Geoscience Reference
In-Depth Information
Another measure based on the eigen values of
X´X
is the use
of
condition indices
of
X´X
which are defined as
λ
λ
max
k
κ
=
,
k
=
12
,
p
,
…
,
(4.3)
Clearly, the largest condition index is nothing but the condi-
tion number defined in Equation 4.2. The number of condition
indices that are large (say >1000) is a useful measure of the
number of near-linear dependencies in
X´X
.
Regression
analysis
using
qualitative
climatic
explanatory
variables
The variables employed in regression analysis are usually
quantitative variables, that is, the variables have a well-defined
scale of measurements. Sometimes the climatic data available
may have nominal or ordinal explanatory variables apart from
various quantitative explanatory variables. For example, data
related to rainfall over a certain period of time for a particu-
lar region are not available; rather, available data indicate the
rainfall as above normal or below normal over that specified
period of time for all the data points. Furthermore, data related
to all other climatic variables such as temperature, atmospheric
pressure and so on are available in usual manner. So, in this
situation, all other explanatory variables apart from rainfall are
quantitative in nature wherein rainfall is a nominal variable
with two levels as above normal or below normal. So it is better
to mention it as an attribute rather than specifying it as a vari-
able. So a question may arise how to incorporate this attribute
into the regression model?
One solution for quantifying such attributes is the use of
indicator variables
or
dummy variables
. A dummy variable is
an artificial variable constructed such that it takes the value
'1' whenever the qualitative phenomenon it represents occurs
(say above normal in the above case), and taking value as '0'
otherwise. Once created, dummies are then used in regression
analyses just like other explanatory variables, yielding standard
ordinary least square results. There is a numeric way in which
one can choose a dummy variable. In general, the most useful
dummy variable setups are simple in form, employing levels
of '0' and '1' or ' − 1' and '1.' Once the qualitative explanatory
variables are quantified by the use of dummy variable, one can
easily employ multiple linear regression on crop yield based on
various explanatory variable in the usual manner. Furthermore,
sometimes a particular qualitative variable may have more
than two levels. Suppose, in the available dataset, there are p
