ohbm19

name:opening

## Statistical Methods for Population of Connectomes

Jaewon Chung | Johns Hopkins University

- talk - [bit.ly/graphstat](https://bit.ly/graphstat)
- repo - [github.com/neurodata/ohbm](https://github.com/neurodata/ohbm)

.foot[[j1c@jhu.edu](mailto:j1c@jhu.edu) | <http://neurodata.io/talks/> | [@neuro_data](https://twitter.com/neuro_data)]

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### Brains as connectomes

1. Vertices = region of interest
2. Edges = connectivity measure (e.g. # of fibers, correlation)

connectomes = graphs = adjacency matrix

.footnote[Talk Link = [bit.ly/graphstat](https://bit.ly/graphstat)]

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### Statistical inference on graphs

If edges in a graph are random,

1. model the randomness via a distribution
2. estimate the parameters of such distribution from data
3. use estimates in subsequent inference tasks

.footnote[Talk Link = [bit.ly/graphstat](https://bit.ly/graphstat)]

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#### A classical example

Collect height data of {males, females} from a population

.footnote[Talk Link = [bit.ly/graphstat](https://bit.ly/graphstat)]

1. model via Gaussian (normal) distribution - $\mathcal{N}(\mu, \sigma^2)$
2. estimate mean $(\mu)$ and variance $(\sigma^2)$ for males and females
3. inference tasks?
  - hypothesis testing: mean of male and female different?
  - classification: naive bayes

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### Two models for population of graphs

- .r[Random Dot Product Graph (RDPG)]
- .purple[Common Subspace Independent Edge Graph (COSIE)]

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### Random Dot Product Graphs (RDPG)

- All nodes have a *latent position* in d-dimensional space
- Probability of an edge between a pair of nodes is equal to the dot product of their latent positions
- Each graph has a latent position matrix $(X_i \in \mathbb{R}^{n\times d})$.

.footnote[[Athreya et al. (JMLR) 2018](http://jmlr.org/papers/v18/17-448.html)]

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### Estimating parameters of RDPG: Omnibus Embedding
(omni for short)

- m = number of connectomes
- n = number of vertices
- d = embedding dimensions

.footnote[[Athreya et al. (JMLR) 2018](http://jmlr.org/papers/v18/17-448.html)]

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### Common Subspace Independent Edge Graph (COSIE)

- Common *latent position* matrix shared across all graphs $(V\in \mathbb{R}^{n\times d})$.
- Individual graphs are transformation of the common matrix $(R_i \in \mathbb{R}^{d\times d})$.

.footnote[Arroyo et al. (In Prep)]

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### Estimating parameters of COSIE: Multiple Adjacency Spectral Embedding
(mase for short)

- m = number of connectomes
- n = number of vertices
- d, d' = embedding dimensions

.footnote[Arroyo et al. (In Prep)]

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### Graph Statistics in Python (GraSPy)

- python package
- scikit-learn API

- website: https://neurodata.io/graspy/
- github: https://github.com/neurodata/graspy

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### HNU1 Dataset

- dMRI processed via [ndmg](https://neurodata.io/ndmg/)
- Connectomes generated via deterministic tractography
- modified Desikan atlas (70 ROIs)
- 30 subjects
- scanned once every 5 days, 10 scans total

.footnote[[Data description](http://fcon_1000.projects.nitrc.org/indi/CoRR/html/hnu_1.html)]

Subsample 4 subjects (40 connectomes total)

Tasks

1. Cluster
2. Classify

---

class: middle, inverse

# .center[[Live Demo!](https://nbviewer.jupyter.org/github/neurodata/ohbm/blob/master/demo.ipynb)]

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### Acknowledgements
<div class="small-container">
  <img src="faces/jovo.png" />
  <div class="centered">Joshua Vogelstein</div>
</div>

<div class="small-container">
  <img src="faces/cep.png"/>
  <div class="centered">Carey Priebe</div>
</div>

<div class="small-container">
  <img src="faces/pedigo.jpg" />
  <div class="centered">Ben Pedigo</div>
</div>

<div class="small-container">
  <img src="faces/hayden.png" />
  <div class="centered">Hayden Helm</div>
</div>

<div class="small-container">
  <img src="faces/ebridge.jpg"/>
  <div class="centered">Eric Bridgeford</div>
</div>

<div class="small-container">
  <img src="faces/vikram.jpg"/>
  <div class="centered">Vikram Chandrashekhar</div>
</div>

<div class="small-container">
  <img src="faces/drishti.jpg"/>
  <div class="centered">Drishti Mannan</div>
</div>

<div class="small-container">
  <img src="faces/jesse.jpg"/>
  <div class="centered">Jesse Patsolic</div>
</div>

<div class="small-container">
  <img src="faces/falk_ben.jpg"/>
  <div class="centered">Benjamin Falk</div>
</div>

<div class="small-container">
  <img src="faces/loftus.jpg"/>
  <div class="centered">Alex Loftus</div>
</div>

<div class="small-container">
  <img src="faces/avanti.jpg"/>
  <div class="centered">Avanti Athreya</div>
</div>

---

## NeuroData Workshop
- August 19-23, 2019
- Baltimore, MD USA

[https://neurodata.devpost.com](https://neurodata.devpost.com)

.footnote[Talk Link = [bit.ly/graphstat](https://bit.ly/graphstat)]

---

## Contact
- email: j1c@jhu.edu
- mattermost: @j1c
- poster: W768 - Clustering Multimodal Connectomes

.footnote[Talk Link = [bit.ly/graphstat](https://bit.ly/graphstat)]

---

class: middle, inverse

# .center[Questions?]

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### Why embed graphs?
(backup slide)

Under RDPG or COSIE model, omni or mase, respectivly, will provide:

1. consistent estimates of parameters 
2. normal limiting distribution of parameters (central limit theorem)

.footnote[[Levin et al. (IEEE) 2017](https://ieeexplore.ieee.org/document/8215766)]

---
### Generative Model for RDPG

Latent position matrix, $X \in \mathbb{R}^{n\times d}$

Probability matrix, $P = XX^T$ and $P_{ij} \in [0, 1]$

Sampled graph, $A_{ij} = \text{Bernoulli}$  $(P_ij)$

- n = # of nodes
- d = latent dimension

---
### Generative Model for COSIE

Common latent position matrix, $V \in \mathbb{R}^{n\times d}$.

Individual matrix, $R \in \mathbb{R}^{d\times d}$.

Probability matrix, $P = VRV^T$ and $P_{ij} \in [0, 1]$.

Sampled graph, $A_{ij} = \text{Bernoulli}$ $(P_ij)$

- n = # of nodes
- d = latent dimension