Unformatted text preview: 3697, 5440, 6081, and 6681 are shown (leftright, topdown). nhd+2 i=1 i=1 xi nh i=1 h (A.6)
^
^
Since fK (j + 1) ; fK (j ) converges to zero, (A.6)
implies that kyj+1 ; yj k also converges to zero, i.e.,
yj j=1 2::: is a Cauchy sequence. This completes the
proof, since any Cauchy sequence is convergent in the
Euclidean space.
p ^^
Proof that the distance d(^ q) = 1 ; (p q) is a
p^
metric i The proof is based on the properties of the Bhattacharyya coe cient (17). According to the Jensen's
inequality 8, p.25] we have
v
um
m
m sq
Xp
X
^u uX q = 1
^^
(p q) =
pu qu = pu p t ^u
^^
^^
u
u=1
u=1
u=1
(A.7) 2 2y>+1 xi ; kyj+1 k2 = nh1+2
j
d
#
n
xi 2 ; ky k2 X g xi 2
j +1
h
h
i=1 (A.4)
7 ^
^
with equality i p = q. Therefore, d(^ q) =
p^
p
^^
^
1 ; (p q) exists for all discrete distributions p and
^
^^
q, is positive, symmetric, and is equal to zero i p = q.
The triangle inequality can be proven as follows.
^^
Let us consider the discrete distributions p, q, and ^,
r
and de ne the associated mdimensional points p =
;p
;p
p^
p^
p1 : : : pm > , q = q1 : : : qm > , and r =
^
^
;p
p^
r1 : : : rm > on the unit hypersphere, centered at
^
the origin. By taking into account the geometric interpretation of the Bhattacharyya coe cient, the triangle
inequality d(^ ^) + d(^ ^) d(^ q)
pr
qr
p^
(A.8)
is equivalent to q
q
q
1 ; cos( p r )+ 1 ; cos( q r )
1 ; cos( p q ):
(A.9)
If we x the points p and q , and the angle between
p and r , the left side of inequality (A.9) is minimized when the vectors p , q , and r lie in the same
plane. Thus, the inequality (A.9) can be reduced to a 2dimensional problem that can be easily demonstrated
by employing the halfangle sinus formula and a few
trigonometric manipulations. 11] A. Djouadi, O. Snorrason, F.D. Garber, \The Quality of TrainingSample Estimates of the Bhattacharyya
Coe cient," IEEE Trans. Pattern Analysis Machine Intell., 12:92{97, 1990.
12] A. Eleftheriadis, A. Jacquin, \Automatic Face Location Detection and Tracking for Mo...
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This note was uploaded on 11/27/2011 for the course MATH 3484 taught by Professor Staff during the Fall '10 term at University of Central Florida.
 Fall '10
 Staff
 Math, The Land

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