Geoscience Reference
In-Depth Information
Fig. 4.4
Linear regression. Whereas classical regression minimizes the ʔ
y
deviations, reduced
major axis regression minimizes the triangular area 0.5*(ʔ
x
ʔ
y
) between the data points and
the regression line, where ʔ
x
and ʔ
y
are the distances between the predicted and the true
x
- and
y
-values. h e intercept of the line with the
y
-axis is
b
0, and the slope is
b
1. h ese two
parameters dei ne the equation of the regression line.
and are ot en regarded as being free of errors. An example is the location
x
within a sediment core from which the variable
y
has been measured. h e
dependent variable
y
contains errors as its magnitude cannot be determined
accurately. Linear regression minimizes the deviations ʔ
y
between the data
points
xy
and the value
y
predicted by the best-i t line
y=b
0
+b
1
x
using a least-
squares criterion. h e basic equation for a general linear model is
where
b
0
and
b
1
are the regression coei cients. h e value of
b
0
is the intercept
with the
y
-axis and
b
1
is the slope of the line. h e squared sum of the ʔ
y
deviations to be minimized is
Partial dif erentiation of the right-hand term in the equation and setting it to
zero yields a simple equation for the regression coei cient
b
1
:
