Hidden Process Modelswith Applications to fMRI Data

Rebecca A. HutchinsonMarch 24, 2010

Biostatistics and Biomathematics SeminarFred Hutchinson Cancer Research Center

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Introduction

• Hidden Process Models (HPMs): – A probabilistic model for time series data.– Designed for data generated by a collection of latent

processes.

• Motivating problem:– Modeling mental processes (e.g. making a decision)

in functional Magnetic Resonance Imaging time series.

• Characteristics of potential domains:– Processes with spatial-temporal signatures.– Uncertainty about temporal location of processes.– High-dimensional, sparse, noisy.

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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 participants, 40 trials per participant.• 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

Keller01

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Motivation

• To track mental 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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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 (overview in Haynes06)

• Does not typically model overlapping hemodynamic responses.

– Dynamic Bayes Networks (Murphy02, Ghahramani97)• HPM assumptions/constraints can be encoded by extending

factorial HMMs with links between the Markov chains.

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Outline• Overview of HPMs

– Generative model– Formalism– Graphical model– Algorithms

• Synthetic data experiments– Accurately estimate parameters– Choose correct model from alternatives with different numbers of processes

• Real data experiments– Evaluation methodology– Extensions to standard HPMs– Model comparison via classification accuracy and data log-likelihood

• Visualizing HPMs• Conclusions

– Summary of contributions– Future work

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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 i1, i2, … ik:

Predicted mean:

Input stimulus :

i1

Timing landmarks : 21

i2

Process instance: i2 Process : 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(0,12)

+ N(0,22)

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HPM FormalismHPM = <,C,C,>

= <1,…,>, a set of processes (e.g. ReadSentence)

= <W,d,,>, a processW = response signature

d = process duration

= allowable offsets

= multinomial parameters over values in

C = <c1,…, cC>, a set of possible configurations

c = <i1,…,iI>, a set of process instances i = <,,O>, a process instance (e.g. ReadSentence(S1))

= process ID = timing landmark (e.g. stimulus presentation of S1)

O = offset (takes values in )

C= a latent variable indicating the correct configuration

= <1,…,V>, standard deviation for each voxel10

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HPMs: the graphical model

Offset o

Process Type

Start Time s

observed

unobserved

Timing Landmark

Yt,v

i1,…,iI

t=[1,T], v=[1,V]

The set C of configurations constrains the joint distribution on {(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:

(i1) = {1,2}(i2) = {1,2}(i1) != (i2)o(i1) = 0o(i2) = 0(i3) = 3o(i3) = {1,2}

Processes:

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ViewPicture = 2

ReadSentence = 1

Decide = 3

ViewPicture = 2

ReadSentence = 1

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Inference• Over C, the latent indicator of the correct

configuration

• Choose the most likely configuration Cn for each trial (n=[1,N]) where:

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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:– E step: estimate the probability distribution over C.– M step: update estimates of W (using reweighted

least squares), , and (using standard MLEs) based on the E step.

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Synthetic Data

Approximate timings for ReadSentence, ViewPicture, Decide

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Synthetic Experiments

MSE of responses (averaged over all voxels, processes, timepoints) = 0.4427.MSE of timing parameters (averaged over all processes, offsets) ~0.01.Estimated standard deviations are 2.4316 and 2.4226 (true value is 2.5).

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Model Selection All numbers *10^5

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Synthetic Data Results

• Tracking processes: HPMs can recover the parameters of the model used to generate the data.

• Comparing models: HPMs can use held-out data log-likelihood to identify the model with the correct number of latent processes.

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Evaluating HPMs on real data

• No ground truth for the problems HPMs were designed for.

• Can use data log-likelihood to compare– Baseline: average of all training trials

• Can do classification of known entities (like the stimuli), but HPMs are not optimized for this.

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Models

• HPM-GNB: ReadSentence and ViewPicture, duration=8sec. (no overlap)– an approximation of Gaussian Naïve Bayes classifier,

with HPM assumptions and noise model

• HPM-2: ReadSentence and ViewPicture, duration=12sec. (temporal overlap)

• HPM-3: HPM-2 + Decide (offsets=[0,7] images following second stimulus)

• HPM-4: HPM-3 + PressButton (offsets = {-1,0} following button press)

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Configurations for HPM-3

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Held-out log-likelihood (improvement over baseline)

• 1000 most active voxels per participant• 5-fold cross-validation per participant; mean over 13

participants. Standard HPMs

HPM-GNB -293

HPM-2 -1150

HPM-3 -2000

HPM-4 -449022

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Extension 1: Regularization

• Subtract a term from the objective function penalizing deviations from:– Temporal smoothness

– Spatial smoothness (based on adjacency matrix A)

– Other possibilities: spatial sparsity and spatial priors.

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Extension 2: Basis Functions

• Re-parameterize process response signatures in terms of a basis set

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The basis set

• Generated as in Hossein-Zadeh03– Create Q (10,000 x 24): 10,000 realizations of 24

timepoints of h(t) varying a in [0.05,0.21] and b in [3,7]

• Basis set = first 3 principal components of Q’Q

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Held-out log-likelihood (improvement over baseline)

• 1000 most active voxels per participant• 5-fold cross-validation per participant; mean over 13

participants.

Standard Regularized Basis functions

HPM-GNB -293 2590 2010

HPM-2 -1150 3910 3740

HPM-3 -2000 4960 4710

HPM-4 -4490 4810 477026

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Classification Accuracies• ReadSentence vs. ViewPicture for first 2 processes• 1000 most active voxels per participant, 5-fold cross-

validation per participant; mean over 13 participants.

Standard Regularized Basis functions

HPM-GNB 85.8 86.5 90.4

HPM-2 87.1 90.0 90.6

HPM-3 86.7 88.7 90.4

HPM-4 83.8 84.4 86.2

GNB = 93.1

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Interpretation and Visualization• Focus in on HPM-3 for a single participant,

trained on all trials, all voxels.

• Timing for the third (Decide) process in HPM-3:

• (Values have been rounded.)

Offset: 0 1 2 3 4 5 6 7

Stand. 0.3 0.08 0.1 0.05 0.05 0.2 0.08 0.15

Reg. 0.3 0.08 0.1 0.05 0.05 0.2 0.08 0.15

Basis 0.5 0.1 0.1 0.08 0.05 0.03 0.05 0.08

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Standard

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Regularized

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Basis functions

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Time courses

Standard

Regularized

Basis functions

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Trial 1: observedRDLPFCZ-slice 5

Trial 1: predictedStandard HPMFull brain, trained on all trials

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ViewPicture

ReadSentence

Decide

Standard HPM

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Trial 1: observedRDLPFCZ-slice 5

Trial 1: predictedRegularized HPMFull brain, trained on all trials

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ViewPicture

ReadSentence

Decide

Regularized HPM

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Trial 1: observedRDLPFCZ-slice 5

Trial 1: predictedBasis function HPMFull brain, trained on all trials

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ViewPicture

ReadSentence

Decide

Basis Function HPM

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Caveats

• While visualizing these parameters can help us understand the model, it is important to remember that they are specific to the design choices of the particular HPM.

• These are parameters – not the results of statistical significance tests.

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Summary of Results

• Synthetic data results – HPMs can recover the parameters of the model

used to generate the data in an ideal situation.– We can use held-out data log-likelihood to

identify the model with the correct number of latent processes.

• Real data results – Standard HPMs can overfit on real fMRI data.– Regularization and HPMs parameterized with

basis functions consistently outperform the baseline in terms of held-out data log-likelihood.

– Example comparison of 4 models.

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Contributions

• Estimates for Decide!

• To our knowledge, HPMs are the first probabilistic model for fMRI data that can estimate the hemodynamic response for overlapping mental processes with unknown onset while simultaneously estimating a distribution over the timing of the processes.

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Future Directions

• Combine regularization and basis functions.• Develop a better noise model.• Relax the linearity assumption.• Automatically discover the number of latent

processes.• Learn process durations.• Continuous offsets.• Leverage DBN algorithms.

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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/.

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.

John-Dylan Haynes and Geraint Rees. Decoding mental states from brain activity in humans. Nature Reviews Neuroscience, 7:523–534, July 2006.

Gholam-Ali Hossein-Zadeh, Babak A. Ardekani, and Hamid Soltanian-Zadeh. A signal subspace approach for modeling the hemodynamic response function in fmri. Magnetic Resonance Imaging, 21:835–843, 2003.

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.

Tim A. Keller, Marcel Adam Just, and V. Andrew Stenger. Reading span and the timecourse of cortical activation in sentence-picture verification. In Annual Convention of the Psychonomic Society, 2001.

Kevin P. Murphy. Dynamic bayesian networks. To appear in Probabilistic Graphical Models, M. Jordan, November 2002.

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Thank you!

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Questions?

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(end of talk)

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ReadSentence Landmarks

ReadSentence Process(=1, = {0})

ViewPicture Landmarks

ViewPicture Process(=2, = {0})

DecideLandmarks

Decide Process(=3, = {0,1,2})

fMRI

I3(1)

S3(1)

S3(2)

I3(2)

I3(3)

S3(3)

Y3

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PressButtonLandmarks

PressButton Process(=4, = {0,1,2})

DecideLandmarks

fMRI

Decide Process(=3, = {0,1,2})