the BB1 copula and is also a two-parameter Archimedean copula. Its generating function and tail dependence functions can be expressed as follows:

ϕθ1,θ2t=1−1−tθ1−θ2−1;λU=2−21θ1;λL=2−1θ2(11.11a)

The limiting copulas for the BB7 copula are the Clayton copula when θ₁ = 1θ1=1 and the Joe copula when θ₂→0θ2→0.

Parameter Estimation and the Best-Fitted Copula for T₁