Axial Data Analysis
Random Vector
X is random vector if X :  , A, P   R m , B m
B  B it follows that X  B   A
m
1
Axial Data
the spherical representation of RP m for axial data is
[ x]  {x,  x}  RP m  S m / x ~  x
axial data can also be writen as
R m1 \ 0/ x ~  x , 
note: this is the homomorphous condtion
x  R m 1 x   x 0 , x1 ,..., x m 1 , x m  and y   y 0 , y1 ,..., y m 1 , y m 
[ y ]  [ x]
y 0 y1
y m 1 y m
 1  ...  m 1  m
0
x
x
x
x
Properties of Axial data
Sometime the observations are not direction but
axes, that is, the unit vector and – are
indistinguishable, so that it is which is observed.
In this context it is appropriate to consider
probability density functions for onwhich are
anitpodally symmetric (diametrically opposite
<an antipodal point on a sphere>)
• i.e. f  x  f  x
• in such cases the observations can be
regrarded as being on the projective space ,
which is obtained by identifying opposite points
on the sphere .
Random axis
Maps to a Projection
Y :  , A, P   RP m ,BRPm
A random axis
where BRPm    algebra generated by open sets in RP m
B  BRP m , Y
1
 B A
Distance
.
d  a1 , a2    where  is the accute angle
B  a,    b  R m | d  a, b    
for a 3-dimensional shpere
volume  RP 2  
volume  S 2 
2
4 

2
3


2
3
Uniform distribtuion
.
X has a density with respect to the volume probability measure
f X  p   lim
 
QX  B  p ,   
U  B  p,   
  1  t
1
1
2

p 3
2
dt  2 1  t 2 
1
0
p 3
2
dt
Density for the Uniform
.
X has a density with respect to the volume probability measure
f X  p   lim
 
QX  B  p ,   
U  B  p,   
  1  t
1
1
2

p 3
2
dt  2 1  t 2 
1
0
p 3
2
dt
Finding the Mean
.
Question : What is the mean axis of an arbitrary distribtuion on RP m
recall for non-axial X  R m we have
  E  X    x  f  x  dx
where x   x1 , x2 ,..., xm 
T
Intrinsic Mean
.
In RP
m
the Frechet of Y minimizer of
E  d Y , p    F  p 
2
For axial data the distance is induced by
a distance in a space of symmetric matrices.
Distance between axes
.
j
 xxT
 x  RP m 
note : xT x =1 and x ~  x
j  x   j   x 


d  x  ,  y   d 0 j  x  , j  y   d 0  xxT , yy T 
m 1
d
2
0
 A, B     aij  bi j 
i, j
2
 Tr
 A  B   A  B 
T


d  x  ,  y   Tr  xxT  yy T   xxT  yy T 
T

 Tr  x  x x  x

 Tr  xxT xxT  2xxT yy T  yy T yy T 
T
.
T
 2xxT yy T  y  y T y  y T
note :  xT x    y T y   1 therefore



d  x  ,  y   Tr  xxT  2xxT yy T  yy T 
 Tr  xxT   2Tr  xxT yy T   Tr  yy T 
note : Tr  ab   Tr  ba  therefore
Tr  xxT  =TR  xT x  =TR (1)  1  Tr  yy T  =TR  y T y  =TR (1)
so
d  x  ,  y   2  2Tr  xxT yy T   0
Hence, d  x  ,  y  is minimized when Tr  xxT yy T  is maximized
The minimum of
expected distance squared
.


E d 2  X  , p   F  p 
min
 d  X  , p  dQ  d  X   min 
2
RP
So
max

RP
m



m



Tr  xxT  2xxT μμT  μμT  dQ  d  X 
Tr  xxT μμT  dQ  d  X   max

RP
m



Tr  μT xxT μ  dQ  d  X  =G μ 
note: μT μ  1 Therefore

RP
m
xxT dQ  d  X   E  XXT   K
note : G μ   μT E  XXT  μ is maximized if
if μ is the eigenvector corresponding to the largest eigenvalue
Finding the Sample Mean
.
 x1  ,...,  x m  is a sample of axes

1
The empical distibution Qˆ p   x1   ...   xn 
n

the sample mean axes is the empirical
n
1
E  XXT    xxT Qˆ p  X    xi    xxT  K
n i 1
i 1
n
Central Limit Theorem
.
let S  G  j , X ab  n
1
 N a   N b   X
1
1
a
r
X X
a
r

N 2
r
r
then
d
T  ν   nνT μ1 , μ 2 ,..., μ m  S 1 μ1 , μ 2 ,..., μ m  ν 
  N2 1 distribution
T
Watson Distribution
• One of the simplest models for axial data is
the Dimroth-Scheidegger-Watson model,
which has densities
• Where

1
2
1 p 
f  x; μ, k   M  , , k  exp k  μT x 
2 2 
1 p 
 p 1 1 
M  , ,k   B
, 
2 2 
 2 2
 p 1 
1
2

 p  1 1  1 
B
,  t
2
2 0

1  t 
1

1 
 1
2 
1
1
e
kt 2
1  t 
 p 3 
1 

 2 
 t
0

2
 p  3 / 2
1  t 
dt
1
 
2
Note: the density is rotationally symmetric
about μ
Bingham distribution
.
1
1 p 
f   x, A   1 F1  , , A  exp  xT Ax 
2 2

1 p 
xT Ax
where 1 F1  , , A    p1 e dx
2 2
 S
Where the integration is with respect to the
uniform distribution on