▁the ▁difference ▁between ▁a ▁measured ▁value ▁of ▁the ▁dependent ▁variable , ▁ ▁and ▁the ▁value ▁predicted ▁by ▁the ▁model , ▁: ▁▁ ▁If ▁the ▁errors ▁are ▁un cor related ▁and ▁have ▁equal ▁variance , ▁then ▁the ▁minimum ▁of ▁the ▁function ▁, ▁is ▁found ▁when ▁ ▁( def ining ▁). ▁ ▁The ▁G auss – Mark ov ▁theorem ▁shows ▁that , ▁when ▁this ▁is ▁so , ▁ ▁is ▁a ▁best ▁linear ▁un bi ased ▁estim ator ▁( BL UE ). ▁If , ▁however , ▁the ▁measurements ▁are ▁un cor related ▁but ▁have ▁different ▁un cert aint ies , ▁a ▁modified ▁approach ▁might ▁be ▁adopted . ▁A it ken ▁showed ▁that ▁when ▁a ▁weight ed ▁sum ▁of ▁squ ared ▁resid uals ▁is ▁minim ized , ▁ ▁is ▁ ▁the ▁B L UE ▁if ▁each ▁weight ▁is ▁equal ▁to ▁the ▁reci pro cal ▁of ▁the ▁variance ▁of ▁the ▁measurement ▁ ▁The ▁gradient ▁equations ▁for ▁this ▁sum ▁of ▁squares ▁are ▁▁▁▁ ▁which , ▁in ▁a ▁linear ▁least ▁squares ▁system ▁give ▁the ▁modified ▁normal ▁equations , ▁▁▁▁ ▁When ▁the ▁observ ational ▁errors ▁are ▁un cor related ▁and ▁the ▁weight ▁matrix , ▁W , ▁is ▁diagonal , ▁these ▁may ▁be ▁written ▁as ▁ ▁If ▁the ▁errors ▁are ▁correl ated , ▁the ▁resulting ▁estim ator ▁is ▁the ▁B L UE ▁if ▁the ▁weight ▁matrix ▁is ▁equal ▁to ▁the ▁inverse ▁of ▁the ▁variance - cov ar iance ▁matrix ▁of ▁the ▁observations . ▁ ▁When ▁the ▁errors ▁are ▁un cor related , ▁it ▁is ▁convenient ▁to ▁simplify ▁the ▁calculations ▁to ▁factor ▁the