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I have some data from neuroscience, to be more precise a set of data which represents the connections between brain cells (so called "axons"). These connections form a binary tree as depicted below. Interestingly when I look at the lengths of these edges (i.e. axons) at each depth, the lengths are lognormally distributed. I.e. the black edges length has a lognormal distribution, the orange edges length has a lognormal distribution and so on.

More then that, even if I look at the distribution of the lengths of all edges from all depths, still I do observe a lognormal distribution. I dont add the edges up but just again consider each edges length as the Random Variables Value and plot its distribution.

So my question is: How can it be explained analytically, that from lognormals at each depth also a lognormal over all depths arise?

enter image description here

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Why do you think they are Log-normal distributed? If it's the case that you are plotting density estimates of your data and assuming that the distribution of lengths follows that distribution, I'd like to suggest an alternative explanation for what you are seeing.

The Log-normal and Gamma distributions can easily be made to look similar. They are defined on the same support $(0, \infty)$, and can be "mound-shaped" given the appropriate parameters. Is it possible that a Gamma distribution can model your data?

If so, there is a straightforward answer to your question. The Gamma distribution has the additive property, that is, if $$ X_i \sim \text{Gamma}(\alpha_i, \beta), \text{ for } i = 1, 2, ..., n, $$ then $$ \sum_{i = 1}^n X_i \sim \text{Gamma}(\sum_{i = 1}^n \alpha_i, \beta), $$ which can be proven using characteristic functions. Perhaps this explains the interesting behavior you see.

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  • $\begingroup$ Thank you first, I will think about it. My reasons for assuming the data follows a lognormal: i) We are several people, and each of us has checked the distribution independently and with various means (e.g. fitting tools by matlab) and have comme to the conlcusion that the distribution is a lognormal ii) Lognormals are not uncommon for natural growth processes iii) The Central Limit Theorem for LogNormals which states that the multiplication of i.i.d. Random Variables is approximately LogN distributed and indeed certain microbiological processes can be modelled as multiplications. $\endgroup$
    – Pugl
    Commented Mar 8, 2013 at 18:01
  • $\begingroup$ Btw, can you explain to me why the additive property is of relevance here? We are nowhere adding up random variables when checking for best-fitting distribution. $\endgroup$
    – Pugl
    Commented Mar 8, 2013 at 18:03

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