Dynamic neural field model links neural and computational
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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)
…..
Psychophysical Functions
Sekuler & Blake
(1994)
Classical Methods of
Psychophysics
Fechner’s Classical Methods
1.
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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
n1
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)