# Suggestions for improving the fit of probability density

I am comparing the error between wind turbine production and predictions from 24 hours ahead. The graphs below are using the normal distribution.

But can I do better?

• Your two sets of graphs, the bottom one fitting better than the top... what are they? Are you trying to say that the ones on the bottom are from simulated data? – John Jul 22 '11 at 15:09

Maybe use the laplace distribution, which has density

$$\frac{1}{2s} {\rm exp} \left( -\frac{|x-\mu|}{s} \right )$$

where $\mu \in \mathbb{R}, s > 0$ are parameters. As Wolfgang pointed out, your distribution appears to be symmetric but has higher kurtosis than the normal distribution; the laplace distribution has these properties and the QQ plot looks pretty similar to what you're observed from your data:

library(VGAM)
x=rlaplace(1000)
qqnorm(x)
qqline(x)


(you will need to download the package VGAM). The logistic distribution could also work but the QQ plots don't agree with what you've observed as well.

You could look at Box-Cox transformations. The boxcox function in the MASS package for R will compute a confidence interval on the value of lambda that gives the best transformation. Combine that with what you know of the science to choose an appropriate transform.