Environmental Engineering Reference
In-Depth Information
hypothesized marginal distributions for the com-
ponents are well `fit'. I
(
u
is minimized when the
components are independent.
An alternate, though philosophically conso-
nant, approach is to:
2. Estimate univariate distributions
u
=
ˆ
(
=
ˆ
(
n
n
F
Wx
,
v
)
F
Wx
via the em-
)
i
i
i
j
j
i
pirical CDF.
3. Choose copula families at each bivariate
pair:
C
1 1
.
4. The bivariate mutual information, or,
E log dC u vi
i
−
−
( )
u
=
η
(
η
( )
u
+
η
( ))
v
θ θ
,
θ θ
,
θ θ
,
1
2
1
2
1
2
1. Still exploit the empirical distribution, setting
u
(
(
( ,
)))
are the elements of the
=
ˆ
( )
, treating the univariate margin-
als as observed or unparameterized, but…
Fit copulas pairwise, say, and minimize I
(
u
by diagonalization of a
mutual information
matrix
n
F x
j
i
i
i
`scatter' matrix.
5. C o n s t r u c t ` s c a t t e r ' m a t r i x
Γ
C
=
((
C u u
ij
( ,
)))
,
θ 1
6. Compute SVD of Γ
C
, λ
i
j
i j
=
..
k
λ
1
, ...,
k
7. Yield
y
=
b
x
=
r w x
with
y
⊥ , \:
y
MI X X
Θ
(
,
)
=
((
dC
( )
u
log dC
(
( ))])
u
=
((
MI
))
k
k
k
k
k
k
i
i
j
θ
θ
θ
ij
ij
ij
∀
i
, via
C
Θ
(18)
When the bivariate copula are well fit:
This approach permits dependencies that may
be restricted or inaccessible in many multivariate
copulas, where the index sets for the dependence
parameters must be hierarchical or nested (see
Joe 1997, Simon 1986). As well, the number
of families of bivariate copula is much larger
than the those for multivariate copula - as many
bivariate copula cannot be extended into greater
dimensions [Joe 1997].
Partite copula estimation, in this manner via
bivariate pairs, models the k-independent mar-
ginal dependence without the restrictions inherent
in k-independent full joint models, with the sac-
rifice that I
(
u
not estimated beyond the second
order.
Set
y RW
= , with
W
a `whitening' matrix
- the PCA transformation - and
R
the ICA trans-
formation. This allows diagonalization of the final
mutual information matrix via ordinary, quick,
Singular Value Decomposition (SVD). The partite
algorithm is:
(
)
C
→
MI C
≥ 0 for all
i
, . Thus
ˆ
ˆ
θ
θ
ij
ij
R
=
((
MI C
ij
(
)))
is positive semi-definite, by
ˆ
θ
exchangeability and the Singular Value Decom-
position of
R
yields an orthogonal basis, with
respect to the mutual information.
The partite approach permits copula model
selection at each index in the partitioned index
set; we fit bivariate copula to the
64
2
pairs. The
candidate copulae at each pair are two-parameter
extensions of
Laplace
type copulae, a subset of
the
Archimedean
family for copulas (see Abayo-
mi
[2008a]). Two-parameter families have the
advantage of allowing multiple types of depen-
dence, including some non-monotone dependence.
Archimedean copulas are exchangeable and have
a direct generating function representation [Joe
1997]. These properties are attractive for this
partite approach: we trade for model flexibility,
in a sense, at each of the bivariate margins with
a full model on the entire data. The exchange-
ability of the Archimedean family, with the non-
negativity of the (copula) mutual information,
1. Compute
Wx
, the PCA output or whitened
data.
