REQUISITOS PARA EL ASCENSO DE OFICIALES DE ARMA Y DE SERVICIOS O TÉCNICOS

In the case of indirect observations, the signal is detectable only through s sensors on the sensors space. Let {Ki} be a collection of s × p real matrices, representing the subject

specific forward operators relating the signal at p pre-defined points {vj : j = 1,..., p}

on the cortical surfaceM with the signal captured by the s sensors. Moreover, define the evaluation operatorΨ : F → Rp to be a vector-valued functional that evaluates a function f ∈ F at the p pre-specified points {vj} ⊂ M , returning the p dimensional

vector ( f (v1), . . . , f (vp))T. The operatorsΨ and {Ki} are known. However, in the described

problem the random function X can be observed only through indirect measurements {yi∈ Rs: i = 1,...,n} generated from the model

   xi = µ +P∞_{r =1}ζi ,rψr yi = KiΨxi+ εi, i = 1,...,n (3.5)

where {xi} are n independent realizations of X , and thus expandible in terms of the PC

functions {ψr} and the coefficients {ζi ,r} given byζi ,r =

R

M{xi(v) − µ(v)}ψr(v)d v. The

terms {εi} represent observational errors drawn independently from an s-dimensional

normal random vector, with mean the zero vector and varianceσ2Ip, where Ipdenotes

the p-dimensional identity matrix. Model (3.5) is an implementation of the idealized Problem 3.2. In Figure 3.3 we give an illustration of the introduced setting.

Note that it would not be necessary to define an evaluation operator if the forward operators were defined to be functionals {Ki :F → Rs}, relating directly the functional

3.2 Mathematical description of the problem 65

objects on the brain space to the real vectors on the sensors space. It is however the case that the operators {Ki} are computed in a matrix form by third part software (see

Section 3.6 for details) for a pre-specified set of points {vj} ⊂ M and it is thus convenient

to take this into account in the model, through the introduction of an evaluation operator Ψ.

Sensors space

Brain Space

Latent objects

Figure 3.3 Illustration of the setting introduced with Model 3.5.

In the case of single subject studies, the surfaceM is the subject’s reconstructed cortical surface, an example of which is shown on the right panel of Figure 3.1. In this case, it is natural to assume that there is one common forward operator K for all the detected signals. In the more general case of multi-subject studies,M is assumed to be a template cortical surface. We are thus assuming that the individual cortical surfaces have been registered to the templateM , which means that there is a smooth and one-to-one correspondence between the points on each individual brain surface and the template surfaceM , where the PC functions are defined.

As mentioned in Section 1.1.4, a Reconstruct-then-Estimate approach is suboptimal in this setting. For this reason, different alternatives have been proposed in the literature. In the simplified setting of a fixed forward operator K := K1= . . . = Kn, Amini and Wainwright

(2012) propose to estimate the space spanned by the first R PC functions in a Regularized- Estimate fashion. The PC functions {ψr} are modelled as elements of a Reproducing

Kernel Hilbert Space (RKHS). On the sensors space, they define a smoothing matrix S−1 translating the smoothness assumption on the PC functions to the sensors space. The

66 Functional Principal Component Analysis in the inverse problem setting

PC loadings on the sensors space {zr} ⊂ Rsare in practice estimated as eigenvectors of a

regularized version of the empirical covariance ˆCY =_n1PiyiyT_i , namely

ˆ

CY − λS−1,

withλ a weighting coefficient. The PC functions of X are estimated as the minimum norm functions {ψr} satisfying

(KΨ)( ˆψr) = ˆzr.

The estimated PC functions ˆψr are such that their images on the sensors space (KΨ)( ˆψr)

interpolate ˆzrand have minimum norm in the RKHS where the PC functions are defined.

An extension to the case of subject specific forward operators has been proposed in Katsevich et al. (2015). The authors formulate a least square estimator for the discretized covariance function (CX(vj, vl))j ,l, which in the notation of this chapter takes the form

ˆ CX= arg min C ∈Rp×p 1 n n X i =1

∥(yi− ¯y)(yi− ¯y)T− KiC K_iT− σ2Is∥2_F,

with ¯y =_n1Pn

i =1yi. The termσ

2_I

s, withσ > 0 a constant and Isthe identity matrix of size

s, captures a diagonal structure, due for instance, to observational error. Intuitively, the

‘covariance’ CX (note that CX is not constrained to be positive semi-definite) is such that

its projections on the sensors space KiCXK_iT match as closely as possible the covariances

in the sensors space (yi− ¯y)(yi− ¯y)T.

In Dobriban et al. (2017), in the context of optimal prediction, the point-wise evalu- ations of the PC functions {ψr}, on v1. . . , vp, are estimated from the eigenvectors of the

empirical covariance of the backprojected data, i.e. 1 n n X i =1 K_iTyiyT_i Ki.

The approach in Amini and Wainwright (2012) cannot be immediately extended to the case of subject-specific forward operators and, to make use of the RKHS machinery, would require the definition of a kernel on the non-linear domainM , which is not a trivial task. The approaches in Dobriban et al. (2017); Katsevich et al. (2015) both lack of a regularization step, fundamental in the inverse problem setting considered here. Also, in Katsevich et al. (2015) the estimated eigenvectors could potentially be associated to negative eigenvalues, as the estimated covariance is not constrained to be positive semi- definite. This motivates a novel estimator proposed in Section 3.3, where we formulate an

3.2 Mathematical description of the problem 67

extension of the Regularized-Estimate fPCA model introduced in Chapter 2 to the inverse problem setting described with model (3.5).