Graphics Programs Reference
In-Depth Information
where
n
is the number of
xy
pairs of data points,
s
x
and
s
y
the univariate
standard deviations. The numerator of Pearson·s correlation coeffi cient is
known as
corrected sum of products
of the bivariate data set. Dividing the
numerator by (
n
-1) yields the
covariance
which is the summed products of deviations of the data from the sample
means, divided by (
n
-1). The covariance is a widely-used measure in bivari-
ate statistics, although it has the disadvantage of depending on the dimen-
sion of the data. We will use the covariance in time-series analysis, which
is a special case of bivariate statistics with time as one of the two variables.
Dividing the covariance by the univariate standard deviations removes this
effect and leads to Pearson·s correlation coeffi cient.
Pearson·s correlation coeffi cient is very sensitive to various disturbances
in the bivariate data set. The following example illustrates the use of the
correlation coeffi cients, highlights the potential pitfalls when using this
measure of linear trends. It also describes the resampling methods that can
be used to explore the confi dence of the estimate for
. The synthetic data
consist of two variables, the age of a sediment in kiloyears before present
and the depth below the sediment-water interface in meters. The use of syn-
thetic data sets has the advantage that we fully understand the linear model
behind the data.
The data are represented as two columns contained in fi le
agedepth.txt
.
These data have been generated using a series of thirty random levels (in
me-
ters
) below the sediment surface. The linear relationship
age
=5.6*
meters
+1.2
was used to compute noisefree values for the variable
age
. This is the equa-
tion of a straight line with slope 5.6 and an intercept with the
y
-axis of 1.2.
Finally, some gaussian noise of amplitude 10 was added to the
age
data. We
load the data from the fi le
agedepth.txt
.
ρ
agedepth = load('agedepth.txt');
We defi ne two new variables,
meters
and
age
, and generate a scatter plot
of the data.
meters = agedepth(:,1);
age = agedepth(:,2);
plot(meters,age,'o')
We observe a strong linear trend suggesting some dependency between the
