![Anscombe stabilized stdev](https://upload.wikimedia.org/wikipedia/commons/b/bb/Anscombe_stabilized_stdev.svg)
In
statistics, the Anscombe transform, named after
Francis Anscombe, is a
variance-stabilizing transformation In applied statistics, a variance-stabilizing transformation is a data transformation that is specifically chosen either to simplify considerations in graphical exploratory data analysis or to allow the application of simple regression-based or anal ...
that transforms a
random variable with a
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
into one with an approximately standard
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 make the
standard deviation approximately constant. Then
denoising
Noise reduction is the process of removing noise from a signal. Noise reduction techniques exist for audio and images. Noise reduction algorithms may distort the signal to some degree. Noise rejection is the ability of a circuit to isolate an un ...
algorithms designed for the framework of
additive white Gaussian noise are used; the final estimate is then obtained by applying an inverse Anscombe transformation to the denoised data.
Definition
For the
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
the mean
and variance
are not independent:
. The Anscombe transform
[
]
:
aims at transforming the data so that the variance is set approximately 1 for large enough mean; for mean zero, the variance is still zero.
It transforms Poissonian data
(with mean
) to approximately Gaussian data of mean
and standard deviation
.
This approximation gets more accurate for larger
,
as can be also seen in the figure.
For a transformed variable of the form
, the expression for the variance has an additional term
; it is reduced to zero at
, which is exactly the reason why this value was picked.
Inversion
When the Anscombe transform is used in denoising (i.e. when the goal is to obtain from
an estimate of
), its inverse transform is also needed
in order to return the variance-stabilized and denoised data
to the original range.
Applying the
algebraic inverse
:
usually introduces undesired
bias
Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group ...
to the estimate of the mean
, because the forward square-root
transform is not
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
. Sometimes using the asymptotically unbiased inverse
:
mitigates the issue of bias, but this is not the case in photon-limited imaging, for which
the exact unbiased inverse given by the implicit mapping
:
should be used. A
closed-form approximation of this exact unbiased inverse is
:
Alternatives
There are many other possible variance-stabilizing transformations for the Poisson distribution. Bar-Lev and Enis report
a family of such transformations which includes the Anscombe transform. Another member of the family is the Freeman-Tukey transformation
:
A simplified transformation, obtained as the
primitive of the reciprocal of the standard deviation of the data, is
:
which, while it is not quite so good at stabilizing the variance, has the advantage of being more easily understood.
Indeed, from the
delta method
In statistics, the delta method is a result concerning the approximate probability distribution for a function of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator.
History
The delta meth ...
,
.
Generalization
While the Anscombe transform is appropriate for pure Poisson data, in many applications the data presents also an additive Gaussian component. These cases are treated by a Generalized Anscombe transform and its asymptotically unbiased or exact unbiased inverses.
See also
*
Variance-stabilizing transformation In applied statistics, a variance-stabilizing transformation is a data transformation that is specifically chosen either to simplify considerations in graphical exploratory data analysis or to allow the application of simple regression-based or anal ...
*
Box–Cox transformation
References
Further reading
*{{Citation , last1=Starck , first1=J.-L. , last2=Murtagh , first2=F. , year=2001 , title=Astronomical image and signal processing: looking at noise, information and scale , periodical=Signal Processing Magazine, IEEE , volume=18 , issue=2 , pages=30–40 , doi=10.1109/79.916319, bibcode=2001ISPM...18...30S , s2cid=13210703
Poisson distribution
Normal distribution
Statistical data transformation