One of the traditional methods for stabilizing the solutions to ill-conditioned linear least squares problems
arising from the linear regression model
has been to truncate the singular value decomposition for the matrix
which can be written
where 
is a diagonal matrix whose elements are the singular
values of 
. The least squares solution
involves the inverse matrix 
so the smallest singular values,
which are also the most poorly determined,
make the largest contribution to the solution. Wild variations in the
calculated solution can be suppressed by setting those singular values
to zero and replacing 
by the generalized inverse of
the truncated 
. This is usually treated as a problem of
determining the ``numerical rank'' of 
, but if the
measurements are stochastically independent, and if good estimates of their
variances are available,
then a more natural way to make the truncation is by comparing those variances
with the elements of the vector 
and zeroing all of the
latter adjudged to be statistically insignificant. Thus the truncation is
made on the rotated right hand side vector rather than on the rotated matrix,
and the number of terms discarded may be different for different measurement
vectors using the same matrix 
. This technique has been incorporated
into a Fortran program and extensively tested. In all cases residual
diagnostics have given good agreement with the standard assumptions about
statistical significance. Future work will be devoted to further testing and
refinements and to documenting the algorithm and computer program.