Homework: Network Structure Inference

Posted: 11/18/2019
Last Update: 11/18/2019

Data

Download timeseries data derived from the ADHD-200 resting fMRI samples (http://fcon_1000.projects.nitrc.org/indi/adhd200/).

https://www.hcbravo.org/networks-across-scales/ data/roi_timeseries.csv.

The data consists of standardized measurements over 176 time points of 39 regions of interest (ROI). The rows of the data correspond to the 39 regions, the first column contains a label indicating the name of the brain ROI, with the remaining columns corresponding to the fMRI timeseries.

Correlation Network

Build a correlation network from the timeseries data as follows:

  • Compute correlation (Pearson) \(\hat{\rho}_{ij}\) for every pair of ROIs
  • Use Fisher’s transform to compute a $z$-value from the estimated correlation
$$ z_{ij} = \frac{1}{2} \log{\frac{(1+\hat{\rho}_{ij})}{(1-\hat{\rho}_{ij})}} $$

  • Compute a $p$-value for the hypothesis test of no correlation between each pair of regions using a normal distribution with mean 0 and variance $1/(n-3)$ with $n$ the number of timepoints (see SAND CH. 7).

  • Use a multiple testing correction procedure to derive a final list of network edges.

Partial Correlation Network

Next, build a partial correlation network from the timeseries data. Use partial correlations between ROIs $i$ and $j$ as follows:

  • For every ROI $k \neq i$ and $k \neq j$, compute partial correlation $\hat{\rho}_{ij|k}$ as the correlation between vectors $r_i$ and $r_j$, where $r_i$ are the residuals of linear model $x_i = b_0 + b_1 x_k$ where $x_i$ and $x_k$ are the timeseries for ROIs $i$ and $k$ respectively.

  • Compute a $p$-value $p_{ij|k}$ for the hypothesis test of no correlation between $i$ and $j$ (now conditioned on $k$) using Fisher’s transformation and a normal distribution with mean $0$ and variance $1/(n-1-3)$.

  • Combine the $p$-values for $i$ and $j$ using the Wille rule

$$ p_{ij} = \max \{ p_{ij|k} | \forall k \neq i,j \} $$
  • Use a multiple testing correction procedure as above to derive the final list of network edges

Graphical Lasso

Implement (or use a library) the graphical lasso procedure to estimate a sparse inverse covariance matrix with entries $\omega_{ij}$.

  • Recall that the sparse inverse covariance estimate depends on a penalty parameter $\lambda$. Build at least two networks with $\lambda$ values that return a very sparse and a very dense network. Additionally, choose a model selection criterion (see SAND Ch.7 pg. 128 for initial ideas), and use the network derived with the optimal $\lambda$ as defined by your chosen criterion.

  • Note, you can derive partial correlation coefficients for $i$ and $j$ from the inverse covariance matrix as

You can find implementations of the graphical lasso in R package glmnet and in python module sklearn.covariance.

$$ \hat{\rho}_{ij|V\setminus\{i,j\}} = \frac{-\omega_{ij}}{\sqrt{\omega_{ii}\omega_{jj}}} $$

Discussion

  1. Compare the networks derived from these procedures using relevant network structure statistics, e.g., vertex degree distribution. Support with appropriate figures.

  2. Provide a qualitative comparison of the derived networks. E.g., are there brain ROIs that are consistently linked in all networks?

Submission

Submit a writeup describing your methods (e.g., multiple testing correction procedure used, etc.) and discussion. Include code as an appendix, or within the writeup if using jupyter notebook, Rmarkdown or similar.