P1: OTA/XYZ P2: ABC
JWST061-09 JWST061-Caers April 6, 2011 13:24 Printer Name: Yet to Come
9.3 VISUALIZING UNCERTAINTY 165
distances and dot-products); we will just discuss one that is convenient here, namely the
radial basis kernel (RBF), which in all generality is given by:
K
ij
= k(x
i
, x
j
) = exp
−
(x
i
− x
j
)
T
(x
i
− x
j
)
2
2
Note that this RBF is function (a single scalar) of two Earth models x
i
and x
j
and of
parameter , called the bandwidth, which needs to be chosen by the modeler. The band-
width acts like a scalar of length. If is very large, then K
ij
will be zero except when
i = j, meaning that all x
i
tend to be very dissimilar; while if is close to zero, then all x
i
are deemed very similar. So it is necessary to choose a that is representative of the differ-
ence between the various x
i
. Practitioners have found that choosing the bandwidth equal
to the standard deviation of values in the L × L distance matrix is a reasonable choice.
Note that the RBF is function of the Euclidean distance (x
i
− x
j
)
T
(x
i
− x
j
). We want
to make this RBF kernel a bit more general, that is, by making it function of any distance,
not just the Euclidean distance. This can be done simply by means of MDS as follows:
1 Specify any distance d(x
i
, x
j
).
2 Use MDS to plot the locations in a low dimension (e.g., 2D or 3D), call these locations
x
d,i
and x
d, j
with d the dimension of the MDS plot.
3 Calculate the Euclidean distance between x
d,i
and x
d, j
.
4 Calculate the kernel function with given .
K
ij
= k(x
i
, x
j
) = exp
−
(x
d,i
− x
d, j
)
T
(x
d,i
− x
d, j
)
2
2
Since we have a new indicator of “distance,” namely K
ij
, we also have a new “metric
space’, which is traditionally called the “feature space” (Figure 9.6). Now, the same MDS
operation can be applied to matrix K. The eigenvalue decomposition of K is calculated
and projections can be mapped in any dimension (Figure 9.9), for example in 2D using
f =2
= V
K, f =2
1/2
K, f =2
where V
K, f =2
contains the eigenvectors of K belonging to the two largest eigenvalues
of K contained in the diagonal matrix
K, f =2
. An illustrative example is provided in
Figure 9.11. One thousand Earth models were mapped into 2D Cartesian space (the same
as Figure 9.6). Shown on the right-hand side of Figure 9.11 are the 2D projections of
models in feature space. Note how the complex cloud of Earth model locations has be-
come more “stretched out” (left-hand side). In the next chapter we will see that this gives
way to better model selection and quantification of response uncertainty.