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Statistical description and dimension reduction of continuous time categorical trajectories with multivariate functional principal components

This paper proposes a multivariate functional principal component analysis framework that transforms categorical trajectories into binary indicator functions within a Hilbert space, enabling consistent dimension reduction and statistical comparison of piecewise constant paths under weak regularity assumptions, as demonstrated on sensory perception data.

Original authors: Hervé Cardot, Caroline Peltier

Published 2026-05-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hervé Cardot, Caroline Peltier

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a group of people taste different foods. Every few seconds, they have to shout out what flavor they are experiencing most strongly: "Sweet!", "Sour!", "Bitter!", or "Salty!". Over the course of a minute, each person creates a unique, jagged timeline of flavors.

In the world of statistics, analyzing these timelines is tricky. Traditional methods usually assume data is smooth and continuous (like a flowing river). But these flavor timelines are more like a series of sudden jumps (like a light switch flipping on and off). If you try to force these "jumping" data into smooth models, you lose important details.

This paper introduces a new, clever way to map and compare these jumping flavor timelines without losing any information. Here is the breakdown of their approach using simple analogies:

1. The Problem: The "Jagged" Timeline

Usually, statisticians look at the "average" flavor at any given second. If 50 people taste a lemon, the average might say "Lemon" at second 1, "Sour" at second 2, and "Sweet" at second 3.

The Flaw: This average hides the individual stories. It doesn't tell you who switched to "Sweet" early or who kept tasting "Sour" late. It also struggles if people can taste two things at once (like "Sweet" and "Salty" together), which happens in some experiments.

2. The Solution: The "Light Switch" Trick

The authors propose a simple but powerful trick: Instead of trying to analyze the complex "flavor" directly, they turn every possible flavor into its own binary light switch.

Imagine a control panel with 8 switches, one for each flavor (Lemon, Sweet, Acid, etc.).
For any specific person at any specific second, the "Lemon" switch is either ON (1) or OFF (0).
If the person is tasting Lemon, the Lemon switch is ON, and all others are OFF (in the standard experiment).
By doing this, they turn one complex, jumping timeline into a set of simple, binary on/off stories.

3. The Map: Finding the "Main Themes"

Now that they have these on/off stories, they use a technique called Multivariate Functional Principal Component Analysis (MFPCA).

The Analogy: Think of this like finding the "main themes" in a song. If you have 150 different recordings of people tasting food, MFPCA asks: "What are the most common ways these timelines vary?"
The Result: They find a few "master patterns" (called Principal Components) that explain most of the differences between the people.
- Pattern 1 might represent the difference between people who taste "Lemon" first versus those who taste "Sweet" first.
- Pattern 2 might represent the difference between those who taste "Salty" at the end versus those who don't.

4. Why This is Better Than Old Methods

The paper compares their new "Light Switch" method to an older method called "Continuous Time Correspondence Analysis" (which is like trying to map the flavors using a complex, multi-dimensional grid).

The Old Method's Weakness: It struggles when a flavor is never chosen by anyone at a specific time. It's like trying to draw a map of a city where some streets don't exist; the math breaks down, and the map has holes. It also uses a "multiplicative" math (multiplying numbers together) which is hard to interpret.
The New Method's Strength: It uses "additive" math (adding numbers). It handles the "holes" (times when no one picks a flavor) perfectly. It treats the data as a collection of simple on/off switches, making it robust even when the data is messy or when people taste multiple things at once.

5. The Real-World Test: The Taste Test

The authors tested this on a real dataset of 150 people tasting three different controlled liquid stimuli (S04, S06, S07).

The Outcome: Their new method successfully grouped the people into three distinct clusters that perfectly matched the three different liquids they tasted.
The Interpretation: They could look at the "Patterns" and say, "Ah, the people in Group A tasted Lemon early and Sweet late," which matched the actual recipe of the liquid they drank. The old method could also separate the groups, but it was much harder to explain why they were different, and it got confused by flavors that no one tasted (like "Mint" in this specific test).

Summary

In short, this paper says: "Don't try to smooth out the jagged jumps in categorical data. Instead, turn every category into a simple on/off switch, and then look for the main patterns in how those switches flip."

This allows statisticians to:

Keep all the original information (no data is lost).
Handle cases where people taste multiple things at once.
Easily interpret the results by seeing which flavors go up and down together.
Reduce a massive amount of complex timeline data into a few simple, understandable "scores" that tell the whole story.

Technical Summary: Statistical Description and Dimension Reduction of Continuous Time Categorical Trajectories

1. Problem Statement

The paper addresses the challenge of statistically describing and comparing sets of continuous-time categorical trajectories. While functional data analysis (FDA) has extensively developed tools for random trajectories taking values in $\mathbb{R}$ , less attention has been paid to trajectories taking values in a finite set of non-numerical states $S = \{S_1, \dots, S_q\}$ .

Existing approaches face specific limitations:

Categorical Functional Data Analysis (CFDA): Based on continuous-time correspondence analysis (Deville, 1982), this method encodes trajectories into real components. However, interpretation of individual trajectories is often delicate, and the method struggles with states having zero probability of occurrence at certain times.
Markov/Semi-Markov Processes: These model the law of trajectories at the population level but are often too simplistic for individual-level analysis. Furthermore, they struggle with frameworks where multiple states can be observed simultaneously (e.g., Temporal Check-All-That-Apply or TCATA), as this requires an exponential increase in the number of system states ( $2^q$ ).

The authors aim to provide a tool for simple representation and comparison of categorical trajectories without losing information, specifically accommodating scenarios where trajectories are piecewise constant with finite jumps and where multiple states may co-occur.

2. Methodology

The proposed approach transforms the problem of statistical description of categorical trajectories into a Multivariate Functional Principal Components Analysis (MFPCA).

Mathematical Framework:

Binary Indicator Transformation: Instead of analyzing the categorical process $Y(t)$ directly, the authors associate each state $S_j$ with a binary random indicator function $X_j(t) = \mathbb{1}_{\{Y(t) \ni S_j\}}$ . This creates a multivariate functional vector $X(t) = (X_1(t), \dots, X_q(t))$ taking values in $\{0, 1\}^q$ .
Functional Space: The trajectories are modeled as random elements in the Hilbert space $H = L^2([0, T], \mathbb{R}^q)$ . The authors establish that under the weak hypothesis that trajectories are continuous in probability (Hypothesis H1) and belong to the space of right-continuous functions with left limits ( $D([0, T], \mathbb{R}^q)$ ), these processes are jointly measurable and possess well-defined mean and covariance operators.
Additive Decomposition: Unlike CFDA, which uses a multiplicative decomposition of joint probabilities, this method relies on an additive decomposition of the covariance structure. The covariance operator $\Gamma$ is defined via the kernel $\gamma(s, t)$ , where off-diagonal terms represent departures from independence.
Dimension Reduction: The method performs MFPCA on the vector $X$ . The Karhunen-Loève expansion allows the representation of the centered process $X - p$ (where $p$ is the mean trajectory) in a low-dimensional subspace spanned by the eigenfunctions $\phi_r$ .

Estimation:

Consistency: The paper provides consistency results for the empirical mean $\hat{p}$ and the empirical covariance operator $\hat{\Gamma}$ under weak regularity assumptions.
Discrete Observation: The authors note that while trajectories can be observed exhaustively in controlled experiments, consistent estimators can also be obtained from a sufficiently dense grid of discretization points using nonparametric smoothing (local linear smoothers), achieving parametric convergence rates up to a logarithmic factor.

3. Key Contributions

Unified Framework: The approach encompasses both Temporal Dominance of Sensations (TDS) (where exactly one state is active at any time) and TCATA (where multiple states can be active simultaneously) without requiring an explosion in the state space dimension.
Theoretical Rigor: The paper provides a rigorous mathematical justification for treating discontinuous, piecewise constant categorical trajectories as random elements in a Hilbert space of square-integrable functions, establishing the continuity of mean and covariance functions under the assumption of continuity in probability.
Interpretability: The resulting principal components are shown to be easily interpretable as variations around the mean probability curves for each state. The method allows for the identification of dominant patterns in individual trajectories.
Robustness to Zero Probabilities: By using an additive decomposition of covariance rather than the multiplicative approach of correspondence analysis, the method avoids the issue of undefined encoding functions when a state has zero probability of occurrence at specific time points.

4. Results

The methodology was illustrated using a dataset of 150 sensory perception experiments involving 50 panelists tasting three different gustometer-controlled stimuli (S04, S06, S07) with $q=8$ possible sensation states.

Dimension Reduction: The first two principal components captured 34% of the total variance (23% and 11% respectively). The eigenvalues decreased slowly, reflecting the discontinuous nature of the trajectories.
Differentiation of Stimuli:
- PC1 effectively distinguished the S04 stimulus (characterized by high probability of Lemon at the start and Sweet at the end) from the S06 stimulus (characterized by the opposite pattern).
- PC2 distinguished the S07 stimulus (Acid $\to$ Sweet $\to$ Salty) from the others.
Comparison with CFDA:
- The MFPCA scores showed a high linear correlation (0.93) with the first dimension of the CFDA scores, successfully separating the stimuli groups.
- However, MFPCA offered a more direct interpretation in terms of probability of occurrence.
- CFDA struggled with the "Mint" descriptor (a distractor), as the encoding functions were undefined when the proportion of occurrence was zero. In contrast, MFPCA handled these zero-probability intervals naturally.
- The second dimension in CFDA primarily captured outliers (rare events like the Mint distractor), whereas the second dimension in MFPCA successfully separated the S07 signal from the others.

5. Significance and Claims

The authors claim that this approach provides a powerful tool for the dimension reduction of categorical functional data that is both statistically consistent and interpretable.

Individual Level Analysis: Unlike Markov-based approaches that focus on population-level laws, this method facilitates the analysis and comparison of individual trajectories.
Handling Simultaneous States: The method naturally handles experimental frameworks like TCATA where multiple states co-occur, a scenario that is difficult to model with standard Markov or correspondence analysis approaches without drastically increasing complexity.
Predictive Utility: The reduced dimension representation (principal component scores) can serve as explanatory variables for predictive models, such as scalar-on-categorical functional data regression. The authors demonstrated that simple discriminant analysis based on these scores could accurately predict the underlying tasting experiment (S04, S06, or S07).
Generality: The framework is presented as an extension of previous work (Cardot and Peltier, 2025), offering a more detailed mathematical description and consistent estimators under weak assumptions, applicable to any continuous-time categorical process with finite jumps.