I have measured data of the following form:
$f(3.2, 2.5) = 10$
$f(3.7, 2.6) = 9$
$f(3.1, 2.8) = 9.1$
That is, I know $f(x, y)$ for certain irregularly spaced values of $x$ and $y$. I want to estimate the integral $\int f(x, y) dx dy$. Is there a standard method to estimate this integral?
I cannot make additional measurements, I have to give my best estimate with the measurements at hand. I do not need especially high accuracy; the data is somewhat noisy anyway. A fast solution would be very helpful, since I will eventually need to repeat this estimation for millions or billions of inputs. If there happens to be a Python solution, that would be excellent.
I should mention that $f(x, y)$ is only nonzero in the local neighborhood that I'm sampling. For some fixed value $a$, if $x^2 + y^2 > a$, then $f(x, y) = 0$.