Geoscience Reference
In-Depth Information
such relationships are expressed mathematically, it will predict
the value of one variable from the knowledge of the other. For
instance, the photosynthetic and transpiration rates of trees are
found to depend on atmospheric conditions, like temperature or
humidity, but it is unusual to expect a reverse relationship. The
dependent variable is usually denoted by Y and the independent
variable by X. When only two variables are involved in regres-
sion, the functional relationship is known as simple regres-
sion. If the relationship between the two variables is linear, it
is known as simple linear regression; otherwise it is known as
non-linear regression. Regression analysis is widely used for
prediction and forecasting.
Multiple regression models
Suppose Y denotes the yield
of a crop over a certain period of time which depends on p
explanatory variables X
1
, X
2
, …, X
p
such as maximum tem-
perature, minimum temperature, atmospheric pressure, rainfall,
CO
2
concentration in the atmosphere and so on over that speci-
fied period of time, and suppose we have n data points. Then a
multiple regression model of crop yield that might describe this
relationship can be written as
y
=+ ++ +
ββ β
x
+x
β
x
ε
,
for i
=
12
,
,
…
,
n
i
0
1
1i
22i
p
pi
i
(4.1)
where β
0
is the intercept term. The parameter β
k
(for k = 1, 2, …,
p) measures the expected change in Y per unit change in X
k
when
all other k − 1 variables are held constant. Here, ε is an identical
and independently distributed (iid) random variable with zero
mean and constant variance. The prediction of the future crop
yield is possible once all the parameters are estimated.
Problem of multi-collinearity in multiple regression
model
The individual regression coefficient in a multiple
regression model determines the interpretation of the model.
However, some of the climatic factors such as rainfall, rela-
tive humidity and so on are not independent of each other
and there exists a close relationship between many of the cli-
matic explanatory variables. So, inference based on the usual
regression model may be erroneous or misleading. When the
explanatory variables are not orthogonal; rather, there exists
near-linear dependencies among them, the problem of multi-
collinearity is said to exist in the regression setup. One of the
major consequences of multi-collinearity is the large vari-
ances and co-variances for the least square estimators of the
