The connectionist inductive learning and logic programming system

(1)

City, University of London Institutional Repository

Citation

:

Garcez, A. and Zaverucha, G. (1999). The connectionist inductive learning and logic programming system. Applied Intelligence Journal, 11(1), pp. 59-77. doi:

10.1023/A:1008328630915

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This version of the publication may differ from the final published

version.

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http://openaccess.city.ac.uk/301/

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:

http://dx.doi.org/10.1023/A:1008328630915

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linked to.

(2)

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(3)

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(4)

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(5)

Symbolic Knowledge Symbolic

Knowledge

Neural Network Examples

Learning

Connectionist System

Inference Machine

Explanation

1 2

3

4

5

Iljxuh 4= Wkh Frqqhfwlrqlvw Lqgxfwlyh Ohduqlqj dqg Orjlf Surjudpplqj V|vwhp1

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5 Iurp Orjlf Surjudpv wr Qhxudo Qhwzrunv

(6)

wkhq vkrzv wkdw wkh qhwzrun rewdlqhg lv htxlydohqw wr wkh ruljlqdo surjudp/ lq wkh vhqvh wkdw zkdw lv frpsxwhg e| wkh surjudp lv frpsxwhg e| wkh qhwzrun dqg ylfh0yhuvd1

Ghqlwlrq 4 D jhqhudo fodxvh lv d uxoh ri wkh irup D # O> ===> O&/ zkhuh D lv dq dwrp dqg O +4 l n, lv d olwhudo +dq dwrp ru wkh qhjdwlrq ri dq dwrp,1 D jhqhudo orjlf surjudp lv d qlwh vhw ri jhqhudo fodxvhv1

Wr lqvhuw wkh edfnjurxqg nqrzohgjh/ ghvfulehg e| d jhqhudo orjlf sur0 judp +S,/ lq wkh qhxudo qhwzrun +Q ,/ zh xvh dq dssurdfk vlplodu wr Kroo0 greohu dqg Ndolqnh*v ^58`1Hdfk jhqhudo fodxvh +F,, ri S lv pdsshg iurp wkh lqsxw od|hu wr wkh rxwsxw od|hu ri Q wkurxjk rqh qhxurq +Q,, lq wkh vlqjoh klgghq od|hu ri Q 1Lqwxlwlyho|/ wkh wudqvodwlrq dojrulwkp iurp S wr Q kdv wr lpsohphqw wkh iroorzlqj frqglwlrqv= +4, Wkh lqsxw srwhqwldo ri d klgghq qhxurq +Q,, fdq rqo| h{fhhg Q,*v wkuhvkrog +,,/ dfwlydwlqj Q,/ zkhq doo wkh srvlwlyh dqwhfhghqwv ri F, duh dvvljqhg wkh wuxwk0ydoxh wuxh zkloh doo wkh qhjdwlyh dqwhfhghqwv ri F,duh dvvljqhg idovh> dqg +5, Wkh lqsxw srwhqwldo ri dq rxwsxw qhxurq +D, fdq rqo| h{fhhg D*v wkuhvkrog +,/ dfwlydwlqj D/ zkhq dw ohdvw rqh klgghq qhxurq Q, wkdw lv frqqhfwhg wr D lv dfwlydwhg1

H{dpsoh 5 Frqvlghu wkh orjlf surjudp S @ ~D # E> F> qrwG> D # H> I > E #1 Wkh wudqvodwlrq dojrulwkp vkrxog ghulyh wkh qhwzrun Q ri jxuh 5/ vhwwlqj zhljkwv +Z_{v, dqg wkuhvkrogv +}_{v, lq vxfk d zd| wkdw frqglwlrqv +4,} dqg +5, deryh duh vdwlvhg1 Qrwh wkdw/ li Q rxjkw wr eh ixoo|0frqqhfwhg/ dq| rwkhu olqn+qrw vkrzq lq jxuh 5, vkrxog uhfhlyh zhljkw }hur lqlwldoo|1

A B

θA θB

W W W

θ1 N1 θ2 N2 θ3 N3

B C D E F

W W

W W -W

Interpretations

Iljxuh 5= Vnhwfk ri d qhxudo qhwzrun iru wkh deryh orjlf surjudp S1

(7)

Qrwh wkdw/ lq h{dpsoh 5/ zh kdyh odehoohg hdfk lqsxw dqg rxwsxw qhx0 urq dv dq dwrp dsshdulqj/ uhvshfwlyho|/ lq wkh erg| dqg lq wkh khdg ri d fodxvh ri S1Wklv doorzv xv wr uhihu wr qhxurqv dqg sursrvlwlrqdo yduldeohv lqwhufkdqjhdeo| dqg wr uhjdug hdfk qhwzrun*v lqsxw yhfwru l @ +l> ===> l6, +lE$$6 5 ^4> 4`, dv dq lqwhusuhwdwlrq iru S51Li l 5 ^D6?> 4` wkhq wkh sursrvlwlrqdo yduldeoh dvvrfldwhg wr wkh m0wk qhxurq lq wkh qhwzrun*v lqsxw od|hu lv dvvljqhg wuxh/ zkloh l 5 ^4> D6?` phdqv wkdw lw lv dvvljqhg idovh/ zkhuh D6? 5 +3> 4, lv d suhghqhg ydoxh dv vkrzq lq wkh qrwdwlrq ehorz1Qrwh dovr wkdw hdfk klgghq qhxurq Q, fruuhvsrqgv wr d fodxvh F, ri S1

Wkh iroorzlqj qrwdwlrq zloo eh xvhg lq rxu wudqvodwlrq dojrulwkp1

Qrwdwlrq = Jlyhq d jhqhudo orjlf surjudp S/ ohw t ghqrwh wkh qxpehu ri fodxvhv F, +4 o t, rffxuulqj lq S>

p> wkh qxpehu ri olwhudov rffxuulqj lq S>

D6?/ wkh plqlpxp dfwlydwlrq iru d qhxurq wr eh frqvlghuhg dfwlyh +ru wuxh,/ 3 ? D6?? 4>

D6@%/ wkh pd{lpxp dfwlydwlrq iru d qhxurq wr eh frqvlghuhg qrw dfwlyh +ru idovh,/ 4 ? D6@% ? 3>

k+{, @ 2

ne3q% 4/ wkh elsrodu vhpl0olqhdu dfwlydwlrq ixqfwlrq/ zkhuh

lv wkh vwhhsqhvv sdudphwhu +wkdw ghqhv wkh vorsh ri k+{,,> j+{, @ { / wkh vwdqgdug olqhdu dfwlydwlrq ixqfwlrq>

Z +uhvs1 0Z ,/ wkh zhljkw ri frqqhfwlrqv dvvrfldwhg zlwk srvlwlyh +uhvs1 qhjdwlyh, olwhudov>

,/ wkh wkuhvkrog ri klgghq qhxurq Q, dvvrfldwhg zlwk fodxvh F,> / wkh wkuhvkrog ri rxwsxw qhxurq D/ zkhuh D lv wkh khdg ri fodxvh F,>

n,/ wkh qxpehu ri olwhudov lq wkh erg| ri fodxvh F,>

s,/ wkh qxpehu ri srvlwlyh olwhudov lq wkh erg| ri fodxvh F,> q,/ wkh qxpehu ri qhjdwlyh olwhudov lq wkh erg| ri fodxvh F,>

,/ wkh qxpehu ri fodxvhv lq S zlwk wkh vdph dwrp lq wkh khdg iru hdfk fodxvh F,>

PD[,+n,> ,,/ wkh juhdwhu hohphqw dprqj n, dqg , iru fodxvh F,/

PD[+n> ===> n^> > ===> ^,/ wkh juhdwhvw hohphqw dprqj doo n*v dqg *v ri S1

5_{Dq lqwhusuhwdwlrq lv d ixqfwlrq iurp sursrvlwlrqdo yduldeohv wr ~R|oec Rs@,re1 D}

(8)

Iru lqvwdqfh/ iru wkh surjudp S ri h{dpsoh 5/ t @ 6/ p @ 9/ n @ 6/ n2@ 5/ n@ 3/ s @ 5/ s2 @ 5/ s@ 3/ q @ 4/ q2 @ 3/ q@ 3/ @ 5/ 2@ 5/ @ 4/ PD[+n> , @ 6/ PD[2+n2> 2, @ 5/ PD[+n> , @ 4/

dqg PD[+n> n2> n> > 2> , @ 6=

Lq wkh wudqvodwlrq dojrulwkp ehorz/ zh ghqh D6? @ +n> ,> Z @ 2+k+{,> n> > D6?,> , @ +n> D6?> Z ,> dqg @ e+> D6?> Z , vxfk wkdw frqglwlrqv +4, dqg +5, duh vdwlvhg/ dv zh zloo vhh odwhu lq wkh surri ri wkhruhp 61

Jlyhq d jhqhudo orjlf surjudp S/ frqvlghu wkdw wkh olwhudov ri S duh qxpehuhg iurp 4 wr p vxfk wkdw wkh lqsxw dqg rxwsxw od|huv ri Q duh yhfwruv ri pd{lpxp ohqjwk p/ zkhuh wkh l0wk qhxurq uhsuhvhqwv wkh l0wk olwhudo ri S1 Dvvxph/ iru pdwkhpdwlfdo frqyhqlhqfh dqg zlwkrxw orvv ri jhqhudolw|/ wkdw D6@% @ D6?1

41 Fdofxodwh PD[+n> ===> n^> > ===> ^, ri S>

51 Fdofxodwh D6?A f_fE&_E&cc&_cc&^_^_c>c>cc>^3

cc>^n>

61 Fdofxodwh Z 2

q fE&,?Encc&^c>4?cc>3,?E3^E4?4?3n 4?n >

71 Iru hdfk fodxvh F, ri S ri wkh irup D # O> ===> O& +n 3,=

+d, Dgg d qhxurq Q, wr wkh klgghq od|hu ri Q >

+e, Frqqhfw hdfk qhxurq O +4 l n, lq wkh lqsxw od|hu wr wkh qhxurq Q, lq wkh klgghq od|hu1 Li O lv d srvlwlyh olwhudo wkhq vhw wkh frqqhfwlrq zhljkw wr Z >rwkhuzlvh/ vhw wkh frqqhfwlrq zhljkw wr Z >

+f, Frqqhfw wkh qhxurq Q, lq wkh klgghq od|hu wr wkh qhxurq D lq wkh rxwsxw od|hu dqg vhw wkh frqqhfwlrq zhljkw wr Z >

+g, Ghqh wkh wkuhvkrog +,, ri wkh qhxurq Q, lq wkh klgghq od|hu dv ,@ En4?₂E&,3Z 1

+h, Ghqh wkh wkuhvkrog +, ri wkh qhxurq D lq wkh rxwsxw od|hu dv @ En4?₂E3>,Z 1

81 Vhw j+{, dv wkh dfwlydwlrq ixqfwlrq ri wkh qhxurqv lq wkh lqsxw od|hu ri Q 1 Lq wklv zd|/ wkh dfwlydwlrq ri wkh qhxurqv lq wkh lqsxw od|hu ri Q / jlyhq e| hdfk lqsxw yhfwru l/ zloo uhsuhvhqw dq lqwhusuhwdwlrq iru S1 91 Vhw k+{, dv wkh dfwlydwlrq ixqfwlrq ri wkh qhxurqv lq wkh klgghq dqg

rxwsxw od|huv ri Q 1 Lq wklv zd|/ d judglhqw ghvfhqw ohduqlqj dojrulwkp/ vxfk dv edfnsursdjdwlrq/ fdq eh dssolhg rq Q h!flhqwo|1

:1 Li Q rxjkw wr eh ixoo|0frqqhfwhg/ vhw doo rwkhu frqqhfwlrqv wr }hur1

(9)

Vlqfh Q frqwdlqv d elsrodu vhpl0olqhdu +glhuhqwldeoh, dfwlydwlrq ixqf0 wlrq k+{,/ lqvwhdg ri d elqdu| wkuhvkrog qrq0olqhdu dfwlydwlrq ixqfwlrq/ wkh qhwzrun*v rxwsxw qhxurqv dfwlydwlrqv duh uhdo qxpehuv lq wkh udqjh ^4> 4`1 Wkhuhiruh/ zh vd| wkdw dq rxwsxw zlwklq wkh udqjh ^D6?> 4` uhsuhvhqwv wkh wuxwk0ydoxh wuxh/ zkloh dq rxwsxw zlwklq ^4> D6?` uhsuhvhqwv idovh1 Zh zloo vhh odwhu lq wkh surri ri wkhruhp 6 wkdw wkh deryh ghqhg zhljkwv dqg wkuhvkrogv gr qrw doorz wkh qhwzrun wr suhvhqw dfwlydwlrqv lq wkh udqjh +D6?> D6?,1

Qrwh wkdw wkh wudqvodwlrq ri idfwv ri S lqwr Q > iru lqvwdqfh E # lq h{dpsoh 5/ lv grqh e| vlpso| wdnlqj n @ 3 lq wkh deryh dojrulwkp1 Dowhu0 qdwlyho|/ hdfk idfw ri wkh irup D # pd| eh frqyhuwhg wr d uxoh ri wkh irup D # W wkdw lv lqvhuwhg lq Q xvlqj n @ 4/ zkhuh W ghqrwhv wuxh/ dqg lv dq h{wud qhxurq wkdw lv dozd|v dfwlyh lq wkh lqsxw od|hu ri Q / l1h1 W kdv lqsxw gdwd {hg dw 41 Iurp wkh srlqw ri ylhz ri wkh frpsxwdwlrq ri S e| Q / wkhuh lv devroxwho| qr glhuhqfh ehwzhhq wkh deryh wzr zd|v ri lqvhuwlqj idfwv ri S lqwr Q 1 Krzhyhu/ frqvlghulqj wkh vxevhtxhqw surfhvv ri lqgxfwlyh ohduqlqj/ uhjduglqj S dv edfnjurxqg nqrzohgjh/ li D # lv lqvhuwhg lq Q wkhq wkh vhw ri h{dpsohv wr eh ohduqhg diwhuzdugv fdq ghihdw wkdw idfw e| fkdqjlqj zhljkwv dqg2ru hvwdeolvklqj qhz frqqhfwlrqv lq Q 1 Rq wkh rwkhu kdqg/ li D # W lv lqvhuwhg lq Q wkhq D fdq qrw eh ghihdwhg e| wkh vhw ri h{dpsohv vlqfh wkh qhxurq W lv fodpshg lq Q 1 Ghihdvleoh dqg qrqghihdvleoh nqrzohgjh fdq wkhuhiruh eh uhvshfwlyho| lqvhuwhg lq wkh qhwzrun e| ghqlqj yduldeoh dqg {hg zhljkwv2qhxurqv1

Wkh deryh wudqvodwlrq dojrulwkp lv edvhg xsrq wkh rqh suhvhqwhg lq ^58`/ zkhuh Q lv ghqhg zlwk elqdu| wkuhvkrog qhxurqv1 Lw lv nqrzq wkdw vxfk qhwzrunv kdyh olplwhg delolw| wr ohduq1 Khuh/ lq rughu wr shuirup lqgxfwlyh ohduqlqj h!flhqwo|/ Q lv ghqhg xvlqj wkh dfwlydwlrq ixqfwlrq k+{,1 Dq lp0 phgldwh uhvxow lv wkdw Q fdq dovr shuirup lqgxfwlyh ohduqlqj iurp h{dpsohv dqg edfnjurxqg nqrzohgjh dv lq ^8:`1 Pruhryhu/ wkh uhvwulfwlrq lpsrvhg ryhu Z lq ^58`/ zkhuh lw lv vkrzq wkdw Q frpsxwhv W iru Z @ 4/ lv zhdnhqhg khuh/ vlqfh wkh zhljkwv pxvw eh deoh wr fkdqjh gxulqj wudlqlqj1

Qhyhuwkhohvv/ lq ^8:`/ dqg pruh fohduo| lq ^88`/ wkh edfnjurxqg nqrzohgjh pxvw kdyh d vx!flhqwo| vpdoo qxpehu ri uxohv dv zhoo dv d vx!flhqwo| vpdoo qxpehu ri dqwhfhghqwv lq hdfk uxoh6 _{lq rughu wr eh dffxudwho| hq0} frghg lq wkh qhxudo qhwzrun1 Xqiruwxqdwho|/ wkhvh uhvwulfwlrqv ehfrph txlwh vwurqj ru hyhq xqihdvleoh li/ iru lqvwdqfh/ D6@% @ ₂ dv lq +^8:`/ Vhfwlrq 8= Hpslulfdo Whvwv ri NEDQQ,1 Frqvhtxhqwo|/ dq lqwhusuhwdwlrq wkdw grhv qrw vdwlvi| d fodxvh fdq zurqjo| dfwlydwh d qhxurq lq wkh rxwsxw od|hu ri Q 1 Wklv uhvxowv iurp wkh xvh ri wkh vwdqgdug +xqlsrodu, vhpl0olqhdu dfwlydwlrq ixqfwlrq/ zkhuh hdfk qhxurq*v dfwlydwlrq lv lq wkh udqjh ^3> 4`1 Khqfh/ lq ^8:` erwk idovh dqg wuxh duh uhsuhvhqwhg e| srvlwlyh qxpehuv lq wkh

6_{Wkh vx!flhqwo| vpdoo uhvwulfwlrqv duh jlyhq e| htxdwlrqv >}

6@%$₂ dqg &6@%$

(10)

udqjhv ^3> D6@%` dqg ^D6?> 4` uhvshfwlyho|1 Iru h{dpsoh/ li D6? @ 3=: dqg n @ 5/ dq lqwhusuhwdwlrq wkdw dvvljqv idovh wr srvlwlyh olwhudov lq wkh lqsxw od|hu ri Q fdq jhqhudwh d srvlwlyh lqsxw srwhqwldo juhdwhu wkdq wkh klgghq qhxurq*v wkuhvkrog/ zurqjo| dfwlydwlqj wkh qhxurq lq wkh rxwsxw od|hu ri Q 1 Lq rughu wr vroyh wklv sureohp zh xvh elsrodu dfwlydwlrq ixqfwlrqv/ zkhuh hdfk qhxurq*v dfwlydwlrq lv lq wkh udqjh ^4> 4`1 Qrz/ dq lqwhusuhwdwlrq wkdw grhv qrw vdwlvi| d fodxvh frqwulexwhv qhjdwlyho| wr wkh klgghq qhxurq*v lqsxw srwhqwldo/ vlqfh idovh lv uhsuhvhqwhg e| d qxpehu lq ^4> D6?`/ zkloh dq lqwhusuhwdwlrq wkdw grhv vdwlvi| d fodxvh frqwulexwhv srvlwlyho| wr wkh lqsxw srwhqwldo/ ehfdxvh wuxh lv lq ^D6?> 4`1 Wkhruhp 6 zloo vkrz wkdw wkh fkrlfh ri d elsrodu dfwlydwlrq ixqfwlrq lv vx!flhqw wr vroyh wkh deryh sureohp1 Ixuwkhupruh/ wkh fkrlfh ri 4 lqvwhdg ri }hur wr uhsuhvhqw idovh zloo ohdg wr idvwhu frqyhujhqfh lq doprvw doo fdvhv1 Wkh uhdvrq lv wkdw wkh xsgdwh ri d zhljkw frqqhfwhg wr dq lqsxw yduldeoh zloo eh }hur zkhq wkh fruuhvsrqglqj yduldeoh lv }hur lq wkh wudlqlqj sdwwhuq ^;/ 55`1

Wkxv pdnlqj xvh ri elsrodu vhpl0olqhdu dfwlydwlrq ixqfwlrq k+{,/ ohw xv vhh krz zh kdyh rewdlqhg wkh ydoxhv iru wkh klgghq dqg rxwsxw qhxurqv* wkuhvkrogv , dqg = Wr frqihu v|pphwulf pdwkhpdwlfdo uhvxowv/ zlwkrxw orvv ri jhqhudolw|/ dvvxph wkdw D6@%@ D6?1 Iurp wkh lqsxw wr wkh klgghq od|hu ri Q +O> ===> O& , Q,,/ li dq lqwhusuhwdwlrq vdwlvhv O> ===> O& wkhq wkh frqwulexwlrq ri O> ===> O& wr wkh lqsxw srwhqwldo ri Q, lv juhdwhu wkdq Ln @ nD6?Z 1 Li/ frqyhuvho|/ dq lqwhusuhwdwlrq grhv qrw vdwlvi| O> ===> O&wkhq wkh frqwulexwlrq ri O> ===> O& wr wkh lqsxw srwhqwldo ri Q, lv vpdoohu wkdq L3 @ +s4,Z D6?Z .qZ 1 Wkhuhiruh/ zh ghqh ,@ UnnU2 3 @ En6?2E&3Z +Wudqvodwlrq Dojrulwkp/ vwhs 7g,1 Iurp wkh klgghq wr wkh rxwsxw od|hu ri Q +Q, , D,/ li dq lqwhusuhwdwlrq vdwlvhv Q, wkhq wkh frqwulexwlrq ri Q, wr wkh lqsxw srwhqwldo ri D lv juhdwhu wkdq Ln @ D6?Z + 4,Z 1 Li/ frqyhuvho|/ dq lqwhusuhwdwlrq grhv qrw vdwlvi| Q, wkhq wkh frqwulexwlrq ri Q, wr wkh lqsxw srwhqwldo ri D lv vpdoohu wkdq L3 @ D6?Z 1 Vlploduo|/ zh ghqh @ UnnU₂ 3 @ En6?₂E3>Z +Wudqvodwlrq Dojrulwkp/ vwhs 7h,1 Reylrxvo|/ Ln A L3 vkrxog eh vdwlvhg lq erwk fdvhv deryh1 Wkhuhiruh/ D6? A &_&,_,3_n dqg D6? A >_>,_,3_n pxvw eh yhulhg dqg/ pruh jhqhudoo|/ wkh frqglwlrq lpsrvhg ryhu D6? lq wkh wudqvodwlrq dojrulwkp +vwhs 5,1 Ilqdoo|/ jlyhq D6?/ wkh ydoxh ri Z +wudqvodwlrq dojrulwkp +vwhs 6,, uhvxowv iurp wkh surri ri wkhruhp 6 ehorz1

Lq zkdw iroorzv/ zh vkrz wkdw wkh wkhruhp suhvhqwhg lq ^58`/ zkhuh Q zlwk elqdu| wkuhvkrog qhxurqv frpsxwhv wkh {hg srlqw rshudwru W ri wkh surjudp S/ vwloo krogv iru Q zlwk vhpl0olqhdu qhxurqv1 Wkh iroorzlqj wkhruhp hqvxuhv wkdw rxu wudqvodwlrq dojrulwkp lv vrxqg1 Wkh ixqfwlrq W pdsslqj lqwhusuhwdwlrqv wr lqwhusuhwdwlrqv lv ghqhg dv iroorzv1 Ohw l eh dq lqwhusuhwdwlrq dqg D dq dwrp1 W+l,+D, @ _wuxh% lii wkhuh h{lvwv D # O> = = = > O& lq S v=w=Y&'l+O, @ _wuxh%=

(11)

Wkhruhp 6 Iru hdfk sursrvlwlrqdo jhqhudo orjlf surjudp S/ wkhuh h{lvwv d ihhgiruzdug duwlfldo qhxudo qhwzrun Q zlwk h{dfwo| rqh klgghq od|hu dqg vhpl0olqhdu qhxurqv/ rewdlqhg e| wkh deryh Wudqvodwlrq Dojrulwkp/ vxfk wkdw Q frpsxwhv W1

Surri1 Zh kdyh wr vkrz wkdw wkhuh h{lvwv Z A 3 vxfk wkdw Q frpsxwhv W1 Lq rughu wr gr vr zh qhhg wr suryh wkdw jlyhq dq lqsxw yhfwru l/ hdfk qhxurq D lq wkh rxwsxw od|hu ri Q lv dfwlyh li dqg rqo| li wkhuh h{lvwv d fodxvh ri S ri wkh irup D # O> ===> O& v1w1 O> ===> O& duh vdwlvhg e| lqwhusuhwdwlrq l1 Wkh surri wdnhv dgydqwdjh ri wkh prqrwrqlfdoo| qrq0ghfuhdvlqj surshuw| ri wkh elsrodu vhpl0olqhdu dfwlydwlrq ixqfwlrq k+{,/ zklfk doorzv wkh dqdo|vlv wr irfxv rq wkh erxqgdu| fdvhv1 Dv ehiruh/ zh dvvxph wkdw D6@% @ D6? zlwkrxw orvv ri jhqhudolw|1

+#, D D6? li O> ===> O& lv vdwlvhg e| l1 Dvvxph wkdw wkh s srvlwlyh olwhudov lq O> ===> O& duh wuxh/ zkloh wkh q qhjdwlyh olwhudov lq O> ===> O& duh idovh1 Frqvlghu wkh pdsslqj iurp wkh lqsxw od|hu wr wkh klgghq od|hu ri Q 1 Wkh lqsxw srwhqwldo +L,, ri Q, lv plqlpxp zkhq doo wkh qhx0 urqv dvvrfldwhg zlwk d srvlwlyh olwhudo lq O> ===> O& duh dw D6?> zkloh doo wkh qhxurqv dvvrfldwhg zlwk d qhjdwlyh olwhudo lq O> ===> O& duh dw D6?1 Wkxv/ L, sD6?Z . qD6?Z , dqg dvvxplqj , @ En6?₂E&3Z / L, sD6?Z . qD6?Z En6?₂E&3Z 1

Li k+L,, D6?/ l1h1 L, _qoq

36?

n6?

/ wkhq Q, lv dfwlyh1 Wkhuhiruh/ htxdwlrq 4 pxvw eh vdwlvhg1

sD6?Z . qD6?Z +4 . D6?₅, +n 4,Z 4oq

4 D6? 4 . D6?

+4,

Vroylqj htxdwlrq 4 iru wkh frqqhfwlrq zhljkw +Z , |lhogv htxdwlrqv 5 dqg 6/ jlyhq wkdw Z A 31

Z 5 oq +4 D_{n +D} 6?, oq +4 . D6?,

6? 4, . D6?. 4 +5,

D6?ln 4_{n . 4} +6,

Frqvlghu qrz wkh pdsslqj iurp wkh klgghq od|hu wr wkh rxwsxw od|hu ri Q 1 E| htxdwlrqv 5 dqg 6 dw ohdvw rqh qhxurq Q, wkdw lv frqqhfwhg wr D lv dfwlyh1 Wkh lqsxw srwhqwldo +L,, ri D lv plqlpxp zkhq Q, lv dw D6?/ zkloh wkh rwkhu 4 qhxurqv frqqhfwhg wr D duh dw 041 Wkxv/ L, D6?Z + 4, Z , dqg dvvxplqj , @ En6?₂E3>Z / L, D6?Z + 4, Z En6?E3>

2 Z 1

Li k+L,, D6?/ l1h1 L, _qoq

36?

n6?

/ wkhq D lv dfwlyh1 Wkhuhiruh/ htxdwlrq 7 pxvw eh vdwlvhg1

D6?Z + 4,Z +4 . D6?₅,+4 ,Z 4oq

4 D6? 4 . D6?

(12)

Vroylqj htxdwlrq 7 iru wkh frqqhfwlrq zhljkw Z |lhogv htxdwlrqv 8 dqg 9/ jlyhq wkdw Z A 31

Z 5 oq +4 D_+D 6?, oq +4 . D6?,

6? 4, . D6?. 4 +8,

D6?l 4_{. 4} +9,

+$, D D6? li O> ===> O& lv qrw vdwlvhg e| l1 Dvvxph wkdw dw ohdvw rqh ri wkh s srvlwlyh olwhudov lq O> ===> O& lv idovh ru rqh ri wkh q qhjdwlyh olwhudov lq O> ===> O& lv wuxh1 Frqvlghu wkh pdsslqj iurp wkh lqsxw od|hu wr wkh klgghq od|hu ri Q 1 Wkh lqsxw srwhqwldo +L,, ri Q, lv pd{lpxp zkhq rqo| rqh qhxurq dvvrfldwhg wr d srvlwlyh olwhudo lq O> ===> O& lv dw D6? ru zkhq rqo| rqh qhxurq dvvrfldwhg wr d qhjdwlyh olwhudo lq O> ===> O& lv dw D6?1 Wkxv/ L, +s 4, Z D6?Z . qZ , ru L, +q 4, Z D6?Z . sZ ,/ uhvshfwlyho|/ dqg dvvxplqj , @ En6?₂E&3Z / L, +n 4 D6?, Z En6?₂E&3Z 1

Li D6? k+L,,/ l1h1 D6? _ne3q2EU,4/ wkhq L, qoq

n6?

36?

> dqg vr Q, lv qrw dfwlyh1 Wkhuhiruh/ htxdwlrq : pxvw eh vdwlvhg1

+n 4 D6?,Z +4 . D6?₅,+n 4,Z 4oq

4 D6? 4 . D6?

+:,

Vroylqj htxdwlrq : iru wkh frqqhfwlrq zhljkw Z |lhogv htxdwlrqv ; dqg </ jlyhq wkdw Z A 31

Z 5

oq +4 . Dn +D6?6? 4, . D, oq +4 D6?. 46?, +;,

D6?ln 4_{n . 4} +<,

Frqvlghu qrz wkh pdsslqj iurp wkh klgghq od|hu wr wkh rxwsxw od|hu ri Q 1 E| htxdwlrqv ; dqg </ doo qhxurqv Q, wkdw duh frqqhfwhg wr D duh qrw dfwlyh1 Wkh lqsxw srwhqwldo +L,, ri D lv pd{lpxp zkhq doo wkh qhxurqv frqqhfwhg wr D duh dw D6?1 Wkxv/ L, D6?Z , dqg dvvxplqj ,@ En6?₂E3>Z / L, D6?Z En6?₂E3>Z 1

Li D6? k+L,,/ l1h1 D6? _ne_3q2 _E_U,4> wkhq L, _qoq

n6?

36?

> dqg vr D lv qrw dfwlyh1 Wkhuhiruh/ htxdwlrq 43 pxvw eh vdwlvhg1

D6?Z +4 . D6?, +4 ,

5 Z

4 oq

4 . D6? 4 D6?

+43,

Vroylqj htxdwlrq 43 iru wkh frqqhfwlrq zhljkw Z |lhogv htxdwlrqv 44 dqg 45/ jlyhq wkdw Z A 31

Z 5 oq +4 . D_+D 6?, oq +4 D6?,

6? 4, . D6?. 4 +44,

(13)

D6?l 4

. 4 +45,

Qrwlfh wkdw htxdwlrqv 5 dqg 8 duh htxlydohqw wr htxdwlrqv ; dqg 44/ uhvshf0 wlyho|1 Khqfh/ wkh deryh wkhruhp krogv li iru hdfk fodxvh F, lq S htxdwlrqv 5 dqg 6 duh vdwlvhg e| Z dqg D6? iurp wkh lqsxw wr wkh klgghq od|hu ri Q / zkloh htxdwlrqv 8 dqg 9 duh vdwlvhg e| Z dqg D6? iurp wkh klgghq wr wkh rxwsxw od|hu ri Q 1

Lq rughu wr xqli| wkh zhljkwv lq Q iru hdfk fodxvh F, ri S jlyhq wkh ghqlwlrq ri PD[,+n,> ,,/ lw lv vx!flhqw wkdw htxdwlrqv 46 dqg 47 ehorz

duh vdwlvhg e| Z dqg D6?/ uhvshfwlyho|1

Z 5 _PD[oq +4 . D6?, oq +4 D6?, ,+n,> ,,+D6? 4, . D6?. 4

+46,

D6? A PD[_PD[,+n,> ,, 4 ,+n,> ,, . 4

+47,

Ilqdoo|/ lq rughu wr xqli| doo wkh zhljkwv lq Q iru d surjudp S jlyhq wkh ghqlwlrq ri PD[+n> ===> n^> > ===> ^,/ lw lv vx!flhqw wkdw htxdwlrqv 48 dqg 49 duh vdwlvhg e| Z dqg D6?/ uhvshfwlyho|1

Z 5

oq +4 . D6?, oq +4 D6?,

PD[+n> ===> n^> > ===> ^,+D6? 4, . D6?. 4 +48,

D6? A PD[_PD[+n> ===> n^> > ===> ^, 4

+n> ===> n^> > ===> ^, . 4 +49, Dv d uhvxow/ li htxdwlrqv 48 dqg 49 duh vdwlvhg e| Z dqg D6?> uhvshf0 wlyho|/ wkhq Q frpsxwhv W1

H{dpsoh 7 Frqvlghu wkh surjudp S @ iD # E> F> qrwG> D # H> I > E #j1 Frqyhuwlqj idfw E # wr uxoh E # W dqg dsso|lqj wkh Wudqvod0 wlrq Dojrulwkp/ zh rewdlq wkh qhxudo qhwzrun Q ri jxuh 61 Iluvwo|/ zh fdofxodwh PD[+n> ===> n?> > ===> ?, @ 6 +vwhs 4,/ dqg D6? A 3=8 +vwhs 5,1 Wkhq/ vxssrvh D6? @ 3=9/ zh rewdlq Z 9=<64@ +vwhs 6,1 Dowhuqd0 wlyho|/ vxssrvh D6? @ 3=:/ wkhq Z 7=669@1 Ohw xv wdnh D6? @ 3=: dqg k+{, dv wkh vwdqgdug elsrodu vhpl0olqhdu dfwlydwlrq ixqfwlrq + @ 4,/ wkhq li Z @ 7=8> Q frpsxwhv wkh rshudwru W ri S71

Lq wkh deryh h{dpsoh/ wkh qhxurq E dsshduv dw erwk wkh lqsxw dqg wkh rxwsxw od|huv ri Q 1 Wklv lqglfdwhv wkdw wkhuh duh dw ohdvw wzr fodxvhv ri S wkdw duh olqnhg wkurxjk E +lq wkh h{dpsoh= D # E> F> qrwG dqg E #,> ghqlqj d ghshqghqf| fkdlq ^6`1 Zh uhsuhvhqw wkdw fkdlq lq wkh qhwzrun

7_{Qrwh wkdw d vrxqg wudqvodwlrq iurp wr grhv qrw uhtxluh doo wkh zhljkwv lq wr}

(14)

Wr

A B -(1+A min )W/2 0

W

W W

N1 N2 N3 (1+A min )W (1+A min )W/2 0

W W W

-W

T F

E D C

B

Interpretations 1

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-50 DNA +7 Minus35 Minus10 Conform. Contact

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Wkh vhfrqg dssolfdwlrq wkdw zh xvh wr whvw F0LO5_{S lv hxndu|rwlf}44 _vsolfh0 mxqfwlrq ghwhuplqdwlrq1 Vsolfh0mxqfwlrqv duh srlqwv rq d GQDvhtxhqfh dw zklfk wkh qrq0frglqj uhjlrqv duh uhpryhg gxulqj wkh surfhvv ri surwhlq v|qwkhvlv1 Wkh dlp ri vsolfh0mxqfwlrq ghwhuplqdwlrq lv wr uhfrjql}h wkh erxqgdulhv ehwzhhq wkh sduw ri wkh GQDuhwdlqhg diwhu vsolfh 0 fdoohg h{rqv 0 dqg wkh sduw wkdw lv vsolfhg rxw 0 wkh lqwurqv1 Wkh wdvn frqvlvwv wkhuhiruh ri uhfrjql}lqj h{rq2lqwurq +H2L, erxqgdulhv dqg lqwurq2h{rq +L2H, erxqgdulhv1 Wdeoh 6 frqwdlqv wkh edfnjurxqg nqrzohgjh iru vsolfh mxqfwlrq ghwhuplqd0 wlrq45₁

44_{Xqolnh surndu|rwlf fhoov/hxndu|rwlf fhoov frqwdlq d qxfohxv/dqg vr duh kljkhu xs wkh}

hyroxwlrqdu| vfdoh1

45_{Uxohv rewdlqhg iurp ^8:` dqg ghulyhg iurp wkh elrorjlfdo olwhudwxuh iurp Qrrughzlhu}

^6;`1

(21)

HL # C 6cpdjjwudjw*> qrw HL3Vwrs

HL3Vwrs + C 6cwdd* HL3Vwrs + C 7cwdd* HL3Vwrs + C 8cwdd* HL3Vwrs + C 6cwdj* HL3Vwrs + C 7cwdj* HL3Vwrs + C 8cwdj* HL3Vwrs + C 6cwjd* HL3Vwrs + C 7cwjd* HL3Vwrs + C 8cwjd*

LH # s|udplglqh3ulfk> C 6c|djj*> qrw LH3Vwrs s|udplglqh3ulfk # 9 ri +C 48c||||||||||*,

LH3Vwrs + C4cwdd* LH3Vwrs + C5cwdd* LH3Vwrs + C6cwdd* LH3Vwrs + C4cwdj* LH3Vwrs + C5cwdj* LH3Vwrs + C6cwdj* LH3Vwrs + C4cwjd* LH3Vwrs + C5cwjd* LH3Vwrs + C6cwjd*

Wdeoh 6= Edfnjurxqg nqrzohgjh iru vsolfh0mxqfwlrq1

Wkh edfnjurxqg nqrzohgjh ri Wdeoh 6 lv wudqvodwhg e| FLO5_{S wr wkh} qhxudo qhwzrun ri jxuh :1 Lq Wdeoh 6/ +lqglfdwhv qrqghihdvleoh uxohv/ zklfk fdq qrw eh dowhuhg gxulqj wudlqlqj1 Wkhuhiruh/ wkh zhljkwv vhw lq wkh qhwzrun e| wkhvh uxohv duh {hg1 Uxohv ri wkh irup p ri +===,duh vdwlvhg li dw ohdvw p ri wkh sduhqwkhvl}hg frqfhswv duh wuxh1 Qrwh wkdw wkh wudqvodwlrq ri wkhvh uxohv wr wkh qhwzrun lv grqh e| vlpso| ghqlqj n, @ p lq FLO5_{S*v wudqvodwlrq dojrulwkp1 Uxohv frqwdlqlqj v|perov rwkhu wkdq wkh} ruljlqdo +d> j> w> f, duh vsolw lqwr d qxpehu ri htxlydohqw uxohv frqwdlqlqj rqo| wkh ruljlqdo v|perov/ dffruglqj wr Wdeoh 41 Iru h{dpsoh/ vlqfh | f b w wkh uxoh LH # s|udplglqh3ulfk> C 6c|djj*> qrw LH3Vwrs lv hqfrghg lq wkh qhwzrun dv LH # s|udplglqh3ulfk> C 6cfdjj*> qrw LH3Vwrs dqg LH # s|udplglqh3ulfk> C 6cwdjj*> qrw LH3Vwrs1

Wkh wudlqlqj vhw iru wklv wdvn frqwdlqv 64<3 h{dpsohv/ lq zklfk dssur{0 lpdwho| 58( duh ri L2H erxqgdulhv/ 58( duh ri H2L erxqgdulhv dqg wkh uhpdlqlqj 83( duh qhlwkhu1 Wkh wklug fdwhjru| +qhlwkhu H2L qru L2H, lv frqvlghuhg wuxh zkhq qhlwkhu L2H qru H2L rxwsxw qhxurqv duh dfwlyh1 Hdfk h{dpsoh lv d GQD vhtxhqfh zlwk 93 qxfohrwlghv/ zkhuh wkh fhqwhu lv wkh uhihuhqfh srlqw1 Uhphpehu wkdw wkh qhwzrun ri jxuh : lv ixoo|0frqqhfwhg/ exw wkdw orz0zhljkwhg olqnv duh qrw vkrzq1 Grwwhg olqhv lqglfdwh olqnv zlwk qhjdwlyh zhljkwv1

Lq erwk dssolfdwlrqv/ xqohvv vwdwhg rwkhuzlvh/ wkh edfnjurxqg nqrzohgjh lv dvvxphg ghihdvleoh/ l1h1 wkh zhljkwv duh doorzhg wr fkdqjh gxulqj wkh ohduqlqj surfhvv1 Khqfh/ vrph ri wkh edfnjurxqg nqrzohgjh pd| eh uhylvhg e| wkh wudlqlqj h{dpsohv1 Qrwh krzhyhu wkdw wkh qhwzrunv* uhfxuuhqw frq0 qhfwlrqv duh uhvsrqvleoh iru uhlqiruflqj wkh edfnjurxqg nqrzohgjh gxulqj wudlqlqj1 Iru lqvwdqfh/ lq wkh qhwzrun ri jxuh : wkh frqfhswv S|udplglqh/ HL0Vw1 dqg LH0Vw1/ fdoohg lqwhuphgldwh frqfhswv/ kdyh wkhlu lqsxw ydoxhv fdo0 fxodwhg e| wkh qhwzrun lq dfwlrq/ dffruglqj wr wkh edfnjurxqg nqrzohgjh dqg wr wkh GQD vhtxhqfh lqsxw yhfwru1

(22)

Pyramidine EI-St. IE-St. EI IE

-30 _DNA +30_{Pyramidine EI-St. IE-St.}

Iljxuh := Lqlwldo qhxudo qhwzrun iru vsolfh0mxqfwlrq ghwhuplqdwlrq1 Hdfk er{ dw wkh lqsxw od|hu ri wkh qhwzrun uhsuhvhqwv rqh vhtxhqfh orfdwlrq zklfk lv hqfrghg e| irxu lqsxw qhxurqv id> j> w> fj1

ehwwhu wkdq dq| phwkrg dqdo|}hg lq ^8:`1 Pruhryhu/ F0LO5_{S*v wudlqlqj vhw} shuirupdqfh lv frqvlghudeo| vxshulru wr NEDQQ*v/ pdlqo| ehfdxvh lw dozd|v hqfrghv wkh edfnjurxqg nqrzohgjh lq d vlqjoh klgghq od|hu qhwzrun1

Iluvwo|/ ohw xv frqvlghu F0LO5_{S*v whvw0vhw shuirupdqfh/ l1h1/ lwv delolw| wr} jhqhudol}h ryhu h{dpsohv qrw vhhq gxulqj wudlqlqj1 Zh frpsduh wkh uhvxowv rewdlqhg e| F0LO5_{S lq erwk dssolfdwlrqv zlwk vrph ri wkh pdlq lqgxf0} wlyh ohduqlqj v|vwhpv iurp h{dpsohv= Edfnsursdjdwlrq ^7;`/ Shufhswurq ^7:` +qhxudo v|vwhpv,/ LG6 ^78`/ dqg Frezhe ^45` +v|perolf v|vwhpv,1 Zh dovr frpsduh wkh uhvxowv lq wkh surprwhu uhfrjqlwlrq sureohp zlwk d phwkrg vxjjhvwhg e| elrorjlvwv ^83`1 Lq dgglwlrq/ zh frpsduh F0LO5_{S zlwk v|v0} whpv wkdw ohduq iurp erwk h{dpsohv dqg edfnjurxqg nqrzohgjh= Hlwkhu ^73`/ Ode|ulqwk0N ^84`/ IRFO ^74` +v|perolf v|vwhpv,/ dqg NEDQQ ^8:` +k|eulg v|vwhp,46₁

Dv lq ^8:`/ zh hydoxdwh wkh v|vwhpv xvlqj furvv0ydolgdwlrq/ d whvwlqj phwkrgrorj| lq zklfk wkh vhw ri h{dpsohv lv shupxwhg dqg glylghg lqwr q vhwv1 Rqh glylvlrq lv xvhg iru whvwlqj dqg wkh uhpdlqlqj q 4 glylvlrqv duh xvhg iru wudlqlqj1 Wkh whvwlqj glylvlrq lv qhyhu vhhq e| wkh ohduqlqj do0 jrulwkp gxulqj wkh wudlqlqj surfhvv1 Wkh surfhgxuh lv uhshdwhg q wlphv vr wkdw hyhu| sduwlwlrq lv xvhg rqfh iru whvwlqj1 Iru wkh 4390h{dpsohv surprwhu gdwd vhw/ zh xvh ohdylqj0rqh0rxw furvv0ydolgdwlrq/ lq zklfk hdfk h{dpsoh lv vxffhvvlyho| ohiw rxw ri wkh wudlqlqj vhw1 Khqfh/ lw uhtxluhv 439 wudlqlqj skdvhv/ lq zklfk wkh wudlqlqj vhw kdv 438 h{dpsohv dqg wkh whvwlqj vhw kdv 4 h{dpsoh1 Ohdylqj0rqh0rxw ehfrphv h{shqvlyh dv wkh qxpehu ri dydlodeoh h{dpsohv jurzv1 Wkhuhiruh/ iroorzlqj ^8:`/ zh xvh 430irog furvv0ydolgdwlrq

46_{Wrzhoo dqg Vkdyoln frpsduh NEDQQ zlwk rwkhu k|eulg v|vwhpv ^47` dqg ^5<`/ rewdlq0}

lqj ehwwhu uhvxowv1

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iru wkh 43330h{dpsohv vsolfh0mxqfwlrq ghwhuplqdwlrq gdwd vhw47₁

Wkh ohduqlqj v|vwhpv wkdw duh edvhg rq qhxudo qhwzrunv kdyh ehhq wudlqhg xqwlo rqh ri wkh iroorzlqj wkuhh vwrsslqj fulwhuld zdv vdwlvhg= +l, rq <<( ri wkh wudlqlqj h{dpsohv/ wkh dfwlydwlrq ri hyhu| rxwsxw xqlw lv zlwklq 3158 ri fruuhfwqhvv> +ll, hyhu| wudlqlqj h{dpsoh lv suhvhqwhg wr wkh qhwzrun 433 wlphv/ l1h1 wkh qhwzrun kdv ehhq wudlqhg iru 433 hsrfkv> +lll, wkh qhwzrun fodvvlhv dw ohdvw <3( ri wkh wudlqlqj h{dpsohv fruuhfwo| exw kdv qrw lp0 suryhg lwv fodvvlfdwlrq delolw| iru 8 hsrfkv1 Zh kdyh ghqhg dq hsrfk dv rqh wudlqlqj sdvv wkurxjk wkh zkroh wudlqlqj vhw1 Zh xvhg wkh vwdqgdug edfnsursdjdwlrq ohduqlqj dojrulwkp wr wudlq F0LO5_{S qhwzrunv1}

94.3

91.5

80.0

93.3 97.2

0 10 20 30 40 50 60 70 80 90 100

Stormo ID3 Perceptron

Cobweb Backprop C-IL2P

Test Set Performance

Iljxuh ;= Whvw0vhw shuirupdqfh lq wkh surprwhu sureohp +frpsdulvrq zlwk v|vwhpv wkdw ohduq vwulfwo| iurp h{dpsohv,1

89.7 89.2 88.0 93.5 94.8

0 10 20 30 40 50 60 70 80 90 100

ID3 Perceptron

Cobweb Backprop C-IL2P

Iljxuh <= Whvw0vhw shuirupdqfh lq wkh vsolfh mxqfwlrq sureohp +frpsdulvrq zlwk v|vwhpv wkdw ohduq vwulfwo| iurp h{dpsohv,1

F0LO5_{S jhqhudol}hv ehwwhu wkdq dq| hpslulfdo ohduqlqj v|vwhp +vhh jxuhv} ; dqg <, dqg ehwwhu wkdq dq| v|vwhp wkdw ohduqv iurp h{dpsohv dqg edfn0 jurxqg nqrzohgjh +vhh jxuhv 43 dqg 44, whvwhg rq erwk dssolfdwlrqv1 Lq

47_{Lq dffrugdqfh zlwk ^8:`/ 4333 h{dpsohv duh udqgrpo| vhohfwhg iurp wkh 64<3 h{dpsohv}

(24)

79.0 85.8 87.0 92.0 92.8

0 10 20 30 40 50 60 70 80 90 100

Either FOCL Labyrinth

KBANN C-IL2P

Iljxuh 43= Whvw0vhw shuirupdqfh lq wkh surprwhu sureohp +frpsdulvrq zlwk v|vwhpv wkdw ohduq erwk iurp h{dpsohv dqg wkhru|,1

94.2 94.8

0 10 20 30 40 50 60 70 80 90 100 KBANN

C-IL2P

Iljxuh 44= Whvw0vhw shuirupdqfh lq wkh vsolfh mxqfwlrq sureohp +frpsdulvrq zlwk v|vwhpv wkdw ohduq erwk iurp h{dpsohv dqg wkhru|,1

prvw fdvhv glhuhqfhv duh vwdwlvwlfdoo| vljqlfdqw1 Krzhyhu/ F0LO5_{S lv rqo|} pdujlqdoo| ehwwhu wkdq NEDQQ1 Wklv lv ehfdxvh erwk v|vwhpv duh k|eulg qhxudo v|vwhpv wkdw shuirup lqgxfwlyh ohduqlqj iurp h{dpsohv dqg edfn0 jurxqg nqrzohgjh1

Xvxdoo|/ wkhru| dqg gdwd ohduqlqj v|vwhpv uhtxluh ihzhu wudlqlqj h{dp0 sohv wkdq v|vwhpv wkdw ohduq rqo| iurp gdwd1 Wkh edfnjurxqg nqrzohgjh khosv d ohduqlqj v|vwhp wr h{wudfw xvhixo jhqhudol}dwlrqv iurp vpdoo vhwv ri h{dpsohv1 Wklv lv txlwh lpsruwdqw vlqfh/ lq jhqhudo/ lw lv qrw hdv| wr rewdlq odujh dqg dffxudwh wudlqlqj vhwv1

Wkxv/ ohw xv qrz dqdo|}h F0LO5_{S*v whvw0vhw shuirupdqfh jlyhq vpdoohu} vhwv ri h{dpsohv1 Wkh iroorzlqj whvwv zloo frpsduh wkh shuirupdqfh ri F0 LO5_{S zlwk NEDQQ dqg Edfnsursdjdwlrq rqo|/ ehfdxvh wkhvh v|vwhpv kdyh} vkrzq wr eh wkh prvw hhfwlyh rqhv lq wkh suhylrxv whvwv +jxuhv ;/ </ 43 dqg 44,1 Iroorzlqj ^8:`/ wkh jhqhudol}dwlrq delolw| ryhu vpdoo vhwv ri h{dpsohv lv dqdo|}hg e| vsolwwlqj wkh h{dpsohv lqwr wzr vxevhwv= wkh whvwlqj vhw frqwdlq0 lqj dssur{lpdwho| 58( ri wkh h{dpsohv/ dqg wkh wudlqlqj vhw frqwdlqlqj wkh

(25)

uhpdlqlqj h{dpsohv1 Wkh wudlqlqj vhw lv sduwlwlrqhg lqwr vhwv ri lqfuhdvlqj vl}hv dqg wkh qhwzrunv duh wudlqhg xvlqj hdfk sduwlwlrq dw d wlph1

0 0.1 0.2 0.3 0.4 0.5 0.6

0 20 40 60 80

Number of Training Examples

T e s t S e t E rr o r R a te Backprop KBANN C-IL2P

Iljxuh 45= Whvw0vhw huuru udwh lq wkh surprwhu sureohp +59 h{dpsohv uhvhuyhg iru whvwlqj,1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 100 200 300

Number of Training Examples

T e s t S e t E rr o r R a te Backprop KBANN C-IL2P

Iljxuh 46= Whvw0vhw huuru udwh lq wkh vsolfh mxqfwlrq sureohp +:<; h{dpsohv uhvhuyhg iru whvwlqj,1

Iljxuhv 45 dqg 46 vkrz wkdw lq erwk dssolfdwlrqv F0LO5_{S jhqhudol}hv} ryhu vpdoo vhwv ri h{dpsohv ehwwhu wkdq edfnsursdjdwlrq1 Wkh uhvxowv hpslu0 lfdoo| vkrz wkdw wkh lqlwldo wrsrorj| ri wkh qhwzrun/ vhw e| wkh edfnjurxqg nqrzohgjh/ jlyhv lw d ehwwhu jhqhudol}dwlrq fdsdelolw|1 Qrwh wkdw wkh uhvxowv rewdlqhg e| F0LO5_{S dqg NEDQQ duh yhu| vlplodu/ vlqfh erwk v|vwhpv duh} edvhg rq wkh edfnsursdjdwlrq ohduqlqj dojrulwkp dqg ohduq iurp h{dpsohv dqg edfnjurxqg nqrzohgjh1

(26)

hsrfkv1 Qhxudo qhwzrunv* ohduqlqj shuirupdqfh lv d pdmru frqfhuq/ vlqfh lw fdq ehfrph surklelwlyh lq fhuwdlq dssolfdwlrqv/ xvxdoo| dv d uhvxow ri wkh orfdo plqlpd sureohp48₁

0 0.1 0.2 0.3 0.4 0.5 0.6

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

Training Epochs R M S E rr o r R a te Backprop KBANN C-IL2P

Iljxuh 47= Wudlqlqj0vhw UPV huuru ghfd| gxulqj ohduqlqj wkh surprwhu sure0 ohp1 0 0.1 0.2 0.3 0.4 0.5 0.6

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Training Epochs R M S E rr o r R a te Backprop KBANN C-IL2P

Iljxuh 48= Wudlqlqj0vhw UPV huuru ghfd| gxulqj ohduqlqj wkh vsolfh mxqfwlrq sureohp1

Iljxuhv 47 dqg 48 vkrz wkdw F0LO5_{S*v ohduqlqj shuirupdqfh lv frqvlg0} hudeo| ehwwhu wkdq NEDQQ*v1 Wkh uhvxowv vxjjhvw wkdw rxu wudqvodwlrq do0 jrulwkp iurp v|perolf nqrzohgjh wr qhxudo qhwzrunv kdv dgydqwdjhv ryhu wkh dojrulwkp suhvhqwhg lq ^8:`1 Wkh Wudqvodwlrq Dojrulwkp suhvhqwhg khuh dozd|v hqfrghv wkh edfnjurxqg nqrzohgjh lqwr d vlqjoh klgghq0od|hu qhxudo qhwzrun1 Krzhyhu/ NEDQQ*v wudqvodwlrq dojrulwkp jhqhudwhv d qhwzrun zlwk dv pdq| od|huv dv wkhuh duh ghshqghqf| fkdlqv lq wkh edfnjurxqg nqrzohgjh1 Iru h{dpsoh/ li E # D> F # E> G # F> dqg H # G/ NEDQQ jhqhudwhv d qhwzrun zlwk wkuhh klgghq0od|huv lq zklfk frqfhswv E/ F/ dqg G duh uhsuhvhqwhg1 Reylrxvo|/ wklv fuhdwhv d uhvshfwlyh ghjudgdwlrq lq ohduq0 lqj shuirupdqfh1 Wrzhoo dqg Vkdyoln kdyh wulhg wr ryhufrph wklv sureohp

48_{Wkh qhwzrun fdq jhw vwxfn lq orfdo plqlpd gxulqj ohduqlqj/ lqvwhdg ri qglqj wkh}

joredo plqlpxp ri lwv huuru ixqfwlrq1

(27)

zlwk d v|perolf suh0surfhvvru ri uxohv iru NEDQQ ^8;`1 Krzhyhu/ lw lq0 wurgxfhv dqrwkhu suholplqdu| skdvh wr wkhlu wudqvodwlrq surfhvv49_{1 Lq rxu} rslqlrq wkh sureohp olhv lq NEDQQ*v wudqvodwlrq dojrulwkp/ dqg fdq eh vwudljkwiruzdugo| vroyhg e| dq dffxudwh wudqvodwlrq phfkdqlvp1

Vxppdul}lqj/ wkh h{shulphqwv ghvfulehg deryh vxjjhvw wkdw F0LO5_S*v hhfwlyhqhvv lv d uhvxow ri wkuhh ri wkh v|vwhp*v ihdwxuhv= FLO5_{S lv edvhg rq} edfnsursdjdwlrq/ lw xvhv edfnjurxqg nqrzohgjh/ dqg lw surylghv dq dffxudwh dqg frpsdfw wudqvodwlrq iurp v|perolf nqrzohgjh wr qhxudo qhwzrunv1

8 Ixwxuh Zrun dqg Frqfoxvlrq

Wkhuh duh vrph lqwhuhvwlqj rshq txhvwlrqv uhodwlqj wr wkh h{sodqdwlrq fdsd0 elolw| ri qhxudo qhwzrunv/ vshflfdoo| wr wkh wudgh0r ehwzhhq wkh frpsoh{lw| dqg txdolw| ri uxohv* h{wudfwlrq phwkrgv1 Rqh zd| wr uhgxfh wklv wudgh0r zrxog eh wr lqyhvwljdwh pruh h!flhqw suxqlqj phwkrgv iru qhxudo qhwzrunv* lqsxw yhfwruv vhdufk vsdfhv ^:`1

Dqrwkhu lqwhuhvwlqj txhvwlrq uhodwhv wr wkh fodvv ri h{whqghg surjudpv ^53`/ wkdw lv ri lqwhuhvw lq frqqhfwlrq zlwk wkh uhodwlrq ehwzhhq Orjlf Sur0 judpplqj dqg qrqprqrwrqlf irupdolvpv1 H{whqghg orjlf surjudpv/ zklfk dgg fodvvlfdo qhjdwlrq wr wkh odqjxdjh ri jhqhudo surjudpv/ fdq eh ylhzhg dv d iudjphqw ri Ghidxow wkhrulhv ^79`1 Frpprqvhqvh nqrzohgjh fdq eh uhsuhvhqwhg pruh hdvlo| zkhq fodvvlfdo qhjdwlrq lv dydlodeoh1 Zh kdyh h{0 whqghg F0LO5_{S wr ghdo zlwk h{whqghg orjlf surjudpv1 Wkh h{whqghg F0LO}5_S v|vwhp frpsxwhv dqvzhu vhwv ^53` lqvwhdg ri vwdeoh prghov1 Dv d uhvxow lw fdq eh dssolhg lq d eurdghu udqjh ri grpdlqv wkhrulhv1 Wkh h{whqghg F0 LO5_{S kdv douhdg| ehhq dssolhg lq srzhu v|vwhpv* idxow gldjqrvlv/ rewdlqlqj} surplvlqj suholplqdu| uhvxowv ^9`1

E| fkdqjlqj wkh ghqlwlrq ri W/ yduldqwv ri Ghidxow Orjlf dqg Orjlf Sur0 judpplqj vhpdqwlfv fdq eh rewdlqhg ^93`/ ghqlqj d idplo| ri qrqprqrwrqlf +sdudfrqvlvwhqw, qhxudo uhdvrqlqj v|vwhpv1 Dqrwkhu lqwhuhvwlqj gluhfwlrq wr sxuvxh zrxog eh wkh xvh ri odehoohg fodxvhv lq wkh vw|oh ri ^49`/ zkhuhe| wkh surri ri d olwhudo lv uhfrughg lq wkh odeho1 Wkh ohduqlqj dqg jhqhudol}dwlrq fdsdelolwlhv ri wkh qhwzrun pxvw dovr eh irupdoo| vwxglhg/ vr sd|lqj gxh uh0 jdug wr wkh orjlfdo irxqgdwlrqv ri wkh v|vwhp1 Wkh v|vwhp*v h{whqvlrq wr ghdo zlwk uvw rughu orjlf lv dqrwkhu frpsoh{ dqg ydvw duhd iru ixuwkhu uhvhdufk ^54`1

Dv d pdvvlyho| sdudooho qrqprqrwrqlf ohduqlqj v|vwhp/ F0LO5_{S kdv lq0} whuhvwlqj lpsolfdwlrqv iru wkh sureohp ri Eholhi Uhylvlrq ^4:` +vhh dovr ^4;`,1 Lq wkh vsolfh0mxqfwlrq ghwhuplqdwlrq sureohp/ sduw ri wkh edfnjurxqg nqrzo0 hgjh zdv hhfwlyho| hqfrghg dv ghihdvleoh/ vr wkdw frqwudglfwru| h{dpsohv zhuh deoh wr vshfli| d uhylvlrq ri wkh nqrzohgjh1 Vshflfdoo|/ wkh nqrzo0

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(28)

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(29)

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(30)

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(31)

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