Probabilistic size-and-shape functional mixed models

Fangyi Wang; Karthik Bharath; Oksana Chkrebtii; Sebastian Kurtek

REVIEW 4 major objections 4 minor 31 references

Probabilistic size-and-shape functional mixed models

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper demonstrates that a Bayesian functional mixed model can recover the size-and-shape of a square-integrable fixed effect function from noisy replicates with phase variation, although pointwise recovery is impossible.

desk verdict A genuinely new Bayesian functional mixed model with sound algebra and one solid independent simulation, but the headline recovery claim is only numerically supported and the missing identifiability theory should shape the review. read the letter →

arxiv 2411.18416 v1 pith:F3UDSY3J submitted 2024-11-27 stat.ME stat.COstat.ML

Fangyi Wang , Karthik Bharath , Oksana Chkrebtii , Sebastian Kurtek This is my paper · ORCID

classification stat.MEstat.COstat.ML MSC 62F1562R1062G08

keywords functionalmixedmodelssize-and-shapeanalysisphasevariabilitynorm-preservingactionBayesianinferenceunitarytransformationsfunctionalignment

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

In functional mixed models, observations are the sum of a population average, individual random deviations, and noise, and when individual time-warps (phase variation) are present, the average function cannot be recovered pointwise. This paper identifies a coarser target that can be recovered: the size-and-shape of the population function, defined as its equivalence class under the norm-preserving action $D_\gamma(f)=(f\circ\gamma)\sqrt{\dot\gamma}$, where $\gamma$ ranges over the group of smooth increasing time-warps of $[0,1]$. The authors build a Bayesian mixed model in $L^2[0,1]$ in which each observation is $f_i=D_{\gamma_i}(\mu+v_i+\epsilon_i)$, so $\gamma_i$ acts as a rotation of the function space rather than a simple time-shift. They report that phase-centering the posterior samples of $\mu$ gives a posterior mean that tracks the true size-and-shape in simulations and real data, and that it improves on the current state-of-the-art estimator. The practical payoff is that meaningful uncertainty quantification survives for a geometric summary of the population function even when the full function is unidentifiable.

What carries the argument

The load-bearing object is the norm-preserving action $D_\gamma(f)=(f\circ\gamma)\sqrt{\dot\gamma}$, where $\gamma$ belongs to the group $\Gamma$ of increasing diffeomorphisms of $[0,1]$; because it preserves the $L^2$ norm, each $D_\gamma$ is a unitary transformation, i.e., an infinite-dimensional rotation of the Hilbert space. The paper defines the size-and-shape of $f$ as the equivalence class $\{D_\gamma(f):\gamma\in\Gamma\}$, treats the random phase $\gamma_i$ as a rotation of the coordinate system in which $\mu$ and $v_i$ are expressed, and carries out inference on the phase-rotated basis coefficients. Two prior families on $\Gamma$---a one-parameter quadratic family and a Dirichlet-increment prior on discretized warps---regularize the rotation, and the marginal likelihood (integrating out $v_i$) is multivariate normal with mean $\Phi_i a$ and covariance $\sigma^2\mathrm{diag}(\dot\gamma_i)+\sigma_c^2\tilde\Phi_i\tilde\Phi_i^T$. Phase centering of posterior draws, using the average posterior phase, converts the posterior into summaries of the size-and-shape orbit.

What would settle it

Simulate $n$ curves from $f_i=D_{\gamma_i}(\mu+v_i+\epsilon_i)$ with phase functions drawn from a flexible family outside the two priors and with a true $\mu$ that is not exactly representable in the chosen basis; compute $d=\inf_{\gamma\in\Gamma}\|\hat\mu_{\mathrm{center}}-D_\gamma\mu\|$ for the centered posterior mean. If $d$ does not decrease as $n$ grows, the claim that the size-and-shape is recoverable is falsified.

Watch

Extended reading notes

Core claim

The paper's central claim is that the size-and-shape of a square-integrable fixed effect $\mu$---its $D_\gamma$-orbit in $L^2[0,1]$---is reliably recoverable under the Bayesian functional mixed model $f_i=D_{\gamma_i}(\mu+v_i+\epsilon_i)$, despite the well-known non-identifiability of $\mu$ itself. The norm-preserving operator $D_\gamma(f)=(f\circ\gamma)\sqrt{\dot\gamma}$ is a unitary rotation of the Hilbert space, so the model separates size-and-shape preserving deviations (the random phase $\gamma_i$) from size-and-shape altering ones (the random function $v_i$ and measurement error $\epsilon_i$). The authors estimate the posterior of $\mu$, then center every posterior draw by the average posterior phase $\bar\gamma$: $(\hat\mu^j\circ\bar\gamma)\sqrt{\dot{\bar\gamma}}$; the centered draws and their pointwise mean are taken as summaries of the size-and-shape class. In simulations the centered posterior mean recovers the true orbit even when the data were generated with value-preserving warping (the mechanism assumed by the state-of-the-art), and in real data examples it preserves topological features such as growth spurts and ECG complexes that the state-of-the-art estimate misses. The paper is explicit that theoretical support for this recovery is not yet provided; the claim is established through posterior sampling and numerical comparisons.

Load-bearing premise

The whole result depends on the assumption that the prior choices and the finite basis used to represent the mean keep time-warping from being confused with amplitude differences or noise; the paper demonstrates this by simulation but provides no proof.

Editorial extensions

If this is right

If the claim holds, practitioners can report a meaningful average function for populations whose members differ in timing, because the centered posterior mean targets the size-and-shape orbit rather than a misaligned pointwise mean.
The model produces posterior credible intervals for that orbit, so uncertainty quantification for the geometric average is available without finite-rank covariance assumptions on the error process.
Because the phase functions rotate the chosen basis, a modest number of fixed Fourier or B-spline basis functions can represent complex fixed effects; the paper reports that this beats an empirical FPCA basis fitted to the same data.
The size-and-shape target remains estimable when the true data generator uses value-preserving warping, which is the assumption of the state-of-the-art approach, so the two modeling traditions are not incompatible at the level of the recovered object.
The authors point out that the same construction extends to sparsely observed or fragmented functions and to curves and surfaces in higher dimensions through the same norm-preserving action.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A likely formal consequence of the numerical findings is a posterior contraction theorem for the orbit $[\mu]$ under the norm-preserving action; the natural next step is to prove that the phase-centered posterior accumulates at the true equivalence class as $n$ grows, with a rate depending on the support of the phase prior.
If the equivalence class under the norm-preserving action is genuinely larger than under the value-preserving action, the method should exhibit faster orbit recovery in high phase-variability regimes; this is a testable prediction that could be checked by comparing centered posterior distances across simulated phase distributions with increasing entropy.
The paper's own loss function for comparing estimators (squared pointwise error on centered representatives) implicitly rewards pointwise fidelity within the orbit; a stricter evaluation would report the distance to the orbit itself, which may change the apparent margin over the state of the art.
The framework suggests a general recipe for mixed effects models with nuisance symmetries: choose a group action whose orbits define the identifiable quantity, put a prior on the group element, and summarize the fixed effect by an orbit-centered posterior; applying this recipe with rotation groups rather than time-warps would yield Bayesian size-and-shape analysis for landmark or curve data.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Reading this made me think the editors should send it out, not desk reject it. The model itself is new: a Bayesian functional mixed model with phase variation through the norm-preserving action, and with the target of inference explicitly the size-and-shape equivalence class of the fixed effect mu. That framing is genuinely different from Claeskens et al., who use value-preserving warping, and from Cheng et al. The derivations in Appendices C and D look correct; the MH ratios are carefully worked out. The warpMix-based simulation is the strongest part: data generated under a value-preserving model, yet the proposed estimator beat warpMix on two of three targets. That is an independent check, not a circle. The real data examples are plausible and the sensitivity analysis in Appendix H is honest about what breaks. The soft spot is exactly where the reader and the stress-test note point: there is no identifiability theorem. Section 5 says 'What is lacking is theoretical support for the same.' That is an honest sentence, but it means the central claim that recovery of the size-and-shape is possible is a numerical demonstration, not an established property. The phase prior is supposed to guard against confounding, but there is no proof that the map from parameters to the sampling distribution is injective on the quotient by the norm-preserving action. If the prior or basis truncation does not fully separate phase from amplitude, the centered posterior mean could be a prior-biased representative rather than the true orbit. Also, the simulations are single runs without repeated-experiment error bars, and the comparison highlights the best of four model variants. No code is released. That said, the paper does not hide its limitations; Appendix H shows that under-specified Bf visibly destroys recovery, which is useful and honest. Who is this for? Statisticians working on functional mixed models or elastic functional data analysis. It is serious work, not a hack. But it should not be accepted on the strength of the abstract's claim as stated. My recommendation: send to peer review, with a referee who knows identifiability theory. The authors have the algebra; they need either a contraction or identifiability result for the quotient, or a weakening of the claim to 'numerical evidence suggests.' A conditional accept is plausible after that.

Referee Report

4 major / 4 minor

Summary. The paper proposes a Bayesian functional mixed model for a fixed effect function μ in the presence of object-level phase and amplitude variability. The observation model is fi = D_{γ_i}(μ + v_i + ε_i), where D_γ is the norm-preserving (square-root slope) action of the phase group Γ. The likelihood is obtained by marginalizing the random-effect coefficients, leading to the multivariate normal model in Eq. (6). Two prior models for phase functions are proposed, one one-parameter and one Dirichlet-based, and posterior inference is carried out by MCMC with Metropolis-Hastings updates. The main claim is that the size-and-shape of μ, i.e., its equivalence class under the norm-preserving action, can be recovered from the posterior, even though pointwise recovery is impossible. Evidence is provided through two simulation studies and several real-data examples, with comparisons to the warpMix estimator and a Bayesian registration approach.

Significance. If the central claim is established, the paper would make a useful contribution: it offers a Bayesian functional mixed model with unrestricted phase variation, avoids finite-rank covariance assumptions on the error process, and provides a principled way to summarize what can be learned about μ under a strong symmetry group. The simulations show competitive or better performance than warpMix in several examples, and the real-data analyses are visually plausible. The paper also ships detailed derivations of the marginal likelihood and Metropolis-Hastings ratios (Appendices C and D), which are valuable. However, the central claim is currently supported only by numerical demonstrations, and the paper explicitly concedes in Section 5 that theoretical support is lacking. The numerical evidence consists of single MCMC runs without repeated-experiment error bars, and the evaluation metric is partially tied to the chosen centering procedure. The significance of the work therefore depends on whether the identifiability and posterior-concentration issue can be resolved or at least sharply characterized.

major comments (4)

[Section 5 and Eq. (6)] The abstract claims to 'demonstrate that it is possible to recover the size-and-shape of a square-integrable μ', but the paper explicitly states in Section 5: 'What is lacking is theoretical support for the same.' This is a load-bearing gap: the model in Eq. (6) involves parameters θ = (a, σ², σ_c², γ_1,...,γ_n) where the phase functions enter both the mean Φ_i a and the covariance σ²diag(γ̇_i) + σ_c² Φ̃_i Φ̃_i^T. No theorem is given that the map from θ to the sampling distribution is identifiable on the quotient by the norm-preserving action, nor that the posterior contracts to the true orbit [μ]. Without such a result, the numerical demonstrations do not establish the general claim for square-integrable μ. A concrete remedy would be to prove identifiability of the orbit under the stated priors, or to state the claim as a conjecture supported by a systematic simulation study with varied μ, n, signal-to-noise ratio, and repeated seeds.
[Section 4.1 and Table 1] The numerical evidence consists of single MCMC runs: Example 1 draws data from the model itself, and Example 2 draws from the warpMix model, but no repeated simulations, Monte Carlo standard errors, or error bars are reported for the entries in Table 1. Consequently, it is unclear whether the reported improvements over warpMix are systematic or within Monte Carlo noise. I recommend reporting repeated-seed summaries (means and standard errors) for the error criterion Δμ, and ideally reporting an orbit-level error that is invariant to the norm-preserving action, rather than only a pointwise criterion applied after the posterior centering step.
[Section 4 and Appendix H] The recovery of μ is evaluated after centering posterior samples by the average posterior phase γ̄, i.e., (μ̂_j ∘ γ̄)√(γ̄̇). This centering is reasonable for visualization, but it means the 'recovered' representative is partly determined by the posterior phase estimates. If the posterior for γ_i is biased, the centered mean may lie in the wrong equivalence class while still looking well-centered. The paper does not provide an orbit-invariant measure of recovery error, so the numerical demonstrations do not fully separate phase recovery from amplitude recovery. A simple additional check would be to report distances between orbits, e.g., inf_{γ∈Γ} ‖μ̂_centered - D_γ(μ_true)‖, or to assess recovery of γ_i and μ jointly under a known generative model.
[Appendix H] Appendix H shows that under-specifying Bf, the number of basis functions for μ, visibly destroys recovery of μ, while the abstract promises recovery for general square-integrable μ. Since the model represents μ in a finite-dimensional basis and no data-driven selection of Bf is provided, the central claim is sensitive to a tuning parameter whose correct value is generally unknown. The paper should either restrict the claim to functions well-approximated by the chosen basis, or provide a practical procedure for choosing Bf and evidence that recovery is robust to reasonable misspecification.

minor comments (4)

[Section 4, Figure 2 caption] The caption contains a typo: 'warpMix esimate' should be 'warpMix estimate'.
[Eq. (7)] The one-parameter phase family γ(t) = t + α t(t−1) with α ∈ (−1,1) is stated to have γ̇(t) ∈ (0,2); please verify and state the exact range of γ̇, since the boundary behavior affects the prior support.
[Appendix D, Eq. (20)] The notation γ̇^{-1}_{cur}(·) in the Jacobian expression is confusing; it would be clearer to write the derivative of γ_cur^{-1} evaluated at the relevant point.
[Section 2, item (iii)] The statement that the measure of an equivalence class under the norm-preserving action is 'larger' than under the value-preserving action is heuristic and is not used in the subsequent theory or experiments; consider clarifying or removing it to avoid overclaiming.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the size-and-shape recovery claim is an empirical demonstration, not an identity; the orbit-centering convention and self-cited phase priors do not reduce the target to the estimator, though identifiability theory is admittedly missing.

full rationale

The central claim is that the posterior mean of µ, after centering with the average estimated phase, recovers the size-and-shape equivalence class [µ] under the norm-preserving action. I could not exhibit an equation in which the target is defined by the estimator or a parameter fitted to the outcome is reused as a prediction. The centering step (ˆµj ◦ ¯γ)√˙¯γ (Section 4) chooses a representative of the inferred orbit; it does not force the representative into the true orbit unless the posterior samples are already there, so the recovery claim retains independent content. The numerical demonstrations include data generated from the warpMix model (Example 2) and are scored against external benchmarks (warpMix, BRFC), so the evaluation is not self-referential. The phase priors PM1/PM2 are drawn from earlier work by the same authors (Bharath & Kurtek 2020; Matuk et al. 2022), but they enter as modeling assumptions, not as an imported uniqueness theorem that forbids alternatives; the paper nowhere invokes a prior theorem to declare its phase-amplitude separation forced. Section 5 concedes the absence of formal support ('What is lacking is theoretical support for the same, and this is work in progress'), and Appendix H shows sensitivity to under-specified Bf — both are correctness/identifiability gaps, not circularity. Per the hard rule, lack of a theorem is not a circular step, so the circularity score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the modeling choices Bf, Br, theta_gamma and on unproved identifiability/regularization assumptions about the phase priors. The size-and-shape target is defined by the norm-preserving action, a known geometric object, so no invented entities are introduced. The 'larger equivalence class helps recovery' statement is an unproved heuristic cited from Bogachev.

free parameters (4)

Bf (number of basis functions for mu) = 6 in most experiments; 12 for PQRST, gait, signature and gene expression
Chosen by the analyst a priori; under-specification prevents recovery of mu (Appendix H). The model's flexibility depends on this truncation.
Br (number of basis functions for random effect v_i) = 6 in all experiments
Chosen a priori; sensitivity analysis in Appendix H shows robustness to over/under-specification, but it still controls the random effect flexibility.
theta_gamma (concentration in PM2 phase prior) = 30
Fixed in all experiments; controls regularization toward identity phase. Sensitivity to theta_gamma is checked in Appendix H but the value is a modeling choice.
T_gamma (number of phase discretization points in PM2) = 7 in Example 2; 5 or 7 generally
Determines the dimension of the phase parameter space; chosen by the user and not data-driven.

assumptions (5)

standard math The norm-preserving action D_gamma(f) = (f composed with gamma) times sqrt(gamma') is a unitary operator and preserves the size-and-shape of f as defined via this action.
Invoked in Section 2 to define the model and the recovery target. This is a definitional/geometric fact.
domain assumption The equivalence class under the norm-preserving action is larger, in the measure-theoretic sense, than under the value-preserving action, and this larger class is what makes recovery of the size-and-shape feasible.
Stated at the end of Section 2 with a citation to Bogachev (1998) Example 6.5.2. The link from 'larger equivalence class' to 'easier reliable recovery' is asserted, not proved.
ad hoc to paper The priors on gamma_i guard against confounding between gamma_i and mu, so the posterior can separate phase from amplitude.
Section 3.2 asserts this; no identifiability theorem is given. This is the main unproved assumption underpinning the numerical recovery.
domain assumption The discretized model with basis truncation (Bf, Br) adequately represents mu and v_i so that the posterior summaries approximate the infinite-dimensional target.
Used throughout the numerical experiments; Appendix H shows under-specified Bf fails, confirming that this assumption matters.
domain assumption MCMC samples converge to the stationary posterior distribution.
Supported by trace plots in Appendix F, but no formal convergence diagnostics are reported.

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Cite this review

Pith. "Pith review of Probabilistic size-and-shape functional mixed models." pith.science (2026). https://pith.science/paper/F3UDSY3J

@misc{pith2026241118416,
  author       = {Pith},
  title        = {Pith review of: Probabilistic size-and-shape functional mixed models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F3UDSY3J}},
  note         = {Machine review of arXiv:2411.18416}
}

abstract

The reliable recovery and uncertainty quantification of a fixed effect function $\mu$ in a functional mixed model, for modelling population- and object-level variability in noisily observed functional data, is a notoriously challenging task: variations along the $x$ and $y$ axes are confounded with additive measurement error, and cannot in general be disentangled. The question then as to what properties of $\mu$ may be reliably recovered becomes important. We demonstrate that it is possible to recover the size-and-shape of a square-integrable $\mu$ under a Bayesian functional mixed model. The size-and-shape of $\mu$ is a geometric property invariant to a family of space-time unitary transformations, viewed as rotations of the Hilbert space, that jointly transform the $x$ and $y$ axes. A random object-level unitary transformation then captures size-and-shape \emph{preserving} deviations of $\mu$ from an individual function, while a random linear term and measurement error capture size-and-shape \emph{altering} deviations. The model is regularized by appropriate priors on the unitary transformations, posterior summaries of which may then be suitably interpreted as optimal data-driven rotations of a fixed orthonormal basis for the Hilbert space. Our numerical experiments demonstrate utility of the proposed model, and superiority over the current state-of-the-art.

Figures

Figures reproduced from arXiv: 2411.18416 by the authors.

**Figure 1.** (a) Berkeley growth rate curves. (b) Convex phase function γ. (c) One example function f from (a) (blue) transformed by value-preserving action f ◦ γ (red) and norm-preserving action (f ◦ γ) √ γ˙ (yellow). Here, f ◦ γ has the same classical notion of shape as f, whereas (f ◦ γ) √ γ˙ has the same size-and-shape as f as described in Section 2. (d) PQRST complexes. (e) PQRST pattern: P wave (first max), QRS complex (sh… view at source ↗

**Figure 2.** Row 1: Phase functions from PM1. Row 2: Phase functions from PM2. (a) Simulated data (n = 30). (b) Estimation of µ: ground truth (black), posterior samples (blue), posterior mean (red), warpMix esimate (yellow). (c)&(d) Histograms of posterior samples for σ 2 and σ 2 c , respectively (posterior mean in red; ground truth in black). (e) Estimation of phase function for a randomly chosen observation: ground truth (blac… view at source ↗

**Figure 3.** Comparison of estimation results based on Model 2-B and warpMix for (a) µ1, (b) µ2 and (c) µ3. In each panel, we show the ground truth (black), centered posterior samples (blue), centered posterior mean (red), and warpMix estimate (yellow) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Estimation results for Berkeley data (row 1) and PQRST complexes (row 2). (a) Posterior samples (blue) and posterior mean (red) of µ, and warpMix estimate (yellow). The warpMix model was unable to yield an estimate of µ for PQRST data. (b)&(c) Histograms of posterior s…

**Figure 5.** Figure 5: (a) Phase function. (b) Six modified Fourier basis functions. (c) The same basis functions [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: Average residual of (i) projection onto modified Fourier basis functions (blue), and (ii) [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: Row 1: Simulated data - row 1 in Figure 2 in Section 4.1. Row 2: Simulated data - row [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]

**Figure 8.** Figure 8: Top to bottom: data simulated using Prior Model 1 for phase functions, data simulated [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]

**Figure 9.** Figure 9: Top to bottom: data simulated using Prior Model 1 for phase functions, data simulated using [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]

**Figure 10.** Figure 10: Centered posterior mean (red) and 95% credible interval (dashed blue) for µ (ground truth, black) for different choices of Bf , Br and θγ in Prior Model 2 (PM 2) on phase functions; PM 1 refers to Prior Model 1 on phase. The data was generated using Bf = Br = 6 and θγ…

**Figure 11.** Figure 11: Estimation results for the pinch force, respiration, gait, signature acceleration and gene [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

[1]

Fixed effect coefficients: acan ∼ N (acur, Σa)

work page
[2]

Variance of error process: (σ2)can ∼ TN((σ2)cur, τ2 σ, 0, ∞)

work page
[3]

Variance of size-and-shape altering random effect: (σ2 c )can ∼ TN((σ2 c )cur, τ2 σc , 0, ∞)

work page
[4]

Size-and-shape preserving random effect (phase functions) under Prior Model 1: αcan i ∼ Uniform(αcur i − δ, αcur i + δ)

work page
[5]

TN stands for the truncated normal distribution

Size-and-shape preserving random effect (phase functions) under Prior Model 2: γcan = γcur ◦ ˜γ, where p(˜γ) ∼ Dirichlet(αt). TN stands for the truncated normal distribution. Notably, the proposal in 5. for phase functions utilizes the group structure of Γ. The proposal covariance matrix Σa is adapted during the burn-in period according to the empirical c...

work page
[6]

, n(Prior Model 2), k = 1,

or (γi)k, i= 1, . . . , n(Prior Model 2), k = 1, . . . , N− Nb. for j in 1:N do if mod(j, Nt) == 0 and j < Nb then

work page
[7]

Update Σa based on the empirical correlation matrix of the last Nt samples of a. end if

work page
[8]

if U < ρa, U ∼ Unif(0, 1) then

Propose acan and compute ρa using (11). if U < ρa, U ∼ Unif(0, 1) then

work page

Show all 31 references

[9]

Set aj+1 = acan. else
[10]

end if if mod(j, Nt) == 0 and j < Nb then

Set aj+1 = aj. end if if mod(j, Nt) == 0 and j < Nb then
[11]

Update τ 2 σ based on the acceptance rate of the last Nt samples of σ2. end if
[12]

if U < pσ2 , U ∼ Unif(0, 1) then

Propose (σ2)can and compute ρσ2 using (12). if U < pσ2 , U ∼ Unif(0, 1) then
[13]

Set (σ2)j+1 = (σ2)can. else
[14]

end if if mod(j, Nt) == 0 and j < Nb then

Set (σ2)j+1 = (σ2)j. end if if mod(j, Nt) == 0 and j < Nb then
[15]

Update τ 2 σc based on the acceptance rate of the last Nt samples of σ2 c . end if
[16]

if U < pσ2 c , U ∼ Unif(0, 1) then

Propose (σ2 c )can and compute ρσ2c using (12). if U < pσ2 c , U ∼ Unif(0, 1) then
[17]

Set (σ2 c )j+1 = (σ2 c )can. else
[18]

end if for i in 1:n do if Prior Model 1 then if mod(j, Nt) == 0 and j < Nb then

Set (σ2 c )j+1 = (σ2 c )j. end if for i in 1:n do if Prior Model 1 then if mod(j, Nt) == 0 and j < Nb then
[19]

Update δ based on the acceptance rate of the last Nt samples of αi. end if
[20]

if U < pαi , U ∼ Unif(0, 1) then

Propose αcan i and compute ραi using (13). if U < pαi , U ∼ Unif(0, 1) then
[21]

Set (αi)j+1 = αcan i . else
[22]

end if end if if Prior Model 2 then if mod(j, Nt) == 0 and j < Nb then

Set (αi)j+1 = (αi)j. end if end if if Prior Model 2 then if mod(j, Nt) == 0 and j < Nb then
[23]

Update α based on the acceptance rate of the last Nt samples of γi. end if
[24]

if U < pγi , U ∼ Unif(0, 1) then

Propose γcan i and compute ργi using (14). if U < pγi , U ∼ Unif(0, 1) then
[25]

Set (γi)j+1 = γcan i . else
[26]

end if end if end for end for 18 (a) (b) (c) (d) (e) Figure 7: Row 1: Simulated data - row 1 in Figure 2 in Section 4.1

Set (γi)j+1 = (γi)j. end if end if end for end for 18 (a) (b) (c) (d) (e) Figure 7: Row 1: Simulated data - row 1 in Figure 2 in Section 4.1. Row 2: Simulated data - row 2 in Figure 2 in Section 4.1. Row 3: Berkeley - row 1 in Figure 4 in Section 4.2. Row 4: PQRST - row 2 in F...
[27]

Pinch force data [Ramsay et al., 1995], which was also analyzed by Claeskens et al. [2021]

1995
[28]

Respiration data [Kurtek et al., 2013]

2013
[29]

Gait data [Kurtek et al., 2013]

2013
[30]

Signature acceleration data [Kneip and Ramsay, 2008]

2008
[31]

22 (a) (b) (c) (d) Figure 11: Estimation results for the pinch force, respiration, gait, signature acceleration and gene expression datasets (top to bottom)

Gene expression data [Srivastava et al., 2011b]. 22 (a) (b) (c) (d) Figure 11: Estimation results for the pinch force, respiration, gait, signature acceleration and gene expression datasets (top to bottom). (a) Data. (b)&(c) Centered posterior mean (black) and 95% credible int...

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Fixed effect coefficients: acan ∼ N (acur, Σa)

work page

[2] [2]

Variance of error process: (σ2)can ∼ TN((σ2)cur, τ2 σ, 0, ∞)

work page

[3] [3]

Variance of size-and-shape altering random effect: (σ2 c )can ∼ TN((σ2 c )cur, τ2 σc , 0, ∞)

work page

[4] [4]

Size-and-shape preserving random effect (phase functions) under Prior Model 1: αcan i ∼ Uniform(αcur i − δ, αcur i + δ)

work page

[5] [5]

TN stands for the truncated normal distribution

Size-and-shape preserving random effect (phase functions) under Prior Model 2: γcan = γcur ◦ ˜γ, where p(˜γ) ∼ Dirichlet(αt). TN stands for the truncated normal distribution. Notably, the proposal in 5. for phase functions utilizes the group structure of Γ. The proposal covariance matrix Σa is adapted during the burn-in period according to the empirical c...

work page

[6] [6]

, n(Prior Model 2), k = 1,

or (γi)k, i= 1, . . . , n(Prior Model 2), k = 1, . . . , N− Nb. for j in 1:N do if mod(j, Nt) == 0 and j < Nb then

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[7] [7]

Update Σa based on the empirical correlation matrix of the last Nt samples of a. end if

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[8] [8]

if U < ρa, U ∼ Unif(0, 1) then

Propose acan and compute ρa using (11). if U < ρa, U ∼ Unif(0, 1) then

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