1 Hidden Process Models Rebecca Hutchinson Joint work with Tom Mitchell and Indra Rustandi

2 Talk Outline fMRI (functional Magnetic Resonance Imaging) data Prior work on analyzing fMRI data HPMs (Hidden Process Models) Preliminary results HPMs and BodyMedia

3 functional MRI

4 fMRI Basics Safe and non-invasive Temporal resolution ~ 1 3D image every second Spatial resolution ~ 1 mm –Voxels: 3mm x 3mm x 3-5mm Measures the BOLD response: Blood Oxygen Level Dependent –Indirect indicator of neural activity

5 The BOLD response Ratio of deoxy-hemoglobin to oxy- hemoglobin (different magnetic properties). Also called hemodynamic response function (HRF). Common working assumption: responses sum linearly.

6 More on BOLD response Signal Amplitude Time (seconds) At left is a typical BOLD response to a brief stimulation. (Here, subject reads a word, decides whether it is a noun or verb, and pushes a button in less than 1 second.)

7

8 Lots of features! 10,000-15,000 voxels per image …

9 Study: Pictures and Sentences 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

10 The star is not below the plus.

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*

13.

14 fMRI Summary High-dimensional time series data. Considerable noise on the data. Typically small number of examples (trials) compared with features (voxels). BOLD responses sum linearly.

15 Talk Outline fMRI (functional Magnetic Resonance Imaging) data Prior work on analyzing fMRI data HPMs (Hidden Process Models) Preliminary results HPMs and BodyMedia

16 It’s not hopeless! Learning setting is tough, but we can do it! Feature selection is key. Learn fMRI(t,t+8)->{Picture,Sentence} Read Sentence View PictureRead Sentence View PictureFixation Press Button 4 sec.8 sec.t=0 Rest

17 Results ABCDEFG 71.3%91.2%76.2%96.3%85.0%66.2%71.3% Gaussian Naïve Bayes Classifier. 95% confidence intervals per subject are +/- 10%-15%. Accuracy of default classifier is 50%. Feature selection: Top 240 most active voxels in brain. Subject: Accuracy: HIJKLMAvg. 95.0%81.2%90.0%85.0%65.0%90.0%81.8% Subject: Accuracy:

18 Why is this interesting? Cognitive architectures like ACT-R and 4CAPS predict cognitive processes involved in tasks, along with cortical regions associated with the processes. Machine learning can contribute to these architectures by linking their predictions to empirical fMRI data.

19 Other Successes We can distinguish between 12 semantic categories of words (e.g. tools vs. buildings). We can train classifiers across multiple subjects.

20 What can’t we do? Take into account that the responses for Picture and Sentence overlap. What does the response for Decide look like and when does it start? Read Sentence View PictureRead Sentence View PictureFixation Press Button 4 sec.8 sec.t=0 Rest

21 Talk Outline fMRI (functional Magnetic Resonance Imaging) data Prior work on analyzing fMRI data HPMs (Hidden Process Models) Preliminary results HPMs and BodyMedia

22 Motivation Overlapping processes –The responses to Picture and Sentence could overlap in space and/or time. Hidden processes –Decide does not directly correspond to the known stimuli. Move to a temporal model.

23 Hidden Markov Models? Can’t do overlapping processes – states are mutually exclusive. Markov assumption: given state t-1, state t is independent of everything before t-1. BOLD response: Not Markov! t-1tt+1t+2 CogProc {Picture, Sentence, Decide} fMRI

24 factorial HMMs? Have more flexibility than we need. –Picture state sequence should not be { …} Still have Markov assumption problem. t-1tt+1t+2 Picture = {0,1} Sentence = {0,1} Decide = {0,1} fMRI

25 Hidden Process Models Process ID = 3 Process ID = 2 Process Instances: Observed fMRI: cortical region 1: cortical region 2: Decide whether consistent View picture Processes: Name: Read sentence Process ID: 1 Response: Name: View Picture Process ID: 2 Response: Name: Decide whether consistent Process ID: 3 Response: Process ID = 1

26 HPM Parameters Set of processes, each of which has: –a process ID –a maximum response duration R –emission weights for each voxel v [W(v,1),…,W(v,t),…,W(v,R)] –a multinomial distribution over possible start times within a trial [  1,…,  t,…,  T ] Set of standard deviations – one for each voxel  1,…,  v,...,  V ] 

27 Interpreting data with HPMs Data Interpretation (int) –Set of process instances, each of which has: a process ID a start time S To predict fMRI data using an HPM and int: –For each active process, add the response associated with its processID to the prediction.

28 Synthetic Data Example Process 1:Process 2:Process 3: Process responses: Process instances: Predicted data ProcessID=1, S=1 ProcessID=2, S=17 ProcessID=3, S=21

29 Our Assumptions Processes, not states. –One hidden variable – process start time. Known number of processes in the model. –e.g. Picture, Sentence, Decide – 3 processes Known number of instantiations of those processes. –e.g. numTrials*3 processes Each process has a unique signature. Contributions of overlapping processes to the same output variable sum linearly.

30 The generative model Together HPM and interpretation (int) define a probability distribution over sequences of fMRI images: where P(y v,t |hpm,int) = N(  v,t,  v )  v,t =  W i.procID (v,t – start(i)) i  active process instances

31 Inference Given: –An HPM –A set of data interpretations (int) of processIDs and start times –Priors over the interpretations P(int=i|Y)  P(Y|int=i)P(int=i) Choose the interpretation i with the highest probability.

32 Synthetic Data Example Interpretation 1: Observed data ProcessID=1, S=1 ProcessID=2, S=17 ProcessID=3, S=21 Interpretation 2: ProcessID=2, S=1 ProcessID=1, S=17 ProcessID=3, S=23 Prediction 1 Prediction 2

33 Learning the Model EM (Expectation-Maximization) algorithm E-step –Estimate a conditional distribution over the start times of the process instances given the observed data, P(S|fMRI). M-step –Use the distribution from the E step to get maximum-likelihood estimates of the HPM parameters { , W,  }.

34 More on the E-step The start times of the process instances are not necessarily conditionally independent given the data. –Must consider joint configurations. –With no constraints, T nInstances configurations. – configurations for typical experiment. Can we consider a smaller set of start time configurations?

35 Reducing complexity Prior knowledge –Landmarks Events with known timing that “trigger” processes. One per process instance. –Offsets The interval of possible delays from a landmark to a process instance onset. One vector of n offsets per process. Conditional independencies –Introduced when no process instance could be active.

36 Before Prior Knowledge Decide whether consistent Read sentence View picture Cognitive processes: Observed fMRI: cortical region 1: cortical region 2:

37 Decide whether consistent Read sentence View picture Cognitive processes: Observed fMRI: cortical region 1: cortical region 2: Landmarks: (Stimuli) Sentence Presentation Picture Presentation Sentence offsets = {0,1} Picture offsets = {0,1} Decide offsets = {0,1,2,3} Landmarks go to process instances. Offset values are determined by process IDs. Prior Knowledge

38 Conditional Independencies Decide whether consistent Read sentence View picture Observed fMRI: cortical region 1: cortical region 2: Landmarks: (Stimuli) Sentence Presentation Picture Presentation Sentence offsets = {0,1} Picture offsets = {0,1} Decide offsets = {0,1,2,3} Decide whether consistent Read sentence View picture Sentence Presentation Picture Presentation Sentence offsets = {0,1} Picture offsets = {0,1} Decide offsets = {0,1,2,3} HERE

39 More on the M-step Weighted least squares procedure –exact, but may become intractable for large problems –weights are the probabilities computed in the E-step Gradient ascent procedure –approximate, but may be necessary when exact method is intractable –derivatives of the expected log likelihood of the data with respect to the parameters

40 Talk Outline fMRI (functional Magnetic Resonance Imaging) data Prior work on analyzing fMRI data HPMs (Hidden Process Models) Preliminary results HPMs and BodyMedia

41 View Picture Or Read Sentence Or View Picture Fixation Press Button 4 sec.8 sec.t=0 Rest picture or sentence? 16 sec. GNB: picture or sentence? HPM: Preliminary Results

42 GNB vs. HPM Classification GNB: non-overlapping processes HPM: simultaneous classification of multiple overlapping processes Average improvement of 15% in classification error using HPM vs GNB E.g., for one subject –GNB classification error: 0.14 –HPM classification error: 0.09

43 trial 25 Learned models Comprehend sentence Comprehend picture

44 Model selection experiments Model with 2 or 3 cognitive processes? –How would we know ground truth? –Cross validated data likelihood P(testData | HPM) Better with 3 processes than 2 –Cross validated classification accuracy Better with 3 processes than 2

45 Current work and challenges Add temporal and/or spatial smoothness constraints. Feature selection for HPMs. Process libraries, hierarchies. Process parameters (e.g. sentence negated or not). Model process interactions. Scaling parameters for response amplitudes to model habituation effects.

46 Talk Outline fMRI (functional Magnetic Resonance Imaging) data Prior work on analyzing fMRI data HPMs (Hidden Process Models) Preliminary results HPMs and BodyMedia

47 One idea… Sensor 1: Sensor 2: Processes: Name: Riding bus Process ID: 1 Response: Name: Eating Process ID: 2 Response: Name: Walking consistent Process ID: 3 Response: Process instances: Observed data: ProcessID=3 ProcessID=2 ProcessID=1

48 Some questions What processes are interesting? What granularity/duration would processes have? What would landmarks be? Variable process durations needed? Better way to parameterize process signatures?