# box-cox

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• Oct 16th 2010, 12:51 PM
MichaelMath
box-cox
I have the BC tranformation:

$\displaystyle y^{(\lambda)}= \left\{ \begin{array}{11} \frac{y^\lambda - 1}{\lambda} & \mbox{ \lambda \neq 0$}\\
log_e (y) & \mbox{ \lambda = 0$}\\ \end{array}$

I need to show that this give a continous famly of transformations in $\displaystyle \lambda$.

How do I go about doing this?
• Oct 17th 2010, 06:28 AM
Guy
I think the question is just asking for you to show that, for fixed y, the function $\displaystyle y^{(\lambda)}$ is continuous in $\displaystyle \lambda$. For $\displaystyle \lambda \ne 0$ this trivial. To show continuity at $\displaystyle \lambda = 0$ it suffices to show that

$\displaystyle \displaystyle \lim_{\lambda \to 0} y^{(\lambda)} = \log y$

which is easy. The simpliest way of going about this is to define $\displaystyle f(\lambda) = y^\lambda$ and write $\displaystyle y^{(\lambda)} = \frac{f(\lambda) - f(0)}{\lambda - 0}$ which is a form that we all recognize.
• Oct 19th 2010, 11:04 AM
MichaelMath
Quote:

Originally Posted by Guy
I think the question is just asking for you to show that, for fixed y, the function $\displaystyle y^{(\lambda)}$ is continuous in $\displaystyle \lambda$. For $\displaystyle \lambda \ne 0$ this trivial. To show continuity at $\displaystyle \lambda = 0$ it suffices to show that

$\displaystyle \displaystyle \lim_{\lambda \to 0} y^{(\lambda)} = \log y$

which is easy. The simpliest way of going about this is to define $\displaystyle f(\lambda) = y^\lambda$ and write $\displaystyle y^{(\lambda)} = \frac{f(\lambda) - f(0)}{\lambda - 0}$ which is a form that we all recognize.

am retarded, can't remember calculus. Can somebody dig me out of the dirt?
• Oct 19th 2010, 11:29 AM
MichaelMath
$\displaystyle f(\lambda)=y^\lambda=\frac{f(\lambda)-f(0)}{\lambda}=\lambda f(\lambda)+1$

$\displaystyle f'(\lambda)=y^\lambda log(y)$

$\displaystyle f'(0)= log(y)$

make sense?
• Oct 19th 2010, 02:38 PM
Guy
The basic argument is

$\displaystyle \displaystyle \lim_{\lambda \to 0} y^{(\lambda)} = \lim_{\lambda \to 0} \frac{f(\lambda) - f(0)}{\lambda - 0} = \frac{d}{d\lambda} y^\lambda \big|_{\lambda = 0} = ... = \log y$