Environmental Engineering Reference
In-Depth Information
In general, let the value of the index, for one
of a collection of administrative units, i, be θ
i
.
Data arrive as
X
of distributions. This method, Copula Based
Component Analysis (CICA), an alternative to
non-parametric Independent Component Analy-
sis (ICA) procedures, offers demonstrably more
descriptive results on an index of environmental
sustainability - the 2002 Environmental Sustain-
ability Index (ESI).
The CICA method accesses non-Gaussian de-
pendence via an information theoretic technique
on the special case of linear mixing models, the
component analysis family. This approach is a
useful
post hoc
procedure for index construction:
the goal in indexing is the, hopefully parsimoni-
ous, description of a multidimensional concept
with a univariate value.
Most useful indexes, perhaps ironically, are
designed to measure concepts and quantities that
are not predictable, have not yet been measured,
and are undefined. Environmental sustainability
is certainly of that type; humanitarian and social
development goals are as well generally ill de-
fined. The statistician's role in these settings is
substantial: it is perversely ironic to avoid exact
elucidation of statistical assumptions and method-
ology when they are dictated by the broad context
of the desired measurements.
Environmental statisticians have a stake
in making the broad concept of sustainability
operational: by providing specific measures by
which decision makers and the public can judge
progress, researchers can justify increased focus.
k
1
, a collection
of ratings/scores with some multivariate, non-
independent, distribution
f
X
. A full (linear) index-
ing scheme would yield: the scores for each unit;
the explicit, perhaps endogenously determined
weights; and confidence intervals for the index
scores as well as the variable weights. That is:
=
(
X
, ...,
X
) ~
f
X
K
∑
1
θ
i
=
c X
- the scores for each unit; the vec-
j
j
j
=
tor,
c
T
the weighting scheme chosen for the index;
confidence intervals for the scores,
P
(
θ
∈
(
L U
,
))
= −
1
α
, and weights,
i
i
i
j i
∈ = 1 α .
Choosing the appropriate weighting scheme
and generating confidence intervals for each
scalar θ
i
are separable tasks. On the other hand,
the confidence intervals are of course affected by
the choice of weighting scheme, even when the
weights themselves are arbitrary in the sense that
they are subject to an exogenous constraint cho-
sen by the indexers.
Essentially this couples the task of definition
and prediction for the indexer: assignment of the
weights is the specification of the index, but the
specification of the index as a proxy for measur-
able idea must influence the estimation of the
weights. Disentangling these tasks is heuristically,
computationally and theoretically non-trivial.
The author, in upcoming work on an index
designed to measure progress towards the United
Nations Development Program Millenium De-
velopment Goals (UNDP-MDG), addresses the
joint prediction and specification problem (UNDP
2009-2010, in progress).
P
(
c
( ,
l u
))
ACKNOWLEDGMENT
Kobi Abayomi thanks Lynne Butler, the Haver-
ford College Mathematics Department, and the
Consortium for Faculty Diversity. Kobi Abayomi
also thanks Jim Berger, Dalene Stangl, the Statisti-
cal and Applied Mathematical Sciences Institute
(SAMSI) and Duke University. This chapter was
completed and revised in pre and post doctoral
fellowships at Haverford and Duke/SAMSI.
CONCLUSION
This chapter illustrates a generalization of Prin-
cipal Component Analysis using copula families
