Hidden Process Models with applications to fMRI data Rebecca Hutchinson August 2, 2009 Joint Statistical Meetings, Washington DC Oregon State University

Introduction Hidden Process Models (HPMs): Example domain: A probabilistic model for time series data. Designed for data generated by a collection of latent processes. Example domain: Modeling cognitive processes (e.g. making a decision) in functional Magnetic Resonance Imaging time series. Characteristics of potential domains: Processes with spatial-temporal signatures. Uncertainty about temporal location of processes. High-dimensional, sparse, noisy.

fMRI Data Features: 5k-15k voxels, imaged every second. Hemodynamic Response Features: 5k-15k voxels, imaged every second. Training examples: 10-40 trials (task repetitions). Signal Amplitude … Neural activity Time (seconds)

Study: Pictures and Sentences Press Button View Picture Read Sentence Read Sentence Fixation View Picture Rest t=0 4 sec. 8 sec. 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.

Goals for fMRI To track cognitive processes over time. Estimate hemodynamic response signatures. Estimate process timings. Modeling processes that do not directly correspond to the stimuli timing is a key contribution of HPMs! To compare hypotheses of cognitive behavior.

v1 v1 v2 v2 1 2 1 2 v1 + N(0,s1) v2 + N(0,s2) 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: 1 2  Predicted mean: v1 v2 + N(0,s1) + N(0,s2)

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 possible 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) C= a latent variable indicating the correct configuration S = <s1,…,sV>, standard deviation for each voxel

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]

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:

Inference Over C, the latent indicator of the correct configuration Choose the most likely configuration, where: Y=observed data, D=input stimuli, HPM=model

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), Q, and s (using standard MLEs) based on the E step.

Process Response Signatures Standard: Each process has a matrix of parameters, one for each point in space and time for the duration of the response (e.g. 24). Regularized: Same as standard, but learned with penalties for deviations from temporal and/or spatial smoothness. Basis functions: Each process has a small number (e.g. 3) weights for each voxel that are combined with a basis to get the response.

Evaluation Select 1000 most active voxels. Compute improvement in test data log-likelihood as compared with predicting the mean training trial for all test trials (a baseline). 5 folds of cross-validation. Average over 13 subjects. Standard Regularized Basis functions HPM-GNB -293 2590 2010 HPM-2 -1150 3910 3740 HPM-3 -2000 4960 4710 HPM-4 -4490 4810 4770

Interpretation and Visualization Timing for the third (Decide) process in HPM-3: (Values have been rounded.) For each subject, average response signatures for each voxel over time, plot result in each spatial location. Compare time courses for the same voxel. Offset: 1 2 3 4 5 6 7 Stand. 0.3 0.08 0.1 0.05 0.2 0.15 Reg. Basis 0.5 0.03

Standard

Regularized

Basis functions

Time courses Basis functions Standard The basis set Regularized

Related Work fMRI Machine Learning 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 can be encoded by extending factorial HMMs with links between the Markov chains.

Conclusions Take-away messages: Future work: 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: Automatically discover the number of latent processes. Learn process durations. Apply to open cognitive science problems.

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