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How Box cox and other transformations convert data into Normal Distributions ?

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Box Cox and other transformations find the exponent needed to transform the data into a normal distribution.

Transformations happen by finding the estimate of $\lambda$ and some values in the neighborhood are chosen to transform the original data.

Once the data is transformed, we assume the data is normally distributed.

The original formula proposed by Box & Cox is:

$$ y(\lambda) = \left\{ \begin{array}{ll} \frac{y^{\lambda} -1 }\lambda \quad if \lambda \neq 0; \\ log(y), \quad if \lambda = 0. \end{array} \right. $$

Which basically turns into choosing $\lambda$ as a model parameter. This value is found by maximizing log-likelihood.

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