Statistical Frameworks in Evolutionary Geometry

• Statistical frameworks for evolutionary geometry integrate metric, Riemannian, and information-geometric methods to represent phylogenetic trees, shapes, and population states.
• They employ advanced models such as tropical, polyhedral, and stochastic approaches to optimize geodesic computations and improve likelihood inference.
• Applications span phylogenetic inference, morphometric analysis, and dynamic population studies, enabling robust evolutionary predictions and quantitative insights.

Statistical frameworks for evolutionary geometry refer to a confluence of geometric, probabilistic, and statistical methodologies designed to rigorously model, analyze, and infer the evolution of biological forms, phylogenetic trees, and more abstract populations under evolutionary forces. These frameworks leverage structures from metric geometry, Riemannian geometry, algebraic geometry (notably tropical and polyhedral), information geometry, and stochastic analysis. The resulting methods address both the representation of evolutionary objects (trees, shapes, population states) as points in suitably constructed geometric spaces and the statistical inference of evolutionary processes from observed data.

1. Geometric Models of Evolutionary Space

A primary pillar of evolutionary geometry is the construction of geometric spaces encoding evolutionary objects—most classically, phylogenetic trees and shape spaces.

Wald Space for Trees. Wald space $\mathcal{W}_N$ is the set of equivalence classes of weighted phylogenetic forests on $N$ labeled taxa, modeled as a union of open cubes parameterized by compatible split-systems $(E, \lambda)$, where each cube is $(0,1)^E$. Each forest is embedded as a point in the manifold of symmetric positive-definite matrices $\operatorname{SPD}_N$, via a mapping $\phi$ determined by path probabilities, conferring a natural geometric structure (Lueg et al., 2022, Garba et al., 2020).

Polyhedral and Tropical Models. The space of tree metrics (additive or ultrametric) is formed as a polyhedral fan—such as the Billera-Holmes-Vogtmann (BHV) cubical complex and the tropical Grassmannian $\operatorname{tropGr}(2,n)$. The tropical model equips tree spaces with the tropical metric (Hilbert projective) and a structure of polyhedral cones facilitating algorithmic sampling and projection (Bhatt et al., 25 Dec 2025, Davidson et al., 2016).

Shape Spaces and Diffeomorphic Models. Morphological variation is modeled in infinite or finite dimensions as the quotient of the space of immersions modulo diffeomorphisms and rigid motions, $$S = \operatorname{Imm}(M, \mathbb{R}^3)/[\operatorname{Diff}(M)\times E(3)]$$, often equipped with an $L^2$-type Riemannian metric (Faigenbaum-Golovin et al., 2024, Arnaudon et al., 2017).

Population Simplex and Information Geometry. The space of population states is the probability simplex $\Delta^n$, endowed with the Fisher-Rao (Shahshahani) metric or its escort deformations, realizing evolutionary game dynamics and statistical divergences as geometric flows (0911.1383, 0911.1764).

2. Riemannian and Information-Geometric Structures

Statistical frameworks impose Riemannian or information-geometric metrics, often with direct statistical interpretations:

Fisher-Rao on Trees and Distributions. In $\mathcal{W}_N$, the intrinsic metric can be induced via the Fisher information of the generative Markov or Gaussian process on the tree, with the infinitesimal line element proportional to $f$-divergences between sequence distributions (Garba et al., 2020). The affine-invariant metric on $\operatorname{SPD}_N$—important for tree geodesics—takes the form

$$d_{\operatorname{SPD}}(X,Y)=\|\log(X^{-1/2}YX^{-1/2})\|_F.$$

Escort Manifolds. Generalized information metrics are realized via the escort function $\varphi$, yielding Riemannian metrics $g^ϕ_{ij}(x) = \delta_{ij}/\varphi(x_i)$ and associated divergences $D_ϕ$ unifying Bregman, Tsallis, and Kullback-Leibler distances (0911.1764).

Curvature and Stratification. Stratified spaces, such as $\mathcal{W}_N$, are unions of cubes glued along boundaries according to split refinement. Sectional curvatures in these models can be computed via pullbacks from ambient space, showing both positive and negative values (mixed curvature), in contrast to the non-positively curved CAT(0) structure of BHV (Lueg et al., 2022, Garba et al., 2020).

3. Stochastic and Dynamic Evolutionary Models

Stochastic Flows on Shapes. In landmark shape spaces, the evolution is modeled as stochastic flows on diffeomorphism groups, descending to stochastic Hamiltonian systems on landmark coordinates. The induced Fokker-Planck equations permit moment closure, leading to efficient parameter inference for spatial correlation of evolutionary noise (Arnaudon et al., 2017).

Spline and SDE Models. Shape splines and their stochastic analogues embed shape evolution as second-order flows (covariant acceleration minimization or stochastic Hamiltonian systems), integrating both deterministic interpolation and probabilistic generative modeling (Trouvé et al., 2010).

Graph-Structured Evolution. Stochastic dynamics on graphs (e.g., invasion, fixation) are approximated using moment-closure techniques, homogenized pair-approximation, unconditioned node-level ODEs, and higher-order closures, allowing quantitative prediction of fixation probabilities and evolutionary dynamics on arbitrary structures (Overton et al., 2019).

Discrete Coalescent Trees. Spaces of rooted binary trees with discrete node times and moves (NNI, rank, length) define a metric supporting efficient geodesic computation and convex subspace properties essential for clock-based phylogenetic inference (Collienne et al., 2021).

4. Statistical and Computational Tools

Geodesics, Means, and Principal Geodesics. Intrinsic or extrinsic geodesic computations are central, with algorithms including projection and mid-point refinement in $\mathcal{W}_N$ and straightening in $\operatorname{SPD}_N$ (Lueg et al., 2022, Garba et al., 2020). Fréchet means are defined as minimizers of summed squared geodesic distances, supporting CLTs and tangent-space analysis; principal geodesics generalize PCA for (stratified) Riemannian settings.

Likelihood and Inference Procedures. Bayesian, variational, and frequentist methods operate within these geometric frameworks:

• Variational inference in continuous or tropical tree space is realized by reparameterizing over geometric latent spaces and employing stochastic gradient estimators (e.g., GeoPhy) (Mimori et al., 2023).
• EM algorithms in stochastic shape models use bridge sampling to compute expectation steps for likelihood maximization (Arnaudon et al., 2017).
• Hypothesis testing (e.g., change-point detection for evolving shapes) uses distribution-free functional central limit theorems in barcode-valued time series, with self-normalized test statistics ensuring pivotal asymptotics (Delft et al., 2023).

Combinatorial and Algebraic Algorithms. Output cones, polyhedral decompositions, and tropical sampling (e.g., hit-and-run, $p$-adic valuation) enable computationally tractable sampling and likelihood optimization in high-dimensional tree spaces (Bhatt et al., 25 Dec 2025, Davidson et al., 2016).

5. Applications and Comparative Analysis

Phylogenetic Inference. Geometric properties of spaces such as $\mathcal{W}_N$, BHV, and tropGr(2,n) influence the behavior of Bayesian inference, frequentist optimization, and the distribution of sample means (e.g., avoidance of stickiness in Wald space vs. BHV) (Lueg et al., 2022, Bhatt et al., 25 Dec 2025).

Morphometric Analysis. Principal geodesic analysis, kernel methods, and phylogenetic generalized least squares (PGLS) in nonlinear shape spaces enable robust association tests for shape evolution across phylogenies (Faigenbaum-Golovin et al., 2024).

Topological Time Series Analysis. Evolving point-cloud data (e.g., from single-cell RNA-seq) are encoded via persistent homology and analyzed using barcode-valued functional time series, with statistical tests for nonstationarity founded in metric geometry (Delft et al., 2023).

Stochastic Population Evolution. Structured population models on graphs connect graph geometry with evolutionary outcomes, bridging statistical physics and population genetics (Overton et al., 2019).

6. Outlook and Open Problems

Future directions identified include:

• Development of biologically informed priors and credible regions within tropical or polyhedral tree spaces (Bhatt et al., 25 Dec 2025).
• Efficient scalable geodesic and mean algorithms for large $N$ within non-Euclidean and stratified geometries (Lueg et al., 2022).
• Robust extension of moment-closure and stochastic differential models to non-Poissonian, multi-type, and highly heterogeneous evolutionary scenarios (Overton et al., 2019).
• Integration of shape and tree geometry with functional data analysis and nonparametric modeling (Delft et al., 2023, Trouvé et al., 2010).
• Formalization of large-sample statistical properties (CLTs, limit theorems, confidence sets) for geometric means and evolutionary test statistics in stratified/nonpositively curved spaces.

The intersection of geometry, statistics, and evolution continues to produce flexible, theoretically grounded, and computationally tractable frameworks for the quantitative analysis of evolutionary processes and structure (Garba et al., 2020, Lueg et al., 2022, Bhatt et al., 25 Dec 2025, Faigenbaum-Golovin et al., 2024).