Outline

1) Motivation

2) Representing/Modeling Causal Systems

3) Estimation and Updating

4) Model Search

5) Linear Latent Variable Models

6) Case Study: fMRI

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Goals

1. From Imaging data, to extract as much information as we can, as accurately as we can, about which brain regions influence which others in the course of psychological tasks.

2. To generalize over tasks

3. To specialize over groups of people.

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What Are the Brain Variables?

In current studies, from20,000 + ………………………………….3

voxels ROIs

ROI = Region of interest

Question: How sensitive are causal inferences to brain variable selection?

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How are ROIs constructed (FSL)?

• Define an experimental variable (box function).• Use a generalized linear model to determine which

voxels “light up” in correlation with the experimental variable.

• Add a group level step if voxels lighting up for the group is desired.

• Cluster the resulting voxels into connected clusters.– Small clusters are eliminated.– Remaining clusters become the ROIs.– Symmetry constraints may be imposed.

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Search Complexity: How Big is the Set of Possible Explanations?

X YX YX YX YX Y

}For N variables:

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2N

For graphical models:

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Statistical Complexity• Graphical models are untestable unless

parameterized into statistical models.• Incomplete models of associations are likely to

fail tests.• Multiple testing problems.• Multiple subjects/Missing ROIs.• No fast scoring method for mixed ancestral

graphs that model feedback and latent common causes.

• Weak time lag information.

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Measurement Complexity• Sampling rate is slower than causal interaction

speed.• Indirect measurement creates spurious

associations of measured variables:N1 N2 N3 X1 X3X1 X2 X3 X2

Neural N, measured X Regression of X3 on X1, X2

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Specification Strategies

1. Guess a model and test it.2. Search the model space or some restriction

of it.a. Search for the full parameterized structureb. Search for graphical structure alonec. Search for graphical features (e.g, adjacencies)

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What Evidence of What Works, and Not?

• Theory.– Limiting correctness of algorithms (PC, FCI, GES,

LiNGAM, etc., under usually incorrect assumptions for fMRI).

• Prior Knowledge– Do automated search results conform with

established relationships?• Animal Experiments (Limited)• Simulation Studies

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Brief Review: Smith’s Simulation Study

• 5 to 50 variables• 28 simulation conditions, 50 subjects/condition.• 38 search methods• Search 1 subject at a time.

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Methods tested by Smith• DCM, SEM excluded; no search. (Not completely true.)• Full correlation in various frequency bands• Partial correlation• Lasso (ICOV)• Mutual Information, Partial MI• Granger Causality• Coherence• Generalized Synchronization• Patel’s Conditional Dependence Measures

– P(x|y) vs P(y|x)• Bayes Net Methods

– CCD, CPC, FCI, PC, GES• LiNGAM

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Smith’s Results

• Adjacencies:– Partial Correlation methods (GLASSO) and several

“Bayes Net” methods from CMU get ~ 90% correct in most simulations.

• Edge Directions– Smith: “None of the methods is very accurate, with

Patel's τ performing best at estimating directionality, reaching nearly 65% d-accuracy, all other methods being close to chance.” (p. 883)

– Most of the adjacencies for Patel’s τ are false.

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Simulation conditions (see handout)…

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SIMULATION 2 (10 variables, 11 edges)

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Simulation 4 (50 variables 61 edges)

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Simulation 7: 250 minutes, 5 variables

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Simulation 8: Shared Inputs

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Simulation 14: 5-Cycle

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Simulation 15: Stronger Connections

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Simulation 16: More Connections

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Simulation 22: Nonstationary Connection Strengths

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Simulation 24: One Strong External Input

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Take Away Conclusion?

• Nothing works!• Methods that get adjacencies (90%) cannot get

directions of influence.• Methods that get directions (60% - 70%) for

normal session lengths cannot tell true adjacencies from false adjacencies.

• Even with unrealistically long sessions (4 hours), the best method gets 90% accuracy for directions but finds very few adjacencies.

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Idea…

• If we could:– Increase sample size (effectively) by using data from

multiple subjects– Focus on a method with strong adjacencies– Combine this with a method with strong orientations

• We may be able to do better (Ramsey, Hanson and Glymour, NeuroImage)– This is the strategy of the PC Lingam algorithm of

Hoyer and several of us, though there are other ways to pursue the same strategy.

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Reminder: If noises are non-Gaussian, we can learn more than a pattern.

Linear Models, Covariance Data,Pattern/CPDAG

Linear Models, Non-Gaussian Noises (LiNG),

Directed Graph

(1) (2)

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Are noises for FMRI models non-Gaussian?• Yes. This is controversial but shouldn’t be.– For word/pseudoword data of Xue and Poldrack (Task 3),

kurtosis ranges up to 39.3 for residuals.• There is a view in the literature that noises are distributed

(empirically) as Gamma—say, with shape 19 and scale 20.

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Are connection functions linear for FMRI data?

• You tell me:

• I’ve not done a thorough survey of studies.

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Coefficients?• One expects them to be positive.– Empirically, in linear models of fMRI data, there are very few

negative coefficients (1 in 200, say).– They’re only slightly negative if so.– This is consistent with negative coefficient occurring due to

small sample regression estimation errors.• For the most part, need to be less than 1.– Brain activations are cyclic and evolving over time.– Empirically, in linear models of fMRI, most coefficients are

less than 1. To the extent that they’re greater than 1, one suspects nonlinearity.

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The IMaGES algorithm• Adaptation for multiple subjects of GES, a Bayes net

method tested by Smith, et al.• Iterative model construction using Bayesian scores

separately on each subject at each step; edge with best average score added.

• Tolerates ROIs missing in various subjects.• Seeks feed forward structure only.• Finds adjacencies between variables with latent

common causes.• Forces sparsity by penalized BIC score to avoid

triangulated variables (see Measurement Complexity)

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IMaGES/LOFS

• Smith (2011): “Future work might look to optimize the use of higher-order statistics specifically for the scenario of estimating directionality from fMRI data.”

• LiNGAM orients edges by non-Normality of higher moments of the distributions of adjacent variables.

• LOFS uses the IMaGES adjacencies, and the LiNGAM idea for directing edges (with a different score for non-Normality, and without independent components).

• Unlike IMaGES, LOFS can find cycles.• LOFS (from our paper) is R1 and/or R2…

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Procedure R1(S)

You don’t have to read these—I’ll describe them!• G <- empty graph over variables of S• For each variable V– Find the combination C of adj(V, S) that maximizes

NG(eV|C).– For each W in C• Add W V to G

• Return G

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Procedure R2(S)• G <- empty graph over variables of S• For each pair of variables X, Y

– Scores<-empty– For each combination of adjacents C for X and Y

• If NG(eX|Y) < NG(X) & NG(eY|X) > NG(Y)– score <- NG(X) + NG(eY|X)– Add <X->Y, score> to Scores

• If NG(eX|Y) > NG(X) & NG(eY|X) < NG(Y)– score <- NG(eX|Y) + NG(Y)– Add <X<-Y, score> to Scores

– If Scores is empty• Add X—Y to G.

– Else• Add to G the edge in Scores with the highest score.

• Return G

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Non-Gaussianity Scores

• Log Cosh – used in ICA • Exp = -e^(-X^2/2) – used in ICA• Kurtosis – ICA (one of the first tried, not great)• Mean absolute – PC LiNGAM• E(e^X) – Cumulant arithmetic

= e^(κ1(X) + 1/(2!) κ2(X) + 1/(3!) κ3(X) + …)

• Anderson Darling A^2 – LOFS – Empirical Distribution Function (EDF) score with heavy

weighting on the tails.– We’re using this one!

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Mixing Residuals

• We are assuming that residuals for ROIs from different subjects are drawn from the same population, so that they can be mixed.

• Sometimes we center residuals from different subjects before mixing, sometimes not.

• For Smith study, doesn’t matter—the data is already centered!

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Precision and Recall

• Precision = True positives / all positives– What fraction of the guys you found were correct?

• Recall = True positives / all true guys– What fraction of the correct guys did you find?

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Some Further Problems

• Discovering nearly canceling 2 cycles is hard (but we will try anyway…)

• Identifying latent latents for acyclic models• Reliability of search may be worse with event

designs than with block designs• Subjects that differ in causal structures will

yield poor results for multi-subject methods.

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Hands On

• Download fmridata.tet.• Attach a Search box to the data and run

IMaGES.• Copy the layout from the layout graph

provided into the search box (using menus).• Attach another Search box with IMaGES and

Data as input and run LOFS.• Try variations!

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Thanks!• S. M. Smith, K L. Miller, G. Salimi-Khorshidi, M. Webster, C. F.

Beckmann, T. E. Nichols, J. D. Ramsey, M. W. Woolrich (2011), Network modelling methods for fMRI, NeuroImage.

• J.D. Ramsey, S.J. Hanson, C. Hanson, Y.O. Halchenko, R.A. Poldrack, and C. Glymour(2010), Six Problems for causal inference from fMRI, NeuroImage.

• J.D. Ramsey, S.J. Hanson, C. Glymour. Multi-subject search correctly identifies causal connections and most causal directions in the DCM models of the Smith et al. simulation study. NeuroImage.

• G. Xue, Poldrack, R., 2007. The neural substrates of visual perceptual learning of words: implications for the visual word form area hypothesis. J. Cogn. Neurosci.

• Thanks to the James S. McDonnell Foundation.