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Statistical inference in high-dimension & application to brain imaging

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Medical imaging involves high-dimensional data, yet their acquisition is obtained for limited samples. Multivariate predictive models have become popular in the last decades to fit some external variables from imaging data, and standard algorithms yield point estimates of the model parameters. It is however challenging to attribute confidence to
these parameter estimates, which makes solutions hardly trustworthy.
In this talk, I will present a new algorithm that assesses parameters statistical significance and that can scale even when the number of predictors p ≥ 10^5 is much higher than the number of samples n ≤ 10^3 , by leveraging structure among features. Our algorithm combines three main ingredients: a powerful inference procedure for linear models –the so-called Desparsified Lasso– feature clustering and an ensembling step. We first establish that Desparsified Lasso alone cannot handle n << p regimes; then we demonstrate that the combination of clustering and ensembling provides an accurate solution, whose specificity is controlled. We also demonstrate stability improvements on two brain imaging datasets.

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Statistical inference in high-dimension & application to brain imaging

  1. 1. 03/04/2019 Bertrand Thirion 1 Statistical inference in high-dimension & application to brain imaging Imaging and machine learning workshop Bertrand Thirion, bertrand.thirion@inria.fr
  2. 2. 03/04/2019 Bertrand Thirion 2 Cognitive neuroscience How are cognitive activities affected or controlled by neural circuits in the brain ?
  3. 3. 03/04/2019 Bertrand Thirion 3 The brain, the mind and the scanner Cognitive theories Brain Scanner FMRI data Brain mapping Experimental paradigm stimuli
  4. 4. 03/04/2019 Bertrand Thirion 4 The brain, the mind and the scanner Cognitive theories Brain Scanner FMRI data Decoding Encoding Brain mapping Experimental paradigm stimuli
  5. 5. 03/04/2019 Bertrand Thirion 5 Encoding: mapping cognitive functions to brain activity right hand- left hand listen-read Button press - reading Computation - instructions expression - intention Guess the gender Face trustworthiness false belief – mechanistic (auditory) False belief – mechanistic (visual) Grasping- orientation judgement Hand – side judgement Intention - random Sentence - checkerboards saccades - fixation Speech-non speech
  6. 6. 03/04/2019 Bertrand Thirion 6 Resolution increases 2007: 3 mm 2014: 1.5 mm 2021: 0.5 mm ? p = 50,000 p = 400,000 p = 107
  7. 7. 03/04/2019 Bertrand Thirion 7 better estimators for large-scale brain imaging ● A causal framework for brain activity decoding ● Dimension reduction for images ● Fast regularized ensembles of models ● Statistical inference for high-dimensional models
  8. 8. 03/04/2019 Bertrand Thirion 8 Causal reasoning on encoding/decoding Task Brain activity Behavior Causal encoding models P(X|T) Causal decoding models P(B|X) Anti-causal decoding models P(T|X) Anti-causal encoding models P(X|B) [Weichwald et al Nimg 2015]
  9. 9. 03/04/2019 Bertrand Thirion 9 Causal interpretation Encoding: causal Decoding: anti-causalTask X1 X2 Xp ... X1 X2 Xp ... Behavior Encoding: anti-causal Decoding: causal
  10. 10. 03/04/2019 Bertrand Thirion 10 Causal reasoning on encoding/decoding [Weichwald et al. NIMG 2015] Task X1 X2 Xp ...
  11. 11. 03/04/2019 Bertrand Thirion 11 Causal reasoning on encoding/decoding [Weichwald et al. NIMG 2015] Task X1 X2 Xp ...
  12. 12. 03/04/2019 Bertrand Thirion 12 Causal reasoning on encoding/decoding [Weichwald et al. NIMG 2015]
  13. 13. 03/04/2019 Bertrand Thirion 13 Causal reasoning on encoding/decoding [Weichwald et al. NIMG 2015]
  14. 14. 03/04/2019 Bertrand Thirion 14 Joint encoding and decoding [Schwartz et al. NIPS 2013, Varoquaux et al. PCB 2018] “Encoding” “Decoding”
  15. 15. 03/04/2019 Bertrand Thirion 15 Decoding maps
  16. 16. 03/04/2019 Bertrand Thirion 16 Joint encoding and decoding [Schwartz et al. NIPS 2013, Varoquaux et al. PCB 2018]
  17. 17. 03/04/2019 Bertrand Thirion 17 Statistical associations and causal reasoning ● Problems: – Establish non-independence based on finite datasets → statistical tests – Large number of conditioning variables – Encoding models: Multiple comparison issues – Decoding problem: statistical tests in multiple regression
  18. 18. 03/04/2019 Bertrand Thirion 18 Brain activity decoding X1 X2 Xp ... y ● behavior = f (brain activity) w
  19. 19. 03/04/2019 Bertrand Thirion 19 Outline ● A causal framework for brain activity decoding ● Dimension reduction for images ● Fast regularized ensembles of Models ● Statistical inference for high-dimensional models
  20. 20. 03/04/2019 Bertrand Thirion 20 Compression in the image domain ● Reduce the complexity of learning algorithms: p→k ≪ p ● Random projections = fast generic solution, but – Sub-optimal for structured signals – Not invertible when p and k are large ● Local redundancy → feature grouping strategies / clustering: “super-pixels” – Fast clustering procedures needed (large-k regime)
  21. 21. 03/04/2019 Bertrand Thirion 21 Superpixels as an image operator
  22. 22. 03/04/2019 Bertrand Thirion 22 Crafting good image compression ● Key assumption: signal of interest L-Lipschitz ● Feature grouping matrix ● almost trivially: ● Worst case Need a fast method to learn balanced clusters
  23. 23. 03/04/2019 Bertrand Thirion 23 Denoising properties ● Noisy signal model ● Denoising ● Equal-size clusters
  24. 24. 03/04/2019 Bertrand Thirion 24 Recursive neighbor Agglomeration Based on local decisions = fast (linear time) – avoid percolation [Thirion et al. Stamlins 2015, Hoyos Idrobo PAMI 2018] ReNA
  25. 25. 03/04/2019 Bertrand Thirion 25 Effect on data analysis tasks Impressive speed-up and increased accuracy with respect to non-compressed representation – Clustering has a denoising effect [Hoyos Idrobo IEEE PAMI 2018]
  26. 26. 03/04/2019 Bertrand Thirion 26 Outline ● A causal framework for brain activity decoding ● Dimension reduction for images ● Fast regularized ensembles of Models ● Statistical inference for high-dimensional models
  27. 27. 03/04/2019 Bertrand Thirion 27 Bagging of clustered models X Clustering (create contiguous regions) y Solve regression on cluster- based representation average ... various clusterings
  28. 28. 03/04/2019 Bertrand Thirion 28 Computationally efficient structure State of the art solution: not very stable, but cheap “fast regularized ensembles of models”
  29. 29. 03/04/2019 Bertrand Thirion 29 Computationally efficient structure
  30. 30. 03/04/2019 Bertrand Thirion 30 Effect on prediction accuracy “fast regularized ensembles of models” [Hoyos Idrobo et al PRNI 2015, Neuroimage 2017, PAMI 2018]
  31. 31. 03/04/2019 Bertrand Thirion 31 More results [Hoyos Idrobo et al PRNI 2015, Neuroimage 2017, PAMI 2018]
  32. 32. 03/04/2019 Bertrand Thirion 32 Outline ● A causal framework for brain activity decoding ● Dimension reduction for images ● Fast regularized ensembles of Models ● Statistical inference for high-dimensional models
  33. 33. 03/04/2019 Bertrand Thirion 33 Statistical inference on w ● Standard solutions for high-dimensional linear models (p ≅ n) – Corrected ridge [Bühlmann 2013] – Desparsified Lasso [Zhang & Zhang 2014, Montanari 2014] – Multi-split [Meinshausen 2009], knockoffs [Candès 2015+] ● Fail for p ≫ n ● Inference: find {j: wj > 0} with some statistical guarantees
  34. 34. 03/04/2019 Bertrand Thirion 34 Desparsified Lasso [Zhang & Zhang 2014 Series B Stat Meth]
  35. 35. 03/04/2019 Bertrand Thirion 35 Desparsified Lasso
  36. 36. 03/04/2019 Bertrand Thirion 36 Preliminary assessment
  37. 37. 03/04/2019 Bertrand Thirion 37 Large p → need dimension reduction Large p kills statistical power CDL tames variance p=2000, n=100 [Chevalier et al. subm. To MICCAI]
  38. 38. 03/04/2019 Bertrand Thirion 38 Adaptation to brain imaging Step 1: compression by clustering Step 2: inference on compressed representations Step 3: ensembling iterate with different parcellations → aggregate p-values (see also FReM) Clustered Desparsified Lasso Ensemble of Clustered Desparsified Lasso
  39. 39. 03/04/2019 Bertrand Thirion 39 From CDL to ECDL DL p-values from different clusterings aggregation
  40. 40. 03/04/2019 Bertrand Thirion 40 ECDL for brain imaging
  41. 41. 03/04/2019 Bertrand Thirion 41 δ-error control
  42. 42. 03/04/2019 Bertrand Thirion 42 δ-error control
  43. 43. 03/04/2019 Bertrand Thirion 43 δ-FWER control
  44. 44. 03/04/2019 Bertrand Thirion 44 δ-FWER-control
  45. 45. 03/04/2019 Bertrand Thirion 45 Simulations: ECDL > CDL [Chevalier et al. MICCAI 2018]
  46. 46. 03/04/2019 Bertrand Thirion 46 Experiments: PR and FWER control Better PR with ECDL + More accurate FWER control [Chevalier et al. MICCAI 2018]
  47. 47. 03/04/2019 Bertrand Thirion 47 Effects on real data [Nguyen et al. IPMI 2019, Chevalier et al. MICCAI 2018] Social cognition Visual feature discrimination Language vs maths HCP dataset, n=900
  48. 48. 03/04/2019 Bertrand Thirion 48 Conclusion ● Causal reasoning → conditional association analysis ● Large-p data bring challenges: – Computation cost – Difficulty of statistical inference ● Solutions: ensembling, subsampling, compression ● Efficient stochastic regularizers ● Ongoing comparison with knockoff WIP ● Classification setting ● Use of bootstrap [Nguyen et al. IPMI 2019] [Aydore et al. subm]
  49. 49. 03/04/2019 Bertrand Thirion 49 From good ideas to good practices: software ● Machine learning in Python ● Machine learning for neuroimaging http://nilearn.github.io ● BSD, Python, OSS – Classification of (neuroimaging) data – Network analysis
  50. 50. 03/04/2019 Bertrand Thirion 50 Acknowledgements Parietal G. Varoquaux, A. Gramfort, P. Ciuciu, D. Wassermann, D. Engemann, B. Nguyen A.L. Grilo Pinho, E. Dohmatob, A. Mensch, J.A. Chevalier, A. Hoyos idrobo, D. Bzdok, J. Dockès, P. Cerda, C. Lazarus D. La Rocca G. Lemaitre L. El Gueddari O. Grisel M. Massias P. Ablin H. Janati J. Massich K. Dadi H. Richard C. Petitot Other collaborators R. Poldrack, J. Haxby C. F. Gorgolevski J. Salmon S. Arlot M. Lerasle

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