Blockwise Formula
Suppose the design matrix can be decomposed by columns as . Define the Hat operator as . Similarly, define the residual operator as . Then the Hat matrix of can be decomposed as follows:
There are a number of applications of such a partitioning. The classical application has a column of all ones, which allows one to analyze the effects of adding an intercept term to a regression. Another use is in the fixed effects model, where is a large sparse matrix of the dummy variables for the fixed effect terms. One can use this partition to compute the hat matrix of without explicitly forming the matrix, which might be too large to fit into computer memory.
Read more about this topic: Hat Matrix
Famous quotes containing the word formula:
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)