Page 75 - Compact Numerical Methods For Computers

64 Compact numerical methods for computers
The method suggested is mainly useful for adding single observations, and other
approaches are better if more than a very few observations are to be included. For
instance, one could update the triangular form which results from the Givens’
reduction if this had been saved, then proceed to the singular-value decomposition
as in algorithm 4.
No methods will be discussed for removing observations, since while methods
exist to accomplish this (see Lawson and Hanson 1974, pp 225-31), the opera-
tion is potentially unstable. See also Bunch and Nielsen (1978).
For instance, suppose we have a Givens’ QR decomposition of a matrix A (or
any other QR decomposition with Q orthogonal and R upper-triangular), then add
T
and delete a row (observation) denoted y . Then after the addition of this row,
the (1, 1) elements of the matrices are related by

(4.31)

where the tilde is used to indicate matrices which have been updated. Deletion of
T
y now requires the subtraction
(4.32)

to be performed in some way or another, an operation which will involve digit
cancellation if y 1 and are close in magnitude. The same difficulty may of
course occur in other columns-the first is simply easier to illustrate. Such cases
T
imply that an element of y dominates the column in which it occurs and as such
should arouse suspicions about the data. Chambers’ (1971) subroutine to delete
rows from a QR decomposition contains a check designed to catch such occur-
rences.
Of interest to those users performing regression calculations are the estimates of
standard errors of the regression coefficients (the least-squares solution elements).
The traditional standard error formula is

2 T -1
SE(b ) = ( s (A A) ii )½ (4.33)
i
2
where s is an estimate of the variance of data about the fitted model calculated by
dividing the sum of squared residuals by the number of degrees of freedom (nRow –
nCol) = (nRow – n). The sum of squared residuals has already been computed in
algorithm 4, and has been adjusted for rank deficiency within the solution phase of
the code.
The diagonal elements of the inverse of the sum of squares and cross-products
matrix may seem to pose a bigger task. However, the singular-value decomposition
leads easily to the expression

T -1 + + T
(A A) = VS S V . (4.34)
In particular, diagonal elements of the inverse of the sum of squares and cross-