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5
GENERALIZATION OF
LINEAR REGRESSION
PROBLEMS
5.1 INTRODUCTION
The three basic problems of linear regression from the point of viw of noisy
data are the OLS, the TLS, and the DLS, which have been described in earlier
chapters. The problem addressed in this chapter is the generalization of these three
cases and a common framework. It will be shown that this analysis will lead to
a global stability study of these cases. In the literature, apart from the technique
introduced in this chapter, the following approach is the most interesting.
5.1.1 Weighted (Scaled) TLS
Here a special class of weighted TLS [146,147,160] is considered. The problem
is the following:
min [ A ; α b ] F
subject to
b + b R ( A + A )
(5.1)
where A and b are perturbations of A and b , respectively and α is a scalar.
When α 0, the solution approaches the OLS solution. When α = 1, it coin-
cides with the TLS formulation. Now consider α →∞ . Perturbing b by even a
small amount will result in a large value of the cost function. Thus, the optimum
solution will be to perturb A such that b lies in the range A + A . This is the
same as the DLS problem posed in eq. (1.53). Hence, the α -parameterization
unifies and compacts the three basic problems. Let A have full rank ( n ). Define
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