class: left, middle, inverse, title-slide # Hybrid Principal Components Analysis ### Emilie Campos ### Department of Biostatistics, UCLA ### 2020-03-10 (updated: 2020-03-10) --- <div class="my-footer"><span>Emilie Campos               Hybrid Principal Component Analysis</span></div> # Outline Aaron Scheffler, Donatello Telesca, Qian Li, Catherine A Sugar, Charlotte Distefano, Shafali Jeste, Damla Şentürk, Hybrid principal components analysis for region-referenced longitudinal functional EEG data, *Biostatistics*, Volume 21, Issue 1, January 2020, Pages 139–157, https://doi.org/10.1093/biostatistics/kxy034 - Introduction - The HPCA decomposition - Group-level inference via bootstrap - Application to the lexical-semantic processing data --- <div class="my-footer"><span>Emilie Campos               Hybrid Principal Component Analysis</span></div> # Introduction - Approximately 30% of children with autism spectrum disorder (ASD) never gain spoken language - Electroencephalography (EEG) gives researchers a unique opportunity to compare and contrast neurocognitive processes involved in language and communication developement without relying on the children's ability to understand directions or provide an overt behavioral response - EEG data possess a complex, high-dimensional structure: regional, functional, and longitudinal dimensions - Standard analysis involves collapsing information along multiple dimensions, i.e. averaging over longitudinal dimension, focusing on one frequency band, or averaging over scalp regions --- <div class="my-footer"><span>Emilie Campos               Hybrid Principal Component Analysis</span></div> # HPCA - Let `\(Y_{di}(r, \omega, s)\)` denote the log principal power for - subject `\(i\)`, `\(i = 1, \ldots, n_d\)` - from group `\(d\)`, `\(d = 1, \ldots, D\)`, - in region `\(r\)`, `\(r = 1, \ldots, R\)`, - at frequency `\(\omega\)`, `\(\omega\in\Omega\)`, and - segment `\(s\)`, `\(s\in S\)` `\begin{align*} Y_{di}(r, \omega, s) = \mu(\omega, s) + \eta_d(r, \omega, s) + Z_{di}(r, \omega, s) + \epsilon_{di}(r, \omega, s) \end{align*}` --- <div class="my-footer"><span>Emilie Campos               Hybrid Principal Component Analysis</span></div> # HPCA Estimation Procedure 1. Estimation fixed effects: `\(\widehat{\mu}(\omega, s)\)`, `\(\widehat{\eta}_d(r, \omega, s)\)` 2. Estimation of marginal covariances and measurement error variance: `\(\widetilde{\Sigma}_{d, \Omega}(\omega, \omega')\)`, `\(\widetilde{\Sigma}_{d, \mathcal{S}}(s, s')\)`, `\(\widetilde{\Sigma}_{d, \mathcal{R}}\)`, `\(\widehat{\sigma}^2_d\)` 3. Estimation of marginal eigencomponents: eigenvalue, eigenvector/function pairs `\(\{\tau_{d\ell, \Omega}, \phi_{d\ell}(\omega): \ell = 1, \ldots, L\}\)`, `\(\{\tau_{dm, \mathcal{S}}, \psi_{dm}(s): m = 1, \ldots, M\}\)`, `\(\{\tau_{dk, \mathcal{R}}, \text{v}_{dk}(r): k = 1, \ldots, K\}\)` 4. Estimation of variance components and subject-specific scores via linear mixed effects models --- <div class="my-footer"><span>Emilie Campos               Hybrid Principal Component Analysis</span></div> # Weak Separability - Weak separability implies that the direction of variation along any one of the dimensions of the EEG data is the same across fixed slices of the other dimensions `\begin{align*} \Sigma_{d, \Omega}(\omega, \omega') &= \sum_r \int_\mathcal{S} \text{cov}\{Z_{di}(r, \omega', s), Z_{di}(r, \omega, s)\}ds \\ &= \sum_{\ell=1}^\infty \tau_{d\ell, \Omega} \phi_{d\ell}(\omega) \phi_{d\ell}(\omega'),\\ \Sigma_{d, \mathcal{S}}(s, s') &= \sum_r \int_\Omega \text{cov}\{Z_{di}(r, \omega, s), Z_{di}(r, \omega, s')\}d\omega \\ & = \sum_{m=1}^\infty \tau_{dm, \mathcal{S}}\psi_{dm}(s)\psi_{dm}(s'),\\ (\Sigma_{d, \mathcal{R}})_{r, r'} &= \int_\mathcal{S}\int_\Omega \text{cov}\{Z_{di}(r, \omega, s), Z_{di}(r', \omega, s)\}d\omega ds \\ &= \sum_{k=1}^R \tau_{dk, \mathcal{R}}\text{v}_{dk}(r)\text{v}_{dk}(r') \end{align*}` --- <div class="my-footer"><span>Emilie Campos               Hybrid Principal Component Analysis</span></div> # HPCA Decomposition - Decomposition of `\(Y_{di(r, \omega, s)}\)` becomes `\begin{align*} Y_{di}(r, \omega, s) &= \mu(\omega, s) + \eta_d(r, \omega, s)\\ &+\sum_{k=1}^R\sum_{\ell=1}^\infty\sum_{m=1}^\infty \xi_{di, k\ell m} \text{v}_{dk}(r) \phi_{d\ell}(\omega)\psi_{dm}(s)\\ &+ \epsilon_{di}(r, \omega, s) \end{align*}` - Decomposition of the total covariance `\(\Sigma_{d, T}\{(r, \omega, s), (r', \omega', s')\}\)` `\begin{align*} &= \text{cov}\{Z_{di}(r, \omega, s), Z_{di}(r', \omega', s')\} + \sigma^2_d\delta\{(r, \omega, s), (r', \omega', s')\}\\ &= \sum_{k=1}^R\sum_{\ell=1}^\infty\sum_{m=1}^\infty \tau_{d, k\ell m} \text{v}_{dk}(r)\phi_{d\ell}(\omega)\psi_{dm}(s) \text{v}_{dk}(r')\phi_{d\ell}(\omega')\psi_{dm}(s') \\ &\;\;\;\;\;+ \sigma^2_d\delta\{(r, \omega, s), (r', \omega', s')\} \end{align*}` --- <div class="my-footer"><span>Emilie Campos               Hybrid Principal Component Analysis</span></div> # Estimate variance components and scores - Denote by `\(\boldsymbol{Y}_{di}\)` the vectorized form of `\(Y_{di}(r, \omega, s)\)` `\begin{align*} \boldsymbol{Y}_{di} &= \boldsymbol{\mu}_i + \boldsymbol{\eta}_{di} + \boldsymbol{Z}_{di} + \boldsymbol{\epsilon}_{di}\\ &=\boldsymbol{\mu}_i + \boldsymbol{\eta}_{di} + \sum_{g=1}^{G'} \zeta_{dig} \boldsymbol{\varphi}_{dig} + \boldsymbol{\epsilon}_{di} \end{align*}` - Given estimates of `\(\boldsymbol{\mu}_i\)`, `\(\boldsymbol{\eta}_{di}\)`, and `\(\boldsymbol{\varphi}_{dig}\)`, estimates of the variance components `\(\kappa_{dg} = var(\zeta_{dig})\)` and `\(\sigma^2_d\)` are obtained using maximum likelihood - Subject-specific scores `\(\zeta_{dig}\)` are estimated using BLUPs --- <div class="my-footer"><span>Emilie Campos               Hybrid Principal Component Analysis</span></div> # Two-dimensional HPCA - The 3D HPCA can be reduced to a 2D HPCA when the longitudinal dimension may not be of interest `\begin{align*} Y_{di}(r, \omega) &= \mu(\omega) + \eta_d(r, \omega) + Z_{di}(r, \omega) + \epsilon_{di}(r, \omega) \\ &= \mu(\omega) + \eta_d(r, \omega)+\sum_{k=1}^R\sum_{\ell=1}^\infty \xi_{di, k\ell} \text{v}_{dk}(r) \phi_{d\ell}(\omega)+ \epsilon_{di}(r, \omega) \end{align*}` --- <div class="my-footer"><span>Emilie Campos               Hybrid Principal Component Analysis</span></div> # Group-level inference - Use a parametric boostrap procedure to test the null hypothesis that all groups have equal means in the scalp region `\(r\)` `\begin{equation*} H_0: \eta_d(r, \omega, s) = \eta(r, \omega, s), \text{ for } d = 1, \ldots, D \end{equation*}` - Generate outcomes under the null in region `\(r\)` as `\begin{align*} Y_{di}^b(r, \omega, s) &= \widehat{\mu}(\omega, s) + \widehat{\eta}(r, \omega, s) + \sum_{g=1}^{G'} \zeta_{dig}^b\widehat{\varphi}_{dig}(r, \omega, s) + \epsilon_{di}^b(r, \omega, s) \end{align*}` and in the other regions as `\begin{align*} Y_{di}^b(r, \omega, s) &= \widehat{\mu}(\omega, s) + \widehat{\eta}_d(r, \omega, s) + \sum_{g=1}^{G'} \zeta_{dig}^b\widehat{\varphi}_{dig}(r, \omega, s) + \epsilon_{di}^b(r, \omega, s) \end{align*}` where `\(\zeta_{di}^b \sim N(0, \widehat{\kappa}_{dg})\)` and `\(\epsilon_{di}^b(r, \omega, s) \sim N(0, \widehat{\sigma}^2_d)\)` --- <div class="my-footer"><span>Emilie Campos               Hybrid Principal Component Analysis</span></div> # Application: Study Cohort - Study cohort: 31 children aged 5-11 years old were recruited; 14 typically developing (TD), 10 verbal ASD (vASD), and 7 minimally verbal ASD (mvASD) - Goal: study the neural mechanisms underlying language impairment in children with ASD (DiStefano, 2019) - Diagnoses made prior to enrollment and confirmed using the Autism Diagnostic Observation Schedule (ADOS) and Social Communication Questionnaire --- <div class="my-footer"><span>Emilie Campos               Hybrid Principal Component Analysis</span></div> # Application: Experiments - Audio paradigm: a picture was presented and an audio recording of the spoken word was played that either matched or did not math - Visual paradigm: a picture was presented and an image of the word appeared on that either matched or did not match - Vocabulary included 60 basic nouns (e.g., bird, dog, bike) .pull-left[ .center[<img src="figures/experiment_sound.png" width=400 height=250>] ] .pull-right[ .center[<img src="figures/experiment_text.png" width=400 height=250>] ] --- <div class="my-footer"><span>Emilie Campos               Hybrid Principal Component Analysis</span></div> # Application: Mean Functions .center[ <img src="BIOS285_final-project-presentation_files/figure-html/Plot mean and eta functions-1.png" width="800px" /> ] --- <div class="my-footer"><span>Emilie Campos               Hybrid Principal Component Analysis</span></div> # Application: Eigenfunctions .center[ <img src="BIOS285_final-project-presentation_files/figure-html/Plot eigenfunctions-1.png" width="400px" /><img src="BIOS285_final-project-presentation_files/figure-html/Plot eigenfunctions-2.png" width="400px" /> ] --- <div class="my-footer"><span>Emilie Campos               Hybrid Principal Component Analysis</span></div> # Application: Eigenvectors .center[ <img src="BIOS285_final-project-presentation_files/figure-html/Plot eigenvectors-1.png" width="400px" /><img src="BIOS285_final-project-presentation_files/figure-html/Plot eigenvectors-2.png" width="400px" /> ] --- <div class="my-footer"><span>Emilie Campos               Hybrid Principal Component Analysis</span></div> # Application: Bootstrap inference <img src="BIOS285_final-project-presentation_files/figure-html/unnamed-chunk-2-1.png" width="800px" /> --- <div class="my-footer"><span>Emilie Campos               Hybrid Principal Component Analysis</span></div> # Future Directions - Current development: adding multilevel dimension to HPCA `\begin{equation*} Y_{dic}(r, \omega, s) = \mu(\omega, s) + \eta_d(r, \omega, s) + Z_{di}(r, \omega, s) + W_{dic}(r, \omega, s) + \epsilon_{dic}(r, \omega, s) \end{equation*}` - Goal: Decompose the variation from the subject and subject-condition levels separately using weak separability on the respective covariances --- class: inverse, center, middle # Questions? <a href="mailto:ejcampos@ucla.edu"><i class="fa fa-paper-plane fa-fw"></i> ejcampos@ucla.edu</a><br> <a href="https://emjcampos.netlify.com"><i class="fa fa-link fa-fw"></i> emjcampos.netlify.com</a><br> <a href="http://twitter.com/emjcampos"><i class="fa fa-twitter fa-fw"></i> @emjcampos</a><br> <a href="http://github.com/emjcampos"><i class="fa fa-github fa-fw"></i> @emjcampos</a><br>