Reminder:!! homework grades and comments are on OAK!! my solutions are on the course web site

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1 PSY318 Week 10

2 Reminder: homework grades and comments are on OAK my solutions are on the course web site

3 Modeling Response Times

4 Nosofsky & Palmeri (1997)

5 Nosofsky & Palmeri (1997)

6 Palmeri (1997)

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8 Response Probabilities A B S1 S2 S Sm

9 Response Probabilities A B A Response Times B S , 432, 675, 434, 754,, 421, , 798, 509, 686, 523,, 602, 776 S , 534, 782, 432, 534,, 873, , 523, 476, 756, 546,, 554, 699 S , 754, 634, 845, 456,, 547, , 873, 888, 934, 687,, 743, Sm , 643, 523, 532, 256, , 633, 623, 587, 677,, 454, 765

10 Response Probabilities A B A Response Times B S , 432, 675, 434, 754,, 421, , 798, 509, 686, 523,, 602, 776 S , 534, 782, 432, 534,, 873, , 523, 476, 756, 546,, 554, 699 S , 754, 634, 845, 456,, 547, , 873, 888, 934, 687,, 743, Sm , 643, 523, 532, 256, , 633, 623, 587, 677,, 454, 765 these can be different stimuli or different conditions

11 Response Probabilities A B A Response Times B S , 432, 675, 434, 754,, 421, , 798, 509, 686, 523,, 602, 776 S , 534, 782, 432, 534,, 873, , 523, 476, 756, 546,, 554, 699 S , 754, 634, 845, 456,, 547, , 873, 888, 934, 687,, 743, Sm , 643, 523, 532, 256, , 633, 623, 587, 677,, 454, 765 how do we summarize the RT data, and what do we actually fit?

12 mean RT RT j = 1 n n i=1 - overall mean RT in a given condition, or for a given stimulus - mean RT conditionalized on the response rt i

13 mean RT RT j = 1 n n i=1 rt i cumulative distribution function (CDF) F j(t) = # elements in sample t n

14 mean RT RT j = 1 n n i=1 rt i cumulative distribution function (CDF) F j(t) = # elements in sample t n probability density function (PDF) histogram kernel density estimation

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16 Histograms vs Kernel Density Estimation

17 pts = 0:.1:20; [pdf_kernel, t_kernel] = ksdensity(data, pts, 'support', 'positive', 'function', pdf'); 'support' 'function' 'positive' 'unbounded' 'pdf' 'cdf'

18 mean RT RT j = 1 n n i=1 rt i cumulative distribution function (CDF) F j(t) = # elements in sample t n probability density function (PDF) histogram kernel density estimation hazard function h( t) = f 1 ( t) F( t) From actuary science: The probability that you will die in the next instant given that you re still alive. h(t) increases as you get older and older. Sigh. probability a process will terminate in the next instant given that it has not terminated yet

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21 mean RT RT j = 1 n n i=1 rt i aren t means enough? cumulative distribution function (CDF) F j(t) = # elements in sample t n probability density function (PDF) histogram kernel density estimation hazard function h( t) = f 1 ( t) F( t)

22 What can Response Time Distributions tell you?

23 What can Response Time Distributions tell you? So why would anyone ever care about a response time distribution?

24 What can Response Time Distributions tell you? Mean response times tell you everything you need to know

25 What can Response Time Distributions tell you? Right?

26 What can Response Time Distributions tell you? Imagine an experiment with three conditions A, B, and C Mean RTs are µ A < µ B < µ C

27 What can Response Time Distributions tell you? Imagine an experiment with three conditions A, B, and C Mean RTs are P(RT<t) 1 0 µ A < µ B < µ C A B C Different qualitative patterns of response time variability can give rise to The same ordering of mean RTs. These different possibilities could imply different mechanisms. P(RT<t) 1 0 A B C

28 1 A B C P(RT<t) 0

29 1 A B C P(RT<t) 0 A animal B bird C sparrow

30 1 A B C P(RT<t) 0 A animal perception Yes B bird C sparrow

31 1 A B C P(RT<t) 0 A animal perception Yes B bird perception Yes C sparrow perception Yes

32 1 A B C P(RT<t) 0 A animal perception Yes B bird perception Yes C sparrow perception Yes

33 1 A B C P(RT<t) 0 A animal B bird C sparrow

34 1 A B C P(RT<t) 0 A animal perception Yes B bird C sparrow

35 1 A B C P(RT<t) 0 A animal perception Yes B bird perception Yes C sparrow perception Yes

36 1 A B C P(RT<t) 0 A animal perception Yes B bird perception Yes C sparrow perception Yes

37 What can Response Time Distributions tell you? Wow. That s actually the kind of differences we d predict. I need to go look at my data again.

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39 Exemplar- Based Random Walk (EBRW) Model Nosofsky & Palmeri 1997, Palmeri 1997

40 A2 B1 A1 B2 A3 B3 From the Generalized Context Model (Nosofsky, 1986) - categories are represented in terms of stored exemplars - exemplars are represented as points in MDS space - similarity is an exponentially decreasing function of distance

41 A2 B1 A1 B2 A3 B3 From the Generalized Context Model (Nosofsky, 1986) - categories are represented in terms of stored exemplars - exemplars are represented as points in MDS space - similarity is an exponentially decreasing function of distance From Instance Theory (Logan, 1988) - each experience is stored as a new instance (exemplar) - exemplars race to be retrieved

42 Race

43 Race

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45 Race

46 Race

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48 Race Logan (1988) I1 I2 I3 Rule

49 Race Logan (1988) I1 I2 I3 Rule early in learning (non- automatic)

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51 Race Logan (1988) I1 I2 I3 I4 I5 I6 I7 Rule

52 Race Logan (1988) I1 I2 I3 I4 I5 I6 I7 Rule later in learning (automatic)

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54 Race Nosofsky & Palmeri (1997) A1 A2 A3 A4 B1 B2 B3 B4

55 Race Nosofsky & Palmeri (1997) A1 A2 A3 A4 B1 B2 B3 B4

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57 Race Nosofsky & Palmeri (1997) A1 A2 A3 A4 B1 B2 B3 B4

58 Race Nosofsky & Palmeri (1997) A1 A2 A3 A4 B1 B2 B3 B4

59 EBRW Nosofsky & Palmeri (1997) d ij = m r i j m m s ij = exp( c d ij p ) f ij (t) = s ij exp( s ij t) 1/r similarity is an exponentially decreasing function of distance retrieval time for exemplar j given probe item i is exponentially distributed with rate sij

60 EBRW Nosofsky & Palmeri (1997) d ij = m r i j m m s ij = exp( c d ij p ) f ij (t) = s ij exp( s ij t) 1/r similarity is an exponentially decreasing function of distance retrieval time for exemplar j given probe item i is exponentially distributed with rate sij

61 Properties of Races of Exponentials Suppose there are n independent runners with exponentially distributed finishing times with rates λ1, λ2, λ3,, λn What is the probability that runner j wins?

62 Properties of Races of Exponentials Suppose there are n independent runners with exponentially distributed finishing times with rates λ1, λ2, λ3,, λn What is the probability that runner j wins? P( j wins) = λ j λ k k Luce s choice model falls out of assumption of racing exponentials

63 Properties of Races of Exponentials P( j wins) = λ j λ k k P(A runner wins) = j Arunners k λ j λ k P(A runner wins) = P(A runner wins) = j Arunners λ k k λ j λ j j Arunners λ j j Arunners j Brunners + λ j

64 Properties of Races of Exponentials P( j wins) = λ j λ k k P(A runner wins) = j Arunners k λ j λ k P(A runner wins) = P(A runner wins) = j Arunners λ k k λ j λ j j Arunners λ j j Arunners j Brunners + λ j

65 Properties of Races of Exponentials P( j wins) = λ j λ k k P(A runner wins) = j Arunners k λ j λ k P(A runner wins) = P(A runner wins) = j Arunners λ k k λ j λ j j Arunners λ j j Arunners j Brunners + λ j

66 Properties of Races of Exponentials P( j wins) = λ j λ k k P(A runner wins) = j Arunners k λ j λ k P(A runner wins) = j Arunners λ k k λ j P(A S i ) = s ij j A s ij + s ij j A j B this is the GCM

67 Properties of Races of Exponentials Suppose there are n independent runners with exponentially distributed finishing times with rates λ1, λ2, λ3,, λn What is the probability that runner j wins? P( j wins) = λ j λ k k What is the average time of the winner?

68 Properties of Races of Exponentials Suppose there are n independent runners with exponentially distributed finishing times with rates λ1, λ2, λ3,, λn What is the probability that runner j wins? P( j wins) = λ j λ k k What is the average time of the winner? E[t] = 1 λ k k the more runners there are, the faster the winning time speed up with experience

69 Logan s (1988) Instance Theory stimulus perceptual processing retrieval race response time - assumes the retrieval times are distributed as Weibulls, generalization of an exponential distribution - only runners in the race are exact matches to the stimulus - predicts response times, speedups in response times - cannot predict accuracy, changes in accuracy with learning

70 Nosofsky and Palmeri (1997) EBRW stimulus perceptual processing retrieval race response time - assumes the retrieval times are distributed as exponentials - runners race according to their similarity* - can predict accuracy and response times - the winner of each race provides incremental evidence that drives a random walk decision process * instance theory assumes that all mismatches have similarity 0

71 Nosofsky and Palmeri (1997) EBRW stimulus perceptual processing response time - assumes the retrieval times are distributed as exponentials - runners race according to their similarity - can predict accuracy and response times - the winner of each race provides incremental evidence that drives a random walk decision process

72 Nosofsky and Palmeri (1997) EBRW stimulus perceptual processing response TR time

73 Nosofsky and Palmeri (1997) EBRW time

74 Nosofsky and Palmeri (1997) EBRW A B time

75 Nosofsky and Palmeri (1997) EBRW A probe exemplar memory with test object i B time

76 Nosofsky and Palmeri (1997) EBRW A exemplars race to be retrieved with rates given by their similarity sij B time

77 Nosofsky and Palmeri (1997) EBRW A Δt Δx imagine exemplar j wins and j is a member of category A B time

78 Nosofsky and Palmeri (1997) EBRW A Δt Δx imagine exemplar j wins and j is a member of category A B Δx = +1 since category A time

79 Nosofsky and Palmeri (1997) EBRW A Δx Δt = α + t w imagine exemplar j wins and j is a member of category A B Δx = +1 since category A Δt = step time t w = retrieval time of winner α = minimum step time

80 Nosofsky and Palmeri (1997) EBRW A B time

81 Nosofsky and Palmeri (1997) EBRW A B time

82 Nosofsky and Palmeri (1997) EBRW A B time

83 Nosofsky and Palmeri (1997) EBRW A B time

84 Nosofsky and Palmeri (1997) EBRW A B time

85 Nosofsky and Palmeri (1997) EBRW A B time

86 Nosofsky and Palmeri (1997) EBRW A B time

87 Nosofsky and Palmeri (1997) EBRW A B time

88 Nosofsky and Palmeri (1997) EBRW A B time

89 Nosofsky and Palmeri (1997) EBRW A A B time

90 Nosofsky and Palmeri (1997) EBRW

91 Easy Stimulus Hard Stimulus perceptual processing perceptual processing TR TR

92 Nosofsky & Palmeri (1997)

93 Early in Learning Later in Learning perceptual processing perceptual processing TR TR

94 Palmeri (1997)

95 perceptual processing race between rules and exemplars TR rule-based algorithmic mechanism following Logan (1988)

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97 Probabilistic versus Deterministic Response Rules timulus perceptual processing category knowledge decision response

98 Probabilistic versus Deterministic Response Rules timulus perceptual processing category knowledge decision response objects are represented as points in multidimensional psychological space

99 Probabilistic versus Deterministic Response Rules timulus perceptual processing category knowledge decision response EA = evidence object in Category A EB = evidence object in Category B

100 Probabilistic versus Deterministic Response Rules timulus perceptual processing category knowledge decision response E A i P(A i) = E A i + E B i probabilistic response rule

101 Probabilistic versus Deterministic Response Rules timulus perceptual processing category knowledge decision response E A i > E B i respond A if else respond B

102 Probabilistic versus Deterministic Response Rules timulus perceptual processing category knowledge decision response respond A if E A i E B i > 0 else respond B

103 Probabilistic versus Deterministic Response Rules timulus perceptual processing category knowledge decision response respond A if E A i E B i > c else respond B deterministic response rule

104 imagine comparing two models model 1 timulus perceptual processing rule-based knowledge deterministic response rule response model 2 stimulus perceptual processing exemplarbased knowledge probabilistic response rule response

105 model 1 if testing category knowledge, equate decision rules timulus perceptual processing rule-based knowledge deterministic response rule response model 2a stimulus perceptual processing exemplarbased knowledge deterministic response rule response

106 Probabilistic versus Deterministic Response Rules timulus perceptual processing category knowledge decision response respond A if E A i E B i > c + noise else respond B deterministic response rule

107 Ashby, F. G., & Maddox, W. T. (1993). Relations between prototype, exemplar, and decision bound models of categorization. Journal of Mathematical Psychology, 37, pdf

108 Properties of Races of Exponentials P( j wins) = λ j λ k k P(A runner wins) = j Arunners k λ j λ k P(A runner wins) = j Arunners λ k k λ j P(A S i ) = s ij j A s ij + s ij j A j B this is the GCM

109 What if racing exponentials feed a random walk (like EBRW)?

110 What if racing exponentials feed a random walk (like EBRW)? P(A S i ) = j A s ij j A K s ij K + s ij j B K EBRW predicts more deterministic decision rules

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112 Diffusion Model Ratcliff (1979), Ratcliff & Rouder (1998)

113 diffusion is a random walk in the limit as Δt approaches zero time

114 evidence time

115 decision A boundary evidence decision B boundary time

116 Yes boundary evidence No boundary time

117 decision A boundary decision time evidence decision B boundary time

118 decision A boundary evidence drift rate diffusion coefficient (noise) starting point decision B boundary time

119 move boundaries in to stress speed over accuracy (homunculus) time

120 move the starting point to bias one response over another response (homunculus) time

121 drift rate diffusion coefficient (noise) determined by stimuli and knowledge time

122 decision time RT = TR + decision time time

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125 Diffusion model: - naturally predicts shapes of observed RT distributions - naturally predicts speed- accuracy tradeoffs - parameters are identifiable - experimental manipulations map onto expected parameters

126 Diffusion model: - naturally predicts shapes of observed RT distributions - naturally predicts speed- accuracy tradeoffs - parameters are identifiable - experimental manipulations map onto expected parameters - there is no theory of drift rates - there is no theory of what happens during TR - there is no theory of how starting point and bounds change but see: Purcell, B.A., Schall, J.D., Logan, G.D., & Palmeri, T.J. (2012). Gated stochastic accumulator model of visual search decisions in FEF. Journal of Neuroscience, 32(10), [PDF] Mack, M.L., & Palmeri, T.J. (2010). Modeling categorization of scenes containing consistent versus inconsistent objects. Journal of Vision, 10(3):11, [PDF] Nosofsky, R.M., & Palmeri, T.J. (2014). Exemplar-based random walk model. To appear in J.R. Busemeyer, J. Townsend, Z.J. Wang, & A. Eidels (Eds.), Mathematical and Computational Models of Cognition, Oxford University Press. [PDF]

127 a evidence z dt dx 0 time evidence = z; while (evidence<a & evidence>0) time = time + dt; r = rand; if r < f(mu,sigma) evidence = evidence + dx; else evidence = evidence dx; end end

128 dx = σ dt dt dx p = 1 " $ 1+ µ 2 # σ dt % ' & q = 1 " $ 1 µ 2 # σ dt % ' &

129 dx = σ dt dt dx p = 1 " $ 1+ µ 2 # σ dt % ' = & 2 2 µ dt σ q = 1 " $ 1 µ 2 # σ dt % ' = 1 1 & 2 2 µ dt σ μ = drift rate σ = noise so μ/σ is a signal-to-noise ratio probability of moving up dx with probability p, or moving down dx with probability q=1-p, is a simple function of the signal-to-noise ratio scaled by the time increment

130 function [time,which]=diffusion_simulation(mu,s2,tr,a,z) time=tr; % start the time at TR tau=.0001; % time per step of the diffusion (more accurate with tau= ) evidence=z; % starting point while(evidence<a && evidence>0) time = time + tau; dx=sqrt(s2.*tau); r=rand; p=0.5.*(1 + mu.*dx./s2); if r < p evidence = evidence + dx; else evidence = evidence - dx; end end if evidence < 0 which = 0; end if evidence > a which = 1; end

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132 a perceptual processing TR z 0 drift core free parameters

133 a perceptual processing TR z 0 drift trial- by- trial variability in TR, z, and drift

134 Homework Assignment explore predictions of diffusion model explore role of trial- to- trial variability

135 How do we simulate trial- to- trial variability? function([time,which]=diffusion_simulation(mu,s2,tr,a,z)( ( time=tr;((((((((%(start(the(time(at(tr( tau=.0001;((((((%(time(per(step(of(the(diffusion((more(accurate(with(tau= )( evidence=z;(((((%(starting(point( ( while(evidence<a(&&(evidence>0)( ((((( ((((time(=(time(+(tau;( ( ((((dx=sqrt(s2.*tau);( ( ((((r=rand;( ( ((((p=0.5.*(1(+(mu.*dx./s2);( ((((( ((((if(r(<(p( ((((((((evidence(=(evidence(+(dx;( ((((else( ((((((((evidence(=(evidence(n(dx;( ((((end( end( ( if(evidence(<(0( ((((which(=(0;( end( ( if(evidence(>(a( ((((which(=(1;( end(

136 How do we simulate trial- to- trial variability? TR is uniform distribution z is uniform distribution mu is normal distribution why? function([time,which]=diffusion_simulation(mu,s2,tr,a,z)( ( time=tr;((((((((%(start(the(time(at(tr( tau=.0001;((((((%(time(per(step(of(the(diffusion((more(accurate(with(tau= )( evidence=z;(((((%(starting(point( ( while(evidence<a(&&(evidence>0)( ((((( ((((time(=(time(+(tau;( ( ((((dx=sqrt(s2.*tau);( ( ((((r=rand;( ( ((((p=0.5.*(1(+(mu.*dx./s2);( ((((( ((((if(r(<(p( ((((((((evidence(=(evidence(+(dx;( ((((else( ((((((((evidence(=(evidence(n(dx;( ((((end( end( ( if(evidence(<(0( ((((which(=(0;( end( ( if(evidence(>(a( ((((which(=(1;( end(

137