Hidden Process Modelsfor Analyzing fMRI Data

Rebecca Hutchinson

Joint work with Tom Mitchell

May 11, 2007Student Seminar Series

In partial fulfillment of the Speaking Requirement

Carnegie Mellon University Computer Science Department

2

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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t

td1

…

dN

…

Process 1:

t

td1

…

dN

…

Process P:

…

d1

…

dN

Prior knowledge:

An instance of Process 1 begins in this window.

An instance of Process P begins in this window.

An instance of either Process 1 OR Process P begins in this window.

There are a total of 6 processes in this window of data.

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t

td1

…

dN

…

Process 1:

t

td1

…

dN

…

Process P:

…

d1

…

dN

Process 1 timings:

Process P timings:

…

More questions:-Can we learn the parameters of these processes from the data (even when we don’t know when they occur)?-Would a different set of processes model the data better?

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Simple Case: Known Timing

• If we know which processes occur when, we can estimate their shapes with the general linear model.

• The timings generate a convolution matrix X:

1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 00 0 1 0 1 0 0 1 00 0 0 0 0 1 0 0 1… … …

p1 p3

t=1t=2t=3t=4…

P

T

p2

6

Simple Case: Known Timing

T

D

=

1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 00 0 1 0 1 0 0 1 00 0 0 0 0 1 0 0 1… … …

p1 p3p2

p1

p3

p2

D

W(1)

W(2)

W(3)

Y

7

Challenge: Unknown Timing

T

D

=

1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 00 0 1 0 1 0 0 1 00 0 0 0 0 1 0 0 1… … …

p1 p3p2

p1

p3

p2

D

W(1)

W(2)

W(3)

Y

Uncertainty about the processes essentially makes the convolution matrix a random variable.

8

Our Approach

• Model of processes contains a probability distribution over when it occurs relative to a known event (called a timing landmark).

• When predicting the underlying processes, use prior knowledge about timing to limit the hypothesis space.

9

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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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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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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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 :

21

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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Uncertain Timings• Convolution matrix models several choices for

each time point.

1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 00 1 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0… … … 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 10 0 0 0 0 1 0 1 00 0 0 0 0 1 0 0 1... … …

P D

t=1t=1t=2t=2…t=18t=18t=18t=18…

T’>T

SConfigurations for each row:

3,41,23,41,2…3412…

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Uncertain Timings

1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 00 1 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0… … …

P D

e1e2e3e4…

S

Y=

W

3,41,23,41,2…

Configurations: Weights:

e1 = P(C=3|Y,Wold,old,old) + P(C=4|Y,Wold,old,old)

• Weight each row with probabilities from E-step.

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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

25

Decide in Visual CortexOffset = P(Offset)0 0.0751 0.0252 0.0253 0.0254 0.2255 0.625

26

ViewPicture

27

ReadSentence

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Decide

0 0.5 1 1.5 2 2.5 3 3.5

0 0 0 0 0.025 0.05 0.075 0.85

Seconds following the second stimulus

Multinomial probabilities on these time points

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Comparing ModelsHPM Avg. Test Set LL

PS -1.0784 * 10^6

PSD -1.0759 * 10^6

PS+S-D -1.0742 * 10^6

PSD+D- -1.0742 * 10^6

PSDB -1.0741 * 10^6

PSDyDn -1.0737 * 10^6

PSDyDnDc** -1.0717 * 10^6

PSDyDnDcB -1.0711 * 10^6

5-fold cross-validation, 1 subject

P = ViewPicture

S = ReadSentence

S+ = ReadAffirmativeSentence

S- = ReadNegatedSentence

D = Decide

D+ = DecideAfterAffirmative

D- = DecideAfterNegated

Dy = DecideYes

Dn = DecideNo

Dc = DecideConfusion

B = Button

** - This HPM can also classify Dy vs. Dn with 92.0% accuracy. GNBC gets 53.9%. (using the window from the second stimulus to the end of the trial)

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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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Future Work

• Incorporate spatial prior knowledge. E.g. share parameters across voxels (extending Niculescu05).

• Smooth hemodynamic responses (e.g. Boynton96).

• Improve algorithm complexities.

• Apply to open cognitive science problems.

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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.

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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/.

Geoffrey M. Boynton, Stephen A. Engel, Gary H. Glover, and David J. Heeger. Linear systems analysis of functional magnetic resonance imaging in human V1. The Journal of Neuroscience, 16(13):4207–4221, 1996.

David D. Cox and Robert L. Savoy. Functional magnetic resonance imaging (fMRI) ”brain reading”: detecting and classifying distributed patterns of fMRI activity in human visual cortex. NeuroImage, 19:261–270, 2003.

Anders M. Dale. Optimal experimental design for event-related fMRI. Human Brain Mapping, 8:109–114, 1999.

Zoubin Ghahramani and Michael I. Jordan. Factorial hidden Markov models. Machine Learning, 29:245–275, 1997.

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.

Marcel Adam Just, Patricia A. Carpenter, and Sashank Varma. Computational modeling of high-level cognition and brain function. Human Brain Mapping, 8:128–136, 1999. http://www.ccbi.cmu.edu/project 10modeling4CAPS.htm.

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.