Transcript Dynamic neural field model links neural and computational

Lecture 8
Detection and Discrimination
Experiments
Martin Giese
What you should learn today
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Perceptual threshold
Detection and discrimination experiments
Psychometric function (PMF)
Classical methods of psychophysics
Models for thresholds
Signal Detection Theory and ROC
Scaling methods
A Detection Experiment
Indicate when you see the stimulus
on the gray background !
Detection Experiment: Results
• Stimulus seen only beyond certain contrast
level
• Different people started to see the stimulus
at different times
• The same subjects see the same stimulus
sometimes and sometimes not
• The number of people who see the stimulus
increases with contrast.
Detection Experiment: Interpretation
• non-trivial relationship:
physical stimulus – percept
• Probabilistic relationship:
P(stimulus seen) = f(contrast)
• Threshold contrast:
P=0 if contrast smaller
Detection and Discrimination
Experiments
Psychophysics
Gustav Fechner (1860):
Elemente der Psychophysik
(Elements of Psychophysics)
Aim: New science that investigates relationship
physical stimulus – sensation
Psychophysics
“Psychophysical function”
P(stimulus seen) = f(contrast)

Determined form measurements!
Possible Measurements
1. Detection experiment:
Absolute threshold: When do subjects start
to perceive the stimulus ?
 Limits of the perceptual system
2. Discrimination experiment:
Difference threshold (just noticeable
difference, JND): how sensitive are subjects
for changes in the stimulus
 Sensitivity to changes
Psychometric Function
(for Detection)
Theshold
Sekuler & Blake (1994)
Psychometric Function:
Quantification
1. 50 % threshold:
S50: S for which P = 0.5
2. Steepness: Steepness of the function at S50
3. Curve fitting:
P(S) = F(S, a)
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Free parameters a
fit a => a*
solve for S50 and
steepness using
F(S50, a*) = 0.5
Psychometric Function:
Quantification
Example functions: P(S) = F(S, a)
Logistic function:
ea 2 ( S a12 )
P( S ,a ) =
1  ea 2 ( S a1 )
Normal function (Gaussian error function):
1
P( S , α ) =
2 a 2
S
e
( v a1 ) 2 /(2a 2 2 )
dv

Weibull function:
S
a1 a1 1 ( x / a 2 )a1
P(S , α) =  a1 a 2 x

e
dx
Difference Thresholds
(for Discrimination)
2 stimuli:
• Reference
• Test
Reference
Presentation:
• At the same time
• Sequential
• Reference can be omitted
(mean as reference)
Test
Did stimulus change ?
Difference Thresholds: Quantification
Just noticeable difference (JND):
minimal physical change of the
stimulus such that change in sensation
is reported (e.g. P75 – P25)
Point of subjective equivalence (PSE):
physical strength of the stimulus that is
perceived as equally strong as reference
Difference Thresholds: Quantification
Difference thresholds depend on the
reference value !
Stimulus strength
Snodgrass et al. (1985)
Weber’s Law
E.H. Weber (~1850):
Experiments on lifting weights
Reference
==
Weber’s law:
Test
?
JND
= const = k
Reference
Fechner’s Law
Fechner (1860):
Aim: finding psychophysical function that maps
stimulus strength S onto “sensation strength” S
Assumption: Change in sensation S = k ' S ~ JND
S
dS
1
S  dS 
= k' 
dS
S
Fechner’s Law
Fechner’s law:
S = k ' log(S )
S
S
Psychophysical Functions
Variation of threshold with parameter
(“family of thresholds”)
Examples:
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Contrast threshold dependent on spatial
frequency
Audibility function depends on temporal
frequency (next lecture)
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Psychophysical Functions
Sekuler & Blake
(1994)
Classical Methods of
Psychophysics
Fechner’s Classical Methods
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Method of constant stimuli:
fixed stimulus intensities
presentation in random order
Ask subject if it perceives the stimulus
plotting of P(S) for several discrete levels
 Subject does not know order of presentation
 Many trials
Fechner’s Classical Methods
2. Method of limits:
• stimulus intensity increased (or decreased)
until subjects starts (or ceases) to see the
stimulus
• threshold intensity recorded
 Few trials sufficient
 Subject knows direction of intensity change
=> systematic errors
Fechner’s Classical Methods
Error of expectancy:
tendency to change response expecting a
stimulus change
Error of habituation:
tendency to persist with previous response
Fechner’s Classical Methods
3. Method of Adjustment:
subject adjusts the stimulus intensity so that
the stimulus is barely perceived
 Very fast
 No information about steepness of PMF
Staircase Method (Cornsweet, 1962)
Sekuler & Blake
(1994)
 Much data near threshold
 Subject may infer rule of presentation
Staircase Method
Interleaved staircases
Sekuler & Blake
(1994)
Forced Choice Methods
Subjective methods: no objective criterion if
subject was right (“seen or not seen”)
Objective methods: (Bregmann, 1852)
Alternative forced choice (AFC):
2 trial types: a) stimulus present
b) stimulus not present
 choice objectively right or false
Detection can be verified ! (Ask for the
grating orientation)
Forced Choice Methods
Threshold
Chance
probability probability
For equal probability
of all alternatives:
2
AFC
50 %
75 %
N
AFC
100 / N
%
(N+1) / N
* 50 %
Forced Choice Methods
Possible response alternatives:
a) 2 AFC (two alternative forced choice)
i. Stimulus present
Choice enforced!
ii. Stimulus not present
b) 3 RC (three response categories)
i. Stimulus present
ii. Stimulus not present
iii. No decision
No choice
enforced!
Forced Choice Methods
Thresholds for 2AFC
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Subject is forced
to decide
Often perception
far below classical
threshold
< Thresholds for classical
PMF and 3RC
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Tendency to persist
with “don’t know”
Dependence on
subject-specific
criterion
Influence of the Response
Criterion
Classical
PMF
Respose
strategy:
P
3RC
P
no ? yes
“risky”
S
P
S
P
no
? yes
“anxious”
S
S
FIVE
Theories about Thresholds
Models for Thresholds
1. Classical (high threshold) theory
Judgement
(binary)
Threshold
Fixed !
Intrinsic response
(random)
Noise
Never detection
without stimulus!
Stimulus
(deterministic)
Models for Thresholds
2. Signal (Sensory) Detection Theory (SDT)
(Green & Swets, 1966)
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No fixed threshold !
Assumption: Stimulus
detected in neural noise
Internal random neural state x
“Response criterion C
(adjusted “dynamically”)
Response criterion C
x
Signal Detection Theory
Conditional responses and probabilities:
Response
“Yes”
“No”
Stimulus
present
Stimulus
Not present
Hit
False
alarm
Miss
Correct
rejection
Signal Detection Theory
Interesting theoretical quantities:
a) Sensitivity:
ms  mn
d'=
sn
fn(x)
b) Bias:
fs(x)
f s (C )
=
f n (C )
sn
mn
ms
x
(likelihood ratio)
Models for Thresholds
Subject “model” of SDT:
Judgement
“yes”
Noise
Intrinsic neural
response X (random)
fs ( X )
>
fn ( X )
?
Stimulus
(deterministic)
Judgement
“no”
Likelihood
Ratio Test
Receiver Operating Characteristics
(ROC)
detectability
Receiver Operating Characteristics
(ROC)
What we need:
Subject says “yes” for x >c.
fs
fn
Hits:

Phit (c) = P( X > c | s) =  f s ( x) dx
c
fs
fn
False alarms:
x
c

Pfa (c) = P( X > c | n) =  f n ( x) dx
c
c
x
Receiver Operating Characteristics
(ROC)
No dectability:
fs
fn
c1
c2 c3 c4
c3
fs
fn
c5
c1
Phit
c5
Ideal detection:
c1
c2
c3
c3
c4
c2
c1
Phit
c2
c4
c4
c5
Pfa
Pfa
c5
Receiver Operating Characteristics
(ROC)
No dectability:
Phit
Ideal detection:
Phit
Pfa
Pfa
Area under ROC is a measure for detectability !
Scaling Methods
Magnitude Estimation
Direct scaling technique (Stevens, 1960)
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Reference stimulus with strength defined as one
Subject says how many times stronger / weaker the
test stimulus is
Similar methods
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Cross-modality matching
Magnitude production (adjustment)
Rating scales
1 2 3
x
4 5
6
Power Laws
Sensation strength varies often with
physical stimulus strength in form of a
power law:
S = k Sn
0<n
Power Laws
n>1
n1
n<1
Power Laws
Linear relationhip:
log S = log k + n log S
Fit by linear regression (Lecture 21).
Power Laws
Typical exponents:
Multi-dimensional Scaling
Idea:
Order data within and N-dimensional
space so that distances are minimally
distorted.
• Distance ratings between data points
• Minimization of a measure for
deviation between distances in Ndimensional space and original
distances
Multi-dimensional Scaling
1D example:
2D example:
Sekuler & Blake
(1994)
Literature
Suggested readings:
Snodgrass, J.G., Levy-Berger, G., Hayden, M. (1985). Human
Experimental Psychology. Oxford University Press, Oxford,
UK. Chapter 4.
Elmes, D.G., Kantowitz, B.H., Roediger III, H.L. (1999).
Research Methods in Psychology. Brooks/Cole Publishing,
Pacific Grove. Chapter 8.
Additional Literature:
Sekuler, R., Blake, R. (1994). Perception. McGraw-Hill, New
York. Appendix.
Signal Detection Theory
Example for Gaussian distributions:
Random internal neural activation: X

1
e
Signal present: f s ( x) = f ( x | s) =
2 s
Noise only:
1
f n ( x) = f ( x | s ) =
e
2 s

( y ms )2
2s 2
( y mn )2
Subject says “yes” if X > c (criterion).
2s 2
Signal Detection Theory


1
e
Signal present: Phit (c) = P( X > c | s) =

2 s c

Noise only:
1
Pfa (c) = P( X > c | n) =
e

2 s c

with Gaussian error function:
x ) =
1
2
x
e


y2
2
dy
( y ms )2
2s 2
( y mn )2
2s 2
 m c
dy =  s

 s 
 mn  c 
dy = 

 s 
Signal Detection Theory
Using the inverse function 1 follows:
ms  c
z hit =  Phit (c) ) =
s
m c
zfa =  1 Pfa (c) ) = n
s
1
Elimination of c:
ms  mn
z hit = zfa 
= zfa  d '
s
Linear function
in d’
 Fit this function by linear regression !
(Lecture 21)