Page 68 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 3

What circumstances will lead to singularity? The condition indicates that two
or more rows (or columns) of the matrix are linear combinations or linear transfor-
mations of other rows; that is, the values in some rows (or columns) are dependent
on values in other rows. For example, the determinant

1 2 3
4 5 6 =O
246

is zero because the third row of the matrix is simply twice the first row. Similarly,
the determinant

1 2 3
4 5 6 =O
579

is zero because the third row is the sum of rows one and two. Of course, in real
problems the source of singularity usually is not so obvious. Consider the data
in file BAL,TIC.TXT, which gives the weight-percent of sand in five successive size
fractions, measured on bottom samples collected in an area of the Baltic Sea. We can
calculate correlations between the five sand size categories and place the results in
a square, symmetric correlation matrix:

1 0.243 -0.301 0.096 -0.261
0.243 1 -0.969 -0.562 -0.422 I
-0.301 -0.969 1 0.340 0.253
0.096 -0.562 0.340 1 0.691
-0.261 -0.422 0.253 0.691 1

It is not obvious that this matrix should be singular with a zero determinant, yet
it is. The linear dependence comes about because the weight-percentages in the
five size categories sum to 100 for each observation, so there are induced negative
correlations between the size categories. (Actually, because of rounding during
computations, you may compute a correlation matrix that is not exactly singular.
Depending upon the numerical precision of the computer program, rather than
exactly 0, you may observe a very small determinant such as -0.0002. A matrix
with a determinant near zero is said to be ill-conditioned.)
Finally, there is another special case of interest. An identity matrix has a de-
terminant equal to 1.0. If several variables are completely independent of each
other, their correlations will be near zero and they will form a correlation matrix
that approximates an identity matrix. The determinant of such a matrix will be
close to one, and its logarithm will be close to zero; this is the basis for one test of
independence between variables.

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