Single Graphical Lasso

Single Graphical Lasso#

The single graphical lasso (SGL) estimates a sparse inverse covariance matrix from a precomputed covariance by solving an L1-penalized maximum likelihood problem. The penalty encourages sparsity in the precision matrix. Each non-zero entry is a conditional dependence between two features — a direct association between two taxa, with the indirect paths through the remaining taxa removed — and its magnitude gives the strength of that association.

The penalty is a single uniform L1 weight, λ₁, applied to every candidate edge, so all pairs are treated alike. Raising λ₁ removes edges. The environmental covariates are absent from this table, so an edge may also record two taxa responding to the same gradient rather than interacting.

Fit SGL first on a new table: it gives an initial view of the network and the core interactions among taxa, before you add weights or a latent block.

Fitting the model#

Estimate the precision matrix from the correlation matrix you computed earlier:

# 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: the number of samples the input covariance was computed from.

  • --p-lambda1-min: lower bound of the sparsity penalty λ₁.

  • --p-lambda1-max: upper bound of the sparsity penalty λ₁.

  • --p-n-lambda1: number of grid points between the two bounds.

  • --p-gamma 0.01: the extended BIC parameter.

  • --p-latent False: fits the standard graphical lasso, with no low-rank component.

  • --i-covariance-matrix: the input covariance, as a QIIME 2 artifact.

  • --o-solution: the output artifact holding the estimated sparse precision matrix.

Visualising the 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:

  • The action writes an interactive QIIME 2 visualization of the estimated network.

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

  • Open the resulting .qzv at QIIME 2 View.