Single Graphical Lasso

Single Graphical Lasso#

This tutorial demonstrates how to estimate a sparse inverse covariance matrix using the Single Graphical Lasso (SGL) method. This approach identifies conditional dependencies between features (e.g., microbial taxa) by solving an L1-penalized maximum likelihood problem, encouraging sparsity in the precision matrix.

Purpose: Estimates sparse direct associations between microbial taxa

Key characteristics:

  • Assumes all edges have equal importance

  • Uses uniform L1 penalty (λ₁) across all potential connections

  • Identifies direct conditional dependencies in the network

  • Best for: Initial network exploration and identifying core microbial interactions

Interpretation:

  • Edges represent direct associations after removing indirect effects

  • Edge thickness indicates association strength

  • Network sparsity controlled by λ₁ parameter

  • May include environment-mediated associations

Step 1: Estimate a Sparse Model#

We begin by estimating the inverse covariance (precision) matrix from a precomputed covariance matrix:

# sparse model
qiime gglasso solve-problem \
     --p-n-samples 50 \
     --p-lambda1-min 0.001 \
     --p-lambda1-max 1 \
     --p-n-lambda1 50 \
     --p-gamma 0.01 \
     --p-latent False \
     --i-covariance-matrix data/atacama-table-corr.qza \
     --o-solution data/atacama-solution-sgl.qza \
     --verbose

Explanation:

  • --p-n-samples 50: Number of individuals used to compute the input covariance matrix.

  • --p-lambda1-min: Lower-bound for the sparsity penalty (λ₁).

  • --p-lambda1-max: Upper-bound for the sparsity penalty (λ₁).

  • --p-n-lambda1: Number of grid points between the min and max lambda values.

  • --p-gamma 0.01: Controls the model selection criterion (e.g., eBIC).

  • --p-latent False: Indicates that this is a standard graphical lasso (not a latent variable model).

  • --i-covariance-matrix: Input covariance matrix in QIIME 2 format.

  • --o-solution: Output artifact containing the estimated sparse inverse covariance matrix.

Step 2: Visualize the Estimated Network#

# visualize the results
qiime gglasso summarize \
    --i-solution data/atacama-solution-sgl.qza \
    --p-label-size 25pt \
    --o-visualization data/sgl-summary.qzv

Explanation:

  • Generates an interactive QIIME 2 visualization of the estimated network.

  • --p-label-size 25pt: Sets the font size of node labels in the network plot.

  • The output .qzv file can be viewed using QIIME 2 View.

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