De Donder–Weyl theory: Difference between revisions
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The '''Greenwood statistic''' is a spacing statistic and can be used to evaluate clustering of events in time or locations in space.<ref name=MG1>[[Major Greenwood|Greenwood, Major]] (1946) The Statistical Study of Infectious Diseases. ''[[Journal of the Royal Statistical Society]]'', 109(2): 85–110. {{jstor|2981176}}</ref> | |||
==Definition== | |||
In general, for a given sequence of events in time or space the statistic is given by:.<ref name=MG1/> | |||
:<math>G(n)=\sum^{n+1}_{i=1}D^2_i ,</math> | |||
where <math>D_i</math> represents the interval between events or points in space and is a number between 0 and 1 such that the sum of all <math>D_i = 1</math>. | |||
Where intervals are given by numbers that do not represent a fraction of the time period or distance, the Greenwood statistic is modified <ref>D'Agostino, Ralph B. and Stephens, Michael A. (1986) ''Goodness-of-fit techniques'', Marcel Dekker, Inc., New York </ref> and is given by: | |||
:<math>G(n)=\frac{\sum^{n+1}_{i=1}X^2_i}{T_n^2} ,</math> | |||
where: | |||
:<math>T_n=\sum^{n+1}_{i=1}X_i,</math> | |||
and <math>X_i</math> represents the length of the '''i''th interval, which is either the time between events or the distances between points in space. | |||
A reformulation of the statistic yields | |||
:<math>G(n)=\tfrac{1}{n+1} (\tfrac{n}{n+1}C_v^2+1) ,</math> | |||
where <math>C_v</math> is the sample [[coefficient of variation]] of the ''n'' + 1 interval lengths. | |||
==Properties== | |||
The Greenwood statistic is a comparative measure that has a range of values between 0 and 1. For example, applying the Greenwood statistic to the arrival of 11 buses in a given time period of say 1 hour, where in the first example all eleven buses arrived at a given point each 6 minutes apart, would give a result of roughly 0.10. However, in the second example if the buses became bunched up or clustered so that 6 buses arrived 10 minutes apart and then 5 buses arrived 2 minutes apart in the last 10 minutes, the result is roughly 0.17. The result for a random distribution of 11 bus arrival times in an hour will fall somewhere between 0.10 and 0.17. So this can be used to tell how well a bus system is running and in a similar way, the Greenwood statistic was also used to determine how and where genes are placed in the chromosomes of living organisms.<ref>Riley, M. C. ''et al.'' (2007) Locational distribution of gene functional classes in ''Arabidopsis thaliana'', ''BMC Bioinformatics''. 8:112</ref> This research showed that there is a definite order to where genes are placed, particularly with regard to what function the genes perform, and this is important in the science of genetics. | |||
== References == | |||
{{reflist}} | |||
[[Category:Spatial data analysis]] | |||
[[Category:Statistical deviation and dispersion]] | |||
Revision as of 00:42, 14 January 2014
The Greenwood statistic is a spacing statistic and can be used to evaluate clustering of events in time or locations in space.[1]
Definition
In general, for a given sequence of events in time or space the statistic is given by:.[1]
where represents the interval between events or points in space and is a number between 0 and 1 such that the sum of all .
Where intervals are given by numbers that do not represent a fraction of the time period or distance, the Greenwood statistic is modified [2] and is given by:
where:
and represents the length of the 'ith interval, which is either the time between events or the distances between points in space.
A reformulation of the statistic yields
where is the sample coefficient of variation of the n + 1 interval lengths.
Properties
The Greenwood statistic is a comparative measure that has a range of values between 0 and 1. For example, applying the Greenwood statistic to the arrival of 11 buses in a given time period of say 1 hour, where in the first example all eleven buses arrived at a given point each 6 minutes apart, would give a result of roughly 0.10. However, in the second example if the buses became bunched up or clustered so that 6 buses arrived 10 minutes apart and then 5 buses arrived 2 minutes apart in the last 10 minutes, the result is roughly 0.17. The result for a random distribution of 11 bus arrival times in an hour will fall somewhere between 0.10 and 0.17. So this can be used to tell how well a bus system is running and in a similar way, the Greenwood statistic was also used to determine how and where genes are placed in the chromosomes of living organisms.[3] This research showed that there is a definite order to where genes are placed, particularly with regard to what function the genes perform, and this is important in the science of genetics.
References
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- ↑ 1.0 1.1 Greenwood, Major (1946) The Statistical Study of Infectious Diseases. Journal of the Royal Statistical Society, 109(2): 85–110. Template:Jstor
- ↑ D'Agostino, Ralph B. and Stephens, Michael A. (1986) Goodness-of-fit techniques, Marcel Dekker, Inc., New York
- ↑ Riley, M. C. et al. (2007) Locational distribution of gene functional classes in Arabidopsis thaliana, BMC Bioinformatics. 8:112