Obtaining ˆ y i and ˆ x i

By (2.1), each function ˆfiis decomposed into amplitude and phase parts (called aligned and warping

functions respectively) by an application of function registration,

ˆ

We choose the alignment method using Fisher-Rao metric proposed in [51] for its good performance. As explained in Section 2.2.2, the warping functions are transformed into the tangent space TµS

by xi = Logµ(Θ(ˆγi)), where µ ≡ 1.

2.3.3 Estimation of FCPCA

2.3.3.1 Estimation of µ and (λC

i , ξCi ) Let the scaling parameter C be given. Evaluate the

functions {ˆyi}n_i=1 and {ˆxi}n_i=1 on a fine and evenly spaced grid {t1, t2, .., tk} of [0, 1] to get their

vectorized versions {ˆyi}ni=1and {ˆxi}ni=1. For each ith pair of vectors (ˆyi, ˆxi), stack them up to form

a single vector ˆgC_i in such a way that (superscript C is attached for every quantity that depends on C), ˆ gC_i =   ˆ yi C ˆxi  , yˆi = [yi(t1) yi(t2) . . . yi(tk)]T, ˆ xi= [xi(t1) xi(t2) . . . xi(tk)]T. Let ˆµ =Pn

i=1ˆgCi /n. The eigendecomposition of the sample covariance matrix bΣgC obtained from {ˆgC_i }n

i=1provides n − 1 pairs of eigenvalues and eigenvectors (ˆλCi , ˆξ C i )’s, b Σ_gC = n X i=1 [ˆgC_i − ˆµ][ˆgi− ˆµ]T = n−1 X i=1 ˆ λC_i ˆξC_i ˆξC_i T ,

where ˆλC₁ ≥ ˆλC₂ ≥ · · · ≥ ˆλC_n−1≥ 0, kˆξC_i k₂ = 1 and hˆξC_i , ˆξC_j i = 0 for i 6= j. Estimates of λC

i are ˆλCi .

Estimates ˆµ and ˆξ_iC of µ and ξ_iC are obtained by interpolating ˆµ and ˆξC_i and normalizing ˆξC_i .

2.3.3.2 Estimation of C The estimates ˆµ, {(ˆλC_i , ˆξ_iC)}n−1_i=1 and {ˆg_iC}n

i=1 are dependent on the

value of C. Our strategy in the estimation of C is to use an empirical minimizer of (2.11). For this, interpolate ˆgC_i ’s to get their functional versions {ˆg_iC}n

i=1and calculate the scores aCij = hˆgCi , ˆξCj i for

i = 1, 2, .., n and j = 1, 2, .., n − 1. Viewing { ˆfi}ni=1 as n realizations of f , the empirical version

ˆ AC

m( ˆfi) of ACm(f ) in (2.9), which approximates ˆfi, is defined by replacing yCm and xCm in (2.10) with

ˆ y_mC and ˆxC_m, where ˆ yC_m( ˆfi)(t) = ˆµ(t) + m X j=1 aC_ijξˆC_j (t), t ∈ [0, 1),

ˆ xC_m( ˆfi)(t) = m X j=1 aC_ij C ˆ ξ_jC(t + 1), t ∈ [0, 1].

For some integer m0 and a suitable interval IC,

ˆ C = argmin C∈IC n X i=1 k ˆAC_m0( ˆf_i) − ˆf_ik2₂ n , (2.17)

which implies that the mean function ˆµ and the first m0 eigenfunctions ˆξ_iCˆ found at C = ˆC recon- struct { ˆfi}ni=1 the most faithfully.

The minimizer ˆC exists for most situations of our simulations and real data analyses, not degenerating to 0 or ∞. Heuristically, this is because, for C > ˆC, { ˆξC

i }m

0

i=1more reflect the variation

of x than y’s so that the residuals from (2.17) start to increase due to the amplitude of ˆfi’s not

being recovered from the approximation ˆAC

m0( ˆfi)’s and, for C < ˆC, temporal mismatching between

ˆ

fi’s and ˆAC_m0( ˆfi)’s begin to raise the residuals.

2.3.4 Estimation of (ρi, ψyi, ψxi) of FCCCA

It is well know that Na¨ıve maximization of the quantity ρ in (2.12) using the pairs of functions {(ˆyi, ˆxi)}ni=1 usually produces pairs of uninterpretable canonical weight functions whose canonical

correlation coefficients are very close to one.

Following [29], we regularize canonical weight functions by introducing a roughness penalty term into the constraints of (2.12) and find estimates ˆρ1, ˆψy,1 and ˆψx,1, which maximize the following

quantity,

ρP( ˆψy1, ˆψx1) = argmaxψy,ψx∈L2[0,1]Cov(hψd y, ˆyii, hψx, ˆxii) (2.18)

subject to dV ar(hψy, ˆyii) + λkD2ψyk22 = dV ar(hψx, ˆxii) + λkD2ψyk22 = 1, where dCov and dV ar are

sample covariance and variance, D2 is a second order differential operator and λ is a smooth- ing parameter. The ˆρ1 is obtained by evaluating (2.18) at ( ˆψy1, ˆψx1). The subsequent triples

{ˆρi, ˆψyi, ˆψxi}n−1i=2 are found as maximizers of (2.12) subject to the constraints (2.13) with a rough-

ness penalty. Refer to [44, Ch 11.] for the estimation algorithm and generalized cross-validation (GCV) method for determining λ.

Figure 3. Parameter settings for ξC₁, ξ₂C, ξ₃C, ξ₄C and µC

.

2.4 SIMULATION STUDY

In this section, we empirically observe the consistency of the estimators of FCPCA and FCCCA. We have tried a range of parameter settings for each of FCPCA and FCCCA and the results are concordant across settings. Below we only choose to present representative cases.