▁written ▁as ▁ ▁where ▁ ▁and ▁ ▁are ▁the ▁resid uals ▁in ▁x ▁and ▁y ▁respectively . ▁Clear ly ▁these ▁resid uals ▁cannot ▁be ▁independent ▁of ▁each ▁other , ▁but ▁they ▁must ▁be ▁const rained ▁by ▁some ▁kind ▁of ▁relationship . ▁Writing ▁the ▁model ▁function ▁as ▁, ▁the ▁constraints ▁are ▁expressed ▁by ▁m ▁condition ▁equations . ▁ ▁Thus , ▁the ▁problem ▁is ▁to ▁minim ize ▁the ▁objective ▁function ▁subject ▁to ▁the ▁m ▁constraints . ▁It ▁is ▁solved ▁by ▁the ▁use ▁of ▁Lag range ▁multip liers . ▁After ▁some ▁algebraic ▁manip ulations , ▁the ▁result ▁is ▁obtained . ▁ ▁or ▁altern atively ▁ ▁where ▁M ▁is ▁the ▁variance - cov ar iance ▁matrix ▁relative ▁to ▁both ▁independent ▁and ▁dependent ▁variables . ▁ ▁Example ▁▁ ▁When ▁the ▁data ▁errors ▁are ▁un cor related , ▁all ▁matrices ▁M ▁and ▁W ▁are ▁diagonal . ▁Then , ▁take ▁the ▁example ▁of ▁straight ▁line ▁fitting . ▁ ▁in ▁this ▁case ▁ ▁showing ▁how ▁the ▁variance ▁at ▁the ▁it h ▁point ▁is ▁determined ▁by ▁the ▁var ian ces ▁of ▁both ▁independent ▁and ▁dependent ▁variables ▁and ▁by ▁the ▁model ▁being ▁used ▁to ▁fit ▁the ▁data . ▁The ▁expression ▁may ▁be ▁generalized ▁by ▁not ing ▁that ▁the ▁parameter ▁ ▁is ▁the ▁slope ▁of ▁the ▁line . ▁ ▁An ▁expression ▁of ▁this ▁type ▁is ▁used ▁in ▁fitting ▁p H ▁tit r ation ▁data ▁where ▁a ▁small ▁error ▁on ▁x ▁transl ates ▁to ▁a ▁large ▁error ▁on ▁y ▁when ▁the ▁slope ▁is ▁large . ▁ ▁Al gebra ic ▁point ▁of ▁view ▁ ▁As ▁was