GPCM Covariance Models¶

GaussianMixture supports all 14 Celeux & Govaert (1995) Gaussian Parsimonious Clustering Models (GPCM) -- the same family mclust implements -- not just the 6 covariance types in the main Gaussian Mixture Model guide.

The idea: a decomposed covariance¶

Each component's covariance is written as

\[ \Sigma_k = \lambda_k D_k A_k D_k^T \]

\(\lambda_k\) (volume): a scalar controlling the component's overall size.
\(A_k\) (shape): a diagonal matrix with determinant 1, decreasing entries -- how stretched the component is along its own axes.
\(D_k\) (orientation): an orthogonal matrix -- which way those axes point.

Each of the three can be **E**qual across components, **V**ary per component, or (volume/shape only) fixed to the **I**dentity (axis-aligned, no rotation). The 14 valid combinations are named by their 3-letter mclust code (Volume-shape-orientation):

Code	tgmm name	Meaning
`EII`	`'tied_spherical'`	one shared variance, spherical
`VII`	`'spherical'`	per-component variance, spherical
`EEI`	`'tied_diag'`	one shared diagonal covariance
`VEI`	(new)	per-component volume, shared diagonal shape
`EVI`	(new)	shared volume, per-component diagonal shape
`VVI`	`'diag'`	per-component diagonal covariance
`EEE`	`'tied_full'`	one shared full covariance
`VEE`	(new)	per-component volume, shared shape & orientation
`EVE`	(new)	shared volume & orientation, per-component shape
`EEV`	(new)	shared volume & shape, per-component orientation
`VVE`	(new)	shared orientation, per-component volume & shape
`VEV`	(new)	shared shape, per-component volume & orientation
`EVV`	(new)	shared volume, per-component shape & orientation
`VVV`	`'full'`	fully unconstrained per-component covariance

Pass either name to covariance_type (mclust codes are case-insensitive):

from tgmm import GaussianMixture

gmm = GaussianMixture(n_components=3, n_features=4, covariance_type='EVE')
gmm.fit(X)

print(gmm.covariance_type_)   # always the canonical mclust code, 'EVE' here
print(gmm.volume_, gmm.shape_, gmm.orientation_)  # the decomposition

covariances_ is always the assembled (n_components, n_features, n_features) matrices regardless of which of the 14 codes you use -- predict/sample/score/plot_gmm/save/load all work unchanged. volume_/shape_/orientation_ expose the decomposition itself (None for the 6 pre-existing types, which don't compute one).

Why bother with 8 more models?¶

More constrained models (fewer free parameters) are less prone to overfitting on small or high-dimensional data, at the cost of flexibility. The 14-model family lets you search this trade-off systematically instead of jumping straight from "fully tied" to "fully free" -- which is exactly what select_model does by default:

from tgmm import select_model

result = select_model(X, n_components=range(1, 8), criterion='bic')
print(result.best_.covariance_type_, result.best_.n_components)

select_model's default covariance_types is all 14 GPCM codes (the mclust workflow); pass an explicit subset to narrow the search.

What's and isn't supported for the 8 new codes¶

MLE only. mean_prior/covariance_prior (MAP/NIW estimation) raise NotImplementedError for the 8 new codes -- a conjugate-prior derivation specific to each parameterization is out of scope. full/diag/ spherical/tied_* still support MAP/NIW as before. weight_concentration_prior (the Dirichlet weight prior) is independent of the covariance parameterization and works with all 14.
No Gibbs mode. GaussianMixture(n_components=None, covariance_type=...) raises NotImplementedError for the 8 new codes, the same way it already does for tied_* -- a covariance that couples components (shared shape or orientation) doesn't factor per component the way Neal's Algorithm 3 needs.
init_covariances='empirical'/'global' project the usual empirical starting point onto the requested GPCM parameterization (the constrained MLE of that starting point); 'eye'/'random' just start from plain identity/random full matrices, since there's nothing to project yet.

The 5 iterative models¶

9 of the 14 codes (the 6 pre-existing ones, plus EVI, EEV, EVV) have a closed-form M-step. VEI, VEE, VEV, EVE, and VVE alternate between sub-steps (volume/shape/orientation, whichever vary) until the model's own complete-data objective stops improving by more than gpcm_tol (default 1e-6), or gpcm_max_iter (default 20) inner iterations are reached:

gmm = GaussianMixture(n_components=3, n_features=4, covariance_type='VEE',
                       gpcm_tol=1e-8, gpcm_max_iter=50)

Each EM iteration's inner alternation is warm-started from the previous EM iteration's decomposition, so in practice it usually takes only 1-2 inner iterations to re-settle, not the full gpcm_max_iter.

EVE and VVE additionally need to choose a shared orientation D that best serves every component's own (differently-shaped) scatter -- there's no closed form for this, so it's solved as a small orthogonal-manifold optimization (gradient-based, on the Stiefel manifold) each inner iteration, accepted only if it actually improves the fit (a generalized EM/GEM step, not a guaranteed-exact M-step) -- still monotone overall across EM iterations, as for every other covariance type.

Reference¶

Celeux, G. & Govaert, G. (1995). Gaussian parsimonious clustering models. Pattern Recognition, 28(5), 781-793. mclust's own naming and model definitions (Scrucca, Fop, Murphy & Raftery, 2016, The R Journal 8(1)) are followed here for compatibility, but every M-step in tgmm is derived independently from the papers' equations -- see docs/superpowers/specs/2026-10-14-phase3-gpcm-covariance-models-design.md for the full derivation (tgmm is MIT-licensed; mclust is GPL, and its source was not read or ported).