THEORIE DE LA DETECTION DU SIGNAL

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1 THEORIE DE LA DETECTION DU SIGNAL
La Théorie de la Détection du Signal (TDS) est à la fois un instrument de mesure de la performance et un cadre théorique dans lequel cette performance est interprétée du point de vue des mécanismes sous-jacents. En tant que telle, la TDS peut être regardée comme étant à la base de la psychophysique moderne. Le cours développera les bases de la TDS, discutera sa relation avec les concepts de seuil sensoriel et de décision et exemplifiera son application dans des situations expérimentales allant depuis le sensoriel jusqu'au cognitif.

2 SIGNAL DETECTION THEORY AND PSYCHOPHYSICS
David M. Green & John A. Swets 1st published in 1966 by Wiley & Sons, Inc., New York Reprint with corrections in 1974 Copyright © 1966 by Wiley & Sons, Inc. Revision Copyright © 1988 by D.M. Green & J. A. Swets 1988 reprint edition published by Peninsula Publishing, Los Altos, CA 94023, USA

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5 LE MODEL STANDARD : THEORIE de la DETECTION du SIGNAL
p(N) p(S) p(‘yes’│N) = p(RN>c) STIMULUS RÉPONSE Bruit Signal Oui FA Hit Non CR Omiss. CR Miss p(‘yes’│S) = p(RS>c) PROBABABILITÉ s1 Hits c FA CONTINUUM SENSORIEL R -

6 Score z Ou « score normal » ou « score standard » -4.0 -3.0 -2.0 -1.0
+1.0 +2.0 +3.0 +4.0 Z scores

7 LE MODEL STANDARD : THEORIE de la DETECTION du SIGNAL
CONTINUUM SENSORIEL fR: PROBABABILITÉ FA Hits The likelihood ratio Le rapport de vraisemblance c R

8 LE MODEL STANDARD : THEORIE de la DETECTION du SIGNAL
b = 1 p(S) = . 50 c = 0 -4 -3 -2 -1 1 2 3 4 5 6 Sensory Continuum (z) 0,1 0,2 0,3 0,4 0,5 P(z) N S d’=1 p(S) = .25 c pN(z=c) pS(z=c) c = 1.10 c cA = c + d’/2 = 1.60 = cA -zFA

9 TDS – « diagramme d’état »
p(H) = p2 + u(1 - p2) p(FA) = p1 + u(1 – p1) DOMAINE DU STIMULUS ETAT INTERNE REPONSE B S RB = « non » RS = « oui » Distracteur (Bruit) Cible (Signal) p2 1 - p2 1 - p1 u 1 - u 1 « vrais » H p1: «vrais» FA

10 High Threshold Seuil Haut Détection du Signal Signal Detection
q 1-q Hit u 1-u p(Hit) p(FA) 1-p(FA) 1-p(Hit) q =(H – F) / (1 – F) [correction for guessing] H = p(“Yes”│S) = q + u(1 – q) F = p(“Yes” │N) = u INTENSITÉ SEUIL p Intensité RÉPONSE PERFORMANCE

11 +z -z +z d’ d’ -z +z z(.5) = 0 z(p < .5) < 0 z(p > .5) > 0
z(FA) 1 - p > .5 1 - p > .5 p > .5 1 - p < .5 p < .5 p > .5 1 - p > .5 +z z(H) -z +z z(.5) = 0 z(p < .5) < 0 z(p > .5) > 0 z(1 - p) = -z(p) d’ d’ -z +z d’ = z(H) – [-z(FA)] d’ = z(H) – z(FA)

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14 d’ computation with Excel
d’ = LOI.NORMALE.STANDARD.INVERSE[p(H)] - LOI.NORMALE.STANDARD.INVERSE[p(FA)] c0 = -0.5*{LOI.NORMALE.STANDARD.INVERSE[p(H)] + LOI.NORMALE.STANDARD.INVERSE[p(FA)]} cA = -LOI.NORMALE.STANDARD.INVERSE[p(FA)] b = LOI.NORMALE(cA;d’;1;FAUX)/LOI.NORMALE(cA;0;1;FAUX)

15 Receiving Operating Characteristics (ROC)
d’ d’ d’

16 ROC & le Rating experiment
1. Faire le choix entre Oui et Non; 375 Tot. 285 90 N 157 218 S Non Oui 2. Donner son niveau de certitude de 1 (peu sûr) à (par ex.) 3 (très sûr); l’on calcule les p pour chaque case en divisant sa fréq. par le no. total 1.00 Tot. N, FA S, H * .301 .059 “3” .200 “2” .160 “1” Non .251 .131 .120 .099 .021 Oui -4 -3 -2 -1 1 2 3 4 5 6 Sensory Continuum (z) 0.1 0.2 0.3 0.4 0.5 P(z) N S d’=.91 c .202-(-.706) = .908 d’ zH-zFA -.706 zFA -.5( ) =.252 -.5(zH + zFA) 90/375 = .24 .202 218/375 = .58 pFA zH pH 2. Calculer le pH, pFA; puis d’ & c. *Notez que l’échelle « Oui/Non » ci-dessus est équivalente à une échelle «Oui» avec 6 graduations, du plus sûr («6») au moins sûr («1»): Tot. “1” “2” “3” “4” “5” “6” Oui 3. Pr chaque case, l’on calcule les p cumulés de gauche à droite, les scores z, les d’ et c FA H 1.00 “1” .701 .942 “2” .400 .742 “3” .582 .382 .131 .240 .120 .021 “4” “5” “6” Oui -1.050 -0.198 0.250 0.738 1.579 c 1.046 0.902 0.913 0.874 0.916 d’ zFA zH 0.527 1.573 -0.253 0.649 0.207 -0.301 -1.121 -0.706 -1.175 -2.037

17 ROC & le Rating experiment
Note that a symmetrical ROC indicates that d’ is independent of c0 0.2 0.4 0.6 0.8 1 pFA pH -4 -3 -2 -1 1 2 3 4 5 6 Sensory Continuum (z) 0.1 0.2 0.3 0.4 0,5 P(z) N S d’=.91

18 ROC & le Rating experiment
0.2 0.4 0.6 0.8 1 pFA pH

19 d’ with unequal variances
CONTINUUM SENSORIEL PROBABABILITÉ FA Hits c s1 s2

20 d’ with unequal variances
0.2 0.4 0.6 0.8 1 pH d’ -2 -1 1 2 zH in sN units in sS units d’ with unequal variances 0.2 0.4 0.6 0.8 1 pFA pH d’ ? d’2 < d’a < d’1 -2 -1 1 2 zFA zH in sN units in sS units d’a

21 d’ with unequal variances
-2 -1 1 2 zFA zH in sN units in sS units d’a The index da has the properties we want. It is intermediate in size between d1 and d‘2. It is equivalent to d' when the ROC slope is 1, for then the perpendicular line of length DY/N coincides with the minor diagonal, and the two lines of length da coincide with the d'1, and d'2 segments. It turns out to be equivalent to the difference between the means in units of the root-mean-square (rms) standard deviation, a kind of average equal to the square root of the mean of the squares of the standard deviations of S1 and S2. d’ with unequal variances To find da from an ROC that is linear in z coordinates, it is easiest to first estimate d’1, d’2 and the slope s = d’1/d’2. Because the standard deviation of the S1 distribution is s times as large as that of S2, we can set the standard deviation of S1 to s and that of S2 to 1. To find da directly from one point on the ROC once s is known:

22 d’ with unequal variances
0.2 0.4 0.6 0.8 1 pFA pH d’ d’ with unequal variances Sometimes a measure of performance expressed as a proportion is preferred to one expressed as a distance. The index Az, is such a proportion and is simply DYN transformed by the cumulative normal distribution function , i.e. (DYN): Az is the area under the normal-model ROC curve, which increases from .5 at zero sensitivity to 1.0 for perfect performance. It is basically equal to %Correct in a 2AFC experiment. It is a truly nonparametric index of sensitivity.

23 q =(H – F) / (1 – F) H = (1 – q)F + q q = 0.6 q = 0.3 q = 0 Hit q 1-q
u 1-u 0.2 0.4 0.6 0.8 1 0.0 pFA pH 1.0 q = 0.6 q = 0.3 q = 0 p(Hit) p(FA) 1-p(FA) 1-p(Hit) 0.2 0.4 0.6 0.8 1 0.0 pFA pH 1.0

24 High-Threshold Sum Absolute Difference (SAD) Max Absolute Difference (MAD) Hits & FA HT SAD MAD H FA Schematic of a single trial used for the set size and the target number experiments. In the orientation and spatial frequency experiments, the colored squares were replaced with Gabor patches. ROCs obtenus avec une méthode de « rating » sur une échelle de 1 à 4. Different symbols represent performance at different set sizes: set-size 2 for color, orientation and SF – red stars; set-size 4 for color and SF, set-size 3 for orientation – green triangles; set-size 6 for color and SF, set-size 4 for orientation – blue circles; set-size 8 for color and SF, set-size 5 for orientation – purple squares. Wilken, P. & Ma, W.J (2004). A detection theory account of change detection. J. Vision 4,

25 LA RELATION ENTRE Y/N & 2AFC
S 1 2 A = « 1 » - « 2 »  -S B = « 1 » - « 2 »  S See MacMillan & Creelman (2005, 2nd ed.) pp

26 LA RELATION ENTRE Y/N & 2AFC
d’ d’2 Réponse Correct Incorrect Droite Gauche Stimulus %Correct  Hits %Incorrect  FA d’ 2AFC d’2AFC = 2d’Oui/Non D G Yes/No See MacMillan & Creelman (2005, 2nd ed.) pp

27 Reasons for “invisibility”
So, objects may be “invisible” for only reasons; because the internal activity they yield upon presentation: is undistinguishable from noise (null sensitivity)  d' = 0, 0,1 0,2 0,3 0,4 0,5 -4 -3 -2 -1 1 2 3 4 5 6 P(z) d' = 3 c = 1.5 is below Sj’s criterion  the objet is ignored, or 0,1 0,2 0,3 0,4 0,5 -4 -3 -2 -1 1 2 3 4 5 6 P(z) because Sj’s report on the presence/absence of the stimulus is solicited within a time delay beyond his/her mnemonic capacity (the case of change blindness). 

28 SENSITIVITY  Failure of discriminating Signal from Noise has a triple edge: Stimuli may be out of the standard window of visibility [] (Signal  Internal noise) e.g. ultraviolet light, SF > 40 c/deg, TF > 50 Hz reduced window of visibility (experimental or pathological – e.g. blindsight) Stimuli may be "masked" by external noise [] so that the S/N ratio is decreased by means of: increasing N (external noise impinges on the detecting mechanism  the classical pedestal effect, or increases spatial/temporal uncertainty  e.g. the "mud-splash effect") decreasing S (external noise activates an inhibitory mechanism  lateral masking/metacontrast) [] Change of the transducer [](experimentally or neurologically induced gain-control effects, normalization failure …) 

29 CRITERION The response criterion can be assimilated to response bias (and everything that goes with it, i.e. predisposition, context effects, etc.) 50% 50% can be manipulated in a number of ways: probability pay-off can be affected by experience in general and, possibly, by some neurological conditions blindsight neglect / extinction attentional deficit syndromes What my message today is going to be is that criterion, as a concept and as a tool, has been largely ignored despite the fact that it may be helpful, to say the least, if not enlightening, when used to address questions on bizarre neurological conditions basic sensory processes (e.g. Weber’s Law) attention and, it goes with it, consciousness

30 AN EXAMPLE: CHANGE BLINDNESS
Dinner Money What my message today is going to be is that criterion, as a concept and as a tool, has been largely ignored despite the fact that it may be helpful, to say the least, if not enlightening, when used to address questions on bizarre neurological conditions basic sensory processes (e.g. Weber’s Law) attention and, it goes with it, consciousness Airplane

31 THÉORIE DE LA DÉTECTION DU SIGNAL
CANAUX, BRUIT et THÉORIE DE LA DÉTECTION DU SIGNAL

32 Le principe de l’UNIVARIANCE
« Qualité » (l) Intensité Intensité ou « qualité » du Stimulus Réponse (s/s ou autre)

33 Le principe de l’UNIVARIANCE
Un autre Réponse d’un filtre (s/s ou autre) Intensité Intensité ou « qualité » du Stimulus « Qualité » (l) Un « canal » ou « filtre » ou « champ récepteur » I = stimulus intensity R = response Rmax = max resp. N = max slope S = semi-saturation cst. Rs = spontaneous R

34 Le principe de l’UNIVARIANCE
Un autre R V Réponse d’un filtre Réponse d’un filtre (s/s ou autre) Intensité ou « qualité » du Stimulus « Qualité » (l) Un Canal, Filtre. Chp Récep. I = stimulus intensity R = response Rmax = max resp. N = max slope S = semi-saturation cst. Rs = spontaneous R

35 RESPONSE NORMALIZATION
Normalization works (here) by dividing each output by the sum of all outputs. Divisive inhibition Heeger, D.J. (1992). Normalization of cell responses in cat striate cortex. Visual Neuroscience, 9,

36 CANAUX & PROBABILITÉ DE RÉPONSE Intensity Discrimination
V RÉPONSE p(s/s) REPONSE (e.g. s/s) pmin pMax pmin pMax p(s/s)

37 Within & Across Channels Discrimination
CANAUX & PROBABILITÉ DE RÉPONSE Within & Across Channels Discrimination R V RÉPONSE p(s/s) REPONSE (e.g. s/s) pmin pMax pmin pMax p(s/s)

38 CANAUX & PROBABILITÉ DE RÉPONSE « OFF-LOOKING » CHANNEL
V p(s/s) REPONSE (e.g. s/s) p(s/s) REPONSE (e.g. s/s) R V

39 TRANSDUCTION s = cst. NO Weber ’s Law DI = k k p(R) DII0 DII1 I0 I1 DR
Input R = kI log (I) log (DI) slope = 1 NO Weber ’s Law DI = k k DII1 DR DII0 DR I0 I1

40 TRANSDUCTION s = cst. Weber ’s Law DI = kI p(R) DII1 DII0 I0 I1 DR
log (I) log (DI) slope = 1 Weber ’s Law DI = kI DII1 DR DII0 DR I0 I1

41 TRANSDUCTION Weber ’s Law DI = kI p(R) DII1 DII0 I0 I1 DR slope > 1
log (I) log (DI) slope > 1 Weber ’s Law DI = kI DII0 DR I0 I1

42 SIGNAL DETECTION THEORY
VISUAL SEARCH and SIGNAL DETECTION THEORY

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45 Treisman A.M. & Gelade G. (1980)
Le paradigme type Temps. Réaction No. Éléments ORIENTATION Parallèle (?!) Temps. Réaction No. Éléments Sériel (?!) ORI + COL

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47 LE MODÉLE STANDARD : LA THÉORIE DE LA DÉTECTION DU SIGNAL
« COULEUR » L R V RESPONSE l CONTINUUM SENSORIEL (R-V) PROBAB. s d’ n’est pas mesurable « COULEUR » L R V RESPONSE l CONTINUUM SENSORIEL (R-V) PROBAB. s DR d’=

48 CANAUX & PROBABILITÉ DE RÉPONSE LE PLUS RÉPONDANT GAGNE
V RÉPONSE p(s/s) REPONSE (e.g. s/s) pmin pMax R V p[R(Y/G) largest] pmin x pmax

49 ORIENTATION COULEUR Temps. Réaction Parallèle (?!) No. Éléments
Sériel (?!) Procs. Coopératifs Temps. Réaction No. Éléments COULEUR Parallèle (?!) Temps. Réaction No. Éléments Sériel (?!) Procs. Coopératifs

50 Window of visibility Hz l cpd Hz l cpd LIMITED WINDOW OF VISIBILITY
Distributed FOURIER Transform LIMITED WINDOW OF VISIBILITY Hz cpd l NON-CIRCUMSCRIBED FOURIER SPECTRUM

51 Noise NOISE

52 Noise

53 Masking “CHANNELS” 1 2 1 2 INHIBITORY “MASKING” MASKING “PROPER” NOISE

54 Transducers? STIMULUS INTENSITY INTERNAL RESPONSE or d’ 1 i j

55 A CAT? Identification UN ANIMAL ?