How to normalize data of unknown distribution

Question

I am trying to find the most appropriate characteristic distribution of repeated measurements data of a certain type.

Essentially, in my branch of geology, we often use radiometric dating of minerals from samples (chunks of rock) in order to find out how long ago an event happened (the rock cooled below a threshold temperature). Typically, several (3-10) measurements will be made from each sample. Then, the mean $\mu$ and standard deviation $\sigma$ are taken. This is geology, so the cooling ages of the samples can scale from $10^5$ to $10^9$ years, depending on the situation.

However, I have reason to believe that the measurements are not Gaussian: 'Outliers', either declared arbitrarily, or through some criterion such as Peirce's criterion [Ross, 2003] or Dixon's Q-test [Dean and Dixon, 1951], are fairly common (say, 1 in 30) and these are almost always older, indicating that these measurements are characteristically skewed right. There are well-understood reasons for this having to do with mineralogical impurities.

Therefore, if I can find a better distribution, that incorporates fat tails and skew, I think that we can construct more meaningful location and scale parameters, and not have to dispense of outliers so quickly. I.e. if it can be shown that these types of measurements are lognormal, or log-Laplacian, or whatever, then more appropriate measures of maximum likelihood can be used than $\mu$ and $\sigma$, which are non-robust and maybe biased in the case of systematically right-skewed data.

I am wondering what the best way to do this is. So far, I have a database with about 600 samples, and 2-10 (or so) replicate measurements per sample. I have tried normalizing the samples by dividing each by the mean or the median, and then looking at histograms of the normalized data. This produces reasonable results, and seems to indicate that the data is sort of characteristically log-Laplacian:

However, I'm not sure if this is the appropriate way of going about it, or if there are caveats that I am unaware of that may be biasing my results so they look like this. Does anyone have experience with this sort of thing, and know of best practices?

Answer 1

Have you considered taking the mean of the (3-10) measurements from each sample? Can you then work with the resulting distribution - which will approximate the t-distribution, which will approximate the normal distribution for larger n?

Answer 2

I don't think you're using normalize to mean what it normally means, which is typically something like normalize the mean and/or variance, and/or whitening, for example.

I think what you're trying to do is find a non-linear reparameterization and/or features that lets you use linear models on your data.

This is non-trivial, and has no simple answer. It's why data scientists are paid lots of money ;-)

One relatively straightforward way to create non-linear features is to use a feed-forward neural network, where the number of layers, and the number of neurons per layer, controls the capacity of the network to generate features. Higher capacity => more non-linearity, more overfitting. Lower capacity => more linearity, higher bias, lower variance.

Another method which gives you slightly more control is to use splines.

Finally, you could create such features by hand, which I think is what you are trying to do, but then, there is no simple 'black box' answer: you'll need to carefully analyze the data, look for patterns and so on.

Answer 3

You can try to use the family of Johnson's (SL, SU, SB, SN) distribution that are four-parameters probability distributions. Each distribution represents the transformation to the normal distribution.