1 hidden process models rebecca hutchinson joint work with tom mitchell and indra rustandi

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Hidden Process Models

Rebecca Hutchinson

Joint work with Tom Mitchell and Indra Rustandi

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Talk Outline

• fMRI (functional Magnetic Resonance Imaging) data

• Prior work on analyzing fMRI data

• HPMs (Hidden Process Models)

• Preliminary results

• HPMs and BodyMedia

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functional MRI

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fMRI Basics

• Safe and non-invasive

• Temporal resolution ~ 1 3D image every second

• Spatial resolution ~ 1 mm– Voxels: 3mm x 3mm x 3-5mm

• Measures the BOLD response: Blood Oxygen Level Dependent– Indirect indicator of neural activity

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The BOLD response

• Ratio of deoxy-hemoglobin to oxy-hemoglobin (different magnetic properties).

• Also called hemodynamic response function (HRF).

• Common working assumption: responses sum linearly.

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More on BOLD response

Sig

nal

Am

plitu

de

Time (seconds)

• At left is a typical BOLD response to a brief stimulation.

• (Here, subject reads a word, decides whether it is a noun or verb, and pushes a button in less than 1 second.)

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Lots of features!• 10,000-15,000 voxels per image

…

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Study: Pictures and Sentences

• 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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The star is not below the plus.

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+

---

*

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fMRI Summary

• High-dimensional time series data.

• Considerable noise on the data.

• Typically small number of examples (trials) compared with features (voxels).

• BOLD responses sum linearly.

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Talk Outline






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It’s not hopeless!

• Learning setting is tough, but we can do it!• Feature selection is key.

• Learn fMRI(t,t+8)->{Picture,Sentence}

Read Sentence



Press Button

4 sec. 8 sec.t=0

Rest

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ResultsA B C D E F G

71.3% 91.2% 76.2% 96.3% 85.0% 66.2% 71.3%

• Gaussian Naïve Bayes Classifier.

• 95% confidence intervals per subject are +/- 10%-15%.

• Accuracy of default classifier is 50%.

• Feature selection: Top 240 most active voxels in brain.

Subject:

Accuracy:

H I J K L M Avg.95.0% 81.2% 90.0% 85.0% 65.0% 90.0% 81.8%

Subject:

Accuracy:

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Why is this interesting?• Cognitive architectures like ACT-R and

4CAPS predict cognitive processes involved in tasks, along with cortical regions associated with the processes.

• Machine learning can contribute to these architectures by linking their predictions to empirical fMRI data.

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Other Successes

• We can distinguish between 12 semantic categories of words (e.g. tools vs. buildings).

• We can train classifiers across multiple subjects.

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What can’t we do?

• Take into account that the responses for Picture and Sentence overlap.

• What does the response for Decide look like and when does it start?

Read Sentence



Press Button

4 sec. 8 sec.t=0

Rest

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Talk Outline






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Motivation

• Overlapping processes– The responses to Picture and Sentence could

overlap in space and/or time.

• Hidden processes– Decide does not directly correspond to the

known stimuli.

• Move to a temporal model.

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Hidden Markov Models?

• Can’t do overlapping processes – states are mutually exclusive.

• Markov assumption: given statet-1, statet is independent of everything before t-1.

• BOLD response: Not Markov!

t-1 t t+1 t+2

CogProc{Picture, Sentence,Decide}

fMRI

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factorial HMMs?

• Have more flexibility than we need.– Picture state sequence should not be {0 1 0 1

0 1 0 1…}

• Still have Markov assumption problem.

t-1 t t+1 t+2

Picture = {0,1}

Sentence = {0,1}

Decide = {0,1}

fMRI

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Hidden Process Models

Process ID = 3

Process ID = 2Process Instances:

Observed fMRI: cortical region 1:

cortical region 2:

Processes:

Name: Read sentenceProcess ID: 1Response:

Name: View PictureProcess ID: 2Response:

Name: Decide whetherconsistentProcess ID: 3Response:

Process ID = 1 Process ID = 1

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HPM Parameters• Set of processes, each of which has:

– a process ID– a maximum response duration R– emission weights for each voxel v [W(v,1),

…,W(v,t),…,W(v,R)] – a multinomial distribution over possible start

times within a trial [1,…,t,…,T]

• Set of standard deviations – one for each voxel 1,…,v,...,V]

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Interpreting data with HPMs

• Data Interpretation (int)– Set of process instances, each of which has:

• a process ID• a start time S

• To predict fMRI data using an HPM and int:– For each active process, add the response

associated with its processID to the prediction.

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Synthetic Data ExampleProcess 1: Process 2: Process 3:

Process responses:

Process instances:

Predicted data

ProcessID=1, S=1

ProcessID=2, S=17

ProcessID=3, S=21

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Our Assumptions

• Processes, not states.– One hidden variable – process start time.

• Known number of processes in the model.– e.g. Picture, Sentence, Decide – 3 processes

• Known number of instantiations of those processes.– e.g. numTrials*3 processes

• Each process has a unique signature.• Contributions of overlapping processes to the

same output variable sum linearly.

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The generative model

• Together HPM and interpretation (int) define a probability distribution over sequences of fMRI images:

where

P(yv,t|hpm,int) = N(v,t,v)

v,t = Wi.procID(v,t – start(i))i active process instances

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Inference

• Given:– An HPM– A set of data interpretations (int) of

processIDs and start times– Priors over the interpretations

• P(int=i|Y) P(Y|int=i)P(int=i)

Choose the interpretation i with the highest probability.

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Synthetic Data ExampleInterpretation 1:

Observed data

ProcessID=1, S=1

ProcessID=2, S=17

ProcessID=3, S=21

Interpretation 2:

ProcessID=2, S=1

ProcessID=1, S=17

ProcessID=3, S=23

Prediction 1

Prediction 2

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Learning the Model

• EM (Expectation-Maximization) algorithm

• E-step– Estimate a conditional distribution over the

start times of the process instances given the observed data, P(S|fMRI).

• M-step– Use the distribution from the E step to get

maximum-likelihood estimates of the HPM parameters {, W, }.

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More on the E-step

• The start times of the process instances are not necessarily conditionally independent given the data.– Must consider joint configurations.– With no constraints, TnInstances configurations. – 2000120 configurations for typical experiment.

• Can we consider a smaller set of start time configurations?

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Reducing complexity

• Prior knowledge– Landmarks

• Events with known timing that “trigger” processes. • One per process instance.

– Offsets• The interval of possible delays from a landmark to

a process instance onset. • One vector of n offsets per process.

• Conditional independencies– Introduced when no process instance could

be active.

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Before Prior Knowledge

Decide whether consistent

Read sentence

View pictureCognitive processes:

Observed fMRI:

cortical region 1:

cortical region 2:

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Read sentence

View pictureCognitive processes:

Observed fMRI:

cortical region 1:

cortical region 2:

Landmarks:(Stimuli)

Sentence Presentation

PicturePresentation

Sentence offsets = {0,1} Picture

offsets = {0,1}Decide offsets = {0,1,2,3}

Landmarks go to process instances.

Offset values aredetermined byprocess IDs.

Prior Knowledge

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Conditional Independencies


Read sentence

View picture

Observed fMRI:

cortical region 1:

cortical region 2:

Landmarks:(Stimuli)


PicturePresentation

Sentence offsets = {0,1} Picture

offsets = {0,1}Decide offsets = {0,1,2,3}

Read sentence

View picture


Sentence offsets = {0,1}

HERE

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More on the M-step

• Weighted least squares procedure – exact, but may become intractable for large

problems– weights are the probabilities computed in the

E-step

• Gradient ascent procedure – approximate, but may be necessary when

exact method is intractable– derivatives of the expected log likelihood of

the data with respect to the parameters

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Talk Outline






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View PictureOr

Read Sentence

Read SentenceOr


Press Button

4 sec. 8 sec.t=0

Rest

picture or sentence? picture or sentence?

16 sec.

GNB:

picture or sentence?

picture or sentence?

HPM:

Preliminary Results

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GNB vs. HPM Classification

• GNB: non-overlapping processes• HPM: simultaneous classification of

multiple overlapping processes

• Average improvement of 15% in classification error using HPM vs GNB

• E.g., for one subject– GNB classification error: 0.14– HPM classification error: 0.09

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Learned models

Comprehend sentence

Comprehend picture

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Model selection experiments

• Model with 2 or 3 cognitive processes?– How would we know ground truth?– Cross validated data likelihood P(testData |

HPM)• Better with 3 processes than 2

– Cross validated classification accuracy• Better with 3 processes than 2

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Current work and challenges

• Add temporal and/or spatial smoothness constraints.

• Feature selection for HPMs.• Process libraries, hierarchies.• Process parameters (e.g. sentence

negated or not).• Model process interactions.• Scaling parameters for response

amplitudes to model habituation effects.

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Talk Outline






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One idea…

Sensor 1:

Sensor 2:

Processes:

Name: Riding busProcess ID: 1Response:

Name: Eating Process ID: 2Response:

Name: WalkingconsistentProcess ID: 3Response:

Process instances:

Observed data:

ProcessID=3

ProcessID=2

ProcessID=1

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Some questions

• What processes are interesting?

• What granularity/duration would processes have?

• What would landmarks be?

• Variable process durations needed?

• Better way to parameterize process signatures?

1 hidden process models rebecca hutchinson joint work with tom mitchell and indra rustandi

Documents

rest slide

bodymedia slide

image slide

functional mri slide

indra rustandi slide

empirical fmri data

sentence overlap

typical bold response