Hidden Process Models

1. Hidden Process Models Rebecca Hutchinson Tom M. Mitchell Indrayana Rustandi June 28, 2006 ICML 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 • 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 Fixation View Picture Rest t=0 4 sec. 8 sec.

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 = <H,C,F,S> H = <h1,…,hH>, a set of processes (e.g. ReadSentence) h = <W,d,W,Q>, a process W = response signature d = process duration W = allowable offsets Q = multinomial parameters over values in W C = <c1,…, cC>, a set of configurations c = <p1,…,pL>, a set of process instances • = <h,l,O>, a process instance (e.g. ReadSentence(S1)) h = process ID • = timing landmark (e.g. stimulus presentation of S1) O = offset (takes values in Wh) • = <f1,…,fC>, priors over C S = <s1,…,sV>, 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 Configuration c Timing Landmark l The set C of configurations constrains the joint distribution on {h(k),o(k)} " k. Process Type h Offset o Start Time s S p1,…,pk observed unobserved Yt,v t=[1,T], v=[1,V]

10. Encoding Experiment Design Processes: Input stimulus : Constraints Encoded: h(p1) = {1,2} h(p2) = {1,2} h(p1) != h(p2) o(p1) = 0 o(p2) = 0 h(p3) = 3 o(p3) = {1,2} ReadSentence = 1 ViewPicture = 2 Timing landmarks : 1 2 Decide = 3 Configuration 1: Configuration 2: Configuration 3: Configuration 4:

11. Inference • Over configurations • Choose the most likely configuration, where: • C=configuration, Y=observed data, D=input stimuli, HPM=model

12. 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. • E step: estimate a probability distribution over configurations. • M step: update estimates of W (using reweighted least squares) and Q (using standard MLEs) based on the E step. • After convergence, use standard MLEs for s.

13. Learned models: S S P P D D S P D predicted D start time chosen by program as t+18 Learned HPM with 3 processes (S,P,D), and d=13sec. S S P P D? D? observed

14. ViewPicture in Visual Cortex Offset q = P(Offset) 0 0.725 1 0.275

15. ReadSentence in Visual Cortex Offset q = P(Offset) 0 0.625 1 0.375

16. 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

17. Comparing Cognitive Hypotheses • Use cross-validation to choose a model. • GNB = HPM w/ ViewPicture, ReadSentence w/ d=8s. • HPM-2 = HPM w/ ViewPicture, ReadSentence w/ d=13s. • HPM-3 = HPM-2 + Decide Accuracy predicting picture vs. sentence (random = 0.5) Data log likelihood

18. 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.

19. 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.

20. 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. • Future work: • Share parameters across voxels (extending Niculescu05). • Use parametric hemodynamic responses (e.g. Boynton96). • Improve algorithm complexities. • Learn process durations automatically. • Apply to open cognitive science problems.

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