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Explainable Outlier Detection for Multivariate Functional Data

This paper proposes an integrated framework for robust and interpretable outlier detection in multivariate functional data with separable covariance structures by combining a matrix-variate Minimum Covariance Determinant estimator with a computationally efficient Shapley value-based decomposition of outlyingness.

Original authors: Marcus Mayrhofer, Una Radojičić, Horst Lewitschnig, Peter Filzmoser

Published 2026-05-21

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CC BY 4.0

Original authors: Marcus Mayrhofer, Una Radojičić, Horst Lewitschnig, Peter Filzmoser

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 a quality control manager at a massive factory that produces thousands of different types of continuous streams of data every day. These aren't just single numbers; they are multivariate functional data. Think of them as a bundle of smooth, flowing rivers (functions) where each river represents a different sensor or measurement (like temperature, pressure, and humidity) recorded over time.

Your job is to spot the "bad" rivers—the ones that are flowing strangely compared to the rest. But here's the catch: sometimes a few bad rivers can trick your measuring tools into thinking the entire factory is running differently than it actually is. This is the problem of outliers messing up your covariance (how the rivers relate to each other).

This paper introduces a new, super-smart toolkit to solve two problems at once:

Robustness: How to find the "true" pattern of the rivers without being fooled by the bad ones.
Explainability: Once you find a bad river, you need to know exactly where and why it's bad. Is it the temperature sensor spiking in July? Is it the pressure dropping in March?

Here is how their solution works, broken down into simple concepts:

1. The "Smoothie" Strategy (Basis Representation)

Real-world data is messy and recorded at specific points (like snapshots). To analyze it, the authors first turn these jagged snapshots into smooth, continuous curves (like blending a smoothie from fruit chunks). They represent these curves using a set of building blocks called basis functions (think of them as Lego bricks).

Instead of trying to analyze millions of data points, they analyze the coefficients (the recipe) that tell you how many of each Lego brick to use. This turns a complex, infinite problem into a manageable matrix (a grid of numbers).

2. The "Separable" Shortcut

The authors assume a helpful structure called separable covariance. Imagine the factory has two types of relationships:

Vertical: How the different sensors (temperature, pressure) relate to each other at any single moment.
Horizontal: How a single sensor behaves over time (e.g., does temperature rise smoothly?).

The "separable" assumption says these two relationships are independent and can be multiplied together. It's like saying the "flavor profile" of the smoothie is separate from the "texture profile." This simplifies the math massively, allowing them to use powerful Matrix Minimum Covariance Determinant (MMCD) estimators.

The Analogy: Imagine you are trying to find the average shape of a crowd of dancers. If one dancer is doing a wild, erratic solo, a normal average would look like a blurry mess. The MMCD method is like a smart camera that ignores the top 50% of the most chaotic dancers, calculates the average of the remaining calm crowd, and gives you a clear, true picture of the group's movement.

3. The "Shapley Value" Detective (Explainability)

Once the system flags a specific river as "outlying" (strange), the big question is: Why?
In the past, you might just know "this curve is weird." But this paper uses Shapley values (a concept from game theory) to act like a forensic detective.

The Analogy: Imagine a group of friends (the different sensors) and a time period (the days of the month) are all contributing to a "strangeness score." The Shapley value method asks: "If we remove the temperature sensor for the month of July, how much does the strangeness score drop?" It does this for every sensor and every time slot, then averages the results.

This allows the system to say: "This observation is an outlier because the Temperature sensor was 20% higher than usual specifically during the winter months, while the Pressure sensor was normal."

The Magic Trick: Usually, doing this kind of detailed breakdown is computationally impossible (it would take longer than the age of the universe for large datasets). The authors found a mathematical shortcut that reduces this complexity from exponential (impossible) to linear (fast), making it practical for real-world use.

4. Real-World Tests

The authors tested their toolkit on two real-world scenarios:

El Niño (Climate Data): They analyzed sea surface temperatures across four different regions of the Pacific Ocean. Their method successfully identified extreme El Niño and La Niña years that other methods missed. More importantly, the "detective" part showed exactly which region (e.g., Niño 3.4) and which season (e.g., autumn) was driving the anomaly.
Resistance Spot Welding (Manufacturing): They analyzed electrical resistance curves from car manufacturing. They found defective welds (outliers) and pinpointed exactly which part of the welding process and which specific electrode was causing the problem.

Summary

In short, this paper builds a robust, detective-style system for analyzing complex, multi-sensor data streams.

It ignores the noise to find the true pattern (Robustness).
It breaks down the pattern into Lego-like building blocks to make calculations fast (Basis Representation).
It uses a game-theory detective to explain exactly which sensor and which time period caused a problem (Shapley Values).

The result is a method that doesn't just say "Something is wrong," but tells you "The Temperature sensor was too high in July," making it incredibly useful for engineers and scientists who need to fix the root cause of anomalies.

Technical Summary: Explainable Outlier Detection for Multivariate Functional Data

Problem Statement

Multivariate Functional Data Analysis (FDA) involves analyzing data represented as continuous functions across multiple variables (e.g., time series of several sensors). While Functional Principal Component Analysis (FPCA) is a dominant tool, robust estimation of mean and covariance functions remains a critical challenge, particularly in the presence of outliers. Outliers can severely distort non-robust estimates, leading to unreliable conclusions. Furthermore, existing outlier detection methods often lack interpretability; they identify that an observation is anomalous but fail to explain why—specifically, which components (variables) and at what time points the deviation occurs.

Current literature offers robust methods for univariate functional data and depth-based approaches for multivariate settings, but there is a gap in providing a unified framework that combines robust distance-based estimation with interpretable, component- and time-specific outlier explanations for multivariate functional data, particularly under separable covariance structures.

Methodology

The proposed framework integrates three core components: robust covariance estimation, a truncated multivariate functional Mahalanobis distance, and Shapley value-based decomposition.

1. Theoretical Framework: Separable Covariance and Basis Representation

The method assumes the multivariate stochastic process $X(t)$ possesses a separable covariance structure, where the covariance kernel $K(s, t)$ decomposes into a cross-covariance matrix $\Sigma_{row}$ (between components) and a temporal kernel $\kappa(s, t)$ (within components):
$K(s, t) = \Sigma_{row} \kappa(s, t)$
To handle discrete observations, the data are smoothed using a finite basis representation (e.g., B-splines). A key theoretical contribution is the establishment of a direct link between the functional process and its coefficient matrix $A$ . If $X(t) = A' \phi(t)$ , the paper proves that under separability, the coefficient matrix $A$ follows a matrix-variate distribution with mean $M_A$ and covariances $\Sigma_{row}$ and $\Sigma_{col}$ (where $\Sigma_{col}$ relates to $\kappa$ ). This allows the functional Mahalanobis distance to be computed as a matrix-variate Mahalanobis distance on the coefficient matrices, significantly reducing computational complexity.

2. Robust Estimation: MMCD

To ensure robustness against outliers, the framework employs the Matrix Minimum Covariance Determinant (MMCD) estimator (Mayrhofer et al., 2025) to estimate the mean and covariance of the coefficient matrices. This approach is highly robust, efficient, and consistent for matrix-variate elliptical distributions. The MMCD estimators ( $\hat{M}_A, \hat{\Sigma}_{row}, \hat{\Sigma}_{col}$ ) are used to construct a robust truncated functional Mahalanobis distance.

3. Distance Metric: Truncated Multivariate Functional Mahalanobis Distance (fMMD)

The authors define the squared truncated functional multivariate Mahalanobis semi-distance (fMMD) as:
$fMMD^2(X; M) = \sum_{k=1}^{M} \frac{1}{\pi_k} \langle X - \mu, \psi_k \rangle^2$
where $(\pi_k, \psi_k)$ are the eigenpairs of the covariance operator. Under the separability assumption, this distance is shown to be equivalent to the matrix-variate Mahalanobis distance of the coefficient matrix $A$ . This equivalence enables efficient computation using robust matrix estimators.

4. Explainability: Shapley Value Decomposition

To address the "black box" nature of outlier detection, the paper generalizes Shapley values (from cooperative game theory) to decompose the total outlyingness (fMMD) into additive contributions.

Coordinate-specific: Identifies which variable (component) contributes most to the anomaly.
Time-specific: Identifies which time intervals contribute most.
Time-coordinate-specific: Simultaneously decomposes contributions by variable and time interval.

Crucially, the authors derive a closed-form solution for these contributions that reduces the computational complexity from exponential (relative to the number of components) to linear, while retaining the key properties of Shapley values (efficiency, symmetry, monotonicity).

Key Contributions

Robust Framework for Multivariate FDA: The paper introduces a comprehensive framework for robust mean and covariance estimation in multivariate functional data using MMCD estimators applied to basis coefficients under a separable covariance assumption.
Theoretical Link between Functional and Matrix-Variate Distributions: It rigorously establishes that the separability of the covariance operator in a multivariate functional process translates to a specific separable structure in the distribution of the random coefficient matrix. This bridges functional robustness with matrix-variate robustness theory.
Efficient Explainable Outlier Detection: The work generalizes Shapley values to functional data, providing a principled method to decompose multivariate functional outlyingness into time- and component-specific contributions. The derivation of a linear-complexity algorithm for these decompositions is a significant computational advancement.
Empirical Validation: The method is validated through extensive simulations and real-world applications, demonstrating its effectiveness in detecting various outlier types (shift, shape, isolated, covariance-induced) and its robustness even when the separability assumption is misspecified.

Results

Simulation Studies

Extensive simulations were conducted comparing the proposed method (MMCD-based Mahalanobis distance) against state-of-the-art depth-based methods (e.g., fSDO, fAO, fDO), magnitude-shape plots, and functional isolation forests.

Performance: The proposed robust distance-based approach consistently outperformed competing methods in terms of F-score, AUC, and recall, particularly for shift and shape outliers.
Robustness to Misspecification: Even when the data violated the separability assumption (non-separable processes), the proposed method remained effective, often outperforming unregularized sample covariance estimators due to a favorable bias-variance trade-off.
Non-Gaussian Data: In heavy-tailed settings (t-distributions), the robust method maintained high recall and AUC, though precision was lower due to the difficulty in distinguishing distribution tails from true outliers.
Interpretability: Shapley values successfully identified the specific coordinates and time intervals responsible for outliers, outperforming naive deviation measures and depth-based local scores in localized contamination scenarios.

Real-World Applications

El Niño–Southern Oscillation (ENSO): Analyzed sea surface temperature (SST) data from four Niño regions. The multivariate approach identified more extreme ENSO episodes than univariate analysis. Shapley decomposition revealed that El Niño events were driven by warming in Niño 3.4 and Niño 3, while La Niña events were driven by cooling in Niño 4 and Niño 3.
Resistance Spot Welding: Applied to dynamic resistance curves from automotive manufacturing. The method achieved an AUC of 0.95, outperforming other methods, and successfully identified which welding points and time intervals were anomalous in cases of expulsion.
Fertility Rates: Analyzed age-specific fertility curves for 22 countries. The robust method detected outliers masked by classical estimators (e.g., USA, Germany) and provided interpretable insights into shifts in fertility timing across different age groups and decades.

Significance and Claims

The paper claims to provide a robust and interpretable approach for analyzing multivariate functional data with separable covariance structures. Its significance lies in:

Moving beyond detection to explanation: By integrating Shapley values, the framework does not just flag anomalies but explains them in terms of specific components and time points, which is crucial for domain experts (e.g., climatologists, manufacturing engineers).
Computational Efficiency: The reduction of Shapley value complexity from exponential to linear makes the method scalable for high-dimensional functional data.
Robustness: The use of MMCD estimators ensures that the detection and interpretation are not skewed by the very outliers they aim to find.
Practical Utility: The framework is shown to be effective even when model assumptions (separability, normality) are not perfectly met, making it a viable tool for real-world data analysis where such assumptions are often approximations.

The authors conclude that while the separability assumption simplifies modeling, the proposed structured estimator acts as a robust approximation that often yields more stable diagnostics than unconstrained estimators in moderate sample sizes. Future work is suggested to extend these robust estimation techniques to non-separable covariance structures.