🌿 ecology

The algebra of community temperature indices

This paper proposes a new "Aitchison CTI" that preserves the algebraic structure of compositional data to provide a more robust classification of species' thermal affinities and a reliable measure of community temperature change, addressing the limitations of existing indices through theoretical derivation and empirical validation.

Original authors: Spencer, M.

Published 2026-08-06

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Spencer, M.

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine you are a detective trying to solve a mystery about how a neighborhood of living things is changing over time. In the world of ecology, this neighborhood is called a "community," and it's made up of many different species of plants, animals, or microbes. Scientists have long wanted a simple way to measure if this neighborhood is getting "hotter" or "colder" in terms of the species' preferences. They do this by tracking something called the Community Temperature Index (CTI). Think of the CTI as a single number that acts like a thermometer for the whole group. If the number goes up, it means the community is shifting toward species that love warmth; if it goes down, it's shifting toward species that prefer the cold.

To understand how this works, you need to know about relative abundance. This isn't about counting exactly how many bugs are in a jar; it's about the proportion of each type. If you have a jar with 90% beetles and 10% ants, and you add more beetles, the proportion of ants drops, even if the total number of bugs stays the same. Because these numbers always have to add up to 100% (or 1), they behave differently than normal numbers. They live in a special mathematical space where adding or multiplying them follows strange, specific rules, much like how mixing colors works differently than mixing water. The big question scientists have been asking is: "Are we using the right math to measure these changes?" If we use the wrong math, we might think the neighborhood is changing when it's actually just shifting its proportions in a way that looks confusing.

The Problem with the Old Thermometer

For years, ecologists have used a popular method to calculate this "neighborhood thermometer." They simply took the average temperature preference of every individual in the community, weighted by how common each species was. It's like calculating the average height of a crowd by just adding up everyone's height and dividing by the number of people. It seems simple and logical, right?

But the author of this paper, Matthew Spencer, argues that this simple average is actually a trap. He points out that this old method treats relative abundances like regular numbers, ignoring the special "mixing" rules they actually follow. To prove this, he sets up a scenario where three species are growing at steady, constant rates—like three cars driving down a highway at unchanging speeds. If you use the old method, the "thermometer" reading starts to wiggle up and down, suggesting the community's temperature preference is fluctuating wildly. But in reality, nothing has changed; the cars are just driving steadily. The old math is creating a fake signal, making it look like the neighborhood is having a crisis when it's actually just cruising along.

The New "Aitchison" Solution

Spencer proposes a new way to calculate the index, which he calls the Aitchison CTI. Instead of using a simple average, this new method uses a special kind of math called Aitchison geometry. If the old method is like measuring a smoothie by just adding up the cups of fruit, the Aitchison method is like measuring the ratios of the flavors. It respects the fact that if you add more of one fruit, you automatically have less of the others.

When Spencer applies this new math to the same "steady driving" scenario, the result is perfect: the thermometer shows a smooth, straight line. It correctly tells us that the community is changing at a constant, predictable rate. This new index preserves the "algebraic structure" of the community, meaning it behaves the way population dynamics actually work.

Redefining "Warm" and "Cold" Friends

One of the most interesting side effects of this new math is how it changes the way we label species. Previously, scientists would call a species "warm-affinity" if its presence made the community's temperature index go up. But the old method had a weird flaw: whether a species was "warm" or "cold" depended on what the other species were doing. A species could be "warm" in one neighborhood and "cold" in another, just because the mix of neighbors was different. It was like saying a person is "tall" only if everyone else in the room is short.

The Aitchison CTI fixes this. It defines a relative warm-affinity species as one whose temperature preference is simply higher than the geometric average of all the species in the group. This definition is fixed; it doesn't change based on who else is in the room. If a species is "warm," it's always "warm," no matter how the community shifts. This makes it much easier to understand who is driving the changes.

Does the New Thermometer Tell the Truth?

The paper also introduces a clever new tool to check if the thermometer is actually useful. It's similar to a score called $R^2$ used in statistics, but adapted for this special math. This score tells us how much of the community's change is actually moving in the direction of temperature preferences.

When the author tested this on real data from a hard-substrate community in the Bay of Biscay (involving 151 species of sea creatures), the results were a bit mixed. The new Aitchison CTI was roughly related to the old one, which is good news. However, the "usefulness score" was only 0.12. This suggests that while the new math is more accurate, the temperature preferences of these 151 species only explain a small part of why the community is changing. The author suggests that with so many species, the changes are likely driven by many other factors, not just temperature. It's like trying to explain a complex dance with only one move; the dance is happening, but temperature is just one small part of the story.

The Takeaway

The paper concludes with a clear recommendation: if you are studying how communities change over time, you should switch to the Aitchison CTI. It's the only method that respects the true math of how species proportions work. It gives a consistent definition of warm and cold species, and it avoids the fake signals that the old method creates.

However, the author also warns us to be humble. Even with the perfect math, a single number like the CTI might not capture the whole story of a complex ecosystem. Before trusting the results, scientists should check if their data actually shows a strong link between temperature preferences and community changes. If the link is weak, as it was in the Bay of Biscay example, the thermometer might be telling us more about the limits of the data than the limits of the climate.

Technical Summary: The Algebra of Community Temperature Indices

Problem Statement
The Community Temperature Index (CTI) is widely used to measure temporal or spatial changes in ecological communities based on species' temperature preferences (Species Temperature Indices, or STIs). The most common form, introduced by Devictor et al. (2008), calculates the CTI as a weighted arithmetic mean of STIs, where weights are relative abundances ( $CTI(x) = \sum c_i x_i$ ). However, this paper argues that this standard formulation fails to preserve the fundamental algebraic structure of community dynamics. Specifically, the standard CTI does not behave linearly under the operations that govern changes in relative abundances (proportional growth). Furthermore, the classification of species as "warm- or cold-affinity" based on the standard CTI is problematic because it depends on the relative abundances of all species in the community, requiring arbitrary reference points, rather than relying solely on intrinsic species traits (STIs).

Methodology
The author proposes that CTI calculations should be grounded in the Aitchison geometry of compositional data analysis, which is the natural mathematical setting for relative abundance data. The methodology involves:

Defining Axioms: Establishing three desirable properties for a CTI:
- Intensivity (I1): Independence from total system size (homogeneity of degree zero).
- Additivity (I2): Additivity with respect to consecutive changes (preserving the structure of sequential perturbations).
- Homogeneity (I3): Proportionality to the time or space interval over which a constant rate of change occurs.
Deriving the Aitchison CTI: By requiring the CTI to be a linear functional (homomorphism) from the simplex of relative abundances to the real line under Aitchison operations (perturbation $\oplus$ and powering $\odot$ ), the author derives a new index. Using the Riesz representation theorem, this leads to the definition of the Aitchison CTI as the Aitchison inner product between the vector of relative abundances ( $x$ ) and a vector of STIs ( $c$ ):
$I_c(x) = \langle x, c \rangle_a = \frac{1}{n} \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \log\left(\frac{x_i}{x_j}\right) \log\left(\frac{c_i}{c_j}\right)$
This can also be expressed as $\sum \log(x_i/gm(x)) \log(c_i/gm(c))$ , where $gm$ denotes the geometric mean.
Re-evaluating Species Classification: The paper redefines "warm- and cold-affinity" species based on the Aitchison CTI. A species is defined as relative warm-affinity if its STI is greater than the geometric mean of all STIs in the community, and cold-affinity if it is lower. This classification is shown to be independent of relative abundances.
Quantifying Information Content: A metric analogous to the coefficient of determination ( $R^2$ ) is introduced. This metric, $\cos^2 \theta$ , measures the extent to which the vector of changes in relative abundances aligns with the vector of relative STIs in the Aitchison geometry.
Empirical Illustration: The approach is tested using two examples: a theoretical case of exponential population growth and an observational dataset of hard-substrate macrobenthos in the Bay of Biscay (2002–2020).

Key Contributions and Results

Algebraic Consistency: The paper demonstrates that the standard Devictor CTI is not a linear function of time even when species grow at constant proportional rates (exponential growth). In contrast, the Aitchison CTI increases linearly with time under these conditions, correctly reflecting constant community dynamics.
Abundance-Independent Classification: The new definition of warm/cold affinity relies strictly on the ratio of a species' STI to the geometric mean of community STIs. This eliminates the need for arbitrary reference communities or time points required by the standard CTI.
Consistent Contribution Analysis: The contribution of a species to the change in CTI is derived as $\lambda_i \log(c_i/gm(c))$ , where $\lambda_i$ is the proportional change in abundance. This ensures that a doubling of abundance contributes equally to the index change regardless of whether the species is rare or common, aligning with population dynamics principles.
Approximate Linearity: The paper shows that the standard CTI is approximately linearly related to the Aitchison CTI if relative abundances are approximately equal and STIs are close to their geometric mean. This suggests the standard CTI may be interpretable in specific, constrained scenarios.
Empirical Findings: In the Bay of Biscay macrobenthos dataset, the Aitchison CTI was approximately linearly related to the standard CTI. However, the calculated $\cos^2 \theta$ was low (0.12), indicating that changes in relative abundances were largely orthogonal to the direction of temperature preferences. This suggests the CTI captured only a modest amount of information regarding community dynamics in this 151-species system.

Significance and Claims
The paper claims that the Aitchison CTI is the only form of the index that preserves the algebraic properties of community dynamics, making it the theoretically correct choice for measuring change over time or space. It argues that using the Aitchison geometry resolves ambiguities in classifying species and interpreting their contributions to community shifts. The author emphasizes that while the standard CTI is widely used, its lack of algebraic rigor can lead to misinterpretations of trends (e.g., falsely inferring changes in dynamics when none exist). The paper concludes with four recommendations: use mathematical notation for CTI definitions, prefer the Aitchison CTI for temporal/spatial studies, verify linear relationships if using alternative forms, and calculate the alignment metric ( $\cos^2 \theta$ ) to assess the utility of the index in describing community change.