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An Explicit Link between Extreme Value Theory and Compositional Data Analysis

This paper establishes an explicit algebraic link between Extreme Value Theory and Compositional Data Analysis by demonstrating how their respective covariance and dependence representations are connected through simple transformations, thereby enabling the transfer of statistical methods such as graphical models and dimensionality reduction techniques between the two fields.

Original authors: Manuel Hentschel, Sebastian Engelke

Published 2026-07-13

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Manuel Hentschel, Sebastian Engelke

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 two different groups of scientists trying to solve the same puzzle, but they are speaking completely different languages. On one side, you have Extreme Value Theory experts, who study rare, massive events like once-in-a-century floods or sudden stock market crashes. On the other side, you have Compositional Data Analysis experts, who study things where only the proportions matter, like the chemical mix in a rock or the percentage of different species in a forest.

For a long time, these two groups were working in parallel, unaware that they were actually solving the exact same mathematical riddle. This paper, written by Manuel Hentschel and Sebastian Engelke, acts as a universal translator, revealing a hidden bridge between these two worlds.

The Core Discovery: It's All About Ratios, Not Sizes

Here is the big idea: Both fields are obsessed with relative information.

In the world of extreme events, scientists realized that when a massive storm hits, the total size of the storm is less important than the shape of the damage. Did the wind blow harder in the north or the south? The math separates the "radial size" (how big the event is) from the "relative profile" (how the parts compare to each other).

In the world of compositions, scientists realized that if you double the amount of every ingredient in a cake recipe, the cake tastes exactly the same. The absolute amounts don't matter; only the ratios between the flour, sugar, and eggs do.

The paper proves that the mathematical tools used to describe these "shapes" and "ratios" are actually identical. It's like discovering that the map used by a sailor navigating a storm is drawn on the exact same grid as the map used by a baker measuring ingredients.

The Magic Toolkit: Projections and Inverses

The authors didn't just say, "Hey, these look similar." They built a rigorous algebraic framework to show exactly how the tools match up. They use a few fancy-sounding mathematical moves, which we can think of as:

Oblique Projections: Imagine shining a light on a 3D object to cast a shadow on a wall. In this paper, they show how to "project" complex data onto a flat surface to see the relative relationships clearly, ignoring the overall size.
The Variogram Map: This is a special calculator that turns a list of variances (how much things wiggle) into a map of differences. The paper shows that the "extremal variogram" used for floods is the same as the "compositional variation array" used for chemical mixes.
Moore–Penrose Inverses: Think of this as a "reverse button" for math problems that usually can't be reversed because they have too many variables. The paper proves that if you hit this reverse button on the compositional data, you get the exact same result as hitting it on the extreme value data.

What This Means for Real Life (The Transfers)

Because the math is the same, the authors show we can steal tricks from one field and use them in the other.

1. From Extremes to Compositions: Finding Hidden Connections
In extreme value theory, researchers have developed a way to draw "graphs" that show which parts of a system are connected during a crisis. For example, if a flood hits one river, does it automatically hit the next one? The paper introduces a new method called intrinsic logistic-normal graphical models for compositional data.

The Test: They tried this on a real dataset of geochemical measurements (the "gemas" dataset). They found that their new method could spot connections between elements just like the extreme value methods do. It's not a magic bullet that solves everything instantly, but in their simulations, it successfully identified the correct connections in a 10-dimensional model, catching 25 out of 27 true links.

2. From Compositions to Extremes: Seeing the Big Picture
Compositional data experts have been using clever ways to shrink huge datasets down to a manageable size (dimensionality reduction) for decades. The paper shows that extreme value researchers can use these same tricks.

The Test: They applied a technique called Log-Ratio Analysis to the "Danube" river dataset, which tracks water flow at 38 different stations. By treating the river flow data like a composition, they created a visual map (a biplot) that clearly showed how the river stations were connected. They even tested a "weighted" version, giving more importance to stations with higher water flow, which made the main river path stand out even more clearly.

3. Smoothing the Thresholds
Usually, to study extreme events, scientists have to draw a hard line (a threshold) and say, "Anything below this is normal, anything above is extreme." This is a bit arbitrary.

The Suggestion: Borrowing from weighted analysis in compositions, the authors suggest a weighted variogram estimator. Instead of a hard cut-off, they give "more extreme" observations higher weights. In their simulations with 500 datasets, this weighted approach slightly reduced the error (Mean Squared Error) compared to the standard unweighted method, suggesting it might be a better way to handle the messy reality of real-world data.

What They Don't Claim

It's important to know what this paper doesn't say.

It doesn't claim to solve the "Zero" problem: Both fields struggle with data that has zeros (like a chemical that isn't present, or a river with no flow). The paper explicitly states that they assumed strictly positive data to make the math work. They suggest that handling zeros is a job for future research, not something they solved here.
It doesn't claim the methods work for all distributions: The perfect mathematical link they found works specifically for the Hüsler–Reiss model (a specific type of extreme value distribution). For other, more complex distributions, the connection is messier and involves "exponential tilting," which means the simple translation doesn't work perfectly.
It's not a "Breakthrough" in the sense of a finished product: The authors are careful to say they are "introducing" and "exploring" these transfers. They show it can be done and provide simulations to back it up, but they aren't claiming that every statistical problem in these fields is now solved.

The Bottom Line

This paper is a masterclass in connecting dots. It takes two seemingly unrelated fields—one studying the loudest storms and the other studying the finest recipes—and proves they are using the same underlying geometry. By building a bridge of algebra, the authors allow statisticians to swap tools, potentially making it easier to understand both the rarest disasters and the most complex mixtures. It's a solid, mathematically proven link that opens the door for future exploration, rather than a final destination.

Technical Summary: An Explicit Link between Extreme Value Theory and Compositional Data Analysis

Problem Statement
Extreme value theory (EVT) and compositional data analysis (CoDA) address distinct statistical domains—rare events and relative proportions, respectively—yet both fundamentally rely on relative information. In multivariate EVT, the dependence structure of threshold exceedances is often characterized by homogeneity properties that separate the radial size of an event from its relative profile. In CoDA, data are analyzed up to multiplicative scale, with inference based on ratios or log-ratios. Despite these conceptual parallels, the fields have developed separate covariance and dependence representations (e.g., extremal variograms in EVT versus compositional variation arrays in CoDA) without a unified algebraic framework. This lack of a formal link has hindered the systematic transfer of statistical methods between the two disciplines.

Methodology
The authors establish a rigorous algebraic connection by developing a general framework for the covariance and precision matrices of random vectors supported on general subspaces. The core of this methodology involves:

Oblique Projections: The paper defines oblique projections $P_{UV^\perp}$ along a kernel subspace $U$ onto an image subspace $V^\perp$ . It demonstrates that covariance structures in both fields can be related through these projections and their associated "covariance projections" ( $\pi_{UV^\perp}$ ), defined as $\pi_{UV^\perp}(A) = P_{UV^\perp} A P_{VU^\perp}$ .
Variogram Mapping: A central component is the variogram map $\gamma$ , which transforms a covariance matrix $S$ into a matrix of pairwise variances of differences: $\gamma(S)_{ij} = S_{ii} + S_{jj} - 2S_{ij}$ . The authors prove that for kernels spanned by the one-vector $\mathbf{1}$ , this map is bijective between specific sets of covariance matrices and conditionally negative definite variogram matrices.
Unification of Representations: The framework unifies three representations in EVT (extremal variogram $\Gamma$ , extremal function covariance $\Sigma_v$ , and precision matrices $\Theta_v$ ) with three analogous representations in CoDA (compositional variation array $\Gamma$ , centred log-ratio covariance $\Sigma_1$ , and additive log-ratio covariance $\Sigma_k$ ). The paper shows that these matrices are related by a small set of transformations: oblique projections, Moore–Penrose inverses, and the variogram map.

Key Contributions and Results
The paper's primary contribution is the formalization of the algebraic isomorphism between the covariance structures of the Hüsler–Reiss model in EVT and log-ratio transformed data in CoDA.

Formal Link: The authors prove that the extremal variogram in the Hüsler–Reiss model corresponds exactly to the compositional variation array. Similarly, the covariance of extremal functions corresponds to the covariance of centred log-ratios (CLR), and the singular precision matrices in Hüsler–Reiss graphical models correspond to the precision matrices of intrinsic logistic-normal models in CoDA.
Method Transfer (EVT $\to$ CoDA): Leveraging this link, the authors introduce intrinsic logistic-normal graphical models for compositional data. These models assume the CLR-transformed data follows a degenerate normal distribution with a sparse precision matrix $\Theta_1$ . The paper proposes the cglearn algorithm, an adaptation of the eglearn algorithm from EVT, to estimate the graphical structure (sparsity pattern) of these models by aggregating results from ALR-transformed data across different pivots.
Method Transfer (CoDA $\to$ EVT): Conversely, the authors demonstrate how dimensionality reduction techniques from CoDA can be applied to EVT. Specifically, they apply log-ratio analysis (LRA) and weighted LRA to the extremal functions of the Danube dataset, showing that these methods effectively visualize the dependence structure of extreme river flows. Additionally, they adapt stepwise log-ratio selection and amalgamation-based hierarchical clustering to select informative pairwise differences in extremal functions.
Weighted Estimation: The paper introduces a weighted extremal variogram estimator for EVT, inspired by weighted LRA. By assigning weights based on the radial magnitude of observations, this estimator softens the hard thresholding typically used in EVT, demonstrating improved mean squared error performance in simulations compared to unweighted estimators.

Significance
The paper claims that while parallels between EVT and CoDA have been noted previously, this work is the first to formalize the connection through a shared algebraic framework. This formalization allows for the systematic transfer of statistical models and methods between the fields. The significance lies in:

Algebraic Unification: Providing a common language (oblique projections and variogram maps) that explains why different covariance representations in both fields are mathematically equivalent.
Methodological Innovation: Enabling the direct import of sparse graphical modeling techniques from EVT to CoDA (via intrinsic logistic-normal models) and dimensionality reduction techniques from CoDA to EVT.
Practical Utility: Demonstrating that established algorithms (like cglearn and easyCODA) can be immediately applied to problems in the other domain, offering new tools for analyzing high-dimensional extremes and compositional data.

The authors remain modest regarding the scope, noting that the transfer of methods is currently established for the Hüsler–Reiss model and standard log-ratio transformations, and that handling zeros (in CoDA) and asymptotic independence (in EVT) requires further research. However, the established algebraic backbone provides a clear pathway for future generalizations.

The Core Discovery: It's All About Ratios, Not Sizes

The Magic Toolkit: Projections and Inverses

What This Means for Real Life (The Transfers)

What They Don't Claim

The Bottom Line

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