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funOCLUST: Clustering Functional Data with Outliers

The paper proposes funOCLUST, a robust extension of the OCLUST algorithm designed to cluster functional data and effectively identify outliers by addressing the challenges of infinite dimensionality and sensitivity to anomalies.

Original authors: Katharine M. Clark, Paul D. McNicholas

Published 2026-07-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Katharine M. Clark, Paul D. McNicholas

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 have a giant box of spaghetti. But these aren't just noodles; they are squiggly, wiggly lines representing things like temperature changes over a day, traffic flow, or how a plant grows. In the world of data science, these are called functional data. The problem? These lines are infinite-dimensional (they have infinite points), and they are messy. Sometimes, a noodle gets a weird kink, or a whole batch gets thrown off by a sudden storm, creating "outliers" that ruin the party.

Enter funOCLUST, a new method proposed by Katharine M. Clark and Paul D. McNicholas. Think of funOCLUST as a super-smart, slightly grumpy chef who wants to sort these spaghetti noodles into perfect piles based on their shape, but first, they have to kick out the weird, burnt, or broken pieces that don't fit.

The Big Idea: Turning Squiggles into Vectors

You can't just throw infinite spaghetti into a standard sorting machine; it's too complex. The authors suggest a clever trick: flatten the curves.

They use something called a cubic B-spline basis. Imagine taking every wiggly line and describing it not by its infinite points, but by a short list of numbers (coefficients) that tell you how to build that line using a specific set of building blocks. It's like turning a complex painting into a simple recipe card. Once the curves are turned into these short lists of numbers (vectors), the problem becomes much easier to handle.

The "Outlier" Hunt: The Log-Likelihood Game

Here is where the magic happens. The authors take an existing method called OCLUST (which was designed for regular data) and adapt it for these new "recipe cards."

The algorithm plays a game of "What if we remove this one?"

It looks at the whole group of curves.
It asks, "If I kick out this specific curve, does the remaining group look more like a perfect, tidy cluster?"
It measures this using something called subset log-likelihood. Think of this as a "tidiness score." If removing a curve makes the score jump up significantly, that curve was probably the troublemaker.
The algorithm checks if the "troublemakers" follow a specific mathematical pattern (a shifted and scaled beta distribution). If the weird curves fit this pattern, they are officially kicked out as outliers.

The authors proved (mathematically) that if the curves are generated from a standard Gaussian mixture model, the "tidiness score" changes in a predictable way when you remove a normal curve. If the score changes too much, the curve is an outlier.

What the Paper Says (and Doesn't Say)

The authors ran 100 simulated datasets to test their chef's skills. They created 8 different scenarios, mixing things up:

Clusters: Sometimes there were 2 groups, sometimes 5.
Complexity: Some curves were simple (like a straight line), others were wild (with bumps and wiggles).
Sparsity: Sometimes the data was dense (lots of points), sometimes sparse (lots of missing points).
Outliers: They created two types of troublemakers. Some were "shift-scale" (the whole curve got bigger or moved up), and others were "heavy-tail" (random, wild errors).

The Results:

Heavy-Tail Errors: When the data had wild, random errors (heavy tails), funOCLUST was the clear winner, beating competitors like funHDDC, T-funHDDC, and tkmeans.
Shift-Scale Errors: When the outliers were just shifted or scaled versions of the normal curves, tkmeans (a trimmed k-means method) actually did a slightly better job, though funOCLUST still held its own.
Real-World Test 1 (Pedestrian Traffic): They tested this on hourly pedestrian traffic in Melbourne. The algorithm successfully separated workdays from weekends/holidays. It correctly identified 22 "outlier" days, including New Year's Day, Christmas, and Chinese New Year. These were days where the traffic pattern didn't fit the usual workday or weekend mold.
Real-World Test 2 (NOx Data): They tested it on air pollution data (NOx levels) in Barcelona. The method achieved a Correct Classification Rate (CCR) between 0.51 and 0.86, depending on the model settings. The best setting (EEE covariance structure) hit 0.86, which is on par with other top methods.

What the Paper Rules Out

The authors are careful to say what their method is not.

They explicitly state that while some methods try to cluster data in "subspaces" (reducing dimensions in a specific way), funOCLUST keeps the entire functional domain intact. They argue that sometimes you need the whole picture, not just a slice.
They note that their method relies on the assumption that the "recipe cards" (coefficients) come from a multivariate normal distribution. If the data is wildly skewed or doesn't fit this, the method might struggle (though they suggest it's robust enough for many real-world cases).
They do not claim this is a "solved problem" for all functional data. In fact, in their simulations, when there were 5 clusters with high complexity and sparse data, the false negative rate (missing an outlier) jumped to 51%. They admit that detecting outliers in those specific, messy conditions is inherently difficult.

The Verdict

The paper suggests that funOCLUST is a robust new tool. It's not a magic wand that fixes everything instantly, but it's a very strong contender, especially when the data is noisy or has heavy-tailed errors.

The authors conclude that this is the first extension of OCLUST to functional data. They see it as a stepping stone. They suggest that in the future, this could be extended to handle skewed data or even have the "recipe" (the basis decomposition) estimated inside the clustering algorithm itself, rather than just as a first step.

So, if you have a box of messy, wiggly spaghetti and you need to sort them while kicking out the burnt ones, funOCLUST is a chef worth hiring—just don't expect it to work perfectly if the spaghetti is too weird and the kitchen is too dark!

Technical Summary: funOCLUST – Clustering Functional Data with Outliers

Problem Statement
Functional data analysis (FDA) faces unique challenges due to the infinite-dimensional nature of functions and their sensitivity to outliers. While model-based clustering methods exist for functional data, they often struggle with noisy observations. Existing robust alternatives, such as those using heavy-tailed distributions or trimming approaches, frequently rely on dimension reduction techniques (e.g., subspace clustering) that may discard information from the full functional domain. Furthermore, there is a scarcity of trimming algorithms specifically designed for model-based functional data clustering that can simultaneously identify outliers and group similar curves without sacrificing the integrity of the functional representation.

Methodology
The paper proposes funOCLUST, an extension of the OCLUST algorithm (Clark and McNicholas, 2024) tailored for functional data. The methodology operates in two primary stages:

Functional Decomposition (Filtering):
Instead of treating discretized observations as high-dimensional vectors, the method decomposes raw functional data $Y_i(t)$ into a linear combination of cubic B-spline basis functions:
$Y_i(t) = \sum_{k=1}^{K+4} \beta_{i,k} B_k(t) + \epsilon_i$
This transforms the infinite-dimensional functions into $(K+4)$ -dimensional coefficient vectors $\mathbf{b}_i$ . The authors assume these coefficients follow a finite Gaussian mixture model.
Iterative Trimming and Clustering:
The core of the algorithm relies on the distribution of subset log-likelihoods.
- Theoretical Basis: The paper derives that if the fitted coefficients belong to a specific cluster $h$ , the difference in log-likelihood between the full dataset and a subset with one observation removed ( $D_j = l_{X \setminus b_j} - l_X$ ) follows an approximate shifted and scaled beta distribution.
- Outlier Definition: A candidate outlier is defined as the coefficient vector whose removal yields the greatest increase in the log-likelihood of the remaining data.
- Algorithmic Process: The algorithm iteratively identifies and removes the most outlying coefficient vector. After each removal, the distribution of the remaining subset log-likelihoods is compared to the theoretical beta mixture distribution using Kullback-Leibler (KL) divergence. The process terminates when the KL divergence is minimized, indicating the remaining data conforms to the model assumptions.

Key Contributions

Extension of OCLUST: This work represents the first extension of the OCLUST framework to the functional data domain.
Derivation of Distribution: The paper provides a mathematical derivation (Lemma 1) establishing that the log-likelihood differences of B-spline coefficients follow a shifted and scaled beta distribution under the assumption of a Gaussian mixture model.
Simultaneous Clustering and Trimming: Unlike two-stage approaches that filter then cluster, funOCLUST integrates outlier detection and clustering within a model-based paradigm, retaining information from the full functional representation via the B-spline basis.
Handling of Outlier Types: The method is shown to be capable of detecting various functional outlier types (isolated, persistent, shift, amplitude, and shape) by analyzing the multivariate properties of the spline coefficients.

Results
The performance of funOCLUST was evaluated through simulation studies and real-world applications:

Simulation Study:
- The algorithm was tested against five competitors (funHDDC, T-funHDDC, tkmeans, mocca, and fOutl) across eight scenarios varying in cluster number, functional complexity, sampling sparsity, and outlier type (heavy-tailed vs. shift-scale).
- Clustering Accuracy: funOCLUST achieved the largest mean Adjusted Rand Index (ARI) across scenarios. It performed exceptionally well in scenarios involving heavy-tailed errors (scenarios 1, 4, 6, 7) and demonstrated robust, consistent performance across other conditions.
- Outlier Identification: In shift-scale scenarios, funOCLUST exhibited the lowest false negative rates for moderate complexity data, though performance decreased in high-complexity, sparse scenarios with many clusters.
Real-World Applications:
- Pedestrian Traffic (Melbourne): Applied to hourly traffic data, funOCLUST successfully identified significant holidays (e.g., New Year's, Christmas, Chinese New Year) as outliers. After removing 22 outliers, the algorithm perfectly classified the remaining days into workdays and non-workdays, outperforming funHDDC, T-funHDDC, and tkmeans.
- NOx Benchmark (Barcelona): Applied to air pollution data, funOCLUST achieved a Correct Classification Rate (CCR) ranging from 0.51 to 0.86, depending on the covariance structure. The best-performing model (EEE structure) achieved a CCR of 0.86, placing it on par with existing methods like T-funHDDC and C-funHDDC.

Significance and Claims
The paper claims that funOCLUST provides a robust, model-based approach to clustering functional data that effectively handles outliers without requiring dimension reduction that discards functional information. The authors state that the algorithm demonstrates "strong performance in clustering and outlier identification" across diverse simulated and real-world datasets.

The significance of the work is framed as paving the way for future extensions, including applications to multivariate and skewed functional data, and potential developments where the functional decomposition is estimated within the Expectation-Maximization (EM) algorithm rather than as a pre-processing step. The authors maintain a modest tone, acknowledging that while the method performs well, challenges remain in scenarios combining high complexity, sparse sampling, and limited per-cluster information.