Linear Least Squares (mathematics) - Further Discussion

Further Discussion

The numerical methods for linear least squares are important because linear regression models are among the most important types of model, both as formal statistical models and for exploration of data-sets. The majority of statistical computer packages contain facilities for regression analysis that make use of linear least squares computations. Hence it is appropriate that considerable effort has been devoted to the task of ensuring that these computations are undertaken efficiently and with due regard to round-off error.

Individual statistical analyses are seldom undertaken in isolation, but rather are part of a sequence of investigatory steps. Some of the topics involved in considering numerical methods for linear least squares relate to this point. Thus important topics can be

• Computations where a number of similar, and often nested, models are considered for the same data-set. That is, where models with the same dependent variable but different sets of independent variables are to be considered, for essentially the same set of data-points.
• Computations for analyses that occur in a sequence, as the number of data-points increases.
• Special considerations for very extensive data-sets.

Fitting of linear models by least squares often, but not always, arise in the context of statistical analysis. It can therefore be important that considerations of computation efficiency for such problems extend to all of the auxiliary quantities required for such analyses, and are not restricted to the formal solution of the linear least squares problem.