Biology Reference
In-Depth Information
FIGURE 10.8
An example of
T
∈
T
5
; see Example 10.8
Using the BME vectors,
Pauplin's [
14
]formula
for the balanced tree length esti-
mation (i.e., estimated BME length)
ω(
)
T
, starting with the dissimilarity map
D
,is
given by
T
ω(
T
)
=
j
w
(
i
,
j
)
d
(
i
,
j
).
(10.4)
i
,
j
:
i
<
Equivalently, in the standard language of innner products of vectors, upon rewriting
D
as a vector
d
using the ordering from before, one gets
T
ω(
T
)
=
w
·
d
.TheBME
method proceeds by seeking
T
in Eq.
10.4
is minimal.
Exercise 10.21.
Using the results of Exercise
10.20
and the dissimilarity map from
Example
10.5
, use the BME method to find a tree
T
∈
T
n
so that
ω(
T
)
∈
T
4
that best fits
D
under
Pauplin's branch length estimation scheme. Also find
ω(
e
)
for the interior edge
e
of
T
in this case as well.
T
w
(
,
)
for any phylogenetic
X
-tree
T
(not
necessarily binary) in terms of certain cyclic permutations of (“circular orderings”)
of
X
that respect the structure of
T
. In the case of edge-weighted binary
X
-trees, one
recovers the expression for
In [
28
], the authors in fact defined terms
i
j
T
in the BME vector and Pauplin's formula. In
[
28
] this perspective was used to establish the consistency of the balanced tree length
estimation. The consistency (and statistical consistency) of the BME method as a
whole was established in [
29
].
Like all methods that would require a complete search over all trees
w
(
i
,
j
)
T
∈
T
n
,
combinatorial explosion is a problem. The large number of trees to consider quickly
overwhelms computer capacity, so that in practice, one must come up with a heuristic.
In [
29
,
15
], the theoretical underpinnings for the BMEmethod were further examined,
and a heuristic proposed and implemented for the BME method via the program
FastME
. This program essentially uses only nearest-neighbor interchange (NNI)
moves among trees to seek out a tree
T
/BME vector
T
w
that is a good candidate for
having minimal
. For more details, see [
29
,
15
].
ω(
T
)
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