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Brain Connectivity-Informed Adaptive Regularization for Generalized Outcomes Jaroslaw Harezlak, Ph.D. Professor and Interim Co-Chair Department of Epidemiology and Biostatistics Indiana University School of Public Health Bloomington, IN, USA

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Brain Connectivity-Informed Adaptive Regularization for Generalized Outcomes

Jaroslaw Harezlak, Ph.D.

Professor and Interim Co-Chair Department of Epidemiology and Biostatistics Indiana University School of Public Health Bloomington, IN, USA

May 22, 2020

Jaroslaw Harezlak May 22, 2020 1 / 33

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Outline

1 Motivating application 2 Brain structure and connectivity 3 Regularization methods 4 riPEER - ridgified Partially Empirical Eigenvectors for Regression 5 Simulation study 6 Brain structure and HIV infection 7 Discussion Jaroslaw Harezlak May 22, 2020 2 / 33

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HIV infection study - WUSM

1 N = 299 HIV-infected individuals: 228 males, 71 females

◮ Duration of infection range: 0 - 33y (mean: 10.2, sd: 8.7) ◮ Age range: 18 - 84 y.o. (mean: 42.3, sd: 16)

2 Imaging modalities

◮ T1 - anatomy ◮ DTI - structural connectivity

Jaroslaw Harezlak May 22, 2020 3 / 33

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Anatomy and connectivity

1 Anatomy

◮ MPRAGE protocol ◮ Processing using FreeSurfer software (version 5.1) ◮ Desikan-Killiany atlas - 66 cortical regions

2 Structural connectivity

◮ DTI and maximal diffusion coherence model ◮ Density of connections between each pair of regions

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MRI data

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MRI-derived data: cortical thickness

1 Parcellation of the cortex into 66 regions 2 Average cortical thickness

(a) Parcellation of the brain (b) Cortical thickness

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Connections in the brain

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Connectivity matrices

(a) Connectivity matrix: subject 1 (b) Connectivity matrix: subject 2

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Population connectivity matrix

A = 𝑏𝑗𝑘

WEAK STRONG

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Questions

1 Scientific

◮ Are changes in the brain structure associated with the HIV

infection?

◮ Is there any additional information provided by the structural

connectivity?

2 Statistical

◮ How to deal with the highly correlated predictors in the

regression models?

◮ How to incorporate the structural connectivity information in

the regression models?

Jaroslaw Harezlak May 22, 2020 10 / 33

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Statistical model

1 y – n-dimensional response (e.g. NP domain score) 2 Z ∈ Rn×66 and X ∈ Rn×m 3 ε ∼ N

0, σ2In for some unknown σ2 > 0

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Statistical model

1 y – n-dimensional response (e.g. NP domain score) 2 Z ∈ Rn×66 and X ∈ Rn×m 3 ε ∼ N

0, σ2In for some unknown σ2 > 0

Jaroslaw Harezlak May 22, 2020 12 / 33

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Penalized estimation To find the estimates of b and β, we consider the

  • ptimization problem of the form

argmin

b,β

y − Zb − Xβ

  • 2

2

  • model fit term

+ λ g(b)

  • penalty on b
  • .

1 g(b) =

i b2 i

− → Ridge estimate

2 g(b) =

i |bi|

− → LASSO estimate

3 g(b) = ||Lb||2

2

− → Generalized ridge

  • T. W. Randolph, J. Harezlak, Z. Feng, Structured penalties for functional linear

models – partially empirical eigenvectors for regression, Electronic Journal of Statistics (2012)

Jaroslaw Harezlak May 22, 2020 13 / 33

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Desired property of the estimate, ˆ b “Stronger connections between the brain regions i and j result in more similar coefficients ˆ

bi and ˆ bj.”

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Penalty selection

The natural choice of the penalty is

g(b) =

  • i,j

aij bi − bj 2.

1 di :=

k Aik.

2 D := diagd1, . . . , d66

  • 3 Q := D − A

[Laplacian of A] Then:

  • i,j

aij bi − bj 2 = bTQb.

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Connections with the linear mixed models (LMM)

Our objective function becomes argmin

b,β

  • y − Zb − Xβ
  • 2

2 + λbTQb

  • .

This optimization problem is “equivalent” to the LMM formulation

1 y = Zb + Xβ + ε, where β is a vector of fixed effects and b a

vector of random effects,

2 ε ∼ N

0, σ2I ,

3 b ∼ N

0, σ2

bQ−1

,

4 λ, σ and σb

are connected via λ = σ2/σ2

b.

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Selection of the regularization parameter

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The method

riPEER (ridgified Partially Empirical Eigenvectors for Regression)

  • ˆ

b

rP

ˆ β

rP

  • := argmin

b,β

  • y − Zb − Xβ
  • 2

2 + λQbTQb

  • graph part

+ λRb2

2 ridge part

  • Figure 3: Different shapes of the set
  • b : λQbT Qb + λRb2

2 ≤ 1

  • for p = 2.

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Connections with the linear mixed models (LMM)

riPEER (ridgified Partially Empirical Eigenvectors for Regression)

ˆ b

rP

ˆ β

rP

  • := argmin

b,β

  • y − Zb − Xβ
  • 2

2 + bT

λQQ + λRIb

  • This problem is “equivalent” to the LMM formulation

1 y = Zb + Xβ + ε, where β is a vector of fixed effects and b a

vector of random effects,

2 ε ∼ N

0, σ2I ,

3 b ∼ N

  • 0,

σ2

QQ + σ2 RI−1

,

4 λQ λR, σ, σQ and σR are connected via

λQ = σ2/σ2

Q, λR = σ2/σ2 R.

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Simulation scheme

SIMULATED SIGNAL

1 2 3 5 4

0.1 0.6 0.3 0.4 0.1

Graph given by adjacency matrix A Laplacian : 𝑅𝑢𝑠𝑣𝑓 „Invertible Laplacian” :

𝑅𝑢𝑠𝑣𝑓 ≔ 𝑅𝑢𝑠𝑣𝑓 + 0.001 ∙ 𝐽

True signal used in simulation:

𝑐𝑢𝑠𝑣𝑓~𝑂(0, 𝜏𝑐

2

𝑅𝑢𝑠𝑣𝑓

−1 )

ESTIMATION

1 2 3 5 4

0.1 0.6 0.3 0.4 0.1

Distorted graph Laplacian of distorted graph was used to find the estimate,

𝑐

𝑐 − 𝑐𝑢𝑠𝑣𝑓

2 2

𝑐𝑢𝑠𝑣𝑓

2 2

MSEr: = E

MSEr defined as as a measure of estimation accuracy

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Simulation scheme – distorted connectivity matrices

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Simulation scheme

Three methods compared:

1 ridge: λQ := 0

(connectivity information is not used)

2 naive: λR := 0, Q →

Q (only λQ is selected)

3 riPEER

(both lambdas are selected in an adaptive way) Axis of the plot

1 X axis:

diss(Atrue, Aobs) :=

number of removed/added connections number of all nonzero connections in Atrue

2 Y axis: MSEr := E

  • ˆ

b−btrue2

2

btrue2

2

  • .

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Simulation results

0.0 0.1 0.2 0.3 0.00 0.25 0.50 0.75

dissimilarity between Atrue and Aobs b estimation MSEr

ridge

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Simulation results

0.0 0.1 0.2 0.3 0.00 0.25 0.50 0.75

dissimilarity between Atrue and Aobs b estimation MSEr

ridge naive

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Simulation results

  • 0.0

0.1 0.2 0.3 0.00 0.25 0.50 0.75

dissimilarity between Atrue and Aobs b estimation MSEr

  • ridge

naive riPEER

Jaroslaw Harezlak May 22, 2020 25 / 33

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Results: HIV study

1 Association between cortical thickness and speed of information

processing

2 66 considered brain’s regions 3 N = 199 individuals Jaroslaw Harezlak May 22, 2020 26 / 33

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Results: Speed of Information Processing

bankssts[L] caudalanteriorcingulate[L] caudalmiddlefrontal[L] cuneus[L] entorhinal[L] fusiform[L] inferiorparietal[L] inferiortemporal[L] isthmuscingulate[L] lateraloccipital[L] lateralorbitofrontal[L] lingual[L] medialorbitofrontal[L] middletemporal[L] parahippocampal[L] paracentral[L] parsopercularis[L] parsorbitalis[L] parstriangularis[L] pericalcarine[L] postcentral[L] posteriorcingulate[L] precentral[L] precuneus[L] rostralanteriorcingulate[L] rostralmiddlefrontal[L] superiorfrontal[L] superiorparietal[L] superiortemporal[L] supramarginal[L] frontalpole[L] temporalpole[L] transversetemporal[L] bankssts[R] caudalanteriorcingulate[R] caudalmiddlefrontal[R] cuneus[R] entorhinal[R] fusiform[R] inferiorparietal[R] inferiortemporal[R] isthmuscingulate[R] lateraloccipital[R] lateralorbitofrontal[R] lingual[R] medialorbitofrontal[R] middletemporal[R] parahippocampal[R] paracentral[R] parsopercularis[R] parsorbitalis[R] parstriangularis[R] pericalcarine[R] postcentral[R] posteriorcingulate[R] precentral[R] precuneus[R] rostralanteriorcingulate[R] rostralmiddlefrontal[R] superiorfrontal[R] superiorparietal[R] superiortemporal[R] supramarginal[R] frontalpole[R] temporalpole[R] transversetemporal[R]

  • 0.05

0.05 riPEER estimate of b

Cortical regions: lingual[L], precentral[L], superiorparietal[L], lateralorbitofrontal[R], precentral[R], superiorparietal[R], supramarginal[R], medialorbitofrontal[L], posteriorcingulate[R]

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Non-Gaussian distributions

yi ∼ member of an Exponential family of distribution

Consider an optimization problem of the form argmin

b,β

  • −2 loglik(y; β, b)
  • model fit term

+

gλ(b)

penalty on b

  • .

gλ(b) := λQbTQb + λRb2

2

λQ and λR are selected based on the equivalence with GLMM

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HIV data: HIV+ vs. HIV-

Empty Masked FA Masked DC (L) Banks superior temporal sulcus (L) Caudal anterior−cingulate cortex (L) Caudal middle frontal gyrus (L) Cuneus cortex (L) Entorhinal cortex (L) Fusiform gyrus (L) Inferior parietal cortex (L) Inferior temporal gyrus (L) Isthmus_cingulate cortex (L) Lateral occipital cortex (L) Lateral orbital frontal cortex (L) Lingual gyrus (L) Medial orbital frontal cortex (L) Middle temporal gyrus (L) Parahippocampal gyrus (L) Paracentral lobule (L) Pars opercularis (L) Pars orbitalis (L) Pars triangularis (L) Pericalcarine cortex (L) Postcentral gyrus (L) Posterior−cingulate cortex (L) Precentral gyrus (L) Precuneus cortex (L) Rostral anterior cingulate cortex (L) Rostral middle frontal gyrus (L) Superior frontal gyrus (L) Superior parietal cortex (L) Superior temporal gyrus (L) Supramarginal gyrus (L) Frontal pole (L) Temporal pole (L) Transverse temporal cortex (R) Banks superior temporal sulcus (R) Caudal anterior−cingulate cortex (R) Caudal middle frontal gyrus (R) Cuneus cortex (R) Entorhinal cortex (R) Fusiform gyrus (R) Inferior parietal cortex (R) Inferior temporal gyrus (R) Isthmus_cingulate cortex (R) Lateral occipital cortex (R) Lateral orbital frontal cortex (R) Lingual gyrus (R) Medial orbital frontal cortex (R) Middle temporal gyrus (R) Parahippocampal gyrus (R) Paracentral lobule (R) Pars opercularis (R) Pars orbitalis (R) Pars triangularis (R) Pericalcarine cortex (R) Postcentral gyrus (R) Posterior−cingulate cortex (R) Precentral gyrus (R) Precuneus cortex (R) Rostral anterior cingulate cortex (R) Rostral middle frontal gyrus (R) Superior frontal gyrus (R) Superior parietal cortex (R) Superior temporal gyrus (R) Supramarginal gyrus (R) Frontal pole (R) Temporal pole (R) Transverse temporal cortex −0.04 −0.02 0.00 0.02 −0.050 −0.025 0.000 0.025 −0.04 −0.02 0.00 0.02

Brain regions griPEER estimate Connectivity Matrix:

Empty (logistic Ridge) Masked FA Masked DC

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Contributions: riPEER and griPEER

1 Brain connectivity informed regularization

methods for regression, Statistics in Biosciences, April 2019, Volume 11, Issue 1, 47–90 mdpeer R package is available online at https://CRAN.R-project.org/package=mdpeer

2 Connectivity-Informed Adaptive Regularization

Under non-Gaussian Design, doi: https://doi.org/10.1101/322420 griPEER Matlab software is available online at https://github.com/dbrzyski/griPEER

Jaroslaw Harezlak May 22, 2020 30 / 33

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Conclusions and Discussion

1 Regularization methods using a priori information in brain imaging

setting in a principled way.

2 Additional (external) information quantified via a Laplacian

matrix.

3 Penalty parameters chosen in an adaptive way via ML/REML. 4 The more knowledge one has about informative structure, the

more specific one can be in defining the penalty matrix.

5 Use of structural connectivity to inform the associations between

cortical thickness and either NP outcomes (continuous) or HIV status (binary). Funding: NIMH R01MH108467

Jaroslaw Harezlak May 22, 2020 31 / 33

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Collaborators

Marta Karas Timothy Randolph Mario Dzemidzic Joaquin Goni Beau Ances

  • Brain Connectivity-Informed Regularization

Methods for Regression, Statistics in Biosciences (2017),

https://doi.org/10.1007/s12561-017-9208-x

  • mdpeer R package is available online at

https://CRAN.R-project.org/package=mdpeer

Washington University Johns Hopkins University Damian Brzyski Technical Univ. of Wroclaw, Poland Fred Hutchinson

  • C. R. Center

Indiana University Purdue University

Jaroslaw Harezlak May 22, 2020 32 / 33

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References

1

  • M. Karas, D. Brzyski, M. Dzemidzic, J. Goni, D. A. Kareken, T. W. Randolph, J.

Harezlak, Brain connectivity–informed regularization methods for regression, Stat Biosci (2019). https://doi.org/10.1007/s12561-017-9208-x.

2

  • T. W. Randolph, J. Harezlak, Z. Feng, Structured penalties for functional linear

models – partially empirical eigenvectors for regression, Electronic Journal of Statistics 6 (2012), 323–353.

3

  • C. Li, H. Li, Network–constrained Regularization and Variable Selection for Analysis
  • f Genomic Data, JBioinformatics 24 (2008), no. 9, 1175–1182.

4

  • F. Chung, Laplacians and the Cheeger Inequality for Directed Graphs, Annals of

Combinatorics 9 (2005), no. 1, 1–19.

5

  • E. Demidenko, Mixed Models: Theory and Applications, Wiley 9 (2004).

Pictures sources: FreeSurfer, http://www.opensourceimaging.org/project/freesurfer/ Athinoula A. Martinos Center, http://www.martinos.org/neurorecovery/technology.htm PsyPost, http://www.psypost.org/2014/04/atypical-brain-connectivity-associated-with- autism-spectrum-disorder-24472

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