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Covariate Assisted Principal (CAP) Regression for Matrix Outcomes


Xi (Rossi) LUO


The University of Texas
Health Science Center
School of Public Health
Dept of Biostatistics
and Data Science
ABCD Research Group 


ICSA China, Tianjin, CHINA
July 4, 2019

Funding: NIH R01EB022911, P01AA019072, P20GM103645, P30AI042853; NSF/DMS (BD2K) 1557467

Co-Authors

Xuefei Cao

Yi Zhao
Johns Hopkins Biostat

B Sandstede

Bingkai Wang
Johns Hopkins Biostat

B Sandstede

Stewart Mostofsky
Johns Hopkins Medicine

B Sandstede

Brian Caffo
Johns Hopkins Biostat

Slides viewable on web:
bit.ly/icsa2019

Statistics/Data Science Focuses

Motivating Example

Brain network connections vary by covariates (e.g. age/sex)


Goal: model how covariates change network connections

$$\textrm{function}(\textbf{graph}) = \textbf{age}\times \beta_1 + \textbf{sex}\times \beta_2 + \cdots $$

Resting-state fMRI Networks

  • fMRI measures brain activities over time
  • Resting-state: "do nothing" during scanning


  • Brain networks constructed using cov/cor matrices of time series

Mathematical Problem

  • Given $n$ (semi-)positive matrix outcomes, $\Sigma_i\in \real^{p\times p}$
  • Given $n$ corresponding vector covariates, $x_i \in \real^{q}$
  • Find function $g(\Sigma_i) = x_i \beta$, $i=1,\dotsc, n$
  • In essense, regress positive matrices on vectors

Some Related Problems

  • Heterogeneous regression or weighted LS:
    • Usually for scalar variance $\sigma_i$, find $g(\sigma_i) = f(x_i)$
    • Goal: to improve efficiency, not to interpret $x_i \beta$
  • Covariance models Anderson, 73; Pourahmadi, 99; Hoff, Niu, 12; Fox, Dunson, 15; Zou, 17
    • Model $\Sigma_i = g(x_i)$, sometimes $n=i=1$
    • Goal: better models for $\Sigma_i$
  • Multi-group PCA Flury, 84, 88; Boik 02; Hoff 09; Franks, Hoff, 16
    • No regression model, cannot handle vector $x_i$
    • Goal: find common/uncommon parts of multiple $\Sigma_i$
  • Tensor-on-scalar regression Li, Zhang, 17; Sun, Li, 17
    • No guarantees for positive matrix outcomes

Massive Edgewise Regressions

  • Intuitive method by mostly neuroscientists
  • Try $g_{j,k}(\Sigma_i) = \Sigma_{i}[j,k] = x_i \beta$
  • Repeat for all $(j,k) \in \{1,\dotsc, p\}^2$ pairs
  • Essentially $O(p^2)$ regressions for each connection
  • Limitations: multiple testing $O(p^2)$, failure to accout for dependencies between regressions

Our CAP in a Nutshell


$\mbox{PCA}(\Sigma_i) = x_i \beta$

  • Essentially, we aim to turn unsupervised PCA to a supervised PCA
  • Ours differs from existing PCA methods:
    • Supervised PCA Bair et al, 06 models scalar-on-vector

Model and Method

Model

  • Find principal direction (PD) $\gamma \in \real^p$, such that: $$ \log({\gamma}^\top\Sigma_{i}{\gamma})=\beta_{0}+x_{i}^\top{\beta}_{1}, \quad i =1,\dotsc, n$$

Example (p=2): PD1 largest variation but not related to $x$

PCA selects PD1, Ours selects PD2

Advantages

  • Scalability: potentially for $p \sim 10^6$ or larger
  • Interpretation: covariate assisted PCA
    • Turn unsupervised PCA into supervised
  • Sensitivity: target those covariate-related variations
    • Covariate assisted SVD?
  • Applicability: other big data problems besides fMRI

Method

  • MLE with constraints: $$\scriptsize \begin{eqnarray}\label{eq:obj_func} \underset{\boldsymbol{\beta},\boldsymbol{\gamma}}{\text{minimize}} && \ell(\boldsymbol{\beta},\boldsymbol{\gamma}) := \frac{1}{2}\sum_{i=1}^{n}(x_{i}^\top\boldsymbol{\beta}) \cdot T_{i} +\frac{1}{2}\sum_{i=1}^{n}\boldsymbol{\gamma}^\top \Sigma_{i}\boldsymbol{\gamma} \cdot \exp(-x_{i}^\top\boldsymbol{\beta}) , \nonumber \\ \text{such that} && \boldsymbol{\gamma}^\top H \boldsymbol{\gamma}=1 \end{eqnarray}$$
  • Two obvious constriants:
    • C1: $H = I$
    • C2: $H = n^{-1} (\Sigma_1 + \cdots + \Sigma_n) $

Choice of $H$

Proposition: When (C1) $H=\boldsymbol{\mathrm{I}}$ in the optimization problem, for any fixed $\boldsymbol{\beta}$, the solution of $\boldsymbol{\gamma}$ is the eigenvector corresponding to the minimum eigenvalue of matrix $$ \sum_{i=1}^{n}\frac{\Sigma_{i}}{\exp(x_{i}^\top\boldsymbol{\beta})} $$

Will focus on the constraint (C2)

Algoirthm

  • Iteratively update $\beta$ and then $\gamma$
  • Prove explicit updates
  • Extension to multiple $\gamma$:
    • After finding $\gamma^{(1)}$, we will update $\Sigma_i$ by removing its effect
    • Search for the next PD $\gamma^{(k)}$, $k=2, \dotsc$
    • Impose the orthogonal constraints such that $\gamma^{k}$ is orthogonal to all $\gamma^{(t)}$ for $t\lt k$

Theory for $\beta$

Theorem: Assume $\sum_{i=1}^{n}x_{i}x_{i}^\top/n\rightarrow Q$ as $n\rightarrow\infty$. Let $T=\min_{i}T_{i}$, $M_{n}=\sum_{i=1}^{n}T_{i}$, under the true $\boldsymbol{\gamma}$, we have \begin{equation} \sqrt{M_{n}}\left(\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right)\overset{\mathcal{D}}{\longrightarrow}\mathcal{N}\left(\boldsymbol{\mathrm{0}},2 Q^{-1}\right),\quad \text{as } n,T\rightarrow\infty, \end{equation} where $\hat{\boldsymbol{\beta}}$ is the maximum likelihood estimator when the true $\boldsymbol{\gamma}$ is known.

Theory for $\gamma$

Theorem: Assume $\Sigma_{i}=\Gamma\Lambda_{i}\Gamma^\top$, where $\Gamma=(\boldsymbol{\gamma}_{1},\dots,\boldsymbol{\gamma}_{p})$ is an orthogonal matrix and $\Lambda_{i}=\mathrm{diag}\{\lambda_{i1},\dots,\lambda_{ip}\}$ with $\lambda_{ik}\neq\lambda_{il}$ ($k\neq l$), for at least one $i\in\{1,\dots,n\}$. There exists $k\in\{1,\dots,p\}$ such that for $\forall~i\in\{1,\dots,n\}$, $\boldsymbol{\gamma}_{k}^\top\Sigma_{i}\boldsymbol{\gamma}_{k}=\exp(x_{i}^\top\boldsymbol{\beta})$. Let $\hat{\boldsymbol{\gamma}}$ be the maximum likelihood estimator of $\boldsymbol{\gamma}_{k}$ in Flury, 84. Then assuming that the assumptions are satisfied, $\hat{ \boldsymbol{\beta}}$ from our algorithm is $\sqrt{M_{n}}$-consistent estimator of $\boldsymbol{\beta}$.

Simulations


PCA and common PCA do not find the first principal direction, because they don't model covariates

Resting-state fMRI

Regression Coefficients


Age

Sex

Age*Sex



No statistical significant changes were found by massive edgewise regression

Brain Map of $\gamma$

Discussion

  • Regress matrices on vectors
  • Method to identify covariate-related directions
  • Theorectical justification
  • Manuscript: DOI: 10.1101/425033
  • R pkg: cap

Thank you!


Comments? Questions?




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