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 • 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 Constraints from experiment design Timing Landmark l Process Type h Offset o Start Time s S p1,…,pk observed unobserved Yt,v t=[1,T], v=[1,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/ 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

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.