Delta clarification

[sarahcrisley]

Can you explain the delta in Brownian Motion a bit more? Is it unitless, a rate, a probability? If you were to graph the distribution of delta, is time on the y-axis? Thanks!

[ajrominger]

Thanks for this clarifying question!

If you measured a trait (say body size in grams) then the units of delta would just be the units of the trait (in this case body size in grams). That's because delta is just the difference in trait values between nodes or tips in a phylogeny. In the video we presented delta as

$r_i = r_0 + d_i$

but you could re-arrange this to be

$d_i = r_i - r_0$

emphasizing that delta really is just the literal arithmetic difference between trait values.

The way I like to think about a graph for delta (I'm wishing now I made this more clear in the lecture!) would be like a series of histograms turned on their sides. Each histogram in the series (from left to right across the x axis) represents a different probability distribution of delta values taken at different amounts of temporal separation between nodes and/or tips in the phylogeny. Something like this:

[sarahcrisley]

Hi Andy,

(I'm bad about checking my personal email account!)

Thanks for this explanation--this was very helpful, particularly the figure. Thanks again for a great semester!

Sarah

On Tue, Nov 29, 2022 at 10:46 PM Andy Rominger @.***> wrote:

Thanks for this clarifying question!

If you measured a trait (say body size in grams) then the units of delta would just be the units of the trait (in this case body size in grams). That's because delta is just the difference in trait values between nodes or tips in a phylogeny. In the video we presented delta as

$r_i = r_0 + d_i$

but you could re-arrange this to be

$d_i = r_i - r_0$

emphasizing that delta really is just the literal arithmetic difference between trait values.

The way I like to think about a graph for delta (I'm wishing now I made this more clear in the lecture!) would be like a series of histograms turned on their sides. Each histogram in the series (from left to right across the x axis) represents a different probability distribution of delta values taken at different amounts of temporal separation between nodes and/or tips in the phylogeny. Something like this:

[image: image] https://user-images.githubusercontent.com/2481785/204702563-b2a16953-5fb7-4156-8171-8abffb03b213.png

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