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I want to apply a filter to my data that assume its normality. The data comes from different sensors measuring the same quantity over the same period of time.

Using a simple histogram plot, It seems that the data of most of the sensors follows a lognormal distribution, so my idea was to apply the following process:

  • fit a lognormal distribution to the data to estimate the shift of the distribution (as I asked in a previous post).
  • shift the data according to the estimated parameter, and apply the logarithm to transform the data to normal
  • apply the filter to data that should now better follow a normal distribution.

What I would like to do is:

  1. to have a way to quantify and compare the "improvement" of normality of the data before and after the transformation, to see if the transformation is significant (at least in some cases).
  2. test for log-normality of the data to see whether to apply this transformation (could I transform and test for normality?).
  3. test for log-normality at different time aggregations/time intervals and compare them

Looking to a previous question, I identified three different statistical tests, but I'm not sure whether these can be meaningfully applied to my case, and used for making these kinds of comparisons:

  • Kolmogorov–Smirnov test
  • Lilliefors test (normality testing without assuming distribution parameters)
  • Akaike information criterion
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