PointCountDistribution

PointCountDistribution[pproc,reg]

represents the distribution of point counts for the point process pproc in the region reg.

PointCountDistribution[pproc,{reg1,,regn}]

represents the joint distribution of point counts in regions regi.

Details

  • PointCountDistribution is also known as count distribution.
  • PointCountDistribution is distribution of point counts in point patterns resulting from the point process pproc.
  • PointCountDistribution is typically used to estimate the total number of points in an observation such as the number of trees in a forest, number of cells in sample or number of defects on a surface.
  • For a point process pproc, the point count in a full-dimensional region reg is a random variable whose distribution is represented by PointCountDistribution[pproc,reg].
  • The mean of the PointCountDistribution[pproc,reg] is the mean measure of the point process pproc in the region reg.
  • The regions regi and reg should be full dimensional and bounded as tested by SpatialObservationRegionQ, but in special cases can be defined with parameters.
  • PointCountDistribution will simplify to known special distributions whenever possible.
  • PointCountDistribution can be used with such functions as Mean, CDF and RandomVariate, etc.

Examples

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Basic Examples  (3)

Distribution of the number of points for a PoissonPointProcess in a unit disk:

Find mean and variance of the number of points:

PointCountDistribution for a HardcorePointProcess in a unit ball:

Simulate the distribution:

PointCountDistribution for a GeoDisk:

Scope  (4)

PointCountDistribution of a BinomialPointProcess on a unit disk:

The distribution of the number of points in a subregion:

Compute the mean number of points:

PointCountDistribution of a InhomogeneousPoissonPointProcess over a unit disk:

Over the union of two disjoint regions:

Simulate:

Over two intersecting regions:

Simulate:

The mean:

PointCountDistribution for cluster processes of NeymanScott type is a CompoundPoissonDistribution:

Cauchy point process:

Matérn point process:

Thomas point process:

Variance gamma point process:

In general, the point count distribution of a NeymanScott point process does not depend on the spatial distribution of the cluster points:

Compute probabilities and expectations in the absence of the closed form of PointCountDistribution:

Compute mean using simulations:

Compute event probability using simulations:

Applications  (5)

Compute the average number of points in a unit disk for a Strauss point process:

Suppose flaws in plywood occur on an average of one flaw per 50 square feet. Simulate the process of finding flaws on a per-square-foot basis:

The number of flaws in a given region:

Find the probability that a 4-foot-by-8-foot sheet will have no flaws:

For a round mirror with area 7.54 cm^2, the probability of no flaws is 0.91. Using the same polishing process, another round mirror with an area of 19.50 cm^2 is fabricated. Assuming the flaws are independent and randomly located, find the probability of no flaws on the larger mirror:

The distribution of the number of flaws in the mirror:

Find the intensity of the flaw point process:

The distribution of the number of flaws in the second mirror:

The probability of no errors in 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:

Simulate the faulty pixel configuration:

The distribution of the number of faulty pixels in the display:

Find the probability of no more than 15 faulty pixels in the display:

Find the pixel failure rate required to produce 4000×2000 pixel displays and still have an acceptance rate of at least 90%:

Plot the acceptance rate as a function of the pixel failure rate:

Find the maximal acceptable pixel failure rate:

Check the result:

Incidents of car-related thefts in Chicago:

Define inhomogeneous Poisson point process with this intensity:

Find the mean number of crimes:

Compute the probability that the number of crimes is less than 1200:

Compute the expected number of incidents in ZIP code 60639:

Properties & Relations  (4)

PointCountDistribution over pairwise disjoint region is a ProductDistribution:

PointCountDistribution over each region:

Multivariate PointCountDistribution over the list of regions:

PointCountDistribution over overlapping regions:

PointCountDistribution over each region:

Multivariate PointCountDistribution over the list of regions:

The components are correlated where the regions intersect:

PointCountDistribution over region covering for BinomialPointProcess:

Define a three-set covering:

Point count distribution for the covering:

PointCountDistribution for a region with non-numeric parameters:

Wolfram Research (2020), PointCountDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/PointCountDistribution.html.

Text

Wolfram Research (2020), PointCountDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/PointCountDistribution.html.

BibTeX

@misc{reference.wolfram_2021_pointcountdistribution, author="Wolfram Research", title="{PointCountDistribution}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/PointCountDistribution.html}", note=[Accessed: 22-October-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_pointcountdistribution, organization={Wolfram Research}, title={PointCountDistribution}, year={2020}, url={https://reference.wolfram.com/language/ref/PointCountDistribution.html}, note=[Accessed: 22-October-2021 ]}

CMS

Wolfram Language. 2020. "PointCountDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PointCountDistribution.html.

APA

Wolfram Language. (2020). PointCountDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PointCountDistribution.html