Does anyone know what the intuition of H-volume is? For a 2-place real function $H$, H-volume of $[x_1,x_2]\times[y_1,y_2]$ is $H(x_2,y_2)-H(x_2,y_1)-H(x_1,y_2)+H(x_1,y_1)$. What is really the intuition of the H-volume? 
 A: The H-Volume is the volume contained by the rectangle $[x_1,x_2] \times [y_1,y_2]$ of a 3-dimensional function $H(x,y)$.  To visualize this, see the Figure 
which is the contour plot of the independence copula (which is simply a 3-dimensional function with some special properties that make it a copula function).
The H-Volume is the volume contained within the box labeled $R3$.  However, remember that the Copula function is defined as the H-Volume of the copula function $H$ from $[0,u] \times [0,v]$.  Thus, $H(x_2,y_2)$ in reference to the figure would be the volume contained by $R1+R2+R3+R4$.  To get the region of interest, which is just $[x_1,x_2] \times [y_1,y_2]$, we must subtract out $R2$ and $R4$.  However, by subtracting out $R2$ and $R4$, we have also subtracted out $R1$ twice. We thus add $R1$ back into the equation (recall that $R1$ is included when computing $H(x_2,y_2)$.  
To think about it in 3-D terms, see the Figure .  The H-Volume of this 3-D function, which happens to be the independence copula density, is the volume enclosed under the blue shaded area, where the points are given by the rectangle $[x_1,x_2] \times [y_1,y_2]$. 
A: The H-volume gives the the volume of some n-box (or hyper-rectangle) from some measure. Volume being generalised here for higher dimensions. When the measure is the Lebesgue measure, then the H-volume is our usual notion of length, area and volume in Euclidian space:

*

*$V_{Lebesgue}([0,0.5]) = 0.5$ (length)

*$V_{Lebesgue}([0,0.5]\times[0,0.5]) = 0.25$ (area)

*$V_{Lebesgue}([0,0.5]\times[0,0.5]\times[0,0.5]) = 0.125$ (volume)

*$V_{Lebesgue}([0,0.5]\times[0,0.5]\times[0,0.5]\times[0,0.5]) = 0.0625$ (volume)
It's useful in probability theory because it can be used to compute the probability mass in some n-box given a multidimensional cdf H.
The two dimensional calculation is the one you mentioned above:
$V_{H}([x_1,x_2] \times [y_1, y_2]) = H(x_2, y_2) - H(x_2, y_1) - H(x_1, y_2) + H(x_1, y_1)$
Pictorially:

Where the contoured lines are a bivariate cdf H.
The general formula is:
$V_{H}(J) = \sum_{\textbf{c}\; \in \; \text{vertices}(J)} \text{sign}_J(\textbf{c}) H(\textbf{c})$
where the sum is performed over all the vertices of the hyper-rectangle J, and with the sign of the sum element being:
$\text{sign}_J(\textbf{c}) = \begin{cases}
                \;\;1, & \text{if c has an even number of lower bounds}.\\
                -1, & \text{if c has an odd number of lower bounds}.
              \end{cases}$
The independence or $\pi$ copula ($\pi(u,v) = uv$) is the same as the Lebesgue measure.
There is a matlab function for performing this calculation in any dimension: https://github.com/AnderGray/Hvolume-Matlab
