Chapter 3 Statistical Techniques for Functional Data Analysis
3.2 Registration
3.2.4 Self-modelling warping functions
The idea of self-modelling warping function is rooted in the work for Kneip & Gasser and the concept of structural meanµ[168; 95]. In effect there one assumes that the “structural points”, local extrema and inflection points, define a structural mean curve and that the deviations from that curve are what one perceives in a sample’s realization. Based on this the generative model of the sample of curvesy becomes:
yi(t) =aiµ{hi−1(t)}+i(t) (3.26)
6
where as beforeh−i 1 is the inverse warping functionhi andare random errors such
that E{i(t)} = 0. An additional caveat being a 6= 0 and E(a) = 1. Given this
generative model of a sample curvesy, the idea of self-modelling warping functions is that one can decompose a warping functionhin terms of spectral-like decomposition:
hi(t) =t+ q
X
j=1
sijφj(t) (3.27)
where as before i= 1, . . . , n,t ∈T and this time si are the zero-meaned score (or
weight vectors) dictating the effect carried from each componentφi(t):
φi(t) =cTjβ(t). (3.28)
Howeverφis not an eigenbasis becauseβ(t) is a vector of B-spline basis functions; thus allowing φ’s to effectively account of variability in different segments of T. This acts as an attempt to “back-engineer landmark registration”[98]. This reverse- engineering happens because intuitively one would expect that the landmarks follow roughly at the same points and the variation (phase or amplitudal) between these time points isirrelevant to the actual warping. Following that rationale, estimating the spline knot-locations is actually related to estimating the landmark locations. That is why E(s) = 0 after all; on average one would assume that the landmark location would be stable, and the deviations from that location would be due to the phase variations. If “nothing happened”, ie. sij = 0 for a fixedi, then the warping
function associated with that indexi should be the identity function,t.
As before, a number of conditions are put forward to ensure identifiability of Eq. 3.26 & 3.27:
• cjk ≥0, k= 1, . . . , p
• ||cj||= 1, j = 1, . . . , q
• C = (cjk)Rq×p has a block structure such that: 1≤K1 ≤K2 ≤ · · · ≤p+ 1
wherecjk = 0k<Kj,k≥Kj=1
• cj1 =cjp = 0∀j
In effect those conditions ensure the sign of the resulting components, its norm, its support and its boundary conditions respectively. With these restrictions in place the actual cost function minimized by the time-registration step is the average integrated square error between the estimated structural mean µ and the warped
instance of the functionyiw(t) defined as: AISEn= 1 n n X i=1 Z b a [yi{hi(t)} −aiµ(t)]2h0i(t)dt (3.29)
where the functional structural mean is defined as : ˆ µ(t) = Pn i=1ˆaiˆh0iyi{hˆi(t)} Pn i=1ˆa2iˆh0i(t) (3.30) 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 350 Unwarped Data T mg 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Self−Modeling Synchronization Warping functions T TNORM 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 350 Warped Data T mg
Figure 3.6: Illustration of self-modelling warping of 60 beetle growth curves7. The left subplot shows the unwarped sample; the middle subplot the warped sample and the right subplot the corresponding warping functions.
The main critiques regarding the performance of self-modelling warping func- tions stem from their use ofB-splines and their overall idea of a common structural mean. Additionally as AUC synchronization beforehand, this framework follows the concept that “all” variability is due to phase variations and that can prove quite restrictive. In particular looking at Fig. 3.6 where essentially the first 60% of func- tion space is collapsed on the structural mean, all of these issues are exemplified immediately. First one sees that as there is an obvious inflection point in approx- imately at .7T, we have no reason to believe or disbelieve that such an inflection point actually exists in our data; especially given that most the raw data seem not to exhibit such a point. Additionally almost the entire first half of sample Y ap- pears collapsed on a common structural mean, again this might not be wrong but it seems highly unlikely that all growth curves evolve on exactly the same way only to reach an inflection point and then “fan out”. One might even argue that this is an artefact of the whole warping process. This happening due to the fact that all amplitude variation is assumed to be “phase related” and the structural mean
7Warping was implemented using the MATLAB functions provided by D. Gervini : https://
is modelled by usingB-splines: essentially what happens is that up until the inflec- tion point all data are collapsed to the structural mean and then they “fan out” by necessity because of their different values at times t = T. Let us stress that this framework critique, and any other warping-related issue can not be assumed a priori to falsify a framework (or justify it for that matter). Each framework has its own different modelling assumptions; given one meets them, any difference is completely justifiable.