Biology Reference
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numbers on the number line. Likewis e, for two points x
= (
x 1 ,
x 2 ),
y
= (
y 1 ,
y 2 )
2 , setting d
in the plane
2 gives the usual dis-
tance between the two points (length of line segment joining x and y ). For two points
x
(
x
,
y
) =
(
x 1
y 1 )
2
+ (
x 2
y 2 )
R
3 , by reducing to a picture with
two triangles, each in their own plane, it i s not hard to generalize the Pythagorea n the-
orem and show that setting d
= (
x 1 ,
x 2 ,
x 3 )
and y
= (
y 1 ,
y 2 ,
y 3 )
in three-space
R
2 gives
the standard distance between the two points (length of line segment joining x and
y )in
2
2
(
x
,
y
) =
(
x 1
y 1 )
+ (
x 2
y 2 )
+ (
x 3
y 3 )
3 . Distances can take on other forms, but going back to general metrics, we
have, by definition, that they are dissimilarity maps with the triangle inequality as an
additional property. Metrics are prominent in the study of mathematical real analysis,
as they generalize distance measures like the absolute value, as we have just seen.
Exercise 10.11.
R
a. G ive an argument
to show that
the standard distance map d
(
x
,
y
)
=
2 truly is a metric.
b. Given the comments above, generalize the distance D
2
2 for x
(
x 1
y 1 )
+ (
x 2
y 2 )
,
y
R
=[
d
(
x
,
y
) ]
given by
2 and its analog in
3
the Pythagorean theorem for
y with four
coordinates, and then five. It is a very important point that the notion of the
Cartesian coordinate plane
to points x
,
R
R
2 generalizes to spaces
4
5 , and so on for
n ,
, R
R
R
R
for any integer n
1, where these spaces exist as collections of all the ordered
n -tuples with n entries drawn from
R
.
c.
If you have not (or not recently) seen a proof of the Pythagorean theorem for
points in
3 , look for one in any linear algebra book, multivariable calcu-
lus text, or find an online source. Generalizing to
R
n
for any positive integer
R
n
1 is a common exercise in courses on linear algebra and advanced calcu-
lus. For a suggestive picture, see http://en.wikipedia.org/wiki/
File:Cube_diagonals.svg .
Turning back to biology, starting with a labeled X -tree T
= (
V
,
E
),
X
V ( T
rooted or unrooted) with edge-weighting
arises
naturally. As demonstrated in the example below, simply sum up the edge-weightings
ω(
ω :
E
R
, a dissimilarity map D T
e
)
, over all distinct edges e on the path
P x , y in T joining any two leaves x
,
y in X ,
) := e P x , y ω(
to get each value d T (
x
,
y
e
)
.
Example 10.5.
For the quartet tree T with leaves X
={
a
,
b
,
c
,
d
}
and the edge-
weighting
ω
indicated above (see Figure 10.4 ),
d T ,w (
a
,
a
)
d T ,w (
a
,
b
)
d T ,w (
a
,
c
)
d T ,w (
a
,
d
)
d T ,w (
b
,
a
)
d T ,w (
b
,
b
)
d T ,w (
b
,
c
)
d T ,w (
b
,
d
)
=
D T ,w
d T ,w (
c
,
a
)
d T ,w (
c
,
b
)
d T ,w (
c
,
c
)
d T ,w (
c
,
d
)
d T ,w (
d
,
a
)
d T ,w (
d
,
b
)
d T ,w (
d
,
c
)
d T ,w (
d
,
d
)
02
.
31
.
63
.
6
.
2
.
301
.
52
.
9
=
(10.1)
1
.
61
.
502
.
8
3
.
62
.
92
.
80
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