Hidden Process Models

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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

• 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.)

Signal Amplitude
Time (seconds)
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Lots of features!

• 10,000-15,000 voxels per image

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Study Pictures and Sentences
Press Button
View Picture
Read Sentence
Read Sentence
View Picture
Fixation
Rest
4 sec.
8 sec.
t0

• 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.

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• The star is not below the plus.

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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

• 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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Its not hopeless!

• Learning setting is tough, but we can do it!
• Feature selection is key.
• Learn fMRI(t,t8)-gtPicture,Sentence

Press Button
View Picture
Read Sentence
Read Sentence
View Picture
Fixation
Rest
4 sec.
8 sec.
t0
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Results
Subject
Accuracy
Subject
Accuracy

• 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.

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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 cant we do?
Press Button
View Picture
Read Sentence
Read Sentence
View Picture
Fixation
Rest
4 sec.
8 sec.
t0

• Take into account that the responses for Picture
  and Sentence overlap.
• What does the response for Decide look like and
  when does it start?

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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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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?
t-1
t
t1
t2
CogProc Picture, Sentence, Decide
fMRI

• Cant do overlapping processes states are
  mutually exclusive.
• Markov assumption given statet-1, statet is
  independent of everything before t-1.
• BOLD response Not Markov!

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factorial HMMs?
t-1
t
t1
t2
Picture 0,1
Sentence 0,1
Decide 0,1
fMRI

• 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.

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Hidden Process Models
Name Read sentence Process ID 1 Response
Name View Picture Process ID 2 Response
Name Decide whether consistent Process ID
3 Response
Processes
Process ID 1
Process ID 1
Process Instances
Process ID 2
View picture
Process ID 3
Decide whether consistent
Observed fMRI cortical region 1 cortical
region 2
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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 q1,,qt,,qT
• Set of standard deviations one for each voxel
  s1,,sv,...,sV.

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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 Example
Process 1
Process 2
Process 3
Process responses
ProcessID1, S1
Process instances
ProcessID2, S17
ProcessID3, S21
Predicted data
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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. numTrials3 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

P(yv,thpm,int) N(mv,t,sv)
where
mv,t S Wi.procID(v,t
start(i))
i Î 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(intiY) a P(Yinti)P(inti)
• Choose the interpretation i with the highest
  probability.

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Synthetic Data Example
ProcessID1, S1
Interpretation 1
ProcessID2, S17
ProcessID3, S21
ProcessID2, S1
Interpretation 2
ProcessID1, S17
ProcessID3, S23
Observed data
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(SfMRI).
• M-step
• Use the distribution from the E step to get
  maximum-likelihood estimates of the HPM
  parameters q, W, s.

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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
Read sentence
Cognitive processes
View picture
Decide whether consistent
Observed fMRI cortical region 1 cortical
region 2
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Prior Knowledge
Sentence Presentation
Picture Presentation
Landmarks (Stimuli)
Landmarks go to process instances. Offset
values are determined by process IDs.
Sentence offsets 0,1
Picture offsets 0,1
Read sentence
Decide offsets 0,1,2,3
Cognitive processes
View picture
Decide whether consistent
Observed fMRI cortical region 1 cortical
region 2
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Conditional Independencies
Sentence Presentation
Picture Presentation
Sentence Presentation
Picture Presentation
Landmarks (Stimuli)
Sentence offsets 0,1
Sentence offsets 0,1
Picture offsets 0,1
Picture offsets 0,1
Read sentence
Read sentence
Decide offsets 0,1,2,3
Decide offsets 0,1,2,3
View picture
View picture
Decide whether consistent
Decide whether consistent
HERE
Observed fMRI cortical region 1 cortical
region 2
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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

• 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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Preliminary Results
Press Button
View Picture Or Read Sentence
Read Sentence Or View Picture
Fixation
Rest
4 sec.
8 sec.
t0
16 sec.
picture or sentence?
GNB
picture or sentence?
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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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trial 25
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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

• 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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One idea
Name Riding bus Process ID 1 Response
Name Eating Process ID 2 Response
Name Walking consistent Process ID 3 Response
Processes
ProcessID3
Process instances
ProcessID2
ProcessID1
Observed data
Sensor 1
Sensor 2
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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?