2.3.2. Functional Analysis

The similarity between the di fferent samples of ageing technology and the di fferent samples of ageing time was contrasted. This analysis was carried out from a vectorial and functional approach. Vectorially, the tests used were the classical ANOVA [25], comparing the mean levels of the groups, and Kruskal's non-parametric test [26], which studies whether the observations of each group come from the same distribution. In addition, from the functional approach, the functional ANOVA (FANOVA) was performed.

### Functional Data Analysis (FDA)

The analyses from a functional approach study functions, based on sets of observations, were defined in a determined interval I = [a,b]. One of its strengths is its structure of infinite dimensions that allows to extend the possibilities of data analysis [29,30]. A random variable, measured at a set of discrete points - *tg G g*=1 ∈ [*a*, *b*], has to take values in metric or semi-metric spaces to be considered functional. Thus, functional data take the form of a matrix with *n* rows, one for each individual studied, and *G* columns representing the points of evaluation of the functions [31,32].

Smoothing is the most used process to convert discrete observations into continuous functions, *<sup>x</sup>*(*t*), *t* ∈X⊂F ; where F is the functional space. Specifically, assuming that the functions are observed with error, a functional basis expansion can be adopted as follows:

$$\mathbf{x}(t) = \sum\_{w=1}^{W} c\_w \phi\_w(t) \tag{1}$$

where {*cw*} *W <sup>w</sup>*=1 is the *w*-th basis coe fficients, - φ*w*(*t*) *W <sup>w</sup>*=1 is the *w*-th basis function, and *W* is the number of basis functions under consideration [29,32]. Thus, the basis functions used in this work were splines [33] due to their specific properties such as the possibility of generating large basis sets easily or their flexibility [34]. On the other hand, to select the number of bases of each sample, the determination coe fficient R<sup>2</sup> was taken into account. The number of bases is the minimum number at which R<sup>2</sup>

stops improving significantly or surpasses the value of 0.99 (see Martínez et al. [15]). Moreover, the smoothing process involves solving the following problem:

$$\min\_{\mathbf{x}\in F} \sum\_{\mathcal{S}'=1}^{G} \|\mathbf{z}\_{\mathcal{S}} - \mathbf{x}(\mathbf{t}\_{\mathcal{S}})\|^2 + \lambda \Gamma(\mathbf{x}) \tag{2}$$

where *zg* = *x*(*tg*) + *g* is the value obtained by evaluating *x* at point *tg* with *g* being a random noise with zero mean, λ is a parameter controlling the intensity of regularisations, and Γ is a parameter that makes it costly to reach complex solutions. Then, the basis coefficients can be expressed as the solution of the smoothing process as follows [29,35]:

$$\mathbf{z} = \left(\Phi^t \Phi + \lambda \mathbf{R}\right)^{-1} \Phi^T \mathbf{z} \tag{3}$$

being **Φ** a *GxW* matrix formed by <sup>Φ</sup>*gw* = φ*w*(*tg*) and **R** being a *WxW* matrix of the elements **<sup>R</sup>***wg* = *T<sup>D</sup>*<sup>2</sup>φ*w*(*t*)*D*<sup>2</sup>φ*g*(*t*)*dt* where *Dn*φ*w*(*t*) is the *n*th-order differential operator of φ*<sup>w</sup>*.
