▁of ▁the ▁training ▁set ▁to ▁be ▁ 1 0 0 ▁exempl ars ▁generated ▁by ▁the ▁cha otic ▁time ▁series . ▁The ▁weight ▁ ▁is ▁taken ▁to ▁be ▁a ▁constant ▁equal ▁to ▁ 6 . ▁The ▁weights ▁ ▁are ▁five ▁exempl ars ▁from ▁the ▁time ▁series . ▁The ▁weights ▁ ▁are ▁trained ▁with ▁projection ▁operator ▁training : ▁ ▁where ▁the ▁learning ▁rate ▁ ▁is ▁again ▁taken ▁to ▁be ▁ 0 . 3 . ▁The ▁training ▁is ▁performed ▁with ▁one ▁pass ▁through ▁the ▁ 1 0 0 ▁training ▁points . ▁The ▁r ms ▁error ▁on ▁a ▁test ▁set ▁of ▁ 1 0 0 ▁exempl ars ▁is ▁ 0 . 0 8 4 , ▁smaller ▁than ▁the ▁un normal ized ▁error . ▁Normal ization ▁yields ▁accuracy ▁improvement . ▁Typ ically ▁accuracy ▁with ▁normal ized ▁basis ▁functions ▁increases ▁even ▁more ▁over ▁un normal ized ▁functions ▁as ▁input ▁dimensional ity ▁increases . ▁ ▁Time ▁series ▁prediction ▁ ▁Once ▁the ▁underlying ▁geometry ▁of ▁the ▁time ▁series ▁is ▁estimated ▁as ▁in ▁the ▁previous ▁examples , ▁a ▁prediction ▁for ▁the ▁time ▁series ▁can ▁be ▁made ▁by ▁iteration : ▁ ▁. ▁ ▁A ▁comparison ▁of ▁the ▁actual ▁and ▁estimated ▁time ▁series ▁is ▁displayed ▁in ▁the ▁figure . ▁The ▁estimated ▁times ▁series ▁starts ▁out ▁at ▁time ▁zero ▁with ▁an ▁exact ▁knowledge ▁of ▁x ( 0 ). ▁It ▁then ▁uses ▁the ▁estimate ▁ ▁of ▁the ▁dynamics ▁to ▁update ▁the ▁time ▁series ▁estimate ▁for ▁several ▁time ▁steps . ▁ ▁Note ▁that ▁the ▁estimate ▁is ▁accurate ▁for ▁only ▁a ▁few ▁time ▁steps . ▁This ▁is ▁a