3.7 Dependence measures
3.7.4 Tail dependence
The two nonparametric measures of dependence (ρ and τ ) in ranks introduced above measure the average of the dependence. Another measure of the dependence is so-called tail dependence, which measures the dependence between the variables in the upper-right quadrant and in the lower-left quadrant of I2.
Let X and Y be continuous random variables with distribution functions F and G, respectively. The upper tail dependence parameter λU is the limit (if it exists) of the
conditional probability that Y is greater than the 100t-th percentile of G given that X is greater than the 100t-th percentile of F as t approaches 1, i.e.
λU = lim
x→+1−P[Y > G
(−1)(t)|X > F(−1)(t)] (3.33)
Similarly, the lower tail dependence parameter λL is the limit (if it exists) of the condi-
tional probability that Y is less than or equal to the 100t-th percentile of G given that X is less than or equal to the 100t-th percentile of F as t approaches 0, i.e.
λL= lim
x→+0+P[Y ≤ G
(−1)(t)|X ≤ F(−1)(t)] (3.34)
These parameters (λU and λL) are also nonparametric and depend only on the Copula
of X and Y , since they are rank based measure of the dependence. Therefore, the upper and lower tail dependence parameters of the random vector (X, Y ) with the Copula C, can be defined as follows (Joe,1997):
λU = lim
x→+1−
1 − 2u + C(u, u)
Chapter 3. Copula Theory 28 and λL= lim x→+0+ C(u, u) u (3.36)
The upper tail dependence expresses the probability occurrence of positive large values (outliers) at multiple locations jointly, while the lower tail dependence expresses the the probability occurrence of positive small values.
Chapter 4
Copula-based stochastic bias
correction framework
The bias correction framework used in this study is based on Copula theory. A bivariate Copula model forms the basis of this stochastic bias correction algorithm. The Copula model consists of two respective marginal distributions and a bivariate Copula function and is then used to generate bias corrected WRF data by conditional stochastic sampling. As already mentioned above in Sec. 3.1, Sklar’s theorem allows to separate the multi- variate joint distribution estimation into individual marginal distribution estimation and the Copula (dependence structure) estimation independently. Which is rather flexible to describe the joint behavior between variables with full freedom to the choice of the uni- variate marginal distributions and the Copulas. This is especially advantageous in cases where the dependence structure between the variates is too complex to be modelled by e.g. a multivariate Gaussian distribution, as it is often the case for hydrometeorological variables (Salvadori and Michele, 2007;Dupuis, 2007). In this study, following Sklar’s theorem, a so called bivariate Copula model is structured to describe the joint behavior between REGNIE and WRF data. It is then used to generate bias corrected WRF data by Copula based conditional stochastic sampling.
4.1
A bivariate Copula model
Suppose that the realizations (x1, y1), · · · , (xn, yn) are given from a pair of random
variates (X, Y ), and that it is desired to identify the bivariate distribution FXY(x, y)
Chapter 4. Copula-based stochastic bias correction framework 30 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 x y
=
0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 x u Theoretical Empirical 0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 y v Theoretical Empirical Observations RCM+
0.00 0.25 0.50 0.75 1.00 v 0.00 0.25 0.50 0.75 1.00 u 2 4 6 8 10 c(u,v)Figure 4.1: Visualization of a bivariate Copula model consisting of two marginal distributions and a theoretical Copula function that describes the pure dependence.
that characterizes their joint behavior. In a view of Sklars theorem, a bivariate Copula model can be applied. The bivariate Copula model of the variates X and Y consists of two univariate marginal distributions (FX(x) and FY(y)) and a Copula function C(u, v).
The marginal distributions describe the statistical properties of the variates (X and Y ) and the Copula captures the dependence structure between them. The Copula model (FX(x), FY(y) and C(u, v)) can be estimated separately based on the realizations x, y.
Figure4.1visualizes the process of estimating a Copula model with a bivariate exemplary data set, i.e. realizations (x, y) of the two random variates X and Y .
A scatter plot of the two realizations (x, y) is shown in Fig.4.1(left). To model the joint behavior by using a Copula model, the first step is to fit a marginal distribution function for the two variates X and Y , respectively (see Fig.4.1, middle). The realizations (x, y) are then transformed from the data space to the rank space (u, v) based on the fitted marginal distributions. The next step is to estimate the Copula function C from the ranked values (u, v) (see Fig.4.1, right). Here a Copula PDF is used instead of Copula CDF as the PDF is more illustrative. Finally, the unknown joint distribution FXY(x, y)
is fully determined by the marginal distributions and the Copula function, i.e. the dependence structure itself (Gr´egoire et al., 2008). Figure 4.1 visualizes the fact that different marginal distributions and Copula functions can be combined independently allowing to model highly complex dependence structures between the variables X and Y . This is especially beneficial if these dependence structures are non-linear, asymmetric or the data show heavy-tail behavior.
Chapter 4. Copula-based stochastic bias correction framework 31