Context:
Poisson point processes (PPP) are widely discussed in the literature.
In the following figure a framework to generate two-dimensional PPP is demonstrated. First the area being studied (part of space which can be in 1D, 2D, 3D, …, in our example is a 2D shape i.e., square) is divided into cells (gridding). Second, for each cell a random number n is drawn from a Poisson distribution with density of $\lambda$. Then within each cell n points are uniformly distributed. The resulting point pattern is a homogeneous point process.
Questions:
- Is the described method correct?
- If we shuffle the numbers within area, then is it still a valid PPP?
- Is it valid for further dimensions?
Please check whether the following steps are the same as I illustrated.
note
Please read the following invaluable answers and comments which are appreciable. I don't repeat them here. However I put a confusing point here hopefully to be solved by a gentle explanation.
The second point is if we do realization only once then why should we call it Poisson where it is simply a uniform-random-values?
Best Answer
You don't need to grid. You could just draw a Poisson count, $n$, for the total number of points and then simulate $n$ iid points uniform on the region. For a square, you can draw $x_i$ and $y_i$ as independent uniforms. (For a more complex region, the simplest thing would be to embed it in a square, simulate for the square, and then just keep the points in the target region.)
The same thing can be done in higher dimensions. You draw the count from a Poisson with mean $\lambda V$ where $\lambda$ is the rate for the Poisson process and $V$ is the volume of the region, and then simulate that number of iid uniform points from the region.
And so 1: yes, 2: yes (if the grid regions are the same area), and 3: yes.