8.4 Anisotropic Covariances
207
X
Y
1
2
31
32
33
34
1024
1023
993
994
961
962
991
992
64
63
Fig. 8.6 Showing the ordering of the 1024-dimension eigenvector on the 32 × 32 spatial grid
Double-Exponential : C(x, x ; y, y ) = s 2 exp[−
|x − x |
L x
+
|y − y |
L y
]
= s 2 exp[−
|x − x |
L x
] exp[−
|y − y |
L y
]
Gaussian :
C(x, x ; y, y ) = s 2 exp[−
(x − x ) 2
L 2
x
+
(y − y ) 2
L 2
y
]
= s 2 exp[−
(x − x ) 2
L 2
x
] exp[−
(y − y ) 2
L 2
y
] ,
(8.25)
where L x and L y are the correlation lengths in the x and y directions, respectively.
Note that these anisotropic covariances remain separable.
The form of the exponential for both covariances is
|x − x | p
L
p
x
+
|y − y | p
L
p
y
, where p =
1 for the double-exponential
2 for the Gaussian ,
(8.26)
and level curves for these functions are shown in Fig. 8.7.