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Origins of Signal Detection Theory. Problem in Psychophysics Thresholds: is sensitivity discrete or continuous? Sensitivity confounded with response bias. Thresholds. Solution: detection theory (engineering ). Signal Detection Theory. Response. Yes. No. Signal.

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origins of signal detection theory
Origins of Signal Detection Theory
  • Problem in Psychophysics
      • Thresholds: is sensitivity discrete or continuous?
      • Sensitivity confounded with response bias
thresholds
Thresholds
  • Solution: detection theory (engineering)
signal detection theory
Signal Detection Theory

Response

Yes

No

Signal

State of the World

Noise

assumptions of signal detection theory
Assumptions of Signal Detection Theory
  • Noise is always present (i.e. in the nervous system and/or in the signal generating system)
  • The noise is normally distributed with σ2 = 1
    • For Gaussian model
  • When a signal is added to the noise, the distribution is shifted upward along the sensory dimension. Variance remains constant (equal variance model).
assumptions of signal detection theory5
Assumptions of Signal Detection Theory
  • Observers are both sensors and decision makers
  • To evaluate the occurrence of an event, observers adopt a decision criterion
  • Sensitivity and Response Bias are independent
    • Statistical
    • Theoretical
    • Empirical
sensitivity
Sensitivity

d’ = zH - zF

d’ Task Person

>3.5 very easy very sensitive

2.6-3.5 moderately easy moderately sensitive

1.6-2.5 moderately difficult moderately insensitive

<1.5 very difficult very insensitive

relation of d to other statistics
Relation of d’ to Other Statistics

If μn=0 and σn=1 (i.e., if the N distribution is unit normal) then the

ROC function, in its most general form, is

response bias
Response Bias

Lenient: 0-1

Unbiased: 1

Conservative: 1-

8

 = f(SN)/f(N)

c = -.5(zH + zF)

Lenient: <0

Unbiased: 0

Conservative: >0

three values of
Three values of 

2

1

3

P(event|x)

N

SN

Sensory magnitude (X)

what is independence
What is Independence?
  • Statistical: P(A|B)=P(A)

PB|A)=P(B)

  • Theoretical/Logical: β can vary independently of d’
  • Empirical: experimental evidence is consistent with the independence assumption (e.g. Form of empirical ROC)
three values of21
Three values of 

2

1

3

P(event|x)

N

SN

Sensory magnitude (X)

alternative sensitivity measures
Alternative Sensitivity Measures

Az: Area under the ROC (e.g., see Swets,1995, ch. 2-3; Swets & Pickett, 1982)

Range: from .5—1.0

Underlying distributions can have unequal variances

Assumes that the underlying distributions can be monotonically transformed to normality

ZH= a + bZF

non parametric measures sensitivity
‘Non-parametric’ Measures: Sensitivity

Not really non-parametric: No distribution assumed, but follows a logistic distribution (Macmillan & Creelman, 1990)

non parametric measures response bias
‘Non-parametric’ Measures: Response Bias

For applications to vigilance, see

See, Warm, Dember, & Howe (1997)

what if the situation is more complex
What if the Situation is More Complex?

Response

State of the World

identification and categorization
Identification and Categorization

1

 5

 2

 6

 4

 3

Response

6

5

4

2

3

1

7

x

fuzzy logic
Fuzzy Logic

Traditional Set Theory: A ∩ A = 0

Fuzzy Set Theory: A ∩ Ā ≠ 0

One assigns non-binary membership, or degrees of

membership, to classes of events (fuzzification).

elements of fuzzy signal detection theory
Elements of Fuzzy Signal Detection Theory
  • Events can belong to the set “signal” (s) to a degree ranging

from 0 to 1

  • Events can belong to the set “response” (r) to a degree ranging

from 0 to 1

computation of fsdt measures
Computation of FSDT Measures
  • Select mapping functions for signal & response dimensions
  • Assignment of degrees of membership to the four outcomes

(H, M, FA, CR) using mixed implication functions.

  • Compute fuzzy Hit, Miss, False Alarm, and Correct Rejection

Rates

  • Compute detection theory measures of sensitivity and response

bias

2 assignment of set membership to categories
2. Assignment of Set Membership to Categories
          • Mixed Implication Functions
  • H = min (s,r)
  • M = max (s-r, 0)
  • FA = max (r-s, 0)
  • CR = min (1-s, 1-r)
3 computation of fuzzy hit and false alarm rate
3. Computation of Fuzzy Hit and False Alarm Rate

H= Σ(Hi)/ Σ(si) for i=1 to N

M = Σ(Mi)/ Σ(si) for i =1 to N

FA = Σ(FAi)/ Σ(1-si) for i=1to N

CR = Σ(CRi)/ Σ(1-si) for i= 1 to N

response time as a function of degree of stimulus criticality
Response Time as a Function of Degree of Stimulus Criticality

1100

1000

900

Response Time (msec)

Transition

800

hh

700

hl

600

lh

500

ll

1

2

3

4

5

6

7

1

0

Stimuli