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<!DOCTYPE html>
<html>
<head>
<title>ohbm19</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8" />
<link rel="stylesheet" href="fonts/quadon/quadon.css">
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<script type="text/javascript" src="assets/plotly/plotly-latest.min.js"></script>
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<body>
<textarea id="source">
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)
<img src="images/neurodata_purple.png" style="height:250px; float:right;"/>
<br><br><br><br><br><br><br><br><br>
<!--
<img src="images/funding/jhu_bme_blue.png" STYLE="HEIGHT:95px;"/>
-->
.foot[[j1c@jhu.edu](mailto:[email protected]) | <http://neurodata.io/talks/> | [@neuro_data](https://twitter.com/neuro_data)]
---
### 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)]
---
### 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)]
---
#### 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
---
### Two models for population of graphs
- .r[Random Dot Product Graph (RDPG)]
- .purple[Common Subspace Independent Edge Graph (COSIE)]
---
### 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)]
---
### Estimating parameters of RDPG: Omnibus Embedding
(omni for short)
<img src="images/omni_method.png" STYLE="width:100%" />
- 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)]
---
### 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)]
---
### Estimating parameters of COSIE: Multiple Adjacency Spectral Embedding
(mase for short)
<img src="images/mase_method.png" STYLE="width:100%" />
- m = number of connectomes
- n = number of vertices
- d, d' = embedding dimensions
.footnote[Arroyo et al. (In Prep)]
---
### Graph Statistics in Python (GraSPy)
- python package
- scikit-learn API
- website: https://neurodata.io/graspy/
- github: https://github.com/neurodata/graspy
---
### 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)]
---
### 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/minh.jpg"/>
<div class="centered">Minh Tang</div>
</div>
<div class="small-container">
<img src="faces/avanti.jpg"/>
<div class="centered">Avanti Athreya</div>
</div>
<div class="small-container">
<img src="faces/gkiar.jpg"/>
<div class="centered">Greg Kiar</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: [email protected]
- mattermost: @j1c
- poster: W768 - Clustering Multimodal Connectomes
.footnote[Talk Link = [bit.ly/graphstat](https://bit.ly/graphstat)]
---
class: middle, inverse
# .center[Questions?]
---
### 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
</textarea>
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