represents a homogeneous Poisson point process with constant intensity μ in .
- PoissonPointProcess is also known as homogeneous Poisson point process, stationary Poisson point process and complete spatial randomness (CSR).
- PoissonPointProcess generates points that are uniformly distributed in a region, with the average number of points per unit volume equal to μ.
- Typical uses are to model and test point collections that are completely spatially random. They are frequently used as building blocks for more complicated point processes such as clustered point processes.
- With intensity μ, the number of points in the observation region with volume follows the distribution PoissonDistribution[μ ν].
- The number of points in disjoint regions for a Poisson point process consists of independent random variables so . This property is also referred to as complete spatial randomness (CSR).
- A point configuration with intensity μ in an observation region with volume has density function with respect to PoissonPointProcess[1,d].
- The Papangelou conditional density for adding a point to a point configuration is for a Poisson point process with intensity μ.
- PoissonPointProcess allows μ to be any positive real number and d to be any positive integer.
- Possible PointProcessEstimator settings in EstimatedPointProcess for PoissonPointProcess are:
Automatic automatically choose the parameter estimator "MaximumPseudoLikelihood" maximize the pseudo-likelihood
- PoissonPointProcess can be used with such functions as RipleyK and RandomPointConfiguration.
Examplesopen allclose all
Basic Examples (3)
Sample from PoissonPointProcess using the different methods:
Estimate PoissonPointProcess over a geo region:
For a round mirror with area 7.54 cm, the probability of no flaws is 0.91. Using the same polishing process, another round mirror with an area of 19.50 cm is fabricated. Assuming the flaws are independent and randomly located, find the probability of no flaws on the larger mirror:
An LCD display has 1920×1080 pixels. A display is accepted if it has 15 or fewer faulty pixels. The probability that a pixel is faulty from production is and the faulty pixel positions are independent and random. Find the proportion of displays that are accepted:
Properties & Relations (11)
The number of points in a PoissonPointProcess is Poisson distributed:
Simulate a PoissonPointProcess over a unit disk:
Fit a PoissonDistribution to the point counts:
PoissonPointProcess has the property of complete spatial randomness:
PairCorrelationG for the Poisson point process is constant:
They both are equivalent to the CDF of an ExponentialDistribution:
Wolfram Research (2020), PoissonPointProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/PoissonPointProcess.html.
Wolfram Language. 2020. "PoissonPointProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PoissonPointProcess.html.
Wolfram Language. (2020). PoissonPointProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PoissonPointProcess.html