The model

Let now {vi : i = 1,...,n} be a set of random samples of a zero-mean and finite second

momentV -valued random function V and {Zi : i = 1,...,n} be a set of random smooth

samples of a zero-mean and finite second moment random real function Z with values in

L2(M0). We assume the following generative model for the i th observation (Mi, Yi):

           Mi = ϕvi◦ M0, Xi = µ + δZi, Yi = Xi◦ ϕ−1vi , (4.1)

96 Functions on Surfaces

whereµ ∈ L2(M0) is a fixed function, modelling the common function behavior between

the different samples, andδ is a coefficient representing the magnitude of the function variations around the meanµ. In addition, we assume the objects in Model 4.1 are subject to a discretization error, which is considered in the estimation process. This formulation generalizes an often used model for the one-dimensional functional registration problem (see, e.g. Tang and Müller (2008)).

Model 4.1 achieves the goal of representing FoSs as a collection of more tractable objects, decomposing the generation of the i th FoS into three main steps. In the first step, the geometryMi of the i th object is generated by the deformationϕvi applied to the

templateM0, where vi is a random sample of V . In the second step, a random function Xi,

on the template, is generated as the sum of the fixed functionµ and a stochastic term δZi.

In the third step the generated function Xi is transported on the manifoldMi, defining

Yi. This is done through the equation Yi = Xi◦ ϕ−1vi , which means that for all x ∈ M0,

Yi(ϕvi(x)) = Xi(x), or informally that the functional value Xi(x) is ‘transported’ with the

deformation to the locationϕvi(x) ∈ Mi.

We now describe the FoSs generation process from Model 4.1, for different choices of the deformation operator:

• Shift operator: LetV = R3, we defineϕvto be such thatϕv(x) = x +v for all v ∈ V , x ∈

R3_{. Clearly, in this case, {M}

i = ϕvi◦ M0} in Model 4.1 would generate a collection of

surfaces shifted in the directions specified by {vi}.

• Identity operator: Let V be the space of smooth functions v : M0 → R3 and let

ϕv(x) = x + v(x) for all x ∈ M0. In this case, {Mi} would be a collection of smoothly

deformed versions of the templateM0. Note however, that the maps being only

smooth and not homeomorphic, it cannot be guaranteed that every choice of v ∈ V preserves the topology ofM0. Nevertheless, this choice might still represent a valid

option in a small deformations setting.

To solve this problem, we could think of restrictingV to contain only smooth and homeomorphic functions, however, in this way, the linearity of the spaceV is lost, and this is a property of fundamental importance to the subsequent analysis, given that we want to apply linear statistics on the random function V , which takes values onV .

• Diffeomorphic operator: LetV be a Sobolev space of sufficiently smooth vector fields fromR3toR3vanishing, with their derivatives, at infinity. Letϕ be a diffeomorphic

4.2 Model for Functions on Surfaces 97

deformation operator, i.e. an operator such thatϕv is a diffeomorphism ofR3for

all v ∈ V . Then, for different choices of v, Model 4.1 would generate a collection of surfaces that are diffeomorphic (and thus homeomorphic) deformations of the templateM0. More importantly, these deformations are parametrized by the linear

spaceV , where linear statistics can be applied. For this choice, an illustration of the generative process is shown in Figure 4.2. The diffeomorphic deformation operator can be defined by means of an Ordinary Differential Equation (ODE). Details of this are described in Section 4.2.4.

Figure 4.2 An illustration of the generation of a FoS through Model 4.1 withϕ the diffeo- morphic deformation operator. From left to right, in the first panel we have a functional sample Xi on the geometric templateM0. In the second panel we have a vector field

vi ∈ V , a sample of the random function V , evaluated on a uniform grid in R3. This is

shown together with (M0, Xi). In the third panel we have the diffeomorphic deformation

ϕvi, obtained from vi as described in Section 4.2.4, here displayed as the set of vectors

{ϕvi(ξk)} ⊂ R

3_{with {ξ}

k} the nodes of the triangulated surface representing the template

M0. In the fourth panel, we have the FoS (Mi, Yi) obtained by applying the deformation

ϕvi toM0and ‘transporting’ the functional values with it.

More complicated generative models could be built from Model 4.1. For example, the functions {vi} and {Xi}, representing respectively geometries and functions, could

be modelled in terms of conditional expectation of different sources of information on the subjects such as age, disease status or other subject-specific explanatory variables, as done, in the case of functional data located on 1D domains, in Hadjipantelis et al. (2015). However, Model 4.1 is the simplest model enabling a comprehensive study of the relation between geometric and functional variability.