(a) (b) 2 input s(t) input s(t) rs o s x*(t) X X .roya ls x*(t) x*(t) ociety p Y ub lis y(t) lliokge-lihood hing.o ratio rg AND computation R .
(a) (b) 2 input s(t) input s(t) rs o s x*(t) X X .roya ls x*(t) x*(t) ociety p Y ub lis y(t) lliokge-lihood hing.o ratio rg AND computation R .
1.0 3 ] ()Xt*0.5 rs s(), [t 0 os.roya 0 20 40 60 80 100 120 lso c ie ty 1.0 p u b ] ()Yt 0.5 lishin [ 0 g.o rg 0 20 40 60 80 100 120 R .
of transient and persistent signals. Our aim is to decide which hypothesis is more likely to hold. We 4 define thelog-likelihood ratio R: jH rs R¼log PP[[mmeeaassuurreedd ddaattaajH1]] , ð2:3Þ os.ro 0 y a whereP[measureddatajHi]isHtheconditional probabilitythatthemeasureddataaregeneratedbythe lsocie signal specified in hypothesis i. Note that we have chosen to use the log-likelihood ratio, rather than typ the likelihood ratio, because it will enable usto build aconnection with C1-FFL lateron. Intuitively, if ub the log-likelihood ratio R is positive, then the measured data are more likely to have been generated lish H in by a persistent signal or hypothesis , and vice versa. Therefore, the key idea of detection theory is g 1 .o to usethe measureddatato compute the log-likelihood ratio and then useit to makeadecision. rg R .
of transient and persistent signals. Our aim is to decide which hypothesis is more likely to hold. We 4 define thelog-likelihood ratio R: jH rs R¼log PP[[mmeeaassuurreedd ddaattaajH1]] , ð2:3Þ os.ro 0 y a whereP[measureddatajHi]isHtheconditional probabilitythatthemeasureddataaregeneratedbythe lsocie signal specified in hypothesis i. Note that we have chosen to use the log-likelihood ratio, rather than typ the likelihood ratio, because it will enable usto build aconnection with C1-FFL lateron. Intuitively, if ub the log-likelihood ratio R is positive, then the measured data are more likely to have been generated lish H in by a persistent signal or hypothesis , and vice versa. Therefore, the key idea of detection theory is g 1 .o to usethe measureddatato compute the log-likelihood ratio and then useit to makeadecision. rg R .
of transient and persistent signals. Our aim is to decide which hypothesis is more likely to hold. We 4 define thelog-likelihood ratio R: jH rs R¼log PP[[mmeeaassuurreedd ddaattaajH1]] , ð2:3Þ os.ro 0 y a whereP[measureddatajHi]isHtheconditional probabilitythatthemeasureddataaregeneratedbythe lsocie signal specified in hypothesis i. Note that we have chosen to use the log-likelihood ratio, rather than typ the likelihood ratio, because it will enable usto build aconnection with C1-FFL lateron. Intuitively, if ub the log-likelihood ratio R is positive, then the measured data are more likely to have been generated lish H in by a persistent signal or hypothesis , and vice versa. Therefore, the key idea of detection theory is g 1 .o to usethe measureddatato compute the log-likelihood ratio and then useit to makeadecision. rg R .
3 5 2 rs o ()xt* 1 s.royals o c ie 0 typ u 0 2 4 6 8 10 12 blis h in g .o rg + R.
(a) (b) 6 40 12 ()st0 1024680 ()/mean ()xtxt**1230000 xm*(eta)n x*(t) rsos.royalsoc 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 iety p 50 u 12 b ()st1 1024680 ()/mean ()xtxt**123400000 xm*(eta)n x*(t) lishing.org R 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 .
Wefirstusethex*(t)generatedbys0(t),togetherwiththetimeprofilesofc0(t)andc1(t),tocompute 7 thelog-likelihoodratioL(t)bynumericallyintegratingequation(4.3).TheresultingL(t)istheredcurvein figure4d.Similarly,thebluecurveinfigure4dshowstheL(t)correspondingtotheinputs1(t).Wecansee rso dtimisteinicnttebrevhaalvsi[o0u,r1s0i)natnhdettw4o0Li(st)s’isminpltehetotiemxpelianinterbveaclasu[s0e,1d0L)/,[d1t0¼,400)inantdhetse4t0im.Tehientbeerhvaavlsi.ourinthe s.roya ls Wenextfocusonthetimeinterval[10,40).Wefirstconsiders1(t)astheinput.Inthistimeinterval,a ocie larges1(t)meanstheactivationXcontinuestohappen:seethebottomplotoffigure4b.Theactivationof typ Xcontributestoanincrease inL(t)duetothefirsttermontheright-handside(RHS) ofequation(4.3). ub Althoughthesecondtermofequation(4.3)contributestoadecreaseinL(t)via(M2x*(t)),whichisthe lishin numberofinactiveX,thecontributioniscomparativelysmall.Therefore,wesee thatthelog-likelihood g .o ratioL(t),whichisthebluecurveinfigure4d,becomesmorepositive.Sinceapositivelog-likelihoodratio rg means that the input signal is more likely to be similar to the reference signal c (t), this is a correct 1 R detection. In a similar way, we can explain the behaviour of the red curve in figure 4d when s (t) is .
Wefirstusethex*(t)generatedbys0(t),togetherwiththetimeprofilesofc0(t)andc1(t),tocompute 7 thelog-likelihoodratioL(t)bynumericallyintegratingequation(4.3).TheresultingL(t)istheredcurvein figure4d.Similarly,thebluecurveinfigure4dshowstheL(t)correspondingtotheinputs1(t).Wecansee rso dtimisteinicnttebrevhaalvsi[o0u,r1s0i)natnhdettw4o0Li(st)s’isminpltehetotiemxpelianinterbveaclasu[s0e,1d0L)/,[d1t0¼,400)inantdhetse4t0im.Tehientbeerhvaavlsi.ourinthe s.roya ls Wenextfocusonthetimeinterval[10,40).Wefirstconsiders1(t)astheinput.Inthistimeinterval,a ocie larges1(t)meanstheactivationXcontinuestohappen:seethebottomplotoffigure4b.Theactivationof typ Xcontributestoanincrease inL(t)duetothefirsttermontheright-handside(RHS) ofequation(4.3). ub Althoughthesecondtermofequation(4.3)contributestoadecreaseinL(t)via(M2x*(t)),whichisthe lishin numberofinactiveX,thecontributioniscomparativelysmall.Therefore,wesee thatthelog-likelihood g .o ratioL(t),whichisthebluecurveinfigure4d,becomesmorepositive.Sinceapositivelog-likelihoodratio rg means that the input signal is more likely to be similar to the reference signal c (t), this is a correct 1 R detection. In a similar way, we can explain the behaviour of the red curve in figure 4d when s (t) is .