Anscombe transform
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In statistics, the Anscombe transform (1948) is a variance-stabilizing transformation that transforms a random variable with a Poisson distribution into one with an approximately Gaussian distribution. The Anscombe transform is widely used in photon-limited imaging (astronomy, X-ray) where images naturally follow the Poisson law. The Anscombe transform is usually used to pre-process the data in order to have the noise of constant standard deviation so as to apply denoising algorithms in the extensively studied framework of Gaussian additive noise.
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[edit] Definition
For the Poisson distribution the mean m, and variance v, are not independent: m = v. The Anscombe transform
aims at transforming the data so that the variance is set approximately 1 whatever the mean. It transforms Poissonian data to approximately Gaussian data of standard deviation 1 and is valid provided that the mean value of the Poissonian data is more than 20.
[edit] Alternatives
There are many other possible variance stabilising transformations for the Poisson distribution. Bar-Lev and Enis (1988) report a family of such tranformations which includes the Anscombe transform. Another member of the family is (Freeman & Tukey, 1950)
A simplified transformation is
which, while it is not quite so good at stabilising the variance, has the advantage of being more easily understood.
[edit] See also
[edit] References
- F.J. Anscombe, The transformation of Poisson, binomial and negative-binomial data. Biometrika, vol. 35, pp. 246-254, 1948.
- An article about astronomical image and signal processing
- Bar-Lev S.K., Enis P. (1988). On the classical choice of variance stabilising transformations and an application for a Poisson variate. Biometrika, 75 (4), 803-4.
- Freeman, M.F., Tukey, J.W. (1950). Transformations related to the angular and the square root. Ann. Math. Statist., 21, 607-11.
