Variance stabilizing transformation¶
It is a transformation, that will give a random variable approximately constant variance.
Suppose \(E[X] = \mu\) and \(var[X] = \sigma^2(\mu)\). Let \(Y = f(x)\) we perform a taylor series expansion:
\[
Y \approx f(\mu) + (X - \mu)f'(\mu)
\]
Hence $\( var[Y] = f'(\mu)^2 var[X - \mu] = f'(\mu)^2 \sigma^2(\mu) \)$
A variance stabilizing transformation is a function \(f\) such that \(f'(\mu)^2 \sigma^2(\mu)\) is independent of \(\mu\)