Numerically determine the global property of the positively quadrant dependency (PQD) characteristic of a copula as described by Nelsen (2006, p. 188). The random variables $X$ and $Y$ are PQD if for all $(x,y)$ in $\mathcal{R}^2$ when
$H(x,y) \ge F(x)G(x)$ for all $(x,y)$ in $\mathcal{R}^2$
and thus by the copula $\mathbf{C}(u,v) \ge uv$ for all $(u,v)$ in $\mathcal{I}^2$. Alternatively, this means that $\mathbf{C}(u,v) \ge \mathbf{\Pi}$ or globally greater than independence ($uv$).Nelsen (2006) shows that a copula is PQD when
$$0 \le \beta_\mathbf{C} \mbox{,\ } 0 \le \gamma_\mathbf{C}\mbox{,\ and\ } 0 \le \rho_\mathbf{C} \le 3\tau_\mathbf{C}\mbox{,}$$
where $\beta_\mathbf{C}$, $\gamma_\mathbf{C}$, $\rho_\mathbf{C}$, and $\tau_\mathbf{C}$ are various copula measures of association or concordance that are described in blomCOP, giniCOP, rhoCOP, and tauCOP.
The concept of negatively quadrant dependency (NQD) is the reverse: $\mathbf{C}(u,v) \le \mathbf{\Pi}$ for all $(u,v)$ in $\mathcal{I}^2$; so NQD is globally smaller than independence.
Conceptually, PQD is related to the probability that two random variables are simultaneously small (or simultaneously large) is at least as great as it would be if they were independent. The graph of a PQD copula lies on or above the copulatic surface of the independence copula $\mathbf{\Pi}$, and conversely a NQD copula lies on or below $\mathbf{\Pi}$.
Albeit a global property of a copula, there can be local variations in the PQD/NQD state. Points in $(u,v)$ where $\mathbf{C}(u,v) - \mathbf{\Pi} \ge 0$ are locally PQD, whereas points in $(u,v)$ where $\mathbf{C}(u,v) - \mathbf{\Pi} \le 0$ and locally NQD. The second example shown is a dramatic case where a switch over from NQD to PQD is made about 0.25 joint probability up the diagonal.
Readers are directed to the last examples wolfCOP because as those examples involve the copulatic difference from independence $\mathbf{C}(u,v) - \mathbf{\Pi} = \mathbf{C}(u,v) - \mathbf{\Pi}$ with 3-D renderings.