Variable Domain Multivariate Functional Principal Component Analysis

This paper proposes a novel Multivariate Functional Principal Component Analysis (MFPCA) method that accommodates variable observation domains by unifying univariate variable-domain scores and smoothing their covariance, demonstrating superior performance over existing approaches through simulations and a real-world application to COVID-19 patient monitoring data.

The Gist

The Problem: The "Mismatched Movie" Dilemma

Imagine you are a film critic trying to review a new movie. You have 1,000 different copies of the same film, but there's a catch:

• Some people watched the full 2-hour movie.
• Some people only watched the first 30 minutes because they fell asleep.
• Others watched only the last 15 minutes because they arrived late.

Now, imagine you are trying to analyze two things happening in the movie at the same time: the plot twists (Variable 1) and the background music (Variable 2).

The Old Way (The "Binning" Approach):
Previous methods for analyzing this data were like saying, "Okay, let's only look at the first 30 minutes of everyone's movie."

• The Problem: You throw away all the information from the people who watched the whole thing. You lose the plot twists that happen at the end.
• The Alternative: You could chop the audience into groups: "Group A watched 0–30 mins," "Group B watched 30–60 mins." But this is messy. It treats a 29-minute watcher as totally different from a 31-minute watcher, even though their experience was almost the same. It's like sorting a library by "books with 100 pages" and "books with 101 pages" instead of just reading the story.

The Paper's Solution (VD-MFPCA):
This paper introduces a new, smarter way to analyze these "mismatched movies." Instead of cutting off the data or forcing everyone into rigid boxes, the authors created a method that understands how the length of the movie changes the story.

How the New Method Works: The "Smart Editor"

The authors propose a four-step process that acts like a very smart film editor:

1. Edit Each Scene Separately: First, they look at the "Plot" and the "Music" separately. They figure out the average story and music for people who watched short clips, medium clips, and long clips. They realize that the "average plot" for a short clip looks different than the "average plot" for a long clip.
2. Stack the Notes: They take the "notes" (scores) from the plot analysis and the "notes" from the music analysis and stack them together for each person.
3. The Magic Smoothie (The Key Innovation): Here is the genius part. They realize that the relationship between the plot and the music changes depending on how long the movie is.
   • Analogy: Imagine that in short movies, the plot and music are very tightly linked. But in long movies, they drift apart. The old methods assumed they were linked the same way for everyone. This new method uses a "smoothie blender" (mathematically called penalized splines) to blend these relationships smoothly. It doesn't force a hard cut; it creates a smooth curve that shows how the connection changes as the movie gets longer.
4. The Final Review: Now, they can find the "main themes" (Principal Components) that explain the movie, knowing exactly how those themes shift based on how long the viewer watched.

The Test: Did It Work?

The authors ran a massive simulation (a "virtual movie theater") to test their method against the old "cutting off" method.

• The Setup: They created fake data where some "patients" (or movie watchers) had short observation times and others had long ones.
• The Result: The new method was much better. It reconstructed the "movies" with far less error. The old method was like trying to guess the ending of a mystery novel by only reading the first chapter; the new method read the whole book for those who had it, and the short chapters for those who didn't, and still figured out the whole story perfectly.

The Real-World Application: The Hospital "Vital Signs" Movie

To prove this works in real life, the authors applied their method to COVID-19 patients in a hospital.

• The Data: They tracked two vital signs: Oxygen Saturation (SpO2) and Body Temperature.
• The Variable Domain: Some patients were in the hospital for 3 days; others were there for 3 months. Their "observation movies" were different lengths.
• What They Found:
  • The Mean Story: They could see that patients who stayed longer started with lower oxygen levels that slowly improved, while short-stay patients had stable oxygen. The temperature of almost everyone started high (fever) and went down, regardless of how long they stayed.
  • The "Main Theme" (PC1): The most important pattern they found (called the first principal component) was a specific combination of oxygen and temperature changes.
  • The Prediction: They discovered that patients with a "high score" on this main theme were much more likely to die (25% mortality) compared to those with a low score (7% mortality).
  • Age Factor: Older patients naturally had higher scores on this "dangerous pattern."

The Bottom Line

This paper says: Stop cutting off your data just because people watched for different amounts of time.

By using their new "Variable Domain" method, researchers can analyze multiple changing things (like heart rate and temperature) simultaneously, even if some people are observed for a week and others for a year. It captures the full story without throwing away the ending, leading to much more accurate predictions about patient health.

Technical Summary

Technical Summary: Variable Domain Multivariate Functional Principal Component Analysis

Problem Statement
Multivariate Functional Principal Component Analysis (MFPCA) is a standard technique for dimension reduction in datasets containing multiple functional variables (e.g., time-series of vital signs) observed on the same subjects. However, existing MFPCA frameworks, such as the comprehensive approach by Happ and Greven [2018], rely on a critical assumption: all functional observations must be recorded over a common, fixed domain. In practical applications, particularly in longitudinal biomedical studies, this assumption is frequently violated. Subjects often have varying observation periods due to factors like differential admission times, varying lengths of hospital stay, or early dropout. This results in "variable domain functional data," where the domain length $T_i$ varies across subjects.

Current ad-hoc solutions to this problem include restricting analysis to a common subset of the domain (discarding valuable data from subjects with longer observation periods) or binning subjects into groups with similar domain lengths (which introduces arbitrary discretization and fails to model the continuous dependence of the covariance structure on domain length). While Johns et al. [2019] addressed variable domains in a univariate setting, no existing framework effectively handles the multivariate case where multiple variables are observed over varying, potentially distinct, domains.

Methodology
The authors propose a novel framework, Variable Domain MFPCA (VD-MFPCA), which extends the univariate variable domain FPCA of Johns et al. [2019] to the multivariate setting. The methodology proceeds in four distinct steps:

1. Univariate Variable Domain FPCA: For each functional variable $j$, the authors apply the Johns et al. [2019] approach separately. This involves modeling the mean function $\mu_j(t, T_i)$ and the covariance function $\gamma_j(t, s, T_i)$ as smooth functions of both time $t$ and domain length $T_i$ using penalized thin plate splines (PTPS) within a generalized additive model framework. This yields univariate eigenfunctions $\hat{\psi}^j_k(t, T_i)$ and scores $\hat{\xi}^j_{ik}(T_i)$ that explicitly depend on the subject's domain length.
2. Stacking Univariate Scores: The univariate scores for each subject are stacked into a single vector $\xi_i(T_i)$.
3. Modeling Score Covariance as a Function of Domain Length: This is the core innovation. The authors recognize that the covariance matrix of the stacked scores, $C(T_i) = \text{Cov}(\xi_i | T = T_i)$, depends on the domain length. Instead of assuming a fixed covariance structure, they model each unique element of the empirical covariance matrix as a smooth function of $T$ using penalized splines. This allows for the estimation of a smooth covariance matrix $\hat{C}(T)$ for any domain length.
4. Multivariate Eigendecomposition: For any specific domain length $T$, the estimated covariance matrix $\hat{C}(T)$ is decomposed to obtain multivariate eigenvalues $\nu_m(T)$ and eigenvectors $c_m(T)$. These are used to compute multivariate scores $\rho_{im}(T_i)$ and multivariate eigenfunctions $\Psi^j_m(t, T_i)$, which now depend on both time and the specific domain length of the subject.

Key Contributions

• Novel Framework: The paper presents the first methodology for MFPCA that explicitly accommodates variable observation domains without truncating data or discretizing subjects into bins.
• Theoretical Extension: It extends the univariate variable domain FPCA framework to the multivariate setting, addressing the complex challenge of modeling the dependence structure across multiple variables when those variables are observed over different time spans.
• Smooth Covariance Modeling: By modeling the covariance of stacked scores as a smooth function of domain length, the method captures continuous variations in the dependence structure that binning strategies miss.

Results
The authors validate the method through extensive simulation studies and a real-world application.

• Simulation Study: The proposed VD-MFPCA was compared against a "binning" approach (grouping subjects by domain length and truncating data to the minimum length in each bin).

  • Reconstruction Accuracy: VD-MFPCA consistently achieved substantially lower Average Root Mean Squared Error (ARMSE) for reconstructing functional observations compared to the binning approach, with improvements ranging from 50% to over 80% in various scenarios.
  • Eigenfunction Estimation: VD-MFPCA demonstrated superior accuracy in estimating eigenfunctions, particularly under skewed domain length distributions (e.g., negative binomial) where the binning approach suffered from high errors due to information loss in truncated domains.
  • Robustness: The proposed method remained stable across different sample sizes ($N=100, 500$), noise levels, and domain distributions, whereas the binning approach showed sensitivity to the distribution shape and the number of bins used.

• Application to COVID-19 Data: The method was applied to body temperature and capillary oxygen saturation (SpO2) trajectories of 782 hospitalized COVID-19 patients with varying lengths of stay (ranging from ~3 days to ~125 days).

  • Domain-Dependent Patterns: The analysis revealed that mean trajectories and variance structures depend on hospitalization length. For instance, patients with longer stays exhibited initially lower SpO2 levels that gradually improved, a pattern obscured by fixed-domain methods.
  • Clinical Relevance: The first principal component (PC1) scores were found to be strongly associated with patient mortality and age, but not with the length of the observation period itself. This confirms that the method successfully separates domain-related artifacts from intrinsic physiological variation.
  • Prognostic Value: Patients in the highest PC1 tercile had a mortality rate of 25.3%, compared to ~7.5% in lower terciles, demonstrating the method's ability to capture prognostic information from joint vital sign trajectories.

Significance and Claims
The paper claims that VD-MFPCA fills a critical gap in functional data analysis by providing a principled approach for dimension reduction in multivariate settings with variable domains. The authors assert that their method offers "substantial gains" in both reconstruction accuracy and eigenfunction estimation compared to existing ad-hoc strategies.

The significance of the work lies in its ability to utilize the full information content of longitudinal data without arbitrary truncation or discretization. In the context of the COVID-19 application, the authors highlight that the method captures complex, time-varying physiological patterns that are predictive of clinical outcomes (mortality and age-related severity), which would likely be missed or biased by traditional fixed-domain MFPCA. The authors conclude that this methodology is particularly valuable for clinical research involving hospitalization data and longitudinal monitoring where observation periods are inherently variable.

The paper remains modest regarding limitations, acknowledging that the current implementation may be computationally demanding for very large datasets or high numbers of variables, and noting that future work could explore Bayesian uncertainty quantification and the handling of irregular, sparse observations within the variable domain framework.