Optimal Eye Movement Strategies in Visual Search (Supplement)

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Loraine French
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1 Opmal Eye Moveme Sraeges Vsual Search (Suppleme) Jr Naemk ad Wlso S. Gesler Ceer for Percepual Sysems ad Deparme of Psychology, Uversy of exas a Aus, Aus X 787 Here we derve he deal searcher for he case of dyamc (emporally ucorrelaed) exeral ad eral ose, ad we descrbe (whou proof) he deal searcher for he case of sac (emporally correlaed) exeral ose ad dyamc (emporally ucorrelaed) eral ose. As a check o he accuracy of he formulas for hese deal searchers, hey were derved va wo separae mehods: a drec dervao from he o probably dsrbuos (WG) ad a dervao based upo he heory of Kalma flerg (JN). he vsbly map o derve he deal searcher s ecessary o cosder he meag of he vsbly map more deal. o do hs we descrbe he deal deecor for a kow se wave arge preseed a a kow locao a sgle erval forced choce ask. Frs, cosder a deal deecor wh drec access o he real mage; ha s, a deal deecor for a o-foveaed vsual sysem. Each ral of a sgle erval forced choce ask cosss eher of backgroud ose aloe or backgroud ose plus he se wave arge. he deal deecor mulples he real mage wh a emplae of he se wave arge ad he egraes he produc o oba a emplae respose W (.e., he emplae respose s he cross correlao of he arge wh he real smulus). he magude of hs emplae respose s he compared o a crero; he opmal behavor s o respod yes f he emplae respose exceeds he crero ad o oherwse. he accuracy of hs deal deecor s deermed by he sgal-o-ose rao, d, whch s he average dfferece he emplae respose o he backgroud plus arge ad backgroud, dvded by he sadard devao of he emplae respose,. he eced value of he emplae respose o backgroud plus arge s proporoal o he arge coras ad he varace of he emplae respose s proporoal o he ose coras power of he backgroud, ad hece Ac d ( c, e ) (S) Be where, c s he RMS coras of he arge, e s he coras power of he ose backgroud, ad A ad B are proporoaly cosas. Now cosder a deal deecor a foveaed vsual sysem. I hs case, he deal deecor does o have drec access o he real mage, bu sead o a represeao ha s degraded by varable spaal resoluo ad eural ose. Reduced spaal resoluo due o spaal flerg wll effecvely reduce he coras of a se wave arge, bu wll have lle effec o he shape of he arge, hus he same emplae ca be used regos of reduced resoluo, alhough he resposes o he arge ad backgroud wll be smaller. (We have verfed ha for our arge ad

2 for eccerces as large as he radus of our dsplay, he approprae emplae shape chages eglgbly for rasfer fucos ha mach huma coras sesvy fucos.) Furhermore, he eural ose wll add a erm C( c, e, ε ) o he varace of he emplae resposes. I geeral, hs eural ose erm may deped o he arge coras, he backgroud ose power, ad he eccercy. herefore, he vsbly map of a deal deecor wh a foveaed vsual sysem s gve by ( c, e, ε) d A( ε) c ( ε) + (,, ε) B e C c e (S) where he proporoaly facors o he emplae resposes o he arge ad backgroud ow vary wh eccercy. Whou loss of geeraly hs formula ca be smplfed by dvdg umeraor ad deomaor by A( ε ): c d ( c, e, ε) αe + β ε ( c, e, ) (S3) where α B( ε) / A( ε) ad β( ce,, ε) C( ce,, ε) / A( ε). Because he same shaped emplae s used a dffere eccerces, he value of α s a cosa ha does o chage wh eccercy. Equao (S3) gves he vsbly map of he deal deecor, oce he values of α ad β are specfed for all eccerces. We oe, however, ha he case of dyamc (emporally ucorrelaed) exeral ad eral ose, he wo ose sources are of he same ype ad so her combed effec s gve drecly by he psychophyscally measured vsbly map d c, e, ε. I he case of sac exeral ose ad dyamc eral ose, he wo oses have ( ) dffere effecs ad so s ecessary o specfy α ad β(,, ) ce ε separaely (see laer). Ideal searcher for dyamc exeral ad eral ose Le be he umber of poeal arge locaos, ad le W k( ) be he Gaussa-dsrbued emplae respose from he h poeal arge locao, a fxao, wh fxao locao k( ). Whou loss of geeraly, we ca se he eced value of W k( ) o 0.5, whe he arge s prese a ha locao, ad o -0.5, oherwse. hs has o affec o he predcos as log as we se he sadard devao of he emplae resposes so as o preserve he value of d specfed by he vsbly map. hus, he mea value of he emplae respose s u k 0.5 arge locao 0.5 arge locao (S4) ad he varace of he emplae respose s

3 k d k (S5) Updag poseror probables. We ow derve he followg formula, whch gves he poseror probably ha he arge s a locao afer fxaos: pror d W k k pror d W k k p (S6) hs formula (whch s equao he ma body of he repor) shows ha o perfecly egrae formao across fxaos s suffce o keep a rug weghed sum of he emplae resposes a each poeal arge locao, where he weghs are he squares of he vsbly map values. Noe frs ha all he emplae resposes colleced a fxao ca be represeed by a vecor, W W,, W k k. Usg Bayes formula, follows mmedaely ha he poseror probably ha he arge s a locao afer fxaos s gve by p pror p ( W,, W( ) ) ( W(, ), W ) pror p (S7) Boh he exeral ose ad eral ose are sascally depede over me, ad for poeal arge locaos separaed suffcely space (as he prese search ermes) he exeral ose ad eral ose are also sascally depede over space. hus, qk q p pror p W qk pror p W q Gve equaos (S4) ad (S5), ad he fac ha he emplae resposes are Gaussa dsrbued, we have 3

4 p / ( ) / ( ) 0.5 π W 0.5 k π W + qk pror q k qk k qk ( π) ( ) W qk pror( ) / / ( W 0.5 k ) π k q qk k qk Dvdg he umeraor ad deomaor by he umeraor gves, p + pror pror ( 0.5) W ( 0.5 k W + k ) k k ( W 0.5) ( W 0.5 k + k ) k k Afer combg erms we have, p pror( ) W W k k + pror k k Fally, equao (S6) s obaed by usg equao (S5) o subsue for he varaces, ad he by mulplyg he umeraor ad deomaor by pror d k W k. Selecg ex fxao locao. o compue he opmal ex fxao po, ( ) k +, he deal searcher cosders each possble ex fxao ad pcks he locao ha, gve s kowledge of he curre poseror probables ad he vsbly map, wll maxmze he probably of correcly defyg he locao of he arge afer he ex fxao s made: k ( + ) ( ( )) k + argmax p C k + op op 4

5 Codog o he arge locao gves he followg equao, whch s equao he ma body of he repor: kop ( + ) argmax p p C, k + k ( + ) ( ) (S8) Here we derve a verso of hs equao ha s praccal o evaluae compuer smulaos. he probably of each possble arge locao s gve by equao (S6), hus our ob s o pck+,, he probably of beg correc gve ha he rue derve a resso for ( ) arge locao s, ad he locao of he ex fxao s k+ ( ). Le Z be he hypohecal h emplae respose from he locao afer he ex fxao s made;.e., Z W k( + ). Afer makg he ex fxao, he decso rule ha would maxmze accuracy would be o pck he locao wh he maxmum poseror probably. If oe uses ha decso rule, he he perce correc s equal o he probably ha he poseror probably a locao wll be greaer ha ha a all oher locaos: (, ( + ) ) ( + ) ( + ),, ( + ) ( + ), ( + ) p C k p p p p p k or equvalely, (, ( + ) ),,, ( + ) pck p L L k (S9) where L s he rao of he poseror probables for locao ad locao. From equao (S6) we have L pror d W pror + d W + k k k k whch ca be paroed o he currely kow poseror probables p ( ) ad p he currely ukow emplae resposes ha wll occur afer he ex fxao. hus,, ad 5

6 L d Z p k d Z p k ( + ) ( + ) Noe ha he L are sascal depede, bu become sascally depede f we codo o he value of Z. hus, d z p( ) k + p( Ck, ( + ) ) p( z) p dz d k( ) Z p( + ) or, (, ( ) ) p + ( ) l d ( ) z k + p d k( + ) p Ck + p z p Z < dz Expressg hs equao erms of he sadard ormal dsrbuo we have p ( ) d ( ) l k + z p ( ) d k( ) d k( ) + + p( Ck, ( + ) ) d ( ) φ( d ( )( z 0.5 k k )) dz + + Φ d k( + ) where φ ( x) s he sadard ormal desy fuco ad ( x) fuco: φ( x) Φ s he sadard ormal egral x x π, Φ x y dy π. Fally, makg he chage w d z 0.5 k +, we have of varable 6

7 ( ) p l d d ( ) w d k + k + k( + ) p p( Ck, ( + ) ) φ( w) Φ dw (S0) d k( + ) Alhough hs formula coas a egral ca be evaluaed rapdly usg umercal egrao, because he sadard ormal desy fuco approaches zero rapdly away from he org. hs formula reduces o he well-kow formula for accuracy he -alerave forced choce ask, he specal case ha all he prors are equal ad all he values of d are equal. I sum, equaos (S6), (S8) ad (S0), ad he measured vsbly map (equao S3) ca be used o smulae opmal vsual search he dyamc ose case. Ideal searcher for sac exeral ose ad dyamc eral ose he deal searcher for sac exeral ose s bascally he same as for dyamc exeral ose, excep ha we mus cosder he exeral ad eral ose separaely. Noe frs ha equao (S3) paros he ose ha lms deeco performace o wo pars, he sac par due o he exeral ose ( α e ) ad he dyamc par due o he eral ose ( β( ce,, ε )). We drecly deermed he value of α (0.08) by measurg he emplae resposes o he arge ad by measurg he varace of he emplae resposes o very large umber of samples of he acual backgroud ose used he ermes. Gve he value of α, he varace of he dyamc eral ose ca be deermed from he measured vsbly maps (he values of d ( c, e, ε) ), usg equao (S3). o smulae he deal searcher s performace, we geeraed a sac ose sample of sadard devao α e c for each poeal arge locao, before he frs fxao. he, o each fxao, for each poeal arge locao, we geeraed a dyamc ose sample of sadard devao β( ce,, ε ) c, whch we added o he sac ose sample a ha locao. Oherwse, he smulao was ru he same fasho as for dyamc case (see he mehods seco he repor). We ow sae (whou proof) he formulas for he deal searcher whe he exeral ose s sac. Updag poseror probables. he opmal egrao of he emplae resposes across fxaos s gve by he followg hree equaos: p pror gk( ) W k pror g W k( ) k (S) 7

8 g ( k ) e αβ k k c v + β (S) v (S3) β r k r hese equaos reflec he fac ha he deal searcher averages ou he eral ose over me (because s dyamc), ad hece obas a ever-mprovg esmae of he sac ose ad sac ose plus arge. Selecg ex fxao locao. Equao (S8) sll apples he sac ose case. However, pck+, are more complcaed. he formulas for ( ) g ( ) w μ h k + k + k + μ k( + ) p( Ck, ( + ) ) φ( w ) Φ dw g ( )( + k ) k( ) + + k( + ) μ k( + ) W 0.5 k ceα βk + + e αv + (S4) (S5) μ k( + ) W k ceα β k, + e αv + (S6) k( + ) k( + ) + eαv + β, + e αv (S7) pror h ( + ) l + ( g ( + ) W g ( + k k k ) Wk ) pror (S8) I sum, equaos (S) - (S8), ad he measured vsbly map (equao S3) ca be used o smulae opmal vsual search he sac ose case. Alhough hese formulas are more complcaed ha for he dyamc ose case, hey compue a almos he same speed. 8

9 Lookg o he fuure. he sub-opmal searcher ha always fxaes he locao wh maxmum poseror probably (he MAP searcher) does o cosder poeal formao ha wll be gaed afer makg eye movemes, ad hece does o look o he fuure. he deal searcher descrbed here cosders he probably of localzg he arge afer he ex eye moveme, ad hece looks oe fxao o he fuure. Because he deal searcher looks oly oe fxao o he fuure s a so-called greedy algorhm. We have derved formulas for deal searchers ha look more fxaos o he fuure. hese algorhms are o praccal because of he combaoral loso of fxao locaos o cosder; however, we have compared (for a sgle search codo) he prese deal searcher wh oe ha s able o look wo fxaos o he fuure. hs searcher performs slghly beer (approxmaely a quarer of a fxao faser) ha he searcher ha looks ahead oe fxao, whch ur performs beer (approxmaely oe fxao faser) ha he MAP searcher ha looks ahead o fxaos. hese resuls demosrae rapdly dmshg reurs for lookg furher o he fuure, ad hus he prese deal searcher appears o be very close o he rue global opmum.. Peerso, W. W., Brdsall,. G. & Fox, W. C. he heory of sgal deecably. rasacos of he Isue of Rado Egeers, Professoal Group o Iformao heory 4, 7- (954).. Gree, D. M. & Swes, J. A. Sgal Deeco heory ad Psychophyscs (Wley, New York, 966). 9

(1) Cov(, ) E[( E( ))( E( ))]

(1) Cov(, ) E[( E( ))( E( ))] Impac of Auocorrelao o OLS Esmaes ECON 3033/Evas Cosder a smple bvarae me-seres model of he form: y 0 x The four key assumpos abou ε hs model are ) E(ε ) = E[ε x ]=0 ) Var(ε ) =Var(ε x ) = ) Cov(ε, ε )