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Depth-based estimation for multivariate functional data with phase variability

Ana Arribas-Gil, Sara López-Pintado

TL;DR

This work develops a depth-based, registration-free approach to estimate a central amplitude pattern in multivariate functional data with cross-component time warping under the latent deformation model $X_{ij}(t) = (\lambda \circ \psi_j \circ h_i)(t)$. By leveraging functional depth measures and their invariance under strictly monotone transformations, the authors extend univariate RobustTW results to the multivariate setting, deriving conditions for consistency and proposing estimators for the component patterns $\gamma_j$ and the common amplitude $\lambda$ directly from pooled data. They introduce the WHyRA plot as a diagnostic tool to assess agreement of individual warping functions across components and demonstrate robustness and computational efficiency through simulations, including outlier contamination, and two real-data applications (Arctic sea-ice extent and European maternity ages). The results show that the depth-based method yields competitive or superior performance relative to registration-based approaches, particularly under moderate warp variability and data contamination, and provides practical insights into phase variability in complex multivariate functional data.

Abstract

In the context of multivariate functional data with individual phase variation, we develop a robust depth-based approach to estimate the main pattern function when cross-component time warping is also present. In particular, we consider the latent deformation model (Carroll and Müller, 2023) in which the different components of a multivariate functional variable are also time-distorted versions of a common template function. Rather than focusing on a particular functional depth measure, we discuss the necessary conditions on a depth function to be able to provide a consistent estimation of the central pattern, considering different model assumptions. We evaluate the method performance and its robustness against atypical observations and violations of the model assumptions through simulations, and illustrate its use on two real data sets.

Depth-based estimation for multivariate functional data with phase variability

TL;DR

This work develops a depth-based, registration-free approach to estimate a central amplitude pattern in multivariate functional data with cross-component time warping under the latent deformation model . By leveraging functional depth measures and their invariance under strictly monotone transformations, the authors extend univariate RobustTW results to the multivariate setting, deriving conditions for consistency and proposing estimators for the component patterns and the common amplitude directly from pooled data. They introduce the WHyRA plot as a diagnostic tool to assess agreement of individual warping functions across components and demonstrate robustness and computational efficiency through simulations, including outlier contamination, and two real-data applications (Arctic sea-ice extent and European maternity ages). The results show that the depth-based method yields competitive or superior performance relative to registration-based approaches, particularly under moderate warp variability and data contamination, and provides practical insights into phase variability in complex multivariate functional data.

Abstract

In the context of multivariate functional data with individual phase variation, we develop a robust depth-based approach to estimate the main pattern function when cross-component time warping is also present. In particular, we consider the latent deformation model (Carroll and Müller, 2023) in which the different components of a multivariate functional variable are also time-distorted versions of a common template function. Rather than focusing on a particular functional depth measure, we discuss the necessary conditions on a depth function to be able to provide a consistent estimation of the central pattern, considering different model assumptions. We evaluate the method performance and its robustness against atypical observations and violations of the model assumptions through simulations, and illustrate its use on two real data sets.
Paper Structure (12 sections, 3 theorems, 26 equations, 11 figures, 6 tables)

This paper contains 12 sections, 3 theorems, 26 equations, 11 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Latent deformation model
  3. Depth-based medians in the latent deformation model
  4. Univariate functional case
  5. Robust estimation in the latent deformation model
  6. Visualizing the agreement of individual warping functions
  7. Simulation study
  8. Simulations under outlier contamination
  9. Analysis of real data
  10. Phase variability in sea ice extent in the Arctic Ocean
  11. Mother age distribution across Europe
  12. Discussion

Key Result

Proposition 1

Let $FD: L_2({\mathcal{I}}) \times {\cal P} (L_2({\mathcal{I}}))\longrightarrow \mathbb{R}:\, (x,P)\mapsto FD(x,P)$ be a functional depth measure with empirical version denoted by $FD_{X_1,\ldots,X_n}(x)$. Let $X$ denote a general random variable with values in $L_2({\mathcal{I}})$ and distribution Then, under the same conditions on $P_X$ required in (P2), for a random sample $X_1,\ldots,X_n$ gen

Figures (11)

  • Figure 1: Illustration of the latent deformation model with $p=4$ and $n=50$. Left: target function (top) and component-based distortion functions ($\psi_j$'s). Center: component target functions ($\gamma_j$'s). Right: Observed data with component target functions as a black solid line.
  • Figure 2: Warping function estimates and WHyRA plots from two synthetic data sets with $p=2$ and $n=50$. In each row, the first two plots present the warping function estimates $\hat{h}_{ij}$ for $j=1,2$, while the third one shows the scatter plot of the MHI values across them. Top: Data set generated according to model (\ref{['model']}), such that $h_i$, $i=1,\ldots,n$ are the same in the two components. Bottom: Data set generated according to (\ref{['sim_model']}) with $\sigma_D=1$, so that the individual warping functions depend on $j=1,2$. Color code is defined as the modified hypograph value for each warping estimate in the first component.
  • Figure 3: Illustration of the elements of the simulation study. Top: target function (left) and component-based distortion functions ($\psi_j$'s) for setting 1 (middle, $p=4$) and setting 2 (right, $p=30$). Bottom: individual warping functions ($h_i$'s) for settings 1 and 2 and different parameter values. In all simulated component-based distortions and individual warping functions a transversal section at $t=0.9$ is presented to help visualize that the resulting univariate distribution is not symmetric around $t=0.9$. The black dashed line represents the diagonal of the interval $\mathcal{I}=[0,1]$.
  • Figure 4: Five simulated data sets under model (\ref{['sim_model']}). For the last one, the number of components is $p=30$ and only four randomly selected components are shown. The fifth plot on each row represents the target function, $\lambda(t)$, as a thick gray solid line, together with the depth-based and the CM estimates (black and red lines, respectively).
  • Figure 5: Daily sea ice extent values in the eight arctic regions defined in Figure \ref{['ocean']}, from 1979 to 2024.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Remark 1
  • Remark 2