SOLUTIONS6MANUAL6FOR A6FIRST6COURS E6IN6MACHINE6 LEARNING6(2ND 6EDITION)6EXER CISE6SOLUTIONS

Chapter61 EX61.1.6A6high6positive6value6of6w06and6a6small6negative6value6for6w1.6These6 reflect6the6high6intercept6on6the6t6axis6(corresponding6to6the6theoretical6tim e6winning6time6at6x6=606and6the6small6decrese6in6winning6time6over6the6 years. EX61.2.6The6following6would6do6the6job: EX61.3.6We6need6to6find6wTXTXw.6We’ll6start6with6XTX.6Multiplying6XT6by6X6gives: N 2 T n=166n1 N n=1 xn1xn2 X6X6= N n=1 xn2 xn1 N 2 n=1666n2 Multiplying6this6by6w6gives: T w06 ΣN 26+6w16Σ N xn1xn2 X6Xw6= w0 n=1 xn2 xn1 +6w 1 N 2 n=1666n2 1 N Σ Σ Σ n=1 x n=1 Σ n1 K26591_SM_C 7 05/04/16 3:37 pm N N Σ × × Σ Σ Σ Σ n x Σ n1 Σ n1 x 2 CHAPTER661. Finally,6pre-multiplying6this6by6wT6gives: wTXTXw =66 w0 w0 n=1 26+6 w1 n Σ =1 xn1xn2 + w1666w0 n Σ =1 xn2xn16+6w 1 2 n2 n=1 N N N =666w26 Σ6 x26+62w0w16 Σ6 xn1xn26+6w26 Σ6 x2 as6required. EX61.4.66Let’s6first6work6ou t6Xw: 0 n1 n=1 n=1 1 n2 n=1 Therefore Xw6=6 w0x116+6w1x126 w0x216+6w1x22 . w0xN616+6w1xN 62 (Xw)T6=6[w0x116+6w1x12,66w0x216+6w1x22,66.6.6.66,66w0xN16+6w1xN2] Finally,6work6out6wTXT: wTXT6=6[w0x116+6w1x12,66w0x216+6w1x22,66.6.6.66,66w0xN16+6w1xN2] as6required. EX61.5.66Starting6with n6xn tn .6The6result6of6this6is6a6column6vector6of6the6same6size6as6x (26×61).6Now,6using6the6definition6of6X, XT6= x11,6x21,6.6.6.6,6xN1 x12,6x22,6.6.6.6,6xN2 (which6is6a62 N6vector).6Multiplying6this6by6t6gives6a62 16vector6that6looks6like6this: N T n=1 xn1tn X6t6= N n=1 xn2tn which6is n6xn tn 6as6required.6The6second6example,6XTXw.66We6already6k