Advanced: Using Concomitant Formulation

Advanced: Using Concomitant Formulation#

The concomitant formulation provides a more robust approach to both regression and classification by jointly estimating the model coefficients and the noise level (σ). This is particularly useful when variance is not homogeneous across your samples.

What is Concomitant Formulation?#

By setting --p-concomitant True, you switch from standard formulation to the concomitant formulation (R3):

Mathematical Formulation#

Standard Formulation (R1):

min ||y - X·β||² / σ + λ·||β||₁    s.t. C·β = 0

Assumes a fixed noise level across all samples.

Concomitant Formulation (R3):

min ||y - X·β||² / σ + e·σ + λ·||β||₁    s.t. C·β = 0

Where e = 20.0 is the penalty parameter for the standard deviation (σ).

Key Differences#

Aspect

Standard (R1)

Concomitant (R3)

Noise level

Fixed

Jointly estimated

Heteroscedasticity

Assumed equal variance

Adaptive to varying noise

Uncertainty estimates

May be inaccurate

More reliable

Computation

Faster

Slightly slower

Output label

“Formulation: R1”

“Formulation: R3 (concomitant with e = 20.0)”

When to Use Concomitant Formulation#

Use --p-concomitant True when:

  • Your residual plots show increasing variance with predicted values (heteroscedasticity)

  • You have samples from different environments with potentially different noise levels

  • You need more reliable confidence intervals for your predictions

  • You want robust feature selection across diverse data

  • Computation time is not a critical constraint

Regression with Concomitant Formulation#

Log-Contrast with Concomitant#

Replace the training step with:

qiime classo regress \
    --i-features data/regress-xtraining_lc.qza \
    --i-c data/ccovariates_lc.qza \
    --i-weights data/wcovariates_lc.qza \
    --m-y-file data/atacama-selected-covariates-veg.tsv \
    --m-y-column average-soil-temperature \
    --p-concomitant True \
    --p-stabsel \
    --p-cv \
    --p-path \
    --p-lamfixed \
    --p-stabsel-threshold 0.5 \
    --p-cv-seed 1 \
    --p-no-cv-one-se \
    --o-result data/regresstaxa_lc_concomitant.qza

Then proceed with prediction and visualization using regresstaxa_lc_concomitant.qza.

TRAC with Concomitant#

Replace the training step with:

qiime classo regress \
    --i-features data/regress-xtraining_trac.qza \
    --i-c data/ccovariates_trac.qza \
    --i-weights data/wcovariates_trac.qza \
    --m-y-file data/atacama-selected-covariates-veg.tsv \
    --m-y-column average-soil-temperature \
    --p-concomitant True \
    --p-stabsel \
    --p-cv \
    --p-path \
    --p-lamfixed \
    --p-stabsel-threshold 0.5 \
    --p-cv-seed 1 \
    --p-no-cv-one-se \
    --o-result data/regresstaxa_trac_concomitant.qza

Then proceed with prediction and visualization using regresstaxa_trac_concomitant.qza.

Classification with Concomitant Formulation#

Log-Contrast with Huber Loss#

For both log-contrast classification and TRAC classification, replace the training step with:

qiime classo classify \
    --i-features data/classify-xtraining_lc.qza \
    --i-c data/ccovariates_lc.qza \
    --i-weights data/wcovariates_lc.qza \
    --m-y-file data/atacama-selected-covariates-veg.tsv \
    --m-y-column vegetation \
    --p-huber False \
    --p-concomitant True \
    --p-stabsel \
    --p-cv \
    --p-path \
    --p-lamfixed \
    --p-stabsel-threshold 0.5 \
    --p-cv-seed 42 \
    --p-no-cv-one-se \
    --o-result data/classifytaxa_lc_concomitant.qza

Then proceed with prediction and visualization using classifytaxa_lc_concomitant.qza.

TRAC with Concomitant#

Replace the training step with:

qiime classo classify \
    --i-features data/classify-xtraining_trac.qza \
    --i-c data/ccovariates_trac.qza \
    --i-weights data/wcovariates_trac.qza \
    --m-y-file data/atacama-selected-covariates-veg.tsv \
    --m-y-column vegetation \
    --p-huber False \
    --p-concomitant True \
    --p-stabsel \
    --p-cv \
    --p-path \
    --p-lamfixed \
    --p-stabsel-threshold 0.5 \
    --p-cv-seed 42 \
    --p-no-cv-one-se \
    --o-result data/classifytaxa_trac_concomitant.qza

Then proceed with prediction and visualization using classifytaxa_trac_concomitant.qza.

Interpreting Results with Concomitant Formulation#

When you run your pipeline with --p-concomitant True, the output will display:

Formulation: R3 (concomitant with e = 20.0)

This indicates:

  • The model is using concomitant formulation

  • The noise level parameter e = 20.0 controls the penalty strength

  • Coefficients and predictions are more robust to heteroscedasticity

Tips for Optimal Results#

  1. Compare Both Formulations: Run your analysis with both --p-concomitant False and True to see which produces better results for your data

  2. Examine Residuals: Plot residuals vs. fitted values before deciding:

    • If variance is constant → Standard formulation is sufficient

    • If variance increases/decreases with fitted values → Use concomitant

  3. Cross-Validation Performance: Compare CV scores between formulations; concomitant often performs better on heteroscedastic data

  4. Computational Budget: Concomitant requires ~1.5-2x more computation time, so consider this for large datasets

  5. Parameter Tuning: The default e = 20.0 works well for most cases, but can be adjusted if needed

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