The Concomitant Formulation

The Concomitant Formulation#

The concomitant formulation estimates the model coefficients and the noise level σ jointly, instead of holding σ fixed. Reach for it when the variance is not homogeneous across your samples.

It applies to qiime classo regress only — see Robust Classification with the Huber Loss below for the classification equivalent.

The two formulations#

Set --p-concomitant True on qiime classo regress and the solver switches from the standard formulation to the concomitant formulation: R3 for least squares, R4 when combined with the Huber loss.

Mathematical formulation#

Standard Formulation (R1):

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

The noise level σ is held fixed rather than estimated.

Concomitant Formulation (R3, least squares):

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

Here σ is estimated jointly with the coefficients β, and there is no tuning parameter for σ. The coefficient n/2, with n the number of samples, is a fixed constant that follows from the perspective function of the squared loss — the plugin does not expose it.

Concomitant Formulation with Huber loss (R4):

min  h_{ρ,σ}(y - X·β) + n·σ + λ·||β||₁    s.t. C·β = 0

Combine --p-concomitant True with --p-huber True for the robust analogue. As in R3, σ is estimated with no tuning parameter; the σ-coefficient is n rather than n/2, again from the perspective function of the Huber loss.

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)”

When to use the concomitant formulation#

Use --p-concomitant True when:

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

  • Your samples come from different environments with potentially different noise levels

  • You need more reliable confidence intervals for your predictions

  • You want feature selection that holds up across diverse data

  • Computation time is not a binding constraint

Regression with the 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.

Robust Classification with the Huber Loss#

Important

The concomitant formulation is not available for classification. qiime classo classify has no --p-concomitant flag — the parameter is absent from its signature, and the underlying classo_problem forces formulation.concomitant = False for classification problems. Passing --p-concomitant to classify fails with an unrecognised-parameter error.

The robust option for classification is the Huber hinge loss, enabled with --p-huber True and tuned through --p-rho (default 0.0 for classification, which is wired to rho_classification).

Log-contrast classification 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 True \
    --p-rho 0.0 \
    --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_huber.qza

Then proceed with prediction and visualization using classifytaxa_lc_huber.qza.

trac classification with Huber loss#

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 True \
    --p-rho 0.0 \
    --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_huber.qza

Then proceed with prediction and visualization using classifytaxa_trac_huber.qza.

Reading the formulation label#

Run a regression with --p-concomitant True and the output reports:

Formulation: R3 (concomitant)

Add --p-huber True and the label becomes Formulation: R4 (concomitant + Huber). Either label tells you that:

  • the fit estimates the noise level σ jointly with the coefficients

  • σ comes out of the fit rather than out of a hyperparameter, so there is nothing to tune

  • the coefficients and predictions are more robust to heteroscedasticity than under the standard formulation

Practical notes#

  1. Compare both formulations. Run the analysis with --p-concomitant False and with True, and see which fits your data better

  2. Examine the residuals. Plot residuals against fitted values before deciding:

    • Constant variance → the standard formulation is sufficient

    • Variance rising or falling with the fitted values → use the concomitant formulation

  3. Cross-validation performance. Compare CV scores between the formulations; the concomitant fit tends to score better on heteroscedastic data

  4. Computational budget. The concomitant formulation takes roughly 1.5-2x more computation time, which is worth weighing on large datasets

  5. Nothing to tune for σ. The concomitant formulation estimates the noise level itself; there is no penalty parameter for σ to set