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Probabilistic size-and-shape functional mixed models
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Proposes a Bayesian functional mixed model that recovers the size-and-shape of a fixed effect function by using norm-preserving (unitary) phase actions on L2, with MCMC over phase functions under two priors.
Problem
Reliable recovery and uncertainty quantification of the population-level fixed effect function in a functional mixed model when unrestricted phase (x-axis) variation and additive measurement error are confounded.
Approach
Model object-level phase as a norm-preserving unitary action D_gamma(f)=(f∘gamma)sqrt(gamma') on L2, so the fixed effect is identifiable only up to size-and-shape; the random phase acts as a data-driven rotation of the basis, with informative Dirichlet or one-parameter priors on phase, and marginalized likelihood with MCMC sampling.
Key results
- On warpMix-generated data, Model 2-F reduces mu_2 estimation error from 0.0394 (warpMix) to 0.0070, an 82% reduction.
- Model 2-B gives the lowest error for mu_3 (0.0033 vs warpMix 0.0077) and Model 1-B for mu_1 (0.0151 vs 0.0179).
- On Berkeley data the model recovers two growth spurts while warpMix recovers only one; on PQRST data warpMix failed to produce an estimate.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Model 1-B instead of warpMix | Fixed-effect estimation · warpMix simulation, μ1 | estimation error: 0.0179 -> 0.0151 | Helped |
| Use Model 2-F instead of warpMix | Fixed-effect estimation · warpMix simulation, μ2 | estimation error: 0.0394 -> 0.0070 | Helped |
| Use Model 2-B instead of warpMix | Fixed-effect estimation · warpMix simulation, μ3 | estimation error: 0.0077 -> 0.0033 | Helped |
| Use PM2 instead of PM1 with a Fourier basis for μ (ablation) | Proposed model · warpMix simulation, μ3 | estimation error: 0.0195 -> 0.0044 | Helped |
| Use a Fourier basis instead of a B-spline basis for μ with PM2 (ablation) | Proposed model · warpMix simulation, μ2 | estimation error: 0.0152 -> 0.0070 | Helped |
Surprising findings
- warpMix failed to yield any estimate on the PQRST data.
Limitations
- No theoretical support for the recovery claims; authors state this is work in progress.
- MCMC is computationally slow and the number of basis functions must be fixed a priori.
- Results are mixed: warpMix beats some Bayesian variants on individual test functions (e.g., Model 1-F on mu_3).
Experimental setup
Datasets: Berkeley growth data PQRST ECG complexes simulated data from proposed model simulated data from warpMix model
Benchmarks: Berkeley growth data PQRST complexes simulated functional mixed model data
Metrics: squared estimation error Delta_mu on fixed effect posterior uncertainty average residual vs basis size
Baselines: warpMix BRFC
Builds on: functional mixed models square-root velocity transformation landmark size-and-shape analysis warpMix
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