gaussian distribution trasform

misi · December 15th, 2004, 3:22 pm

hi all,is there any way to trasform an observed distribution (for example left tailed)into a normal one?please, can anyone help me?

rowntree · December 15th, 2004, 4:56 pm

Misi,If you want to rescale the distribution, try calculating the empirical cumulative distribution function, passing it through the normal distribution function's inverse cdf, and then adjusting for mean and std. Ie, x' = sigma*inverse_normal_cdf(empirical_cdf(x)) + muyou do need to be careful with your last data point however, to avoid infinity problems.

franck · December 15th, 2004, 5:51 pm

Misi,you can use the Box-Cox Transformation (X^lambda-1)/lambda. lambda is the transformationparameter. When lambda=0, you have the log transformation.Given a particular transformation, you can define a measure of the normality of the resulting transformation. One measure is to compute the correlation coefficient of a normal probability plot. You can then choose the optimal transformation parameter.Hope this helpFranck

misi · December 16th, 2004, 10:14 am

really thanks both!misi

Errrb · December 16th, 2004, 5:53 pm

Any unimodal distribution with pdf given by g(y) can be mapped to a normal distribution with pdf n(x) (or any other unimodal), using the following transformationg(y) dy=n(x) dx

misi · December 21st, 2004, 2:57 pm

sorry errrb,I'm really stupid.please, could you help me to understand?

Errrb · December 21st, 2004, 10:00 pm

hi misi,unimodal means that probability distribution function has bell shape, i.e. unique maximum, similar to normal distribution n(x)=Const*exp(-x*x/2). the transformation that I described below is just change of variables ....

misi · December 22nd, 2004, 2:08 pm

suppose to have a log-normal distributed variable,how can I apply the transformation you suggest?

Errrb · December 22nd, 2004, 3:56 pm

in the case of lognormal distribution g(y)=Const * (1/y) * exp(-0.5*(log y)^2) n(x)=Const * exp(-0.5 *x^2), Const=1/sqrt(2*pi)from g(y) dy = n(x) dx one may get (1/y) * exp(-0.5*(log y)^2) dy = exp(-0.5 *x^2) d xand therefore log(y)=x or y(x)=exp(x) This case is too simple in a more complicated cases it is more fun

franck · December 24th, 2004, 3:15 pm

The problem is when you don't know the distribution of your data.In this case Box-Cox transformation