Nested Sets' Metrics

In article <[email hidden]>,

Albretch said:

Nested Sets, NSs, implemented as tree-like, adjacent lists,
are relatively flexible and extensible data structures and
I was wondering how you could define a metric or some sort
of criterion of measurability on them.

...

Quoted message said:

Could you point me to any links on these ideas?

Looking at Nested Sets material on the web, I see that it is
a way of encoding a tree of items. There is an existing
literature on Consensus Trees and a related one on distances
between trees, as well as papers on consensus subtrees and
on supertrees, all of which might apply. My recent book
"Inferring Phylogenies" has coverage of these.

--
Joe Felsenstein [email hidden]
Department of Genome Sciences and Department of
Biology, University of Washington, Box 357730, Seattle,
WA 98195-7730 USA

Quoted message said:

A Nested Sets, NSs, implemented as tree-like, adjacent
lists, are relatively flexible and extensible data
structures and I was wondering how you could define a
metric or some sort of criterion of measurability on them.

...

Quoted message said:

Could you point me to any links on these ideas?

JF:- Looking at Nested Sets material on the web, I see that
it is a way of encoding a tree of items. There is an
existing literature on Consensus Trees and a related one on
distances between trees, as well as papers on consensus
subtrees and on supertrees, all of which might apply. My
recent book "Inferring Phylogenies" has coverage of these.

JE:- Amazingly, nested sets of fitness levels are not even
considered by gene centric Neo Darwinists. In reality,
genomic genes occupy a nested set of fitness entirely within
the Darwinian fertile organism fitness set, forcing all gene
fitnesses to only be sub selected AFTER selection at the
Darwinian fertile organism level of selection has been
completed. This is why genomic gene fitnesses are non
additive. Thus Hamilton's rule cannot support organism
fitness altruism (OFA) because the selfish gene must be
firstly selected at the fertile organism outer nested set of
fitness, interface. It should be noted that populations of
fertile organisms do _not_ form another nested set over the
fertile organism, Darwinian level. The individuals that form
these populations only form fitness set _intersections_
between every other. This is why such a group fitness
remains additive allowing all individuals to remain
independently selectable. Thus Hamilton's rule can only
support organism fitness mutualism (OFM) and not OFA. The
use of Hamilton's rule to support OFA after group selection
failed to be able to do so was logically invalid.

John Edser Independent Researcher

PO Box 266 Church Pt NSW 2105 Australia

[email hidden]

[email hidden] (Albretch) wrote in message news:<[email hidden]>...

Quoted message said:

Then, given a number of nested sets, how would you define
a 'metric' that would measurably give you the distance
between any two pairs of them?

Is this a typo? Do you want a "distance" between two sets,
or between "two pairs" of sets?

The best metric will depend on the properties you want your
distance to have. Is it desirable that the distance between
two sets not change as more sets are added to the
"database"? The metrics that Joe F. suggests will not have
this kind of "stability" (nor should they).

If you want a metric that gives the same answer as more data
becomes available, then there is really only one metric
worth considering - the distance is the size of the symetric
difference between the two sets.

I would suggest that you start with an axiomatic
"requirements analysis" before searching the literature for
examples of metrics. Once you know what properties you want
from your metric, the best definition of the metric will be
obvious to you.

Start with the triangle inequality, and add axioms
from there.

In article <[email hidden]>,

Jim Menegay said:

[email hidden] (Albretch) wrote in message
news:<[email hidden]>...

Quoted message said:

Then, given a number of nested sets, how would you
define a 'metric' that would measurably give you the
distance between any two pairs of them?

Is this a typo? Do you want a "distance" between two sets,
or between "two pairs" of sets?

Yes, I think it would be between two nested sets, not two
pairs of them.

Quoted message said:

The best metric will depend on the properties you want your
distance to have. Is it desirable that the distance between
two sets not change as more sets are added to the
"database"? The metrics that Joe F. suggests will not have
this kind of "stability" (nor should they).

The nested sets referred to are, if I understand properly,
each trees. So the tree distances I discussed are relevant,
and would not change as more trees (nested sets) are added.

Quoted message said:

If you want a metric that gives the same answer as more
data becomes available, then there is really only one
metric worth considering - the distance is the size of the
symetric difference between the two sets.

As these nested sets are not just single sets, but are
nested sets of objects (i.e. hierarchies), the symmetric
distance of sets is not the answer of itself.

--
Joe Felsenstein [email hidden]
Department of Genome Sciences and Department of
Biology, University of Washington, Box 357730, Seattle,
WA 98195-7730 USA