Hidden Process Models

1
Hidden Process Models

• Rebecca Hutchinson
• Tom M. Mitchell
• Indrayana Rustandi
• October 4, 2006
• Women in Machine Learning Workshop
• 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.

3
fMRI Data
Hemodynamic Response
Features 10,000 voxels, imaged every
second. Training examples 10-40 trials (task
repetitions).
Signal Amplitude
Neural activity
Time (seconds)
4
Study Pictures and Sentences
Press Button
View Picture
Read Sentence
Read Sentence
View Picture
Fixation
Rest
4 sec.
8 sec.
t0

• 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.

5
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.

6
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.

7
HPM Formalism

• HPM ltH,C,F,Sgt
• H lth1,,hHgt, a set of processes (e.g.
  ReadSentence)
• h ltW,d,W,Qgt, a process
• W response signature
• d process duration
• W allowable offsets
• Q multinomial parameters over values in W
• C ltc1,, cCgt, a set of configurations
• c ltp1,,pLgt, a set of process instances
• lth,l,Ogt, a process instance (e.g.
  ReadSentence(S1))
• h process ID
• timing landmark (e.g. stimulus presentation of
  S1)
• O offset (takes values in Wh)
• ltf1,,fCgt, priors over C
• S lts1,,sVgt, standard deviation for each voxel

8
Process 1 ReadSentence Response signature
W Duration d 11 sec. Offsets W 0,1
P(?) q0,q1
Process 2 ViewPicture Response signature
W Duration d 11 sec. Offsets W 0,1
P(?) q0,q1
Processes of the HPM
v1 v2
v1 v2
Input stimulus ?
sentence
picture
Timing landmarks ?
Process instance ?2 Process h 2 Timing
landmark ?2 Offset O 1 (Start time ?2 O)
?1
?2
One configuration c of process instances
?1, ?2, ?k (with prior fc)
?1
?2
?
Predicted mean
N(0,s1)
v1 v2
N(0,s2)
9
HPMs the graphical model
Constraints from experiment design
Timing Landmark l
Process Type h
Offset o
Start Time s
S
p1,,pk
observed
unobserved
Yt,v
t1,T, v1,V
10
Algorithms

• Inference
• over configurations of process instances
• choose most likely configuration with
• Learning
• Parameters to learn
• Response signature W for each process
• Timing distribution Q for each process
• Standard deviation s for each voxel
• Expectation-Maximization (EM) algorithm to
  estimate W and Q.
• After convergence, use standard MLEs for s.

11
ViewPicture in Visual Cortex
Offset q P(Offset) 0 0.725 1 0.275
12
ReadSentence in Visual Cortex
Offset q P(Offset) 0 0.625 1 0.375
13
Decide in Visual Cortex
Offset q P(Offset) 0 0.075 1 0.025 2 0.025 3
0.025 4 0.225 5 0.625
14
Comparing Cognitive Hypotheses

• Use cross-validation to choose a model.
• GNB HPM w/ ViewPicture, ReadSentence w/ d8s.
• HPM-2 HPM w/ ViewPicture, ReadSentence w/
  d13s.
• HPM-3 HPM-2 Decide

Accuracy predicting picture vs. sentence (random
0.5)
Data log likelihood
15
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.

16
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.