Environmental Engineering Reference
In-Depth Information
=
ˆ
( )
; the data as trans-
formedby their empirical CDF's. Copulas are estimated, separately, on each bivariate pair. Lower
panel: 3D Plot of PCA
1
vs. PCA
2
vs. PCA
3
; the bivariate plots are the planes in the 3D plot. The ap-
pearance of extreme dependence in the bivariate plots - figures (a) and (b) especially, is a feature of the
imputation procedure. Compare these with the bivariate diagnostic plots in Abayomi et al. [2008]
n
Figure 4. Upper panels: Plots of first three PCA components. Each u
F x
i
i
i
yields a positive semi definite mutual information
matrix Γ
C
which can be orthogonalized via
ordinary SVD methods. See Figure 5.
In analogy with the covariance/correlation
matrix in a PCA procedure, we use the mutual
information matrix Γ
C
- estimated via the bi-
variate copulae - as a representation of the mul-
tivariate scatter. In PCA the covariance for each
bivariate is estimated via
x
T
; in this version of
CICA the mutual information for each bivariate
is estimated via the copula density:
formation, negentropy (distance from Gaussian-
ity), or high order sample moments (usually cu-
mulant) via gradient descent or other iterative
procedure. In this version of CICA we substitute
the iterative optimization with SVD orthogonal-
ization.
Copulae at each pair are selected - separately
- via maximum likelihood from bivariate two-
parameter (Archimedian) Laplace families in Joe
[1997]. Additionally, each copula model is rotated
0, 90, 180 or 270 degrees.
The Scree plot in Figure 6 [Catell 1966] sug-
gests that the majority of `variation' (68 percent)
- as approximated by the cumulative eigenvalues
of the SVD - is explained at about seven compo-
nents. This can be interpreted as an indicator that
a majority of the variation - Gaussian as well as
non-Gaussian, by the CICA procedure - in the
ESI can be explained by a reduced amount of
information.
n
=
∑
1
142
n
n
n
n
dC u u log dC u u
(
,
)
(
(
,
))
(20)
ˆ
ˆ
θ
i
j
θ
i
j
n
=
=
ˆ
( )
is the order statistic of
the `whitened' data,
W
the `whitening' matrix.
ICA methods typically optimize the mutual in-
where each
u
F w x
n
i
i
i
i
