Wasserstein distances are metrics on probability distributions inspired by the problem of optimal mass transportation. Roughly speaking, they measure the minimal effort required to reconfigure the probability mass of one distribution in order to recover the other distribution. They are ubiquitous in mathematics, with a long history that has seen them catalyse core developments in analysis, optimization, and probability. Beyond their intrinsic mathematical richness, they possess attractive features that make them a versatile tool for the statistician: they can be used to derive weak convergence and convergence of moments, and can be easily bounded; they are well-adapted to quantify a natural notion of perturbation of a probability distribution; and they seamlessly incorporate the geometry of the domain of the distributions in question, thus being useful for contrasting complex objects. Consequently, they frequently appear in the development of statistical theory and inferential methodology, and have recently become an object of inference in themselves. In this review, we provide a snapshot of the main concepts involved in Wasserstein distances and optimal transportation, and a succinct overview of some of their many statistical aspects.

We develop a canonical framework for the study of the problem of registration of multiple point processes subjected to warping, known as the problem of separation of amplitude and phase variation. The amplitude variation of a real random function {Y (x) : x ∈ [0, 1]} corresponds to its random oscillations in the y-axis, typically encapsulated by its (co)variation around a mean level. In contrast, its phase variation refers to fluctuations in the x-axis, often caused by random time changes. We formalise similar notions for a point process, and nonparametrically separate them based on realisations of i.i.d. copies {Πi} of the phase-varying point process. A key element in our approach is to demonstrate that when the classical phase variation assumptions of Functional Data Analysis (FDA) are applied to the point process case, they become equivalent to conditions interpretable through the prism of the theory of optimal transportation of measure. We demonstrate that these induce a natural Wasserstein geometry tailored to the warping problem, including a formal notion of bias expressing over-registration. Within this framework, we construct nonparametric estimators that tend to avoid over-registration in finite samples. We show that they consistently estimate the warp maps, consistently estimate the structural mean, and consistently register the warped point processes, even in a sparse sampling regime. We also establish convergence rates, and derive √ n-consistency and a central limit theorem in the Cox process case under dense sampling, showing rate optimality of our structural mean estimator in that case.

We consider two statistical problems at the intersection of functional and non-Euclidean data analysis: the determination of a Fréchet mean in the Wasserstein space of multivariate distributions; and the optimal registration of deformed random measures and point processes. We elucidate how the two problems are linked, each being in a sense dual to the other. We first study the finite sample version of the problem in the continuum. Exploiting the tangent bundle structure of Wasserstein space, we deduce the Fréchet mean via gradient descent. We show that this is equivalent to a Procrustes analysis for the registration maps, thus only requiring successive solutions to pairwise optimal coupling problems. We then study the population version of the problem, focussing on inference and stability: in practice, the data are i.i.d. realisations from a law on Wasserstein space, and indeed their observation is discrete, where one observes a proxy finite sample or point process. We construct regularised nonparametric estimators, and prove their consistency for the population mean, and uniform consistency for the population Procrustes registration maps.

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Covariance operators are fundamental in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen-Loève expansion. These operators may themselves be subject to variation, for instance in contexts where multiple functional populations are to be compared. Statistical techniques to analyse such variation are intimately linked with the choice of metric on covariance operators, and the intrinsic infinitedimensionality of these operators. In this paper, we describe the manifold geometry of the space of trace-class infinite-dimensional covariance operators and associated key statistical properties, under the recently proposed infinite-dimensional version of the Procrustes metric. We identify this space with that of centred Gaussian processes equipped with the Wasserstein metric of optimal transportation. The identification allows us to provide a complete description of those aspects of this manifold geometry that are important in terms of statistical inference, and establish key properties of the Fréchet mean of a random sample of covariances, as well as generative models that are canonical for such metrics and link with the problem of registration of functional data.

Summary Lande and Arnold's approach to quantifying natural selection has become a standard tool in evolutionary biology due to its simplicity and generality. It treats linear and nonlinear selection in two separate frameworks, generating coefficients of selection (e.g. linear and quadratic selection gradients) that are not directly comparable. Due to this somewhat artificial division, the Lande–Arnold approach lacks an integrated measure of the strength of selection that applies across qualitatively different selection regimes (e.g. directional, stabilizing or disruptive selection). We define a unified measure of selection, the distributional selection differential (DSD), which includes both linear and nonlinear selection. The DSD quantifies total selection on a trait, regardless of the underlying selection regime. The DSD can be partitioned into a directional component, representing selection on the trait mean, and a non‐directional component, representing selection on the shape of the trait distribution (e.g. variance, skew or the number of modes). When multiple traits are measured, the DSD can also be separated into direct and correlated effects, analogously to linear selection gradients. As with linear selection differentials, the DSD on a standardized trait is limited in magnitude by the opportunity for selection. The DSD is a general‐purpose measure of the total strength of selection. It is particularly valuable where traditional analyses provide limited insight, such as in comparative studies where the shape of selection is variable. Partitioning the DSD into directional and non‐directional selection allows biologists to assess whether selection acts consistently in one direction, or in opposing directions over different parts of the trait range.

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