Copula Identification for T₁

According to Kendall’s tau correlation coefficient matrix, the proper structure for T₁ is as follows: R330058 − R333780 − R336949 − R331458R330058−R333780−R336949−R331458 (i.e., the bivariate pairs for T₁ are [R330058, R333780]; [R333780, R336949]; [R336949, R331458]). Using the empirical marginals (Weibull plotting position formula), let U₁, U₂, U₃, U₄U1,U2,U3,U4 represent the empirical marginals as follows:

U1=F̂nR330058;U2=F̂nR333780;U3=F̂nR336949;andU4=F̂nR331458.

The D-vine structure for this example is the same as in Figure 10.5. In this case study, we choose Archimedean copulas for dealing with the positive dependence (Gumbel–Hougaard, Clayton, Frank, Joe, and BB1 copulas) as the candidates. Chapter 4 listed the one-parameter Archimedean copulas candidates. Hence we only give the formula for BB1 copula, which is a two-parameter Archimedean copula with the limiting conditions of either the Clayton or Gumbel–Hougaard copula. The BB1 copula (Joe, 1997) can be formulated as follows:

Cuvθ1θ2=1+u−θ1−1θ2+v−θ1−1θ21θ2−1θ1;θ1>0;θ2≥1(10.6)

The BB1 copula converges to (i) the Gumbel–Hougaard copula if θ₁→0θ1→0; and (ii) the Clayton copula if θ₂ = 1θ2=1.

Figure 10.5 D-vine structure for four-dimensional rainfall variables: (1) R330058, (2) R336949, (3) R333780, and (4) R331458.

In addition, the BB1 copula has both upper- and lower-tail dependence coefficients, as follows:

λL=2−1θ1θ2,λU=2−21θ2

The parameters of T₁ are estimated with the pseudo-MLE through the empirical marginals for all the copula candidates (Table 10.5). Table 10.6 also lists the log-likelihood, AIC, and BIC values with the best-fitted copula highlighted. From Table 10.6, we see that the two-parameter BB1 copula is the best-fitted copula for stations (R330058, R333780, R333780, and R336949), and the Gumbel–Hougaard copula is the best-fitted copula for stations R336949 and R331458.

Based on the AIC/BIC model selection criteria, we again find that (1) the BB1 copula reaches the lowest AIC/BIC values for pairs (U₁, U₂U1,U2); (2) the BB1 copula is also selected to model the pairs (U2 and U3) since it yields the compariable AIC/BIC and may capture the lower tail dependence, compared with Gumbel–Houggard copula and (3) the Gumbel–Hougaard copula reaches the lowest AIC/BIC for pair (U₃, U₄U3,U4).

Copula Identification for T₂

Using the best-fitted copulas for T₁, Table 10.7 lists the conditional probability computed for T₂.

From Kendall’s correlation coefficient estimated in Table 10.7, we again have the positive dependence for [U₁|U₂, U₃|U₂]U1U2U3U2 and [U₂|U₃, U₄|U₃]U2U3U4U3. Using all copula candidates for T₁, Table 10.8 lists the results from pseudo-MLE for T₂. Based on AIC and BIC, Frank copula is found as the best fitted copula for T₂ variables as shown in Table 10.8. However the goodness-of-fit study shows that BB1 copula should be applied to model the dependence at T₂ (Table 10.8).

Copula Identification for T₃

Now we can move on to T₃. Similar to T₂, we first need to compute the conditional copula through the BB1 copula as follows:

FU1≤u1U2=u2U3=u3=∂C13∣2Fu1u2Fu3u2∂Fu3u2

FU4≤u4U2=u2U3=u3=∂C24∣3Fu4u3Fu2u3∂Fu2u3

The computed conditional probability distribution using the selected fitted copulas in T₁ and T₂ is listed in Table 10.9.

Using the conditional probability listed in Table 10.9, Kendall’s correlation coefficient is computed as τ =  − 0.06τ=−0.06, and we have negative dependent variables for T₃. Applying the Frank copula, we have θ =  − 0.5062θ=−0.5062.

Applying the goodness-of-fit study for the fitted vine copula, the Rosenblatt transform approach is applied. The test results (Table 10.10) further confirmed the selected vine copula may properly study the dependence structure of the four-dimensional rainfall dataset.

Applying the goodness-of-fit study, we have SnB=0.046,P=0.95. Thus, we have the four-dimensional fitted D-vine copula as BB1-BB1-GH (T₁), BB1-BB1 (T₂) and Frank (T₃) as shown in Figure 10.6.

Figure 10.6 Fitted D-vine copula for four-dimensional rainfall variables.

With the fitted D-vine copula in Figure 10.6, we can simulate the four-dimensional pseudo-rainfall variables (i.e., the marginal CDF of rainfall variables) as shown in Figure 10.7. Here, we will show how to simulate the random variates from the fitted D-vine copula given in Figure 10.6 with a simple example:

1. 
   

   1. Generate four independent uniform random variables in [0,1]

   
   
   
   

   W = [0.7582, 0.6289, 0.9611, 0.2743]

   W=0.75820.62890.96110.2743.
   
   
   
   

   For the independent random variables generated, we set the following:

   
   
   u1=W1=0.7582
   
   
   
   

   C(u₂| U₁ = u₁) = 0.6289

   Cu2U1=u1=0.6289
   
   
   
   

   C(u₃| U₁ = u₁, U₂ = u₂) = 0.9611;

   Cu3U1=u1U2=u2=0.9611;
   
   
   
   

   C(U₄| U₁ = u₁, U₂ = u₂, U₃ = u₃) = 0.2743

   CU4U1=u1U2=u2U3=u3=0.2743.
   
   
   
   
2. 
   

   2. Simulate u₂u2 from C(U₂| U₁ = u₁) = 0.6289CU2U1=u1=0.6289.

   
   

   According to the fitted D-vine copula, we know [U₁, U₂]U1U2 is fitted with the BB1 copula (i.e., Equation (9.1)) with parameters [1.0203,1.9788]. Its conditional copula is then written as follows:

   
   
   Cu2U1=u1=∂Cu1u2∂u1u1=0.6478=S1θ2−1u1−θ1−1θ2−1u1θ1+1S1θ2+11θ1+1(10.7)
   
   
   
   

   where:S=1u1θ1−1θ2+1u2θ1−1θ2.

   
   

   Substituting u₁ = 0.7582u1=0.7582 and C(u₂| U₁ = u₁) = 0.6289Cu2U1=u1=0.6289 into Equation (10.7), we can solve for u₂u2 numerically and obtain the following:

   
   
   u2=0.7755.
   
   
   
   
3. 
   

   3. Simulate u₃u3 from C(u₃| U₁ = 0.7582, U₂ = 0.7755)Cu3U1=0.7582U2=0.7755.

   
   

   According to the fitted D-vine copula, we know U₂U2 is the one of the center variables, and from the probability density composition discussed in Chapter 5, we have the following:

   
   
   Cu3U1=u1U2=u2=∂C13∣2C3∣2C1∣2∂C1∣2(10.8)
   
   
   
   

   As seen in Equation (10.8), we may simulate u₃u3 with the following two steps:

   
   
   
   
   1. 
      

      i. Compute C_3 ∣ 2C3∣2 from C_13 ∣ 2C13∣2. According to Figure 10.6, we know that the BB1 copula with parameter [0.2336, 1.0001] properly models C_13 ∣ 2(C_3 ∣ 2, C_1 ∣ 2)C13∣2C3∣2C1∣2. With this in mind and after computing C_1 ∣ 2C1∣2 using the BB1 conditional copula (i.e., Equation (10.7)), we immediately have the following: C(U₁ ≤ 0.7582| U₂ = 0.7755) = 0.5465CU1≤0.7582U2=0.7755=0.5465. Given {C_3 ∣ 2, C_1 ∣ 2}C3∣2C1∣2 (i.e., one of the bivariate copulas at T₂) again modeled by the BB1 copula, C_3 ∣ 2C3∣2 can then be computed by substituting C_1 ∣ 2 = 0.5465C1∣2=0.5465 as u₁u1 and C_3 ∣ 2C3∣2 as u₂u2, and by equating Equation (10.8) to 0.9383. We can solve for C_3 ∣ 2C3∣2 numerically as C_3 ∣ 2 = 0.9636C3∣2=0.9636.

      
      
   2. 
      

      ii. Compute u₃u3 from C_3 ∣ 2C3∣2. From Figure 10.6, {U₂, U₃}U2U3 is also modeled with the BB1 copula; u₃u3 can then be solved for numerically by substituting u₂ = 0.7755u2=0.7755 as u₁u1, and u₃u3 as u₂u2 into Equation (10.7), and by setting the equation equal to C_3 ∣ 2 = 0.9636C3∣2=0.9636. We then have the following:

      
      
      u3=0.9383.
      
      

   
   
4. 
   

   4. Simulate u₄u4 from C(u₄| U₁ = 0.7582, U₂ = 0.7755, U₃ = 0.6865)Cu4U1=0.7582U2=0.7755U3=0.6865.

   
   

   From the fitted D-vine copula, we know the conditional copula C(u₄| U₁ = u₁, U₂ = u₂, U₃ = u₃)Cu4U1=u1U2=u2U3=u3 may be written using the probability function decomposition discussed in Chapter 5 as follows:

   
   
   Cu4U1=u1U2=u2U3=u3=∂C14∣23C1∣23C4∣23∂C1∣23(10.9)
   
   
   
   

   where:

   
   
   C1∣23=∂C13∣2C1∣2C3∣2∂C3∣2;C4∣23=∂C24∣3C4∣3C2∣3∂C2∣3(10.9a)
   
   
   
   

   We know from Equations (10.9) and (10.9a) that C_14 ∣ 23, C_1 ∣ 23, and C_4 ∣ 23C14∣23,C1∣23,andC4∣23 are modeled by bivariate Frank, BB1, and BB1 copulas, respectively (Figure 10.6). To this end, we can simulate u₄u4 with the steps given in what follows:

   
   
   
   
   1. 
      

      i. Simulate C_4 ∣ 23C4∣23 using C(u₄| U₁ = 0.7582₁, U₂ = 0.7755, U₃ = 0.9383) = 0.2743Cu4U1=0.75821U2=0.7755U3=0.9383=0.2743.

      
      

      With the previously simulated u₁, u₂, u₃u1,u2,u3, we first compute the conditional copula C_1 ∣ 23C1∣23 in Equation (10.9a). Applying the corresponding fitted BB1 copulas, we compute the conditional copula as follows: C_1 ∣ 2 = 0.5465, C_3 ∣ 2 = 0.9636, C_1 ∣ 23 = 0.4764C1∣2=0.5465,C3∣2=0.9636,C1∣23=0.4764. The given Frank copula may be applied to model C_14 ∣ 23C14∣23 of T₃, and Equation (10.9) may be rewritten using the conditional Frank copula as follows:

      
      
      Cu4U1=u1U2=u2U3=u3=e−θC1∣23e−θC4∣23−1e−θC1∣23+C4∣23−e−θC1∣23−e−θC4∣23+e−θ(10.10)
      Substituting θ =  − 0.5062, C_1 ∣ 23 = 0.4764, C(u₄| U₁ = u₁, U₂ = u₂, U₃ = u₃) = 0.2743θ=−0.5062,C1∣23=0.4764,Cu4U1=u1U2=u2U3=u3=0.2743 into Equation (10.10), C_4 ∣ 23C4∣23 is solved for numerically as follows: C_4 ∣ 23 = 0.2777C4∣23=0.2777.
      
      
      
   2. 
      

      ii. Simulate C_4 ∣ 3C4∣3 using C(u₄| U₂ = 0.7755, U₃ = 0.9383) = 0.2777Cu4U2=0.7755U3=0.9383=0.2777.

      
      

      C_4 ∣ 3C4∣3 can be simulated from the conditional copula C_4 ∣ 23C4∣23 through C_24 ∣ 3C24∣3, given as Equation (10.9a). C_24 ∣ 3C24∣3 may be modeled with the bivariate BB1 copula through C_4 ∣ 3, C_2 ∣ 3C4∣3,C2∣3. Applying the BB1 copula to {U₂, U₃U2,U3}, we can easily compute C_2 ∣ 3 = C(U₂ ≤ 0.7755| U₃ = 0.9383) = 0.1736C2∣3=CU2≤0.7755U3=0.9383=0.1736. In model construction (e.g., Figure 10.6), C_24 ∣ 3C24∣3 is also modeled by the BB1 copula. Thus, we can solve for C_4 ∣ 3C4∣3 numerically as C_4 ∣ 3 = 0.1867C4∣3=0.1867.

      
      
   3. 
      

      iii. Simulate u₄u4 using C_4 ∣ 3 = 0.1867C4∣3=0.1867.

      
      

      We know that {U₃, U₄U3,U4} is modeled with the Gumbel–Hougaard copula (Figure 10.6), as shown in Chapter 4; the conditional Gumbel–Hougaard copula can be written as follows:

      
      
      Cu4U3=u3=−lnu3−1+θ(−lnu3)θ+−lnu4θ−1+1θu3exp−lnu3θ+−lnu4θ(10.11)
      
      

   Substituting u₃ = 0.9383, C_4 ∣ 3 = 0.1867u3=0.9383,C4∣3=0.1867 into Equation (10.11), we have the following:
   
   u4=0.6865
   
   

Figure 10.7 Comparison of simulated random variables with the pseudo-rainfall observations and simulated rank-based Kendall correlation coefficient with sample Kendall correlation coefficient.

Finally, with four independent uniform random variables W = [0.7582, 0.6289, 0.9611, 0.2743]W=0.75820.62890.96110.2743; we successfully simulate the pseudo-rainfall variables from the fitted D-vine copula as follows:

Comparison of the simulated copula random variables with the pseudo-rainfall variables (the upper triangle of Figure 10.7) shows that the fitted D-vine copula reasonably preserves the overall dependence. With the use of 200 simulations, the lower triangle of Figure 10.7 compares the Kendall correlation coefficient computed from the simulations with the sample Kendall correlation coefficient computed from the observed four-dimensional rainfall variables. Comparison through the Kendall’s correlation coefficient indicates the following:

1. 
   

   1. The sample correlation coefficient is within 50% bound for all free bivariate variates in T₁, i.e.,

   
   
   
   

   (U₁, U₂) : (R330058, R336949), (U₂, U₃) : (R336949, R333780), (U₃, U₄) : (R333780, R331358)

   U1U2:R330058R336949,U2U3:R336949R333780,U3U4:R333780R331358;
   
   
   
   
2. 
   

   2. The sample correlation coefficient is also within 50% bound for the bivariate variates through conditioning, i.e., (U₁, U₃) : (R330058, R333780), (U₁, U₄) : (R330058, R331358)U1U3:R330058R333780,U1U4:R330058R331358.

   
   
3. 
   

   3. The sample correlation coefficient is very close to the 50% bound for the last pair of the bivariate variate through conditioning: (U₂, U₄) : (R336949, R331358)U2U4:R336949R331358.

The preceding comparison ensures the appropriateness of applying the fitted D-vine copula model to investigate the four-dimensional rainfall variables. In addition, with the closeness of rain gauges, it is reasonable to assume that there may exist the tail dependence among the rainfall variables (i.e., there is the concurrent tendency of extreme weather events, e.g., storm events). The possible tail dependence makes the BB1 copula the best choice for a majority of cases. We will provide a detailed discussion in this regard when we compare the fitted vine copula to meta-elliptical and asymmetric Archimedean copulas later in the chapter.

Flexibility and Complexity of Copula Functions

First, as discussed in Chapter 5, the vine copula is constructed, based on the probability density function decomposition, such that only bivariate copulas are considered as the building blocks either unconditionally (base level T₁) or conditionally (upper levels). The bivariate copulas (i.e., the building blocks) are allowed for free specification (i.e., the copulas do not need to belong to the same family at all). Additionally, there are many choices for model construction. For example, in our four-dimension example illustrated here, we may be able to build 24 different D-vine copula structures through different pairing schemes.

Second, the meta-elliptical copula is only dependent on the correlation matrix for the meta-Gaussian copula, and the correlation matrix and degree of freedom for meta-Student t copula. In addition, its parameter estimation is easier than that of a vine copula.

Third, there are constraints on the asymmetric Archimedean copula. In addition, there are implications for the dependence for indirectly connected bivariate random variates (as discussed in the previous section).

Overall, the vine copula is most complex with the most flexibility of model construction. The meta-elliptical copula may always be able to capture the overall dependence. The asymmetric Archimedean copula has the least flexibility for model construction, and the dependence structure may not be properly captured due the theoretical constraints of the asymmetric copula function.

Comparison of Copula Performances

Applying Equation (5.61) in Chapter 5 to the fitted D-vine copula in Section 10.3.1, we will be able to compute the joint CDF for the four-dimensional rainfall variables. Similar to application of Equation (7.32) and Equation (7.46) in Chapter 7 and Equation (10.12), we can compute the joint CDF fitted by the meta-Gaussian, meta-Student t and asymmetric Gumbel–Hougaard copulas for the four-dimensional rainfall variables. Figure 10.12 compares the fitted parametric four-dimensional copula function with the nonparametric empirical copulas. Table 10.13 lists the RMSE computed between the parametric and empirical copulas. Figure 10.12 shows that (1) there is minimal visual difference between the performance of meta-Gaussian and that of meta-Student T copulas; (2) there is visual difference between the performance of fitted D-vine copula and asymmetric GH copula; and (3) the fitted D-vine copula may underestimate the JCDF for higher orders (>35) more than the asymmetric GH copula. The RMSE results listed in Table 10.13 further confirm the findings visually seen in Figure 10.12.

Figure 10.12 Comparison of vine, meta-elliptical, and asymmetric Archimedean copulas with empirical copula.

Comparing all three types of the copulas, one may directly apply the meta-elliptical copula for higher dimensions as the following:

1. 
   

   1. The variance–covariance structure may be very well preserved (Figures 10.8 and 10.9).

   
   
2. 
   

   2. A meta-elliptical copula is easy to construct, compared to both vine and asymmetric Archimedean copulas.

   
   
3. 
   

   3. A meta-elliptical copula yields the overall best performance.