The true copula modeling the dependence of random variables X and Y is the Gumbel–Hougaard copula with parameter θ = 2.5θ=2.5:

1. 
   

   i. Construct MECC using empirical marginals.

   
   
2. 
   

   ii. Construct MECC using MaxEnt-based marginals.

   
   
3. 
   

   iii. Construct MECC using the true underlying population X~Gamma (3,4), Y~Gaussian (5,3²).

   
   
4. 
   

   iv. Compare the constructed MECC with the underlying copula function as the Gumbel–Hougaard copula with parameter θ = 2.5θ=2.5.

Solution: Before we proceed to build the MECC, we first plot the histograms, the frequency computed from the true population and MaxEnt-based probability distribution in Figure 8.1. The MaxEnt-based univariate distribution (plotted in Figure 8.1) will be further explained in later sections. One purpose of applying empirical, true population and MaxEnt-based univariate distributions is to evaluate the impact of marginals on the derived copula function.

Figure 8.1 Histograms and underlying true probability density functions.

Furthermore, throughout the example, the first two noncentral moments of the marginals (Equations (8.12a) and (8.12b)) and E(UV)EUV, which is one-to-one related to the rank-based correlation coefficient, Spearman’s rho (Equation (8.12c)), will be applied as the constraints for the MECC as follows:

∫01∫01ucuvdudv=EU=∫01∫01vcuvdudv=EV=12(8.12a)

∫01∫01u2cuvdudv=EU2=∫01∫01v2cuvdudv=EV2=13(8.12b)

∫012uvcuvdudv=ρ̂s+312=0.3140;sampleρ̂s=0.7677(8.12c)

In Equation (8.12c), sample ρ̂s is computed using Equation (3.70).

In what follows, we will proceed with constructing MECC with different marginal distributions.

Construct MECC using empirical distribution

The empirical probability is computed with the use of Weibull plotting position formula (Equation (3.103)) that is partially listed in Table 8.2. Minimizing the partition (i.e., objective) function (Equation (8.9a)) using the MATLAB optimization toolbox (e.g., the GA/fminsearch function), Table 8.3 lists the Lagrange multipliers estimated and the relative differences between moment constraints computed from the MECC (Equation (8.12)) and the corresponding sample moments. Figure 8.2 compares the constructed MECC with the empirical copula.

Figure 8.2 Comparison of MaxEnt-based univariate distribution to empirical and underlying distributions.

Construct MECC using MaxEnt-based marginal distribution

To apply the MaxEnt-based marginal distribution, we first transform the random variables X and Y into the range (0, 1). To avoid reaching the lower and upper limit, we use the following equation for the monotone transformation:

Xt=x−1−d*minx1+dmaxx−1−dminx,d=0.01(8.13)

In Equation (8.13), X denotes the random variable that needs to be transformed, d denotes the threshold ratio to avoid the transformed variable reaching the lower and upper limits, and X^tXt denotes the variable after transformation.

Strictly from the sample dataset listed in Table 8.1, we evaluate whether the fourth noncentral moment to derive the MaxEnt-based univariate probability distribution by testing whether the sample kurtosis is significantly different from 3 (i.e., kurtosis = 3 for normal distribution) described in Zhang and Singh (2012). The test statistic is computed using the following:

γ2′=n∑i=1nxi−x¯4∑i=1nxi−x¯22−3(8.14b)

G2=n−1n−2(n−3n+1γ2′+6(8.14c)

SEK=26nn−12n−2n+5n2−9(8.14d)

In Equation (8.14), n is the sample size; γ2′ is the excess kurtosis; G₂G2 is sample excess kurtosis; and SEKSEK is the standard error of kurtosis. The test statistic T follows the standard normal distribution. Applying Equation (8.14), we computed the test statistic (T) for variables X and Y, which were 0.06 and –0.73; and P-values were 0.95 and 0.46, respectively. Thus, the kurtosis was not significantly different from 3 such that we only need to apply the first three noncentral moments to drive the MaxEnt-based distribution with the Lagrange multipliers. The MaxEnt-based univariate distribution for the scaled transformed variable (x^t)xt) is written as follows:

andλ0=ln∫01exp−∑i=13λixtidx(8.14f)

The corresponding MaxEnt-based marginal PDF for the observed random variable can be written as follows:

fx=1Afxt=1Aexp−λ0−λ1xt−λ2xt2−λ3xt3(8.15)

where A = (1 + d) max (x) − (1 − d) min (x)A=1+dmaxx−1−dminx.

The Lagrange multipliers may again be estimated by minimizing the objective function given by Equation (8.14b) using the MATLAB optimization toolbox, as listed in Table 8.4. Results in Table 8.4 indicate that the first three noncentral moments (sample moments) are well preserved. Table 8.2 lists the marginal probabilities computed from the fitted MaxEnt-based distribution. It is worth noting that we may use the transformed variable to compute the marginal probability directly, given the monotone transformation between observed and scale-transformed variables. The MaxEnt-based probability density function is plotted in Figure 8.1, whereas Figure 8.3 compares the MaxEnt-based univariate distribution with the empirical distribution. The comparisons again indicate that the MaxEnt-based distribution matches well the empirical distribution as well as the true population.

Figure 8.3 Comparison of MECC with the empirical and Gumbel–Hougaard copulas.

Using the CDF computed from the constructed MaxEnt-based univariate distribution, Table 8.4 lists the Lagrange multipliers estimated for the MECC. Figure 8.2 compares the MECC using the CDF computed from MaxEnt-based univariate distribution with the empirical copulas.

Construct MECC using the underlying population

In this case, gamma (3, 4) and Gaussian (5, 3²) are applied to random variables X and Y, respectively. The computed CDF is listed in Table 8.2. The MECC is then constructed with the Lagrange multipliers listed in Table 8.4. Figure 8.3 compares the MECC from the underlying population with the empirical copula.

Compare the constructed MECC with the underlying copula function

Applying the Gumbel–Hougaard copula with parameter θ = 2.5θ=2.5 to the marginals computed from the empirical formula, the MaxEnt-based distribution, and underlying populations, Figure 8.3 compares the Gumbel–Hougaard copulas with MECCs. The comparison indicates the following:

1. 
   

   a. The Gumbel–Hougaard copula, computed using the empirical distribution, has a better match with the MECC computed from empirical and MaxEnt-based univariate distributions than the MECC computed from underlying univariate populations.

   
   
2. 
   

   b. The Gumbel-Hougaard copula computed using the underlying univariate populations matches better the MECC computed using the underlying univariate populations than those from empirical and MaxEnt-based univariate distributions.

   
   
3. 
   

   c. It is understandable that we reach the conclusions in a and b. With the sample data, the MaxEnt-based distribution is derived by equating moment constraints to the sample moments. It is seen from Figure 8.1 that there exists the difference in fitting between MaxEnt-based distributions and true populations. It may be explained with the sample size. It is expected that the MaxEnt-based, true underlying, and empirical distributions should match each other better with the increased sample size.

Solution:

1. 
   

   1. Construct MECC by adding two Blest moment constraints

According to Chu (2011), the Blest I and II moment constraints are given as follows:

∫012u2vcuvdudv=2ρs−v̂1+212(8.16a)

∫012v2ucuvdudv=2ρs−v̂2+212(8.16b)

In Equations (8.16a) and (8.16b),

Sample Blest measureI:v̂1=2N+1N−1−12N2−N∑i=1N1−RiN+12Si(8.16c)

Sample Blest measureII:v̂2=2N+1N−1−12N2−N∑i=1N1−SiN+12Ri(8.16d)

In Equations (8.16c) and (8.16d), {R_i, S_i : i = 1…N}RiSi:i=1…N is the rank for {x_i, y_i : i = 1…N}xiyi:i=1…N.

By adding the Blest dependence measure, the MECC can be rewritten as folllows:

The partition (or objective) function (i.e., Equation (8.11a)) can be rewritten as follows:

ZΛ=ln∫012exp∑i=12λiui−∑i=12γivi−λ3uv−λ4u2v−λ5vu2dudv+∑i=12λi1i+1+∑i=12γi1i+1+λ3Êuv+λ4Êu2v+λ5Êuv2(8.18)

In Equation (8.18), Ê⋅ denotes the sample moment, and λ₁ = γ₁; λ₂ = γ₂; λ₄ = λ₅λ1=γ1;λ2=γ2;λ4=λ5.

Minimizing Equation (8.18), Table 8.5 lists the Lagrange multipliers estimated, the sample moment constraints, and those computed from the MECC with additional Blest I and II measures. Again, the MATLAB optimization toolbox is applied to minimize the objective function in order to estimate the Lagrange multipliers. Table 8.5 indicates that the moment constraints are preserved reasonably well with the relative error less than 2.5%.

1. 
   

   2. Compare MECC with the additional Blest I and II dependence measure constraints with the Gumbel–Hougaard copula and MECC constructed in Example 8.1

To compare the MECC with added constraints to the MECC constructed in Example 8.1, Table 8.6 lists the numerical JCDF computed from empirical copula, MECCs in Example 8.1, and the MECC with added Blest I and II constraints of Example 8.2. In this case, we will compare the results from Example 8.2 (i.e., column 7 in Table 8.6) with those from Example 8.1 (i.e., columns 2 and 5 in Table 8.6). Figure 8.4 compares the results graphically for the hypothesized Gumbel–Hougaard copula, MECC (Example 8.1), and MECC (Example 8.2) using empirical marginals. Comparison shows that the MECC constructed in Example 8.1 (i.e., only using Spearman’s rho as the constraints for dependence measure) yields better performance than the MECC constructed in Example 8.2 (i.e., with added Blest I and II dependence measure constraints) compared to the hypothesized Gumbel–Hougaard copula. Comparing Table 8.5 to Table 8.3, it is seen that the moment constraints are better preserved by the MECC constructed in Example 8.1 than in Example 8.2. To this end, it is concluded that we need to be cautious when adding more constraints to derive the MECC. In this sample study, the dependence measure through Spearman’s rho preserves the dependence structure of datasets well.

Figure 8.4 Comparison of the MECC constructed in Examples 8.1 and 8.2 with the hypothesized Gumbel–Hougaard copula in Example 8.1.

For a real-world example using flood data from the Walnut Gulch Experimental Watershed (Flume 1) given in Table 8.7, do the following:

1. 
   

   1. Construct the MaxEnt-based marginal distributions using the first three noncentral moments as constraints.

   
   
2. 
   

   2. Construct the MECC using Equation (8.12) as the constraint with the MaxEnt-based marginals from step 1. Then compare the MECC constructed with the Gumbel–Hougaard copula with the same marginals.

   
   
3. 
   

   3. Construct the MECC and fit the Gumbel–Hougaard copula to the flood data with empirical marginal distributions.

   
   
4. 
   

   4. Compare the MECC and Gumbel–Hougaard copulas fitted in steps 2 and 3 with empirical copulas.

Solution: Flume 1 is located at the most downstream point of the Walnut Gulch Experimental Watershed (i.e., 31^o43’45.32” N and 110^o9’12.06” W). It covers an area of about 150 km². The annual maximum series (AMS) are extracted from the event-based dataset (1957–2012). In this example, flood data of the year 1979 were not used in analysis to avoid uncertainty (i.e., from the dataset, there was no obvious runoff for the entire year).

1. 
   

   1. Construct the MaxEnt-based univariate distributions.

Using the first three noncentral moments as the constraints, we have the constraint equation for the univariate density function f(x)fx:

∫fxdx=1;∫xifxdx=Exi≈xi¯,i=1,…,3.

and the MaxEnt-based univariate density function is given as follows:

where λ0=gλ1λ2λ3=ln∫exp−∑i=13λixidx;λ3>0.

As discussed in the previous examples, parameters Λ = (λ₁, λ₂, λ₃)Λ=λ1λ2λ3 can be estimated by minimizing the partition function: ZΛ=λ0+∑i=13λixi¯. Similar to the previous example, the flood variables are transformed from (0, +∞)0+∞ to (0, 1) using Equation (8.13) with d = 0.1d=0.1.

Applying the Shannon entropy, the Lagrange multipliers estimated for the peak discharge and flood volume are listed in Table 8.8, and comparison of MaxEnt-based PDF and CDF with their empirical form is plotted in Figure 8.5. Results from Table 8.8 indicate that the first three noncentral moments for the transformed flood variables are well preserved. Comparison in Figure 8.5 graphically confirms the good fit between empirical and MaxEnt-based distributions. The MaxEnt-based univariate distributions may be written for discharge and flood volume without transformation as follows:

fdis=11.235⋅104exp2.137−9.584disT+4.146disT2−4.499⋅10−4disT3fvol=13.975⋅107exp1.695−5.456volT+0.17volT2−0.139volT3disT=dis−22.761.235⋅104,volT=vol−361623.975⋅107;disT,volT∈01.

Table 8.8. Results of the MaxEnt-based univariate distributions for the transformed discharge and flood volume variables.

	λ0λ0	λ1 : [E(x)]λ1:Ex	λ2 : [E(x2)]λ2:Ex2	λ3 : [E(x3)]λ3:Ex3
	Transformed discharge variables
Multipliers estimated	−2.137	9.584	−4.146	4.499E−04
xi¯:i=1,2,3		0.138	0.043	0.022
Exi−xi¯xi¯		−2.50E−04	−1.58E−04	1.12E−03
	Transformed volume variables
Multipliers estimated	−1.695	5.456	−0.170	0.139
xi¯:i=1,2,3		0.184	0.067	0.033
Exi−xi¯xi¯		3.58E−03	4.31E−03	3.08E−03

Figure 8.5 Comparison of MaxEnt-based univariate distribution with empirical frequency and CDFs.

2. Construct MECC using Equation (8.12) with given constraints and marginal distributions estimated in step 1.

Using the Lagrange multipliers estimated in step 1, the cumulative probability distributions of discharge and flood volume are computed, as listed in Table 8.9. To apply Equation (8.12) as the constraints to construct MECC, the Spearman’s rho rank-based correlation coefficient is computed as ρ̂s=0.9387. Equating the sample moment to the dependence measure constraint in Equation (8.12), we have Euv≈uv¯=ρ̂s+312=0.3282.

Now the objective function (i.e., Equation (8.9)) can be rewritten as follows:

ZΛ=ln∫012exp−∑i=12λiui−∑i=12γivi−λ3uvdudv+∑i=12λi1i+1+∑i=12γi1i+1+λ3Euv

or equivalently

ZΛ=∫012exp−∑i=12λiui−1i+1−∑i=12γivi−1i+1−λ3uv−Euvdudv

As discussed earlier, we have λ₁ = γ₁; λ₂ = γ₂λ1=γ1;λ2=γ2. Minimizing the objective function, we list the estimated Lagrange multipliers in Table 8.10.

Table 8.10. Lagrange multipliers estimated for MECC and comparison with Gumble–Hougaard copula.

Marginals		λ0λ0	λ1λ1		λ2λ2	γ1γ1	γ2γ2	λ3λ3
			E(U)EU		E(U2)EU2	E(V)EV	E(V2)EV2	E(UV)EUV
	MECC	–1.803	1.163		42.331	1.163	42.331	–86.990
	Moment constraint (comp.)		0.50		0.333	0.50	0.333	0.328
	Relative differences		–3.21E–09		–3.44E–09	4.59E–09	9.14E–09	2.70E–09
	Gumbel MECC vs. Gumbela		θ̂=3.926;ρscomp=0.909 k = 0.9978; Rsquare = 0.9969k=0.9978;Rsquare=0.9969
	MECC	–1.803	1.163		42.331	1.163	42.331	–86.990
Empirical	Moment constraint (comp.)		0.50		0.333	0.50	0.333	0.328
	Relative differences		–3.21E–09		–3.44E–09	4.59E–09	9.14E–09	2.70E–09
	Gumbel MECC vs. Gumbela		θ̂=4.335;ρscomp=0.925 k = 0.9954; Rsquare = 0.9975k=0.9954;Rsquare=0.9975

Notes: ^aRegression MECC on Gumbel using y = kxy=kx.

As shown in Table 8.10, the moment constraints are well preserved with the relative difference of less than 10^–8. Figure 8.6 compares the MECC and Gumbel–Hougaard copula with the use of MaxEnt-based marginals. Comparison shows that the Gumbel–Hougaard copula and MECC yield very similar results. As shown in the scatter plot, the joint CDF computed from two copulas closely follows a 45^o line. Numerical regression ensures a very similar performance of MECC and Gumbel–Hougaard copulas.

Figure 8.6 Comparison of MECC with Gumbel–Hougaard copulas.

3. Construct the MECC and fit the Gumbel–Hougaard copula to the flood data with empirical marginal distributions.

Using the same moment constraints as in step 2, we will obtain exactly the same MECC copula for the MaxEnt-based and empirical CDFs (i.e., Table 8.10). Fitting the Gumbel–Hougaard copula with the use of empirical marginals listed in Table 8.9, the parameters estimated are higher than those from MaxEnt-based marginals. The values of Spearman’s rho computed using the estimated parameters (the Gumbel–Hougaard copulas) are close to each other. The relative differences to the sample Spearman’s rho are about 0.032 and 0.015 for the Gumbel–Hougaard copula with MaxEnt-based and empirical marginals, respectively. This information indicates the advantage of applying empirical marginals to construct copulas, i.e., better avoiding the misidentification of the univariate distributions.

Again, comparing the MECC with the Gumbel–Hougaard copulas fitted to empirical marginals, two copulas yield very similar results, as shown in Table 8.10 and Figure 8.6.

4. Compare the MECC and Gumbel–Houggard copulas fitted in steps 2 and 3 with empirical copulas.

The empirical copula for the bivariate flood variable is computed using Equation (3.65). Table 8.11 lists the JCDF computed from steps 2 and 3, as well as the empirical copulas. Table 8.12 lists the regression results with the use of simple linear regression y = kxy=kx, in which the empirical copula is considered as the independent variable x, with a visual comparison plotted in Figure 8.7. As shown in Table 8.12 and Figure 8.7, the MECC and Gumbel–Hougaard copulas (fitted to the MaxEnt-based and empirical marginals) indicate good fit to the empirical copulas.

Figure 8.7 Comparison of MECC and Gumbel–Hougaard copulas to empirical copulas.