hidden process models
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Hidden Process Models. Rebecca Hutchinson Tom M. Mitchell Indrayana Rustandi June 28, 2006 ICML Carnegie Mellon University Computer Science Department. Introduction. Hidden Process Models (HPMs): A new probabilistic model for time series data. - PowerPoint PPT PresentationTRANSCRIPT
Hidden Process Models
Rebecca HutchinsonTom M. Mitchell
Indrayana Rustandi
June 28, 2006ICML
Carnegie Mellon University Computer Science Department
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Introduction
• Hidden Process Models (HPMs): – A new probabilistic model for time series data.– Designed for data generated by a collection of latent
processes.
• Potential domains:– Biological processes (e.g. synthesizing a protein) in
gene expression time series.– Human processes (e.g. walking through a room) in
distributed sensor network time series.– Cognitive processes (e.g. making a decision) in
functional Magnetic Resonance Imaging time series.
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fMRI Data
…
Sign
al
Am
plitu
de
Time (seconds)
Hemodynamic Response
Neural activity
Features: 10,000 voxels, imaged every second.Training examples: 10-40 trials (task repetitions).
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Study: Pictures and Sentences
• Task: Decide whether sentence describes picture correctly, indicate with button press.
• 13 normal subjects, 40 trials per subject.• Sentences and pictures describe 3 symbols: *,
+, and $, using ‘above’, ‘below’, ‘not above’, ‘not below’.
• Images are acquired every 0.5 seconds.
Read Sentence
View Picture Read Sentence
View PictureFixation
Press Button
4 sec. 8 sec.t=0
Rest
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Goals for fMRI
• To track cognitive processes over time. – Estimate process hemodynamic responses.– Estimate process timings.
• Allowing processes that do not directly correspond to the stimuli timing is a key contribution of HPMs!
• To compare hypotheses of cognitive behavior.
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HPM Modeling Assumptions
• Model latent time series at process-level. • Process instances share parameters
based on their process types. • Use prior knowledge from experiment
design. • Sum process responses linearly.
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HPM FormalismHPM = <H,C,,>
H = <h1,…,hH>, a set of processes (e.g. ReadSentence)
h = <W,d,,>, a processW = response signature
d = process duration
= allowable offsets
= multinomial parameters over values in
C = <c1,…, cC>, a set of configurations
c = <1,…,L>, a set of process instances = <h,,O>, a process instance (e.g. ReadSentence(S1))
h = process ID = timing landmark (e.g. stimulus presentation of S1)
O = offset (takes values in h)
= <1,…,C>, priors over C
= <1,…,V>, standard deviation for each voxel
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Process 1: ReadSentence Response signature W:
Duration d: 11 sec. Offsets : {0,1} P(): {0,1}
One configuration c of process instances 1, 2, … k: (with prior c)
Predicted mean:
Input stimulus :
1
Timing landmarks : 21
2
Process instance: 2 Process h: 2 Timing landmark: 2
Offset O: 1 (Start time: 2+ O)
sentencepicture
v1v2
Process 2: ViewPicture Response signature W:
Duration d: 11 sec. Offsets : {0,1} P(): {0,1}
v1v2
Processes of the HPM:
v1
v2
+ N(0,1)
+ N(0,2)
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HPMs: the graphical model
Offset o
Process Type h
Start Time s
observed
unobserved
Timing Landmark
Yt,v
1,…,k
t=[1,T], v=[1,V]
The set C of configurations constrains the joint distribution on {h(k),o(k)} k.
Configuration c
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Encoding Experiment Design
Configuration 1:
Input stimulus :
Timing landmarks :
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ViewPicture = 2
ReadSentence = 1
Decide = 3
Configuration 2:
Configuration 3:
Configuration 4:
Constraints Encoded:
h(1) = {1,2}h(2) = {1,2}h(1) != h(2)o(1) = 0o(2) = 0h(3) = 3o(3) = {1,2}
Processes:
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Inference• Over configurations
• Choose the most likely configuration, where:
• C=configuration, Y=observed data, =input stimuli, HPM=model
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Learning
• Parameters to learn:– Response signature W for each process– Timing distribution for each process – Standard deviation for each voxel
• Expectation-Maximization (EM) algorithm to estimate W and .– E step: estimate a probability distribution over
configurations.– M step: update estimates of W (using reweighted
least squares) and (using standard MLEs) based on the E step.
– After convergence, use standard MLEs for
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Learned HPM with 3 processes (S,P,D), and d=13sec.
P PS S
D?
observed
Learned models:
S
P
D
D start time chosen by program as t+18
predicted
P PS S
D D
D?
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ViewPicture in Visual Cortex
Offset = P(Offset)0 0.7251 0.275
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ReadSentence in Visual Cortex
Offset = P(Offset)0 0.6251 0.375
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Decide in Visual CortexOffset = P(Offset)0 0.0751 0.0252 0.0253 0.0254 0.2255 0.625
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Comparing Cognitive Hypotheses
• Use cross-validation to choose a model. – GNB = HPM w/ ViewPicture, ReadSentence w/ d=8s.– HPM-2 = HPM w/ ViewPicture, ReadSentence w/ d=13s.– HPM-3 = HPM-2 + Decide
Accuracy predictingpicture vs. sentence(random = 0.5)
Data log likelihood
Subject: A B C
GNB 0.725 0.750 0.750
HPM-2 0.750 0.875 0.787
HPM-3 0.775 0.875 0.812
GNB -896 -786 -476
HPM-2 -876 -751 -466
HPM-3 -864 -713 -447
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Are we learning the right number of processes?
• Use synthetic data where we know ground truth.– Generate training and test sets with 2/3/4 processes.– Train HPMs with 2/3/4 processes on each.– For each test set, select the HPM with the highest data log
likelihood.
Number of processes in the training and test data
Number of times the correct number of
processes was chosen for the test set
2 5/5
3 5/5
4 4/5
Total: 14/15 = 93.3%
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Related Work
• fMRI– General Linear Model (Dale99)
• Must assume timing of process onset to estimate hemodynamic response.
– Computer models of human cognition (Just99, Anderson04)• Predict fMRI data rather than learning parameters of processes from
the data.
• Machine Learning – Classification of windows of fMRI data (Cox03, Haxby01,
Mitchell04)• Does not typically model overlapping hemodynamic responses.
– Dynamic Bayes Networks (Murphy02, Ghahramani97)• HPM assumptions/constraints are difficult to encode in DBNs.
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Conclusions
• Take-away messages:– HPMs are a probabilistic model for time series data
generated by a collection of latent processes.– In the fMRI domain, HPMs can simultaneously
estimate the hemodynamic response and localize the timing of cognitive processes.
• Future work:– Share parameters across voxels (extending
Niculescu05).– Use parametric hemodynamic responses (e.g.
Boynton96).– Improve algorithm complexities.– Learn process durations automatically.– Apply to open cognitive science problems.
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ReferencesJohn R. Anderson, Daniel Bothell, Michael D. Byrne, Scott Douglass, Christian Lebiere, and Yulin Qin. An integrated theory of the mind. Psychological Review, 111(4):1036–1060, 2004. http://act-r.psy.cmu.edu/about/.
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Anders M. Dale. Optimal experimental design for event-related fMRI. Human Brain Mapping, 8:109–114, 1999.
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James V. Haxby, M. Ida Gobbini, Maura L. Furey, Alumit Ishai, Jennifer L. Schouten, and Pietro Pietrini. Distributed and overlapping representations of faces and objects in ventral temporal cortex. Science, 293:2425–2430, September 2001.
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Tom M. Mitchell et al. Learning to decode cognitive states from brain images. Machine Learning, 57:145–175, 2004.
Kevin P. Murphy. Dynamic bayesian networks. To appear in Probabilistic Graphical Models, M. Jordan, November 2002.
Radu Stefan Niculescu. Exploiting Parameter Domain Knowledge for Learning in Bayesian Networks. PhD thesis, Carnegie Mellon University, July 2005. CMU-CS-05-147.