Complete spatial randomness

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Full spatial randomness (CSR) describes a process point at which point occurrences occur within a particular study area in a completely random manner. This is identical to a homogeneous spatial Poisson process. Such a process is modeled using only one parameter ${\ displaystyle \ rho}$ , ie the density of points within the specified area. The term complete spatial complication is usually used in Applied Statistics in the context of examining a particular point pattern, whereas in other statistical contexts, it refers to the concept of a spacial Poisson process.

Video Complete spatial randomness

Model

Data in the form of a set of points, distributed irregularly in a spatial domain, appears in various contexts; examples include the location of trees in forests, birds' nests, nuclei in the tissues, sick people in populations at risk. We call each data set as a spatial point pattern and refer to the location as an event, to distinguish it from the arbitrary points in the region. The complete spatial randomness hypothesis for the spatial point pattern confirms that the number of events in each region follows the Poisson distribution with the average number determined per uniform subdivision. Pattern events are independently and evenly distributed in outer space; in other words, it happens everywhere and does not interact with each other.

"Uniform" is used in the sense of following a uniform probability distribution across the study area, not in the "uniform" sense spread throughout the study area. There was no interaction between the events, because the intensity of events did not vary on the plane. For example, the assumption of independence will be violated if the existence of an event either encourages or inhibits the occurrence of other events in the environment.

Maps Complete spatial randomness

Distribution

Probabilitas untuk menemukan dengan tepat ${\ displaystyle k}  ÂÂ$ menunjuk dalam area ${\ displaystyle V}  ÂÂ$ dengan kepadatan peristiwa ${\ displaystyle \ rho}  ÂÂ$ adalah:

${\ displaystyle P (k, \ rho, V) = {\ frac {(V \ rho) ^ {k} e ^ {- (V \ rho)}} { k!}}. \, \!}  ÂÂ$

Saat pertama, jumlah rata-rata titik di area tersebut, hanyalah ${\ displaystyle \ rho V}  ÂÂ$ . Nilai ini intuitif karena merupakan parameter tingkat Poisson.

Probabilitas untuk menemukan ${\ displaystyle N ^ {\ mathrm {th}}}  ÂÂ$ tetangga dari suatu titik tertentu, pada beberapa jarak radial ${\ displaystyle r}  ÂÂ$ adalah:

${\ displaystyle P_ {N} (r) = {\ frac {D} {(N-1)!}} {\ lambda} ^ {N} r ^ {DN -1} e ^ {- \ lambda r ^ {D}},}  ÂÂ$

di mana ${\ displaystyle D}  ÂÂ$ adalah jumlah dimensi, ${\ displaystyle \ lambda}  ÂÂ$ adalah parameter bergantung-kerapatan yang diberikan oleh ${\ displaystyle \ lambda = {\ frac {\ rho \ pi ^ {\ frac {D} {2}}} {\ Gamma ({\ frac {D} {2 }} 1)}}}  ÂÂ$ dan ${\ displaystyle \ Gamma}  ÂÂ$ adalah fungsi gamma, yang ketika argumennya adalah integer, hanyalah fungsi faktorial.

Nilai yang diharapkan dari ${\ displaystyle P_ {N} (r)}  ÂÂ$ dapat diturunkan melalui penggunaan fungsi gamma menggunakan momen statistik. Saat pertama adalah jarak rata-rata antara partikel yang terdistribusi secara acak di ${\ displaystyle D}  ÂÂ$ dimensi.

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Aplikasi

The study of CSR is critical for the comparison of point data measured from experimental sources. As a method of statistical testing, tests for CSR have many applications in social sciences and in astronomy examinations. CSR is often the standard used to test data sets. Roughly described one approach to test the CSR hypothesis is as follows:

1. Use statistics that are a function of distance from each event to the next closest event.
2. First focus on a specific event and formulate a method to test whether the next event and the nearest event are significantly near (or remote).
3. Next consider all events and formulate methods to test whether the average distance from each event to the next closest event is significantly short (or long).

In cases where computational test statistics are analytically difficult, numerical methods, such as Monte Carlo method simulations are used, by simulating stochastic processes in large quantities.

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References

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Further reading

Diggle, P. J. (2003). Spatial Point Pattern Statistics Analysis (2nd ed.). New York: Academic Press. ISBN: 0340740701.

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External links

• Improving the Events Distance Event Test of Randomness in a Spatial Point Process

Source of the article : Wikipedia