Geoscience Reference
In-Depth Information
regression coefficients. Owing to this, the estimates of the
regression coefficients become unstable which ultimately leads
to imprecise interpretation of the regression model. Thus, a
multi-collinearity problem in the multiple regression setup
must be detected properly before going for the usual multiple
regression. Lots of techniques are available in the literature that
deals with the detection of multi-collinearity. Some of the most
simple and user-friendly techniques are the examination of the
correlation matrix and the detection of multi-collinearity based
on conditional number and conditional indices, which has been
discussed.
Detection of multi-collinearity by the examination of cor-
relation matrix
The simplest measure which is available in
the literature is the inspection of off-diagonal elements ρ
ij
in
X´X
, where
X
is an n × p matrix of the levels of the various cli-
matic explanatory variables (here, one has to consider variables
involved in the regression model as unit length scaled variable).
If two climatic explanatory variables, say Xi
i
and X
j
(for i, j = 1,
2, …, n), are nearly linear related, then |ρ
ij
| will be near to unity.
So, by examining the correlation matrix, one can easily detect
the problem of multi-collinearity.
Detection of multi-collinearity through condition num-
ber and condition indices
Another simple measure is the
measure in which the characteristic roots or eigen values of
X´X
can be used for the detection of multi-collinearity among
the climatic explanatory variables. Let, λ
1
, λ
2
, …, λ
p
be the
eigen values of
X´X
. When one or more near linear relation-
ships exist in the data, one or more eigen values will be small.
Apart from the eigen values, the
condition number
of
X´X
may also be preferred. The conditional number of
X´X
is
defined as
λ
λ
max
min
κ
=
(4.2)
Generally, when the condition number is less than 100, it can
be said that there is no serious problem of multi-collinearity.
Moderate to strong multi-collinearity exists when the condition
number lies between 100 and 1000. But if the condition number
exceeds 1000, then one has to give special consideration to the
problem of multi-collinearity as the condition number of more
than 1000 indicates severe multi-collinearity.
