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. 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. 3 fMRI Data … Signal Amplitude Time (seconds) Hemodynamic Response Neural activity Features: 10,000 voxels, imaged every second. Training examples: 10-40 trials (task repetitions).

4. 4 Study: Pictures and Sentences 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 View PictureRead Sentence View PictureFixation Press Button 4 sec.8 sec.t=0 Rest

5. 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. 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. 7 HPM Formalism HPM = H =, a set of processes (e.g. ReadSentence) h =, a process W = response signature d = process duration  = allowable offsets  = multinomial parameters over values in  C =, a set of configurations c =, a set of process instances  =, a process instance (e.g. ReadSentence(S1)) h = process ID = timing landmark (e.g. stimulus presentation of S1) O = offset (takes values in  h )  =, priors over C  =, standard deviation for each voxel

8. 8 Process 1: ReadSentence Response signature W: Duration d: 11 sec. Offsets  : {0,1} P(  ): {  0,  1 } One configuration c of process instances  1,  2, …  k : (with prior  c ) Predicted mean: Input stimulus  : 11  Timing landmarks : 2 1 22 Process instance:  2 Process h: 2 Timing landmark: 2 Offset O: 1 (Start time: 2 + O) sentence picture v1 v2 Process 2: ViewPicture Response signature W: Duration d: 11 sec. Offsets  : {0,1} P(  ): {  0,  1 } v1 v2 Processes of the HPM: v1 v2 + N(0,  1 ) + N(0,  2 )

9. 9 HPMs: the graphical model Offset o Process Type h Start Time s observed unobserved Timing Landmark Y t,v  1,…,  k t=[1,T], v=[1,V]  Constraints from experiment design

10. 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  for each process Standard deviation  for each voxel –Expectation-Maximization (EM) algorithm to estimate W and . –After convergence, use standard MLEs for 

11. 11 ViewPicture in Visual Cortex Offset  = P(Offset) 00.725 10.275

12. 12 ReadSentence in Visual Cortex Offset  = P(Offset) 00.625 10.375

13. 13 Decide in Visual Cortex Offset  = P(Offset) 00.075 10.025 20.025 30.025 40.225 50.625

14. 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 Subject:ABC GNB0.7250.750 HPM-20.7500.8750.787 HPM-30.7750.8750.812 GNB-896-786-476 HPM-2-876-751-466 HPM-3-864-713-447

15. 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. Number of processes in the training and test data Number of times the correct number of processes was chosen for the test set 25/5 3 44/5 Total:14/15 = 93.3%

16. 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.