Adaptive Graphical Lasso

Adaptive Graphical Lasso#

This tutorial demonstrates how to incorporate environmental covariates into sparse inverse covariance estimation using the Adaptive Graphical Lasso model. This method allows users to assign custom penalty weights to selected covariates, enabling more flexible and biologically informed regularization.

Purpose: Incorporates prior knowledge about environmental covariates

Key characteristics:

  • Applies differential penalties to different types of associations

  • Lower penalties for biologically meaningful connections

  • Higher penalties for environment-mediated associations

  • Integrates environmental metadata directly into the model

Interpretation:

  • Edges represent environment-independent microbial interactions

  • Environmental covariates explicitly modeled rather than confounded

  • Most biologically interpretable results

  • Requires domain knowledge for penalty weight selection

Step 1: Add Metadata#

First, apply a modified centered log-ratio (mCLR) transformation and standardize environmental covariates alongside microbiome features:

# standardise covariates 
qiime gglasso transform-features \
     --p-transformation mclr \
     --p-add-metadata True \
     --i-table data/atacama-counts.qza \
     --i-taxonomy data/classification.qza \
     --m-sample-metadata-file data/selected-atacama-sample-metadata.tsv  \
     --o-transformed-table data/atacama-table-mclr-meta.qza

Explanation:

  • --p-add-metadata True: Appends covariates (e.g., environmental features) to the feature table.

  • --p-transformation mclr: Applies mCLR transformation to address compositionality.

  • The output table contains both standardized microbial and environmental data.

Step 2: Calculate new Covariance Matrix#

Next, compute the scaled covariance matrix for the transformed table:

# calculate correlation including covariates
qiime gglasso calculate-covariance \
     --p-method scaled \
     --i-table data/atacama-table-mclr-meta.qza \
     --o-covariance-matrix data/atacama-table-corr-meta.qza

Explanation:

  • --p-method scaled: Produces a Pearson correlation matrix (i.e., scaled covariance).

  • Covariates are included in the input table and will be treated equally with other features.

Step 3: Fit the Adaptive Graphical Lasso Model#

Use adaptive penalty weights for selected covariates to guide the network estimation:

# sparse model with specific weights for covariates
qiime gglasso solve-problem \
     --p-n-samples 50 \
     --p-lambda1-min 0.001 \
     --p-lambda1-max 1 \
     --p-n-lambda1 2 \
     --p-gamma 0.01 \
     --p-latent False \
     --p-weights elevation 0.01 ph 0.01 average-soil-relative-humidity 0.01 average-soil-temperature 0.01  \
     --i-covariance-matrix data/atacama-table-corr-meta.qza \
     --o-solution data/atacama-solution-adapt.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-weights: Assigns individual penalty weights to selected covariates.

  • --p-latent False: Indicates that this is a standard graphical lasso w/o low-rank.

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

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

Step 4: Visualize the Estimated Network#

Generate a visualization that includes covariates in the network:

# visualize the results
qiime gglasso summarize \
    --i-solution data/atacama-solution-adapt.qza \
    --p-label-size 25pt \
    --p-n-cov 4 \
    --o-visualization data/adapt-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.

  • --p-n-cov 4: Indicates the number of covariates included, so they are treated as separate nodes.

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

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