▁weight ▁matrix ▁as ▁. ▁The ▁normal ▁equations ▁can ▁then ▁be ▁written ▁in ▁the ▁same ▁form ▁as ▁ordinary ▁least ▁squares : ▁ ▁where ▁we ▁define ▁the ▁following ▁scale d ▁matrix ▁and ▁vector : ▁ ▁This ▁is ▁a ▁type ▁of ▁wh iten ing ▁transformation ; ▁the ▁last ▁expression ▁involves ▁an ▁entry wise ▁division . ▁ ▁For ▁non - linear ▁least ▁squares ▁systems ▁a ▁similar ▁argument ▁shows ▁that ▁the ▁normal ▁equations ▁should ▁be ▁modified ▁as ▁follows . ▁ ▁Note ▁that ▁for ▁empir ical ▁tests , ▁the ▁appropriate ▁W ▁is ▁not ▁known ▁for ▁sure ▁and ▁must ▁be ▁estimated . ▁ ▁For ▁this ▁feas ible ▁generalized ▁least ▁squares ▁( F GL S ) ▁techniques ▁may ▁be ▁used ; ▁in ▁this ▁case ▁it ▁is ▁special ized ▁for ▁a ▁diagonal ▁cov ar iance ▁matrix , ▁thus ▁yield ing ▁a ▁feas ible ▁weight ed ▁least ▁squares ▁solution . ▁ ▁If ▁the ▁uncertainty ▁of ▁the ▁observations ▁is ▁not ▁known ▁from ▁external ▁sources , ▁then ▁the ▁weights ▁could ▁be ▁estimated ▁from ▁the ▁given ▁observations . ▁This ▁can ▁be ▁useful , ▁for ▁example , ▁to ▁identify ▁out liers . ▁After ▁the ▁out liers ▁have ▁been ▁removed ▁from ▁the ▁data ▁set , ▁the ▁weights ▁should ▁be ▁reset ▁to ▁one . ▁ ▁Mot iv ation ▁In ▁some ▁cases ▁the ▁observations ▁may ▁be ▁weight ed — for ▁example , ▁they ▁may ▁not ▁be ▁equally ▁reliable . ▁In ▁this ▁case , ▁one ▁can ▁minim ize ▁the ▁weight ed ▁sum ▁of ▁squares : ▁ ▁where ▁w i ▁> ▁ 0 ▁is ▁the ▁weight ▁of ▁the ▁it h ▁observation ,