try delta model

[lizzieinvancouver]

Suggested by JD on 12 June 2022:

You probably don’t want to get into this, but … another approach would be to use a delta transform of branch lengths (versus lambda) – the delta transform is a more direct measure of early vs late burst evolution. I think you only want to do this if it can be implemented by a simple substitution in your Stan code (if not, forget I ever mentioned it).

The lambda model multiplies all off-diagonal elements in the phylogenetic variance-covariance matrix by the value of λ.

In the delta model all elements of the phylogenetic variance-covariance matrix are raised to the power δ.

You could then optimise for delta, which would allow you a more direct interpretation of rate variation (delta of 1 == BM, delta <1 == Early Burst, >1 == Late Burst).

I asked: Is there still a sigma term that all elements are also multiplied by?

Yes, I think so, this is just the BM rate parameter, right?

.. and then I pulled together some notes in October when meeting with Mike Betancourt.

"Delta transform: The node heights of the phylogeny are raised to the power delta (Pagel 1999). Delta > 1 increases the length of external nodes and thus models a scenario where rates of evolution increase through time. Conversely, delta < 1 decreases the length of external nodes and thus models a scenario where rates of evolution decrease through time. Delta = 1 is a Brownian motion model (so the tree is returned unchanged)."

From https://wiki.duke.edu/display/AnthroTree/7.3.2+Branch+length+transformations+in+R

And issues with it being correlated with sigma^2: https://github.com/traitecoevo/modeladequacy/issues/58

See tempo of evolution in "Inferring the historical patterns of biological evolution' (Pagel 1999, Nature).

JD explained (over email during this time):

The delta (δ) transform simply raises node heights to the power delta; delta <1 trait evolution is concentrated early in the tree, delta >1 trait evolution is concentrated towards the tips.

See: https://lukejharmon.github.io/pcm/chapter6_beyondbm/

I asked, "Yes, but I can't tell what that means that I need to do to the matrix. Do you think I multiply all the off diagonal values by 1^delta? That's my best guess so far" ... and he replied:

I think you raise all elements to ^delta - see Eq. 6.3 in the Harmon book.

[lizzieinvancouver]

see also this OSPREE issue

[lizzieinvancouver]

I started on this several weeks okay in an ospree repo commit (see here) ... but I seem just to have recapitulated the issue Luke Harmon mentioned ...

[lizzieinvancouver]

So, next step would be to add the tree transformation to the Stan code, though Will thought you could not pull the whole thing outside of the matrix (as I did, copying Mike B.) ... but I sort of think you can.