▁ ▁in ▁the ▁un normal ized ▁and ▁normal ized ▁cases , ▁respectively . ▁Here ▁ ▁are ▁weights ▁to ▁be ▁determined . ▁High er ▁order ▁linear ▁terms ▁are ▁also ▁possible . ▁ ▁This ▁result ▁can ▁be ▁written ▁ ▁where ▁ ▁and ▁ ▁in ▁the ▁un normal ized ▁case ▁and ▁ ▁in ▁the ▁normal ized ▁case . ▁ ▁Here ▁ ▁is ▁a ▁Kr one cker ▁delta ▁function ▁defined ▁as ▁ ▁. ▁ ▁Training ▁ ▁R BF ▁networks ▁are ▁typically ▁trained ▁from ▁pairs ▁of ▁input ▁and ▁target ▁values ▁, ▁ ▁by ▁a ▁two - step ▁algorithm . ▁▁ ▁In ▁the ▁first ▁step , ▁the ▁center ▁vectors ▁ ▁of ▁the ▁R BF ▁functions ▁in ▁the ▁hidden ▁layer ▁are ▁chosen . ▁This ▁step ▁can ▁be ▁performed ▁in ▁several ▁ways ; ▁cent ers ▁can ▁be ▁randomly ▁sample d ▁from ▁some ▁set ▁of ▁examples , ▁or ▁they ▁can ▁be ▁determined ▁using ▁k - me ans ▁clust ering . ▁Note ▁that ▁this ▁step ▁is ▁un super vised . ▁ ▁The ▁second ▁step ▁simply ▁fits ▁a ▁linear ▁model ▁with ▁coefficients ▁ ▁to ▁the ▁hidden ▁layer ' s ▁outputs ▁with ▁respect ▁to ▁some ▁objective ▁function . ▁A ▁common ▁objective ▁function , ▁at ▁least ▁for ▁regression / function ▁estimation , ▁is ▁the ▁least ▁squares ▁function : ▁ ▁where ▁. ▁We ▁have ▁explicitly ▁included ▁the ▁dependence ▁on ▁the ▁weights . ▁Min im ization ▁of ▁the ▁least ▁squares ▁objective ▁function ▁by ▁optimal ▁choice ▁of ▁weights ▁optim izes ▁accuracy ▁of ▁fit . ▁ ▁There ▁are ▁occasions ▁in ▁which ▁multiple ▁object ives , ▁such ▁as ▁smooth ness ▁as