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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.

Mean vs. median sample age.  Red line indicates mean=median.  Note older means caused by skewed measurements.

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:

enter image description here

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?

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    $\begingroup$ Since 'normalize' is used to mean several different things in contexts like this, precisely what do you mean by "normalize"? What information are you trying to get out of the data? $\endgroup$ – Glen_b Jun 4 '14 at 21:30
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    $\begingroup$ @Glen_b: By 'Normalize' I simply mean scaling things by median (or the mean) all of the measured ages of a sample by the median (or mean, or whatever). There is experimental evidence that the dispersion in the samples increases linearly with age. What I want out of the data is to see whether this type of measurement is best characterized by a normal, or log-normal, or beta, or whatever distribution, so that the most accurate location and scale can be derived, or L1 vs. L2 regression justified, etc. In this post I am asking how I can take data that I have described and investigate this. $\endgroup$ – cossatot Jun 5 '14 at 14:05
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    $\begingroup$ I have no expertise in this field, but your graphs and the thought you put into this looks good. You may have already seen it, but the Wikipedia article on Log-Laplace links to a nice paper, which doesn't directly address your question, but might have some interesting insights: wolfweb.unr.edu/homepage/tkozubow/0_logs.pdf $\endgroup$ – Wayne Sep 14 '16 at 18:17
  • $\begingroup$ I am not sure I completely understand, but maybe bootstrapping might help? If you recover the variance etc. of your distribution using bootstrapping methods, you can use the recovered information to normalize your data. en.wikipedia.org/wiki/Bootstrapping_(statistics) $\endgroup$ – 123 Feb 17 '17 at 15:06
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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?

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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.

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  • $\begingroup$ Normalize has several meanings across mathematics and the sciences; declaring that the one meaning personally most familiar is standard is what most people are tempted to do, but it won't wash with others. More seriously, this starts on-topic but then veers off. Where's the indication of interest in nonlinear models? Neural nets? Splines? What do these have to do with identifying a distribution or family of distributions, which is the question? I can't see the connection, so recommend cutting what is not relevant or expanding it to show how it is relevant. $\endgroup$ – Nick Cox Mar 21 '17 at 9:05
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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.

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