Page 1
Modeling Function-Valued Stochastic Processes,With Applications to Fertility Dynamics
Kehui Chen1, Pedro Delicado2 and Hans-Georg Muller3
1Dept. of Statistics, University of Pittsburgh, Pittsburgh, USA2Dept. d’Estadıstica i Inv. Op., Universitat Politecnica de Catalunya, Barcelona, Spain
3Department of Statistics, University of California, Davis, USA
October 27, 2015
ABSTRACT
We introduce a simple and interpretable model for functional data analysis for situations
where the observations at each location are functional rather than scalar. This new ap-
proach is based on a tensor product representation of the function-valued process and
utilizes eigenfunctions of marginal kernels. The resulting marginal principal components
and product principal components are shown to provide optimal representations in a well-
defined sense. Given a sample of independent realizations of the underlying function-valued
stochastic process, we propose straightforward fitting methods to obtain the components
of this model and to establish asymptotic consistency and rates of convergence for the pro-
posed estimates. The methods are illustrated by modeling the dynamics of annual fertility
profile functions for 17 countries. This analysis demonstrates that the proposed approach
leads to insightful interpretations of the model components and interesting conclusions.
KEY WORDS: Asymptotics, demography, functional data analysis, functional principal
components, product principal component analysis, tensor product representation.
This research was supported by NSF grants DMS-1104426, DMS-1228369, DMS-1407852, by the
Spanish Ministry of Education and Science, and FEDER grant MTM2010-14887. The main part
of this work was done when Pedro Delicado was visiting UC Davis with the financial support
of the Spanish Government (Programa Nacional de Movilidad de Recursos Humanos del Plan
Nacional de I-D+i).U.S.
Page 2
1. INTRODUCTION
In various applications one encounters stochastic processes and random fields that are
defined on temporal, spatial or other domains and take values in a function space, assumed
to be the space of square integrable functions L2. More specifically, for S ⊂ Rd1 and
T ⊂ Rd2 , we consider the stochastic process X : T → L2(S) and denote its value at time
t ∈ T by X(·, t), a square integrable random function with argument s ∈ S. A key feature
of our approach is that we consider the case where one has n independent observations of
the functional stochastic process.
A specific example that we will discuss in detail below (see Section 5) is that of female
fertility profile functions X(·, t), available annually (t = year) for n = 17 countries, with
age as argument s. The starting point is the Age-Specific Fertility Rate (ASFR) X(s, t) for
a specific country, defined as
X(s, t) = ASFR(s, t) =Births during the year t given by women aged s
Person-years lived during the year t by women aged s. (1)
Figure 1 illustrates the ASFR data for the U.S. from 1951 to 2006. The left panel shows
ASFR(·, t) for t = 1960, 1980 and 2000. The image plot representing ASFR(s, t) for all
possible values of s and t in the right panel provides a visualization of the dynamics of
fertility in the U.S. over the whole period.
20 30 40 50
0.0
00
.10
0.2
00
.30
ASFR(.,t) for USA
s=Age
AS
FR
t=1960t=1980t=2000
1960 1970 1980 1990 2000
20
30
40
50
ASFR(s,t) for USA
t=Year
s=
Ag
e
0.00
0.05
0.10
0.15
0.20
0.25
Figure 1: Age Specific Fertility Rate for the U.S. Left: Profiles for three calendar years.
Right: Image plot representing ASFR(s, t) for all possible values of s and t.
1
Page 3
For data structures where one observes only one realization of a function-valued process,
related modeling approaches have been discussed previously (Delicado et al. 2010; Nerini
et al. 2010; Gromenko et al. 2012, 2013; Huang et al. 2009). Similarly, Hyndman and
Ullah (2007) and Hyndman and Shang (2009) considered functional time series in a setting
where only one realization is observed. In related applications such as mortality analysis,
the decomposition into age and year has been studied by Eilers and Marx (2003); Currie
et al. (2004, 2006); Eilers et al. (2006), using P-splines. The case where i.i.d. samples
are available for random fields has been much less studied. Multilevel functional models
and functional mixed effects models have been investigated by Morris and Carroll (2006),
Crainiceanu et al. (2009), Greven et al. (2010), and Yuan et al. (2014), among others, while
Chen and Muller (2012) developed a “double functional principal component” method and
studied its asymptotic properties.
Our approach applies to general dimensions of both the domain of the underlying ran-
dom process, with argument t, as well as of the domain of the observed functions, with
argument s, while we emphasize the case of function-valued observations for stochastic
processes on a one-dimensional time domain. This is the most common case and it often
allows for particularly meaningful interpretations. Consider processes X(s, t) with mean
µ(s, t) = E(X(s, t)) for all s ∈ S ⊆ Rd1 and all t ∈ T ⊆ Rd2 , and covariance function
C((s, t), (u, v)) = E(X(s, t)X(u, v))− µ(s, t)µ(u, v) = E(Xc(s, t)Xc(u, v)), (2)
where here and in the following we denote the centered processes by Xc.
A well-established tool of Functional Data Analysis (FDA) is Functional Principal Com-
ponent Analysis (FPCA) (Ramsay and Silverman 2005) of the random process X(s, t),
which is based on the Karhunen-Loeve expansion
X(s, t) = µ(s, t) +∞∑r=1
Zrγr(s, t), s ∈ S, t ∈ T . (3)
Here {γr : r ≥ 1} is an orthonormal basis of L2(S×T ) that consists of the eigenfunctions of
the covariance operator of X, and {Zr =∫γr(s, t)X
c(s, t)dsdt : r ≥ 1} are the (random)
2
Page 4
coefficients. This expansion has the optimality property that the first K terms form the
K-dimensional representation of X(s, t) with the smallest unexplained variance.
A downside of the two- or higher-dimensional Karhunen-Loeve representation (3) is
that it allows only for a joint symmetric treatment of the arguments and therefore is not
suitable for analyzing the separate (possibly asymmetric) effects of s and t. An additional
technical drawback is that an empirical version of (3) requires the estimation of the covari-
ance function C in (2) that depends on dimension 2(d1 + d2), and for the case of sparse
designs, this then requires to perform non-parametric regression depending on at least four
variables, with associated slow computing, curse of dimensionality and loss of asymptotic
efficiency. Finally, Karhunen-Loeve expansions for functional data depending on more than
one argument are non-standard and suitable software is hard to obtain.
Aiming to address these difficulties and with a view towards interpretability and sim-
plicity of modeling, we propose in this paper the following representation,
X(s, t) = µ(s, t) +∞∑j=1
ξj(t)ψj(s) = µ(s, t) +∞∑k=1
∞∑j=1
χjkφjk(t)ψj(s), (4)
where {ψj : j ≥ 1} are the eigenfunctions of the operator in L2(S) with kernel
GS(s, u) =
∫TC((s, t), (u, t))dt, (5)
while {ξj(t) : j ≥ 1} are the (random) coefficients of the expansion of the centered pro-
cesses Xc(·, t) in ψj(s), and ξj(t) =∑∞
k=1 χjkφjk(t) is the Karhunen-Loeve expansion of the
random functions ξj(t) in L2(T ) with eigenfunctions φjk and FPCs χjk.
We refer to GS as the marginal covariance function, and to (4) as the marginal
Karhunen-Loeve representation of X that leads to the marginal FPCA and note that the
product basis functions φjk(t)ψj(s) are orthogonal to each other. Hence the scores χjk can
be optimally estimated by the inner product of Xc with the corresponding basis. Also,
for each j ≥ 1, we have Eχjkχjk′ = 0 for k 6= k′. In related settings, marginal covari-
ance functions very recently have also been utilized by other researchers (Park and Staicu
2015; Aston et al. 2015). In Theorem 1 below we establish the optimality of the marginal
3
Page 5
eigenfunctions ψj under a well-defined criterion and show in Theorem 2 that the finite
expansion based on the marginal FPCA approach nearly minimizes the variance among all
representations of the same form.
When using the representation (4), the effects of the two arguments s and t can be
analyzed separately, which we will show in greater detail below in Sections 2 and 5. We
also note that the estimation of the marginal representation (4) requires only to estimate
covariance functions that depend on 2d1 or 2d2 real arguments. In particular, when d1 =
d2 = 1, only two-dimensional surfaces need to be estimated and marginal FPCA can be
easily implemented using standard functional data analysis packages.
Motivated by a common principal component perspective, we also introduce a simplified
version of (4), the product FPCA,
X(s, t) = µ(s, t) +∞∑k=1
∞∑j=1
χjkφk(t)ψj(s), (6)
where the φk, k ≥ 1, are the eigenfunctions of the marginal kernel GT (s, u), analogous to
GS(t, v), with supporting theory provided by Theorem 4 and Theorem 5.
Sections 2 and 3 provide further details on model and estimation. Theoretical consider-
ations are in Section 4. In Section 5, we compare the performance of the proposed marginal
FPCA, product FPCA and the conventional two-dimensional FPCA in the context of an
analysis of the fertility data. Simulation results are described in Section 6 and conclusions
can be found in Section 7. Detailed proofs, additional materials and the analysis of an
additional human mortality data example have been relegated to the Online Supplement.
2. MARGINAL FPCA
2.1. Modeling
Consider the standard inner product, 〈f, g〉 =∫S
∫Tf(s, t)g(s, t)dtds in the separable
Hilbert space L2(S × T ) and the corresponding norm ‖ · ‖. In the following, X is in
L2(S × T ) with mean µ(s, t). Using the covariance function C((s, t), (u, v)) as kernel for
4
Page 6
the Hilbert-Schmidt covariance operator Γ(f)(s, t) =∫S
∫T C((s, t), (u, v))f(u, v)dv du with
orthonormal eigenfunctions γr, r ≥ 1, and eigenvalues λ1 ≥ λ2 ≥ . . . then leads to the
Karhunen-Loeve representation of X in (3), where E(Zr) = 0 and cov(Zr, Zl) = λrδrl, with
δrl = 1 for r = l and = 0 otherwise; see Horvath and Kokoszka (2012) and Cuevas (2013).
Since the marginal kernel GS(s, u) as defined in (5) is a continuous symmetric positive
definite function (see Lemma 1 in Online Supplement A), denoting its eigenvalues and
eigenfunctions by τj, ψj, j ≥ 1, respectively, the following representation for X emerges,
X(s, t) = µ(s, t) +∞∑j=1
ξj(t)ψj(s), (7)
where ξj(t) = 〈X(·, t)−µ(·, t), ψj〉S , j ≥ 1, is a sequence of random functions in L2(T ) with
E(ξj(t)) = 0 for t ∈ T , and E(〈ξj, ξk〉T ) = τjδjk (see Lemma 2 in Online Supplement A).
Theorem 1 in Section 4 shows that the above representation has an optimality property.
The marginal Karhunen-Loeve representation (7) provides new functional data, the
score functions ξj(t), which are random functions that depend on only one argument. For
each j ≥ 1, the ξj have their own covariance functions Θj(t, v) = E(ξj(t)ξj(v)), t, v ∈
T , j ≥ 1, with eigencomponents (eigenvalues/eigenfunctions) {ηjk, φjk(t) : k ≥ 1}. The
continuity of the covariance function C implies that the Θj(t, v) are also continuous func-
tions. The random functions ξj(t) then admit their own Karhunen-Loeve expansions,
ξj(t) =∞∑k=1
χjkφjk(t), j ≥ 1, (8)
with E(χjk) = 0 and E(χjkχjr) = ηjkδkr. From (7) and (8) we obtain the representation
for X(s, t) in (4), X(s, t) = µ(s, t) +∑∞
j=1
∑∞k=1 χjkφjk(t)ψj(s). As already mentioned, this
expansion does not coincide with the standard Karhunen-Loeve expansion of X and it is
not guaranteed that χjk and χlr are uncorrelated for j 6= l. But the product functions
φjk(t)ψj(s) remain orthonormal in the sense that∫S,T φjk(t)ψj(s)φlh(t)ψl(s)dsdt = δjk,lh,
where δjk,lh = 1 when j = l and k = h; zero otherwise.
2.2 Estimating Procedures
Time- or space-indexed functional data consist of a sample of n independent subjects or
5
Page 7
units. For the i-th subject, i = 1, . . . , n, random functions Xi(·, t) are recorded at a series of
time points tim, m = 1, . . . ,Mi. Ordinarily, these functions are not continuously observed,
but instead are observed at a grid of functional design points sl, l = 1, . . . , L. In this paper
we focus on the case where the grid of s is dense, regular and the same across all subjects.
The case of sparse designs in s will be discussed in Section 7. Our proposed marginal FPCA
procedure consists of three main steps:
Step 1. Center the data to obtain Xci (s, t) = Xi(s, t) − µ(s, t). Obtain an estimator of
µ(s, t) by pooling all the data together. If the recording points t are densely and regularly
spaced, i.e., tim = tm, an empirical estimator by averaging over n subjects and interpolating
between design points can be used. This scheme is also applicable to dense irregular designs
by adding a pre-smoothing step and sampling smoothed functions at a dense regular grid.
Alternatively, one can recover the mean function µ by smoothing the pooled data (Yao
et al. 2005), for example with a local linear smoother, obtaining a smoothing estimator
µ(s, t) = a0, where
(a0, a1, a2) = arg min1
n
n∑i=1
Mi∑m=1
Lim∑l=1
{[Xi(tim, siml)− a0 − a1(siml − s)− a2(tim − t)]2
×Khs(siml − s)Kht(tim − t)}. (9)
Step 2. Use the centered data Xci (s, t) from Step 1 to obtain estimates of the marginal co-
variance function GS(s, u) as defined in (5), its eigenfunctions ψj(s) and the corresponding
functional principal component (FPC) score functions ξi,j(t). For this, we pool the data
{Xci (·, tim), i = 1, . . . , n, m = 1, . . . ,Mi} and obtain estimates
GS(sj, sl) =|T |∑ni=1Mi
n∑i=1
Mi∑m=1
Xci (sj, tim)Xc
i (sl, tim), (10)
where 1 ≤ j ≤ l ≤ L and |T | is the Lebesgue measure of T , followed by interpolating be-
tween grid points to obtain GS(s, u) for (s, u) ∈ S×S. One then obtains the eigenfunctions
ψj and eigenvalues τj by standard methods (Yao et al. 2005) as implemented in the PACE
package (http://www.stat.ucdavis.edu/PACE) or as in Kneip and Utikal (2001), and the
6
Page 8
FPC function estimates ξi,j(t) by interpolating numerical approximations of the integrals
ξi,j(tim) =∫Xc
i (s, tim)ψj(s)ds. Theorem 3 shows that GS in (10) and ψj are consistent
estimates of the marginal covariance function GS and its eigenfunctions and that estimates
{ξi,j(t), i = 1, . . . , n, } converge uniformly to the target processes {ξi,j(t), j ≥ 1}.
Step 3. This is a standard FPCA of one-dimensional processes {ξi,j(t), j ≥ 1}, where for
each fixed j, one obtains estimates for the FPCs χjk and eigenfunctions {φjk(t) : k ≥ 1};
see for example Ramsay and Silverman (2005); Kneip and Utikal (2001) for designs that
are dense in t and Yao et al. (2005) for designs that are sparse in t.
After selecting appropriate numbers of included components P and Kj, j = 1, . . . , P ,
one obtains the overall representation
Xi(s, t) = µ(s, t) +P∑
j=1
ξi,j(t)ψj(s) = µ(s, t) +P∑
j=1
Kj∑k=1
χi,jkφjk(t)ψj(s). (11)
The included number of components P can be selected via a fraction of variance explained
(FVE) criterion, finding the smallest P such that∑P
j=1 τj/∑M
j=1 τj ≥ 1 − p, where M is
large and we choose p = 0.15 in our application. The number of included components Kj
can be determined by a second application of the FVE criterion, where the variance Vjk
explained by each term (j, k) is defined as
Vjk =1n
∑ni=1 χ
2i,jk
1n
∑ni=1 ||X(s, t)− µ(s, t)||2S×T
. (12)
Note that Vjk does not depend on the choice of P in the first step, since it is the fraction of
total variance explained. Here total variance explained,∑Kj
k=1
∑Pj=1E(χ2
jk), cannot exceed
the variance explained in the first step,∑P
j=1 τj.
We will illustrate these procedures in Section 5. Since the functions ψj(s)× φjk(t) are
orthogonal, the unexplained variance, E‖Xc‖2−∑P
j=1
∑Kj
k=1E(χ2jk), and the reconstruction
loss, E(∫S,T {X
c(s, t)−∑P
j=1
∑Kj
k=1〈Xc, φjkψj〉φjk(t)ψj(s)}2dsdt)
, are equivalent.
3. PRODUCT FPCA
In this section we discuss a simplified version of the marginal Karhunen-Loeve representa-
tion (4). A simplifying assumption is that the eigenfunctions φjk in the Karhunen-Loeve
7
Page 9
expansion of ξj(t) in (4) do not depend on j. This assumption leads to a more compact
representation of X as given in (6), X(s, t) = µ(s, t) +∑∞
j=1
∑∞k=1 χjkφk(t)ψj(s).
To study the properties of this specific product representation, we consider product
representations with general orthogonal basis X(s, t) = µ(s, t) +∑∞
j=1
∑∞k=1 χjkfk(t)gj(s),
where χjk = 〈Xc, fkgj〉. For such general representations, the assumption
cov(χjk, χjl) = 0 for k 6= l, and cov(χjk, χhk) = 0 for j 6= h (13)
implies that the covariance kernel induced by ξj(t) = 〈Xc(t, ·), gj〉S has common eigen-
functions {fk(t), k ≥ 1}, not depending on j, and the covariance kernel induced by
ξk(s) = 〈Xc(·, s), fk〉T has common eigenfunctions {gj(s), j ≥ 1}, not depending on k.
Therefore we refer to (13) as the common principal component assumption. We prove in
Theorem 4 below that if there exists bases {gj(s), j ≥ 1} and {fk(t), k ≥ 1} such that
(13) is satisfied, then gj ≡ ψj and fk ≡ φk, the eigenfunctions of the marginal covariance
GS(s, u) and GT (t, v), respectively, where GT (t, v) is defined as
GT (t, v) =
∫SC((s, t), (s, v)) ds, with t, v ∈ T . (14)
Even without invoking (13), in Theorem 5 we show that the finite expansion based on
the marginal eigenfunctions φk and ψj yields a near-optimal solution in terms of minimiz-
ing the unexplained variance among all possible product expansions. This result provides
additional theoretical support for the use of product FPCA based on the marginal kernels
GS and GT under fairly general situations. While the product functions φk(t)ψj(s) are or-
thonormal, without addtional conditions, the scores χjk in general will not be uncorrelated.
Product FPCA (6) is well suited for situations where the two arguments of X(s, t) play
symmetric roles. This simplified model retains substantial flexibility, as we will demonstrate
in the application to fertility data (see Online Supplement C).
The estimation procedures for this model are analogous to those described in the pre-
vious section. This also applies to the theoretical analysis of these estimates and their
asymptotic properties. A straightforward approach to estimate the eigenfunctions appear-
ing in (6) is to apply the estimation procedure described in Section 2.2 twice, first following
8
Page 10
the description there to obtain estimates of GS and ψj and then changing the roles of the
two arguments in a second step to obtain estimates of GT and φk.
4. THEORETICAL PROPERTIES
Detailed proofs of the results in this section are in Online Supplement A. We show that
the optimal finite-dimensional approximation property of FPCA extends to the proposed
marginal FPCA under well defined criteria. Theorem 1 establishes the optimality of the
basis functions ψj, i.e. the eigenfunctions in (4) derived from the marginal covariance in
(5). Theorem 2 shows the near optimality of the marginal representation (4), based on
the eigenfunctions φjk and ψj, in terms of minimizing the unexplained variance among all
functional expansions of the same form.
Theorem 1. For each P ≥ 1 for which τP > 0, the functions g1, . . . , gP in L2(S) that
provide the best finite-dimensional approximations in the sense of minimizing
E
(∫T‖Xc(·, t)−
P∑j=1
〈Xc(·, t), gj〉Sgj‖2Sdt
)are gj = ψj, j = 1, . . . , P, i.e., the eigenfunctions of GS . The minimizing value is∑∞
j=P+1 τj.
Theorem 2. For P ≥ 1 and Kj ≥ 1, consider the following loss minimization
minfjk,gj
E
∫S,T{Xc(s, t)−
P∑j=1
Kj∑k=1
〈Xc, fjkgj〉fjk(t)gj(s)}2dsdt
,
with minimizing value Q∗, where the gj(s), j ≥ 1, are orthogonal and for each j, the
fjk(t), k ≥ 1 are orthogonal. The marginal eigenfunctions ψj(s), and φjk(t) achieve good
approximation in the sense that
E
∫S,T{Xc(s, t)−
P∑j=1
Kj∑k=1
〈Xc, φjkψj〉φjk(t)ψj(s)}2dsdt
< Q∗ + aE‖Xc‖2,
where a = max1≤j≤P aj, with (1 − aj) denoting the fraction of variance explained by Kj
terms for each process ξj(t) = 〈Xc(·, t), ψj〉S .
9
Page 11
In the following, ‖GS(s, u)‖S = {∫S
∫S(GS(s, u))2dsdu}1/2 is the Hilbert-Schmidt norm
and a � b denotes that a and b are of the same order asymptotically. For the consistency
of marginal FPCA (4) it is important that the covariance kernel GS and its eigenfunctions
ψj and eigenvalues τj can be consistently estimated from the data. Uniform convergence of
the empirical working processes {ξi,j(tim), 1 ≤ i ≤ n, 1 ≤ m ≤ Mi} to the target processes
{ξi,j(t), t ∈ T } then guarantees the consistency of the estimates of the eigenfunctions φjk
and the eigenvalues ηjk (Yao and Lee 2006).
The following assumptions are needed to establish these results. We use 0 < B <∞ as
a generic constant that can take different values at different places.
(A.1) sups,t |µ(s, t)| < B and sups |ψj(s)| < B for all 1 ≤ j ≤ P .
(A.2) E sups,t |X(s, t)| < B and sups,tE|X(s, t)|4 < B.
(A.3) sup(s,u)∈S2,(t1,t2)∈T 2 |C((s, t1), (u, t1))− C((s, t2), (u, t2))| < B|t1 − t2|
(A.4) sup(s1,u1,s2,u2)∈S4 |GS(s1, u1)−GS(s2, u2)| < B(|s1 − s2|+ |u1 − u2|).
(A.5) For all 1 ≤ j ≤ P , δj > 0, where δj = min1≤l≤j(τl − τl+1).
(A.6a) The grid points {tim : m = 1, . . . ,M} are equidistant, and n/M = O(1).
(A.6b) The grid points {tim : m = 1, . . . ,Mi} are independently and identically distributed
with uniform density, and miniMi � maxiMi.
Condition (A.1) generally holds for smooth functions that are defined on finite domains.
Condition (A.2) are commonly used moment conditions for X(s, t). Conditions (A.3) and
(A.4) are Lipschitz conditions for the joint covariance C and the marginal covariance GS
and quantify the smoothness of these covariance surfaces. Condition (A.5) requires non-
zero eigengaps for the first P leading components and is widely adopted in the literature
(Hall et al. 2006; Li and Hsing 2010). Conditions (A.6a) and (A.6b) correspond to two
alternative scenarios for the design at which the underlying random process is sampled
over t. Here (A.6a) reflects the case of a dense regular design, where one observes functions
X(·, tm) at a dense and regular grid of {tm : m = 1, . . . ,M}, with n/M = O(1), while
(A.6b) corresponds to the case of a random design, where one observes functions X(·, tim)
10
Page 12
at a series of random locations corresponding to the time points {tim : m = 1, . . . ,Mi},
where the number of available measurements Mi may vary across subjects.
Theorem 3. If conditions (A.1)-(A.5),(A.6a) or (A.1)-(A.5), (A.6b) hold, max(sl −
sl−1) = O(n−1), and µ(s, t) obtained in Step 1 above satisfies sups,t |µ(s, t) − µ(s, t)| =
Op((log n/n)1/2), one has the following results for 1 ≤ j ≤ P :
‖GS(s, u)−GS(s, u)‖S = Op((log n/n)1/2) (15)
|τj − τj| = Op((log n/n)1/2) (16)
‖ψj(s)− ψj(s)‖S = Op((log n/n)1/2) (17)
1
n
n∑i=1
sup1≤m≤Mi
|ξi,j(tim)− ξi,j(tim)| = Op((log n/n)1/2). (18)
The empirical estimator and the smoothing estimator that are discussed in Step 1 both
satisfy sups,t |µ(s, t)− µ(s, t)| = Op((log n/n)1/2) under appropriate conditions and appro-
priate choice of the bandwidth in the smoothing estimator. We refer to Chen and Muller
(2012), Theorems 1 and 2 for detailed conditions and proofs. The following result estab-
lishes the uniqueness of the product representation with marginal eigenfunctions ψj and φj
derived from (5) and (14) under the common principal component assumption (13). An
important implication of Theorem 4 is that the product FPCA based on marginal eigen-
functions is optimal if the eigenfunctions of kernel C(s, t;u, v) indeed can be written as
products in their arguments.
Theorem 4. If there exist orthogonal bases {gj(s), j ≥ 1} and {fk(t), k ≥ 1}, under which
the common principal component assumption (13) is satisfied, we have gj(s) ≡ ψj(s) and
fk(t) ≡ φk(t), with
GS(s, u) =∑∞
j=1τjψj(s)ψj(u), for all s, u ∈ S, (19)
GT (t, v) =∑∞
k=1ϑkφk(t)φk(v), for all t, v ∈ T , (20)
11
Page 13
where
τj =∑∞
k=1var(χjk), ϑk =∑∞
j=1var(χjk),
χjk =
∫S
∫T
(X(s, t)− µ(s, t))ψj(s)φk(t) dt ds,
E(χjk) = 0, cov(χjk, χjl) = var(χjk)δkl, cov(χjk, χhk) = var(χjk)δjh. (21)
Theorem 5. For P ≥ 1 and K ≥ 1, consider the following loss minimization
minfk,gj
E
(∫S,T{Xc(s, t)−
P∑j=1
K∑k=1
〈Xc, fkgj〉fk(t)gj(s)}2dsdt
),
with minimizing value Q∗, where fk, k ≥ 1 are orthogonal, and gj, j ≥ 1 are orthogonal.
The marginal eigenfunctions ψj(s) of GS(s, u) and φk(t) of GT (t, v) achieve good approxi-
mation in the sense that
E
(∫S,T{Xc(s, t)−
P∑j=1
K∑k=1
〈Xc, φkψj〉φk(t)ψj(s)}2dsdt
)< Q∗ + aE‖Xc‖2,
where a = min(aT , aS), with (1 − aT ) denoting the fraction of variance explained by K
terms for GT (t, v) and analogously for aS .
Similarly to the situation in Theorem 2, the error term aE‖Xc‖2 depends on the loss
involved in truncating just the (marginal) principal component decompositions, which also
imposes a lower bound on Q∗.
5. FUNCTIONAL DATA ANALYSIS OF FERTILITY
Human fertility naturally plays a central role in demography (Preston et al. 2001) and its
analysis recently has garnered much interest due to declining birth rates in many developed
countries and associated sub-replacement fertilities (Takahashi 2004; Ezeh et al. 2012). The
Human Fertility Database (HFD 2013) contains detailed period and cohort fertility annual
data for 22 countries (plus five subdivisions: two for Germany and three for the U.K.). We
are interested in Age-Specific Fertility Rates (ASFR), considered as functions of women’s
age in years (s) and repeatedly measured for each calendar year t for various countries.
These rates (see (1)) constitute the functional data X(s, t) = ASFR(s, t).
12
Page 14
0.00
0.05
0.10
0.15
0.20
0.25
1960 1970 1980 1990 2000
20
30
40
50
ASFR sample mean
Year
Age
Year
1960
1970
1980
1990
2000
Age
20
30
40
50
ASFR sample m
ean
0.00
0.05
0.10
0.15
Figure 2: Sample means of the 17 fertility rate functions by calendar year.
A detailed description how ASFR is calculated from raw demographic data can be found
in the HFD Methods Protocol (Jasilioniene et al. 2012). The specific definition of ASFR we
are using corresponds to period fertility rates by calendar year and age (Lexis squares, age
in completed years). In HFD (2013), ASFR(s, t) is included for mothers of ages s = 12−55
years, thus the domain S is an interval of length L = 44 years. The interval of calendar
years with available ASFR varies by country. Aiming at a compromise between the length
M of the studied period T and the number n of countries that can be included, we choose T
as the interval from 1951 to 2006. There are n = 17 countries (or territories) with available
ASFR data during this time interval (see Table 4 and Figure 5 in Online Supplement B for
the list of n = 17 included countries and heat maps depicting individual functions ASFRi).
The sample mean ASFR(s, t) of the ASFR functions for 17 countries displayed in Figure
2 shows that fertility rates are, on average, highest for women aged between 20 and 30 and
are decreasing with increasing calendar year; this overall decline is interspersed with two
periods of increasing fertility before 1965, corresponding to the baby-boom, and after 1995
with a narrow increase for ages between 30 and 40 years; is narrowing in terms of the
age range with high fertility; and displays an increase in regard to the ages of women
where maximum fertility occurs. We applied marginal FPCA, product FPCA and two-
dimensional FPCA to quantify the variability across individual countries and summarize
the main results here. Additional details can be found in Online Supplement C.
The fertility data include one fertility curve over age per calendar year and per country
13
Page 15
20 30 40 50−
0.2
0.0
0.2
0.4
Eigenfunction 1 (FVE: 61.18%)
Age
Eig
enfu
nctio
n 1
20 30 40 50
0.00
0.10
0.20
Eigenfunction 2 (FVE: 27.73%)
Age
Eig
enfu
nctio
n 2
20 30 40 50
−0.
3−
0.1
0.1
Eigenfunction 3 (FVE: 6.93%)
Age
Eig
enfu
nctio
n 3
1950 1960 1970 1980 1990 2000
−0.
2−
0.1
0.0
0.1
0.2
Functional scores at eigenfunction 1
Year
Sco
res
at e
igen
func
tion
1
AUT AUT
BGR BGR
CANCAN
CZE
CZEFIN FIN
FRA
FRA
HUN
HUN
JPN
JPN
NLD
NLDPRT
PRT
SVKSVK
SWE
SWECHE
CHEGBRTENW
GBRTENW
GBR_SCO
GBR_SCO
USA
USA
ESP
ESP
BGR BGRCZE
CZE
HUN
HUN
SVKSVK
USA
USA
1950 1960 1970 1980 1990 2000
−0.
2−
0.1
0.0
0.1
0.2
Functional scores at eigenfunction 2
Year
Sco
res
at e
igen
func
tion
2
AUT
AUTBGR
BGR
CAN
CANCZE
CZE
FIN FINFRA
FRA
HUN HUN
JPN
JPN
NLD NLDPRT
PRT
SVK
SVK
SWE
SWE
CHE
CHE
GBRTENW
GBRTENW
GBR_SCO
GBR_SCO
USAUSA
ESP ESP
CAN
CAN
HUN HUN
PRT
PRT
USAUSA
ESP ESP
1950 1960 1970 1980 1990 2000
−0.
15−
0.05
0.00
0.05
Functional scores at eigenfunction 3
Year
Sco
res
at e
igen
func
tion
3
AUT
AUTBGR
BGR
CAN CAN
CZE CZE
FINFIN
FRA FRAHUN HUNJPN
JPN
NLD
NLD
PRT
PRT
SVK
SVK
SWESWE
CHE
CHE
GBRTENW
GBRTENW
GBR_SCO
GBR_SCO
USA USA
ESP
ESP
JPN
JPN
Figure 3: Results of the marginal FPCA for the fertility data. First row: Estimated
eigenfunctions ψj(s), j = 1, 2, 3 , where s is age. Second row: Score functions ξi,j(t), where
t is calendar year. Colored lines are used for countries mentioned in the text.
and are observed on a regular grid spaced in years across both coordinates age s and
calendar year t, which means that the empirical estimators described in Section 2 can be
applied to these data. Figure 6 (Online Supplement B) displays the nM = 952 centered
functional data ASFRci(sl, tm) = ASFRi(sl, tm) − ASFR(sl, tm), for l = 1, . . . , L = 44,
m = 1, . . . ,M = 56 and i = 1, . . . , n = 17, demonstrating that there is substantial variation
across countries and calendar years. The results of the proposed marginal FPCA are
summarized in Figures 3 and 4 for the first three eigenfunctions, ψj(s), j = 1, 2, 3, resulting
in a FVE of 95.8%. From Figure 3, the first eigenfunction ψ1(s) can be interpreted as a
contrast between fertility before and after the age of 25 years, representing the direction
from mature fertility (negative scores) to young fertility (positive scores).
The second eigenfunction ψ2(s) takes positive values for all ages s, with a maximum at
age s = 24. The shape of ψ2(s) is similar to that of the mean function ASFR(s, t) for a
fixed year t (see the right panel of Figure 2). Therefore ψ2(s) can be interpreted as a size
component: Country-years with positive score in the direction of this eigenfunction have
higher fertility ratios than the mean function for all ages. The third eigenfunction ψ3(s)
represents a direction from more concentrated fertility around the age of 25 years to a more
14
Page 16
dispersed age distribution of fertility.
Examining the score functions ξi,j(t), t ∈ T , which are country-specific functions of
calendar year, one finds from Figure 3 for ξi,1(t) that there are countries, such as U.S.
(light pink), Bulgaria (red) or Slovakia (green) for which ξ1(t) is positive for all calendar
years t, which implies that these countries always have higher fertility rates for young
women and vice versa for mature women, relative to the mean function. Countries from
Eastern Europe such as Bulgaria, Czech Republic (pink) Hungary (brown) and Slovakia
have high scores until the end of the 1980s when there is a sudden decline, implying that
the relationship of fertility between younger and more mature women has reversed for these
countries. Also notable is a declining trend in the dispersion of the score functions since
1990, implying that the fertility patterns of the 17 countries are converging.
The score functions ξi,2(t) corresponding to the size component indicate that Canada
(purple) and the USA had a particularly strong baby boom in the 1960s, while Portugal
(blue) and Spain (medium gray) had later baby booms during the 1970s. In contrast,
Hungary had a period of relatively low fertility during the 1960s. Again, the dispersion of
these size score functions declines towards 2006. The patterns of the score functions ξi,3(th)
indicate that Japan (dark grey) has by far the largest degree of concentrated fertility at
ages from 22 to 29 years, from 1960-1980, but lost this exceptional status in the 1990s and
beyond. There is also a local anomaly for Japan in 1966. Takahashi (2004) reports that in
1966 the total fertility in Japan declined to the lowest value ever recorded, because 1966
was the year of the Hinoe-Uma (Fire Horse, a calendar event that occurs every 60 years),
associated with the superstitious belief of bad luck for girls born in such years.
Trends over calendar time for particular countries can be visualized by track plots, which
depict the changing vectors of score functions (ξi,1(t), . . . , ξi,K(t)), parametrized in t ∈ T ,
as one-dimensional curves in RK . Track plots are most useful for pairs of score functions
and are shown in the form of planar curves for the pairs (ξi,1(t), ξi,2(t)) and (ξi,1(t), ξi,3(t)),
t ∈ T , in Figure 4 for selected countries and in Figure 7 (Online Supplement B) for all
15
Page 17
−0.2 −0.1 0.0 0.1 0.2
−0.
2−
0.1
0.0
0.1
0.2
Scores at eigenfunctions 2 vs. 1
Scores at eigenfunction 1
Sco
res
at e
igen
func
tion
2
1951.CZE
1966.CZE
1981.CZE
1996.CZE2006.CZE
1951.JPN
1966.JPN
1981.JPN1996.JPN
2006.JPN
1951.NLD
1966.NLD
1981.NLD
1996.NLD
2006.NLD
1951.PRT
1966.PRT
1981.PRT
1996.PRT
2006.PRT
1951.USA
1966.USA
1981.USA
1996.USA
2006.USA
1951.ESP
1966.ESP
1981.ESP
1996.ESP
2006.ESP
1951.CZE
1966.CZE
1981.CZE
1996.CZE2006.CZE
1951.PRT
1966.PRT
1981.PRT
1996.PRT
2006.PRT
1951.USA
1966.USA
1981.USA
1996.USA
2006.USA
1951.ESP
1966.ESP
1981.ESP
1996.ESP
2006.ESP
−0.2 −0.1 0.0 0.1 0.2
−0.
15−
0.10
−0.
050.
000.
05
Scores at eigenfunctions 3 vs. 1
Scores at eigenfunction 1
Sco
res
at e
igen
func
tion
3
1951.CZE
1966.CZE
1981.CZE
1996.CZE2006.CZE1951.JPN
1966.JPN
1981.JPN
1996.JPN 2006.JPN
1951.NLD
1966.NLD
1981.NLD
1996.NLD
2006.NLD
1951.PRT
1966.PRT
1981.PRT
1996.PRT
2006.PRT
1951.USA
1966.USA
1981.USA
1996.USA
2006.USA
1951.ESP
1966.ESP 1981.ESP
1996.ESP
2006.ESP
1951.JPN
1966.JPN
1981.JPN
1996.JPN 2006.JPN
1951.NLD
1966.NLD
1981.NLD
1996.NLD
2006.NLD1951.ESP
1966.ESP 1981.ESP
1996.ESP
2006.ESP
−0.2 −0.1 0.0 0.1 0.2
−0.
2−
0.1
0.0
0.1
0.2
Scores at eigenfunctions 2 vs. 1
Scores at eigenfunction 1
Sco
res
at e
igen
func
tion
2
1951.CZE
1966.CZE
1981.CZE
1996.CZE2006.CZE
1951.JPN
1966.JPN
1981.JPN1996.JPN
2006.JPN
1951.NLD
1966.NLD
1981.NLD
1996.NLD
2006.NLD
1951.PRT
1966.PRT
1981.PRT
1996.PRT
2006.PRT
1951.USA
1966.USA
1981.USA
1996.USA
2006.USA
1951.ESP
1966.ESP
1981.ESP
1996.ESP
2006.ESP
1951.CZE
1966.CZE
1981.CZE
1996.CZE2006.CZE
1951.PRT
1966.PRT
1981.PRT
1996.PRT
2006.PRT
1951.USA
1966.USA
1981.USA
1996.USA
2006.USA
1951.ESP
1966.ESP
1981.ESP
1996.ESP
2006.ESP
−0.2 −0.1 0.0 0.1 0.2
−0.
15−
0.10
−0.
050.
000.
05
Scores at eigenfunctions 3 vs. 1
Scores at eigenfunction 1
Sco
res
at e
igen
func
tion
3
1951.CZE
1966.CZE
1981.CZE
1996.CZE2006.CZE1951.JPN
1966.JPN
1981.JPN
1996.JPN 2006.JPN
1951.NLD
1966.NLD
1981.NLD
1996.NLD
2006.NLD
1951.PRT
1966.PRT
1981.PRT
1996.PRT
2006.PRT
1951.USA
1966.USA
1981.USA
1996.USA
2006.USA
1951.ESP
1966.ESP 1981.ESP
1996.ESP
2006.ESP
1951.JPN
1966.JPN
1981.JPN
1996.JPN 2006.JPN
1951.NLD
1966.NLD
1981.NLD
1996.NLD
2006.NLD1951.ESP
1966.ESP 1981.ESP
1996.ESP
2006.ESP
Figure 4: Track-plots {(ξi,1(t), ξi,2(t)) : t = 1951, . . . , 2006} (left panel) and
{(ξi,1(t), ξi,3(t)) : t = 1951, . . . , 2006} (right panel), indexed by calendar time t, where
ξi,j(t) is the j-th score function for country i (for selected countries) as in (4).
countries. The left panel with the track plot illustrating the evolution in calendar time of
first and second FPCs shows predominantly vertical movements: From 1951 to 2006 for
most countries there are more changes in total fertility than changes in the distribution of
fertility over the different ages of mothers. Exceptions to this are Portugal (blue), Spain
(medium gray), Czech Republic (pink) and the U.S. (light pink), with considerable variation
over the years in the first FPC score. There was more variation in fertility patterns between
the countries included in this analysis in 1951 than in 2006, indicating a “globalization” of
fertility patterns. In the track plot corresponding to the first and third eigenfunctions in
the right panel of Figure 4, the anomalous behavior of Japan (dark gray) stands out. The
third step of the marginal FPCA described in Section 2 consists of performing a separate
FPCA for the estimated score functions ξi,j(t), i = 1, . . . , n, for j = 1, 2, 3, with estimated
eigenfunctions φjk shown in Figure 8 (Online Supplement B). The interpretation of these
eigenfunctions is relative to the shape of the ψj(s).
The results in Table 1 for estimated representations (11) justify to include only the six
terms with the highest FVE in the final model, leading to a cumulative FVE of 87.49%,
where the FVE for each term (j, k) is estimated by (12). The corresponding 6 product
functions φjk(t)ψj(s) are shown in Figure 9 (Appendix B). Regarding the comparative
performance of standard two-dimensional FPCA, product FPCA (with detailed results in
16
Page 18
Table 1: Fraction of Variance Explained (FVE) of ASFR(s, t) for the leading terms in the
proposed marginal FPCA, product FPCA and two-dimensional FPCA. Number of terms
in each case is selected to achieve fraction of variance explained (FVE) of more than 85%.
marginal FPCA FVE in % product FPCA FVE in % 2d FPCA FVE in %
Six terms 87.49 Seven terms 87.38 Four terms 89.73
φ11(t)ψ1(s) 54.33 φ1(t)ψ1(s) 53.69 γ1(s, t) 58.93
φ21(t)ψ2(s) 13.04 φ2(t)ψ2(s) 8.10 γ2(s, t) 13.71
φ22(t)ψ2(s) 6.88 φ1(t)ψ2(s) 8.08 γ3(s, t) 11.04
φ12(t)ψ1(s) 4.62 φ3(t)ψ2(s) 5.51 γ4(s, t) 6.05
φ23(t)ψ2(s) 4.40 φ2(t)ψ1(s) 4.47
φ31(t)ψ3(s) 4.22 φ4(t)ψ2(s) 3.85
φ1(t)ψ3(s) 3.68
Online Supplement C) and marginal FPCA, we find: (1) As expected, standard FPCA
based on the two-dimensional Karhunen-Loeve expansion requires fewer components to
explain a given amount of variance, as 4 eigenfunctions lead to a FVE of 89.73% (see
Table 1), while marginal FPCA representation achieves a FVE of 87.49% with 6 terms,
and product FPCA needs 7 terms to explain 87.38%. (2) Product FPCA and Marginal
FPCA represent the functional data as a sum of terms that are products of two functions,
each depending on only one argument. This provides for much better interpretability
and makes it possible to discover patterns in functional data that are not found when
using standard FPCA. For instance, the second eigenfunction ψ2 in the first step of the
marginal FPCA could be characterized as a fertility size component, with a country-specific
time-varying multiplier ξ2(t). Standard FPCA does not pinpoint this feature, which is an
essential characteristic of demographic changes in fertility. (3) Marginal FPCA makes it
much easier than standard FPCA to analyze the time dynamics of the fertility process.
Specifically, the plots in the second row of Figure 3 or the track plots in Figure 4
are informative about the fertility evolution over calendar years: (a) The relative balance
between young and mature fertility at each country changes over the years. The graphical
17
Page 19
representation of functional score functions ξi,1(t) allows to characterize and quantify this
phenomenon. (b) The track plot in the left panel of Figure 4 indicates that in general it is
much more common that fertility rates rise or decline across all ages compared to transfers
of fertility between different age groups. (c) The fertility patterns of the various countries
are much more similar in 2006 than in 1951.
All three approaches to FPCA for function-valued stochastic processes, namely standard
FPCA (3), marginal FPCA (4) and the product FPCA (6), can be used to produce country
scores which can be plotted against each other. They turn out to be similar for these
approaches; as an example the standard FPCA scores are shown in Figure 12 (Appendix
C). We conclude that standard FPCA, marginal FPCA and product FPCA complement
each other. Our recommendation is to perform all whenever feasible, in order to gain as
much insight about complex functional data as possible.
6. SIMULATIONS
We conducted two simulation studies, one to investigate the estimation procedure for
marginal FPCA, and a second study to evaluate the performance of product FPCA. Both
were conducted in a scenario that mimicks the fertility data. For simulation 1, we gener-
ated data following a truncated version of (4), where we used the estimated mean function
ASFR(s, t) from the country fertility data (Section 5) as mean function and the estimated
product functions φjk(t)ψj(s), 1 ≤ j, k ≤ 4, as base functions in (4). Random scores χjk
were generated as independent normal random variables with variances λjk, corresponding
to the estimates derived from the fertility data, λjk = 1n
∑ni=1 χ
2i,jk. We also added i.i.d.
noise to the actual observations Yi(sl, th) = X(sl, th) + εi,lh, l = 1, . . . , 44, h = 1, . . . , 56,
where εi,lh ∼ N(0, σ2) with σ = 0.005 to mimic the noise level of the fertility data.
Estimated and true functions ψj(s) and φjk(t) obtained for one sample run with n = 50
are shown in Figure 10 (Online Supplement B), demonstrating very good recovery of the
true basis functions. To quantify the quality of the estimates of µ(s, t), we use the relative
18
Page 20
Table 2: Results for simulation 1, reporting median relative errors (RE), as defined in (22)
(with median absolute deviation in parentheses), for various components of the model and
varying sample sizes n.
RE FVE in % n = 50 n = 100 n = 200
µ 0.0012 (0.0008) 0.0006 (0.0004) 0.0003 (0.0002)
Xc 0.1523 (0.0228) 0.1483 (0.0168) 0.1435 (0.0091)
φ11(t)ψ1(s) 53.6967 0.0092 (0.0071) 0.0045 (0.0040) 0.002 (0.0016)
φ21(t)ψ2(s) 12.9333 0.0584 (0.0538) 0.0280 (0.0243) 0.0133 (0.0110)
φ22(t)ψ2(s) 6.7450 0.1306 (0.1267) 0.0660 (0.0619) 0.0311 (0.0287)
φ12(t)ψ1(s) 4.5367 0.0222 (0.0178) 0.0129 (0.0091) 0.005 (0.0037)
φ23(t)ψ2(s) 4.1917 0.0999 (0.0904) 0.0469 (0.0417) 0.0296 (0.0238)
φ31(t)ψ3(s) 4.0400 0.0283 (0.0238) 0.0127 (0.0100) 0.0077 (0.0062)
squared error
RE =||µ(s, t)− µ(s, t)||2
||µ(s, t)||2, (22)
where ||µ(s, t)||2 =∫ ∫
µ(s, t)2dsdt, analogously for Xci (s, t) and φjk(t)ψj(s). The relative
squared errors over 200 simulation runs, reported in Table 2, were found to be quite small
for µ, Xci and for the six product functions φjk(t)ψj(s) with largest FVEs, which are the
same six functions as in Figure 9. The errors decline with increasing sample size n, as
expected. The FVEs for each term (j, k) are also in Table 2, averaged over simulation runs
and over the different sample sizes, as they were similar across varying sample sizes.
For simulation 2, data were generated according to a truncated product FPC model
X(s, t) = µ(s, t) +4∑
j=1
4∑k=1
χjkφk(t)ψj(s),
where µ(s, t) and φk(t)ψj(s) for 1 ≤ j, k ≤ 4 are substituted by the estimates obtained from
the fertility data. As in simulation 1, the random scores χjk were generated as independent
normal random variables with variances estimated from the data. Estimated and true
functions ψj(s) and φk(t) obtained for one sample run with n = 50 are shown in Figure
11 (Online Supplement B). The relative squared errors over 200 simulation runs, for µ, Xci
19
Page 21
Table 3: Results for simulation 2, reporting median relative errors (RE), as defined in (22)
(with median absolute deviations in parentheses) for various components of the model and
varying sample sizes n.
RE FVE in % n = 50 n = 100 n = 200
µ 0.0011 (0.0006) 0.0006 (0.0003) 0.0003 (0.0002)
Xc 0.1464 (0.0216) 0.1390 (0.0138) 0.1324 (0.0080)
φ1(t)ψ1(s) 53.5400 0.0099 (0.0075) 0.0053 (0.0048) 0.0022 (0.0017)
φ2(t)ψ2(s) 8.1500 0.0559 (0.0525) 0.0342 (0.0275) 0.0174 (0.0180)
φ1(t)ψ2(s) 8.0817 0.0109 (0.0078) 0.0064 (0.0051) 0.0026 (0.0018)
φ3(t)ψ2(s) 5.4300 0.0776 (0.0635) 0.0389 (0.0331) 0.0208 (0.0204)
φ2(t)ψ1(s) 4.2783 0.0543 (0.0502) 0.0328 (0.0271) 0.0173 (0.0180)
φ4(t)ψ2(s) 3.7817 0.0368 (0.0293) 0.0167 (0.0131) 0.0089 (0.0072)
φ1(t)ψ3(s) 3.5917 0.0108 (0.0075) 0.0056 (0.0039) 0.0028 (0.0019)
and for the seven product functions φk(t)ψj(s) with largest FVEs (among 16 total product
functions), which are the same seven functions as in Figure 18 (Online Supplement C),
are reported in Table 3. Both figure and numbers demonstrate good performance of the
method.
7. DISCUSSION
The proposed marginal FPCA and product FPCA provide a simple and straightforward
representation of function-valued stochastic processes. This holds especially in comparison
with a previously proposed two-step expansion for repeatedly observed functional data
(Chen and Muller 2012), in which processes X are represented as
X(s, t) = µ(s, t) +∞∑j=1
νj(t)ρj(s|t) = µ(s, t) +∞∑j=1
∞∑k=1
θjkωjk(t)ρj(s|t), (23)
where ρj(·|t) is the j-th eigenfunction of the operator in L2(S) with kernel GS(s, u|t) =
C((s, t), (u, t)), νj(t) = 〈X(·, t), ρj(·|t)〉S and∑∞
k=1 θjkωjk(t) is the Karhunen-Loeve ex-
pansion of νj(t) as a random function in L2(T ). This method can be characterized as a
20
Page 22
conditional FPCA approach (we note that in Chen and Muller (2012) the notation of s
and t is reversed as compared to the present paper). Similarly to the proposed marginal
approach this conditional method provides for asymmetric handling of arguments s and t
of X and is a two-step procedure which is composed of iterated one-dimensional FPCAs.
Key differences between the marginal FPCA and the conditional FPCA are as follows:
(1) The first step of the conditional FPCA approach (23) requires to perform a separate
FPCA for each t ∈ T , while in the marginal approach (4) only one FPCA is required, with
lower computational cost, and, most importantly, using all the data rather than the data
in a window around t. (2) In (23), the eigenfunctions ρj(s|t) depend on both arguments,
making it difficult to separate and interpret the effects of s and t in conditional FPCA,
in contrast to marginal FPCA, where the eigenfunctions in (4) only depend on s. (3) For
sparse designs, conditional FPCA requires a smoothing estimator of the function GS(s, u|t)
that depends on 2d1 + d2 univariate arguments. This improves upon the standard two-
argument FPCA (3), where the corresponding covariance functions depend on 2d1 + 2d2
arguments. The improvement is however even greater for marginal FPCA, where the
covariance function depends on only 2d1 or 2d2 arguments, leading to faster convergence.
The proposed marginal FPCA improves upon standard FPCA by providing inter-
pretable components and making it possible to treat the index of the stochastic process
asymmetrically in the arguments of the random functions that constitute the values of the
process. While we have discussed in detail the case of time-indexed function-valued pro-
cesses, and our example also conforms with this simplest setting, extensions to spatially
indexed function-valued processes or processes which are indexed on a rectangular subdo-
main ofRp are straightforward. Marginal FPCA also is supported by theoretical optimality
properties as per Theorem 1 and Theorem 2.
A promising simplified version of the marginal FPCA is product FPCA, motivated by a
common principal component assumption, see Theorem 4. Additional motivation is its near
optimality even without the common principal component assumption, as per Theorem 5.
21
Page 23
In our fertility data example, the loss of flexibility is quite limited and may be outweighed
by the simplicity and interpretability of this model. In general, the explanatory power of the
product FPCA model depends on the structure of the double-indexed array ηjk = var(χjk).
When one of the marginal kernels does not have fast decaying eigenvalues, relatively large
values of ηjk might show up in large j or large k and in such situations the product FPCA
might have limited explanatory power, and it would be better to apply marginal FPCA or
two-dimensional FPCA. The eigenvalues of the marginal kernels can be directly estimated
and can be used to diagnose this situation in data applications.
In this paper we mainly focus on the case where the argument of the functional values s
is densely and regularly sampled. In practical applications with designs that are sparse in
s, one may obtain GS by pooling the data {Xci (·, tim), i = 1, . . . , n, m = 1, . . . ,Mi}, and
utilizing a two-dimensional smoothing estimator of the covariance (Yao et al. 2005). The
FPCs can be obtained through conditional expectation (PACE) under Gaussian assump-
tions; software is available at http://www.stat.ucdavis.edu/PACE/. For this case, one
can only show that ξi,j(t)→a.s. E(ξi,j(t)|Data) under Gaussian assumptions.
REFERENCES
Aston, J. A., Pigoli, D. and Tavakoli, S. (2015) Tests for separability in nonparametric
covariance operators of random surfaces. arXiv preprint arXiv:1505.02023.
Bosq, D. (2000) Linear Processes in Function Spaces: Theory and Applications. New York:
Springer.
Chen, K. and Muller, H.-G. (2012) Modeling repeated functional observations. Journal of
the American Statistical Association, 107, 1599–1609.
Crainiceanu, C., Staicu, A. and Di, C. (2009) Generalized multilevel functional regression.
Journal of the American Statistical Association, 104, 1550–1561.
Cuevas, A. (2013) A partial overview of the theory of statistics with functional data. Journal
of Statistical Planning and Inference, 147, 1–23.
22
Page 24
Currie, I. D., Durban, M. and Eilers, P. H. (2004) Smoothing and forecasting mortality
rates. Statistical modelling, 4, 279–298.
— (2006) Generalized linear array models with applications to multidimensional smoothing.
Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68, 259–280.
Delicado, P., Giraldo, R., Comas, C. and Mateu, J. (2010) Statistics for spatial functional
data: some recent contributions. Environmetrics, 21, 224–239.
Eilers, P. H., Currie, I. D. and Durban, M. (2006) Fast and compact smoothing on large
multidimensional grids. Computational Statistics & Data Analysis, 50, 61–76.
Eilers, P. H. and Marx, B. D. (2003) Multivariate calibration with temperature interac-
tion using two-dimensional penalized signal regression. Chemometrics and Intelligent
Laboratory Systems, 66, 159–174.
Ezeh, A. C., Bongaarts, J. and Mberu, B. (2012) Global population trends and policy
options. The Lancet, 380, 142–148.
Greven, S., Crainiceanu, C., Caffo, B., Reich, D. et al. (2010) Longitudinal functional
principal component analysis. Electronic Journal of Statistics, 4, 1022–1054.
Gromenko, O., Kokoszka, P., Zhu, L. and Sojka, J. (2012) Estimation and testing for
spatially indexed curves with application to ionospheric and magnetic field trends. The
Annals of Applied Statistics, 6, 669–696.
— (2013) Nonparametric estimation in small data sets of spatially indexed curves with ap-
plication to temporal trend determination. Computational Statistics and Data Analysis,
59, 82–94.
Hall, P., Muller, H.-G. and Wang, J.-L. (2006) Properties of principal component methods
for functional and longitudinal data analysis. Annals of Statistics, 34, 1493–1517.
HFD (2013) Human Fertility Database. Max Planck Institute for Demographic Research
(Germany) and Vienna Institute of Demography (Austria). (Version March 18, 2013).
URL http://www.humanfertility.org.
Horvath, L. and Kokoszka, P. (2012) Inference for Functional Data with Applications. New
23
Page 25
York: Springer.
Huang, J. Z., Shen, H. and Buja, A. (2009) The analysis of two-way functional data using
two-way regularized singular value decompositions. Journal of the American Statistical
Association, 104, 1609–1620.
Hyndman, R. J. and Shang, H. L. (2009) Forecasting functional time series. Journal of the
Korean Statistical Society, 38, 199–211.
Hyndman, R. J. and Ullah, S. (2007) Robust forecasting of mortality and fertility rates: a
functional data approach. Computational Statistics & Data Analysis, 51, 4942–4956.
Jasilioniene, A., Jdanov, D. A., Sobotka, T., Andreev, E. M., Zeman, K. and Shkolnikov,
V. M. (2012) Methods Protocol for the Human Fertility Database. Max Planck Institute
for Demographic Research (Germany) and Vienna Institute of Demography (Austria).
Kneip, A. and Utikal, K. (2001) Inference for density families using functional principal
component analysis. Journal of the American Statistical Association, 96, 519–542.
Li, Y. and Hsing, T. (2010) Uniform convergence rates for nonparametric regression and
principal component analysis in functional/longitudinal data. Annals of Statistics, 38,
3321–3351.
Morris, J. S. and Carroll, R. J. (2006) Wavelet-based functional mixed models. Journal of
the Royal Statistical Society: Series B, 68, 179–199.
Nerini, D., Monestiez, P. and Mante, C. (2010) Cokriging for spatial functional data. Jour-
nal of Multivariate Analysis, 101, 409–418.
Park, S. and Staicu, A. (2015) Longitudinal functional data analysis. STAT, 4, 212–226.
Preston, S. H., Heuveline, P. and Guillot, M. (2001) Demography: Measuring and modeling
population processes. Blackwell Publishing.
Ramsay, J. O. and Silverman, B. W. (2005) Functional Data Analysis. Springer Series in
Statistics. New York: Springer, second edn.
Takahashi, S. (2004) Demographic investigation of the declining fertility process in japan.
The Japanese Journal of Population, 2, 93–116.
24
Page 26
WPP (2012) World population prospects: The 2012 revision, dvd edition.
Yao, F. and Lee, T. C. M. (2006) Penalized spline models for functional principal component
analysis. Journal of the Royal Statistical Society: Series B, 68, 3–25.
Yao, F., Muller, H.-G. and Wang, J.-L. (2005) Functional data analysis for sparse longitu-
dinal data. Journal of the American Statistical Association, 100, 577–590.
Yuan, Y., Gilmore, J. H., Geng, X., Martin, S., Chen, K., Wang, J.-l. and Zhu, H. (2014)
Fmem: Functional mixed effects modeling for the analysis of longitudinal white matter
tract data. NeuroImage, 84, 753–764.
25
Page 27
Online Supplement
Supplement A: Auxiliary Results and Proofs
Lemma 1. The function GS defined in (5) is a continuous, symmetric and positive definite
function. Moreover it is a valid covariance function.
Proof. Symmetry is obvious. Continuity follows from the continuity of the covariance
function C. We have that GS(s, u) =∫T cov(X(s, t), X(u, t))dt. Observing that for each
fixed t, cov(X(s, t), X(u, t)) is positive definite, it holds that for any function f in L2(S) we
have that∫S×S cov(X(s, t), X(u, t))f(u)f(s)duds ≥ 0 for all t ∈ T . Therefore, by Fubini,∫
S×SGS(u, s)f(u)f(s)duds =
∫T
∫S×S
cov(X(s, t), X(u, t))f(u)f(s)dudsdt ≥ 0.
To see that GS is a valid covariance function, remember that this is equivalent to say
that∫S GS(s, s)ds <∞ (see, for instance, Horvath and Kokoszka 2012, page 24). Observe
that ∫SGS(s, s)ds =
∫S
∫TC((s, t), (s, t))dtds
and the last integral is finite because C is a valid covariance function.
Lemma 2. Let ξj(t) = 〈X(·, t)− µ(·, t), ψj〉S , j ≥ 1, be the random functional coefficients
in the marginal Karhunen-Loeve representation (7). Then E(ξj(t)) = 0 for almost all
t ∈ T , and E(〈ξj, ξk〉T ) = τjδjk, where δjk = 1 if j = k and = 0 otherwise.
Proof. First, for almost all t ∈ T , X(·, t) is a random element of L2(S) because X is in
L2(S ×T ). Then there exists a unique (in the L2 sense) function µ(·, t) in L2(S) such that
E(〈X(·, t), y〉S) = 〈µ(·, t), y〉S for all y ∈ L2(S) and it follows that µ(s, t) = E(X(s, t)) =
µ(s, t) for almost all s ∈ S (see, for instance, Horvath and Kokoszka (2012), Section 2.3),
so that µ(·, t) = µ(·, t) in the L2 sense. Then taking y = ψj we have that
E(ξj(t)) = E (〈X(·, t), ψj〉S)− 〈µ(·, t), ψj〉S = 0.
26
Page 28
Furthermore,
E(〈ξj, ξk〉T ) = E
(∫Tξj(t)ξk(t)dt
)=E
(∫T
(∫SXc(s, t)ψj(s)ds
)(∫SXc(u, t)ψk(u)du
)dt
)=
∫S
∫S
∫TE (Xc(s, t)Xc(u, t)) dt ψj(s)ψk(u) ds du
=
∫S
(∫SGS(s, u)ψj(s) ds
)ψk(u) du
=〈Ψ(ψj), ψk〉S = τj〈ψj, ψk〉S = τjδjk,
where we have used that τj, ψj, j ≥ 1, are, respectively, the eigenvalues and eigenfunctions
of the operator Ψ defined as Ψ(f)(s) =∫GS(s, u) f(u) du.
Proof of Theorem 1:
Observe that
E
(∫T‖Xc(·, t)−
P∑j=1
〈Xc(·, t), gj〉Sgj‖2Sdt
)
=E
(∫T〈Xc(·, t)−
P∑j=1
〈Xc(·, t), gj〉Sgj, Xc(·, t)−P∑
j=1
〈Xc(·, t), gj〉Sgj〉Sdt
)
=E
(∫T〈Xc(·, t), Xc(·, t)〉Sdt
)−
P∑j=1
E
(∫T〈Xc(·, t), gj〉2Sdt
)
=E
(∫T
∫S
(Xc(s, t))2 dsdt
)−
P∑j=1
E
(∫T
(∫SXc(s, t)gj(s)ds
)2
dt
)
=E(‖Xc‖2
)−
P∑j=1
∫S
∫S
∫TE (Xc(s, t)Xc(u, t)) dt gj(u)du gj(s)ds
=E(‖Xc‖2
)−
P∑j=1
∫S
∫S
∫TC((s, t), (u, t))dt gj(u)du gj(s)ds
=E(‖Xc‖2
)−
P∑j=1
∫S
∫SGS(s, u)gj(u)du gj(s)ds
=E(‖Xc‖2
)−
P∑j=1
〈Ψ(gj), gj〉S ,
27
Page 29
where Ψ is the operator in L2(S) with kernel GS . Then minimizing
E
(∫T‖Xc(·, t)−
P∑j=1
〈Xc(·, t), gj〉Sgj‖2Sdt
)
is equivalent to maximizing∑P
j=1
∫S〈Ψ(gj), gj〉S . Given that Ψ is a symmetric, positive
definite Hilbert-Schmidt operator (see Lemma 1), standard arguments (see, for instance,
Theorem 3.2 in Horvath and Kokoszka (2012)) complete the proof.
Proof of Theorem 2:
For fjk(t) and gj(s) that satisfy the orthogonal conditions, we have
E
∫S,T{Xc(s, t)−
P∑j=1
Kj∑k=1
〈Xc, fjkgj〉fjk(t)gj(s)}2dsdt
=E‖Xc‖2 − 2×
P∑j=1
Kj∑k=1
E
(∫S,T
Xc(s, t)〈Xc, fjkgj〉fjk(t)gj(s)
)dsdt
+ E
∫S,T
P∑j=1
Kj∑k=1
P∑l=1
Kj∑h=1
〈Xc, fjkgj〉fjk(t)gj(s)〈Xc, flhgl〉flh(t)gl(s)dsdt
=E‖Xc‖2 − 2×
P∑j=1
Kj∑k=1
∫E (Xc(s, t)Xc(u, v)) fjk(t)fjk(v)gj(s)gj(u)dsdtdudv
+P∑
j=1
Kj∑k=1
∫E (Xc(s, t)Xc(u, v)) fjk(t)fjk(v)gj(s)gj(u)dsdtdudv
=E‖Xc‖2 −P∑
j=1
Kj∑k=1
∫E (Xc(s, t)Xc(u, v)) fjk(t)fjk(v)gj(s)gj(u)dsdtdudv
(24)
Let fjk(t) and gj(s) denote the optimal basis to achieve the minimum reconstruction
error Q∗, and define
(I) =P∑
j=1
Kj∑k=1
∫E (Xc(s, t)Xc(u, v))φjk(t)φjk(v)ψj(s)ψj(u)dsdtdudv
and
(II) =P∑
j=1
Kj∑k=1
∫E (Xc(s, t)Xc(u, v)) fjk(t)fjk(v)gj(s)gj(u)dsdtdudv.
28
Page 30
By Eq. (24), to prove the theorem, we only need to show that (II)− (I) < aE‖Xc‖2.
We further define
(III) =P∑
j=1
∫S×S
∫TE (Xc(s, t)Xc(u, t)) dtgj(s)gj(u)dsdu,
and
(IV ) =P∑
j=1
∫S×S
∫TE (Xc(s, t)Xc(u, t)) dtψj(s)ψj(u)dsdu.
We will prove that (II) < (III) < (IV ) and (IV )− (I) < aE‖Xc‖2, which implies that
(II)− (I) < aE‖Xc‖2.
By definition, ψj are the leading eigenfunctions of the marginal kernel GS(s, u) so that
(III) < (IV ).
To show (II) < (III), let ξj(t) = 〈Xc(s, t), gj(s)〉. Then,
(II) =P∑
j=1
Kj∑k=1
∫E (Xc(s, t)Xc(u, v)) fjk(t)fjk(v)gj(s)gj(u)dsdtdudv
=P∑
j=1
Kj∑k=1
∫E(ξj(t)ξj(v)
)fjk(t)fjk(v)dtdv
<P∑
j=1
∞∑k=1
∫E(ξj(t)ξj(v)
)fjk(t)fjk(v)dtdv
=P∑
j=1
∫E(ξj(t)ξj(t)
)dt
=P∑
j=1
∫E
(∫Xc(s, t)gj(s)ds
∫Xc(u, t)gj(u)du
)dt
=(III) (25)
29
Page 31
Finally, we prove (IV )− (I) < aE‖Xc‖2,
(IV )− (I) =P∑
j=1
∫E (ξj(t)ξj(t)) dt
−P∑
j=1
Kj∑k=1
∫E (Xc(s, t)Xc(u, v))φjk(t)φjk(v)ψj(s)ψj(u)dsdtdudv
=P∑
j=1
∞∑k=1
∫E (ξj(t)ξj(v))φjk(t)φjk(v)dtdv
−P∑
j=1
Kj∑k=1
∫E (ξj(t)ξj(v))φjk(t)φjk(v)dtdv
< aE‖Xc‖2,
(26)
where a = max1≤j≤P aj, with (1− aj)% denoting the fraction of variance explained by Kj
terms in each process ξj(t) = 〈Xc(·, t), ψj〉.
Proof of Theorem 3:
Recall that
GS(s, u) =
∫TC((s, t), (u, t))dt,
where C((s, t), (u, t)) = E[(X(s, t)−µ(s, t))(X(u, t)−µ(u, t))]. For (s, u) on the grid points,
we have
GS(s, u) =|T |∑ni=1Mi
n∑i=1
Mi∑m=1
Xc(s, tim)Xc(u, tim),
where Xc(s, tim) = X(s, tim)− µ(s, tim) and |T | is the Lebesgue measure of T . We define
GS(s, u) =|T |∑ni=1Mi
n∑i=1
Mi∑m=1
Xc(s, tim)Xc(u, tim).
Using sups,t |µ(s, t) − µ(s, t)| = Op((log n/n)1/2), it is easy to show that ‖GS(s, u) −
GS(s, u)‖S = Op((log n/n)1/2). Next we show
‖GS(s, u)−GS(s, u)‖S = Op((1/n)1/2). (27)
30
Page 32
We first prove (27) under assumption (A.6a). By (A.6a), we have Mi ≡ M , and the
grid of t is {t1, . . . , tM}. Therefore,
sup(s,u)∈S2
|EGS(s, u)−GS(s, u)| (28)
= sup(s,u)∈S2
∣∣∣∣∣ |T |MM∑
m=1
C((s, tm), (u, tm))−∫TC((s, t), (u, t))dt
∣∣∣∣∣A.3=O(
1
M) = O(1/n),
and
sup(s,u)∈S2
var(GS(s, u))
= sup(s,u)∈S2
|T |2
(nM)2
n∑i=1
var(M∑
m=1
Xc(s, tm)Xc(u, tm)) (29)
≤ sup(s,u)∈S2
|T |2
(nM)2
n∑i=1
M∑m,m′=1
E(Xc(s, tm)Xc(u, tm)Xc(s, tm′))Xc(u, tm′))
A.1,A.2
≤ |T |2
n2M2
n∑i=1
M2B = O(1/n).
Combining (28) and (29) we have
sup(s,u)∈S2
E|GS(s, u)−GS(s, u)|2 = O(1/n).
Therefore, by (A.4) and (sl − sl−1) = O(n−1),
E‖GS(s, u)−GS(s, u)‖2S =
∫S
∫S|GS(s, u)−GS(s, u)|2dsdu (30)
=|S|L2
L∑j=1
L∑l=1
E|GS(sj, sl)−GS(sj, sl)|2 +O(1/n)
= O(1/n),
which implies (27). The same result can be derived using a similar argument under (A.6b),
31
Page 33
by substituting (28) with
sup(s,u)∈S2
|EGS(s, u)−GS(s, u)| (31)
=
∣∣∣∣∣E |T |∑ni=1Mi
n∑i=1
Mi∑m=1
C((s, tim), (u, tim))−∫TC((s, t), (u, t))dt
∣∣∣∣∣ = 0,
and substituting (29) with
sup(s,u)∈S2
var(GS(s, u)) =|T |2
(∑n
i=1Mi)2
n∑i=1
var(
Mi∑m=1
Xc(s, tim)Xc(u, tim)) (32)
A.1,A.2
≤ |T |2
(∑n
i=1Mi)2
n∑i=1
M2i B = O(1/n).
This completes the proof for Eq. (15).
For a fixed j, Lemma 4.3 in Bosq (2000) implies that
|τj−τj| ≤ ||GS(s, u)−GS(s, u)||S , ||ψj(s)−ψj(s)||S ≤ 2√
2δ−1j ||GS(s, u)−GS(s, u)||, (33)
where δj is defined in (A.5). By (A.5), Eq. (16) and Eq. (17) directly follow.
In the following, we establish Eq. (18) as follows,
1
n
n∑i=1
sup1≤m≤Mi
|ξij(tim)− ξij(tim)|
≤ 1
n
n∑i=1
sup1≤m≤Mi
|∫
(Xi(s, tim)− µ(s, tim))(ψj(s)− ψj(s))ds|
+1
n
n∑i=1
sup1≤m≤Mi
|∫
(µ(s, tim)− µ(s, tim))ψj(s)ds|
+1
n
n∑i=1
sup1≤m≤Mi
|∫
(µ(s, tim)− µ(s, tim))(ψj(s)− ψj(s))ds|
≤ 1
n
n∑i=1
sups,t|Xi(s, tim)− µ(s, t)|‖ψj(s)− ψj(s)‖S
+ sups,t|µ(s, t)− µ(s, t)| sup
s|ψj(s)|+ sup
s,t|µ(s, t)− µ(s, t)|‖ψj(s)− ψj(s)‖S
= Op((log n/n)1/2) +Op((log n/n)1/2) +Op((log n/n)1/2)
= Op((log n/n)1/2), (34)
32
Page 34
where we used (A.1),(A.2), sups,t |µ(s, t) − µ(s, t)| = Op((log n/n)1/2), and the previous
result sups |φj(s)− φj(s)| = Op((log n/n)1/2). This completes the proof.
Proof of Theorem 4:
C((s, t), (u, v)) = cov(X(s, t), X(u, v))
=cov
(∞∑j=1
∞∑k=1
χjkfk(t)gj(s),∞∑l=1
∞∑h=1
χlhfh(v)gl(u)
)
=∞∑j=1
∞∑k=1
∞∑l=1
∞∑h=1
cov(χjkχlh)fk(t)gj(s)fh(v)gl(u).
Furthermore, by the orthogonality of fk and cov(χjk, χlk) = 0 for j 6= l, we have
CS(s, u) =
∫TC((s, t), (u, t))dt =
∞∑j=1
∞∑l=1
∞∑k=1
∞∑h=1
cov(χjkχlh)gj(s)gl(u)
∫Tfk(t)fh(t)dt
=∞∑j=1
(∞∑k=1
cov(χjk, χjk))gj(s)gj(u).
Therefore, gj(s) are the unique eigenfunctions of CS(s, u), and τj =∑∞
k=1 var(χjk). By
symmetry, one obtains the analogous result fk(t) ≡ φk(t).
Proof of Theorem 5:
For fk(t) and gj(s) that satisfy the orthogonality conditions,
E
(∫S,T{Xc(s, t)−
P∑j=1
K∑k=1
〈Xc, fkgj〉fk(t)gj(s)}2dsdt
)
=E‖Xc‖2 − 2×P∑
j=1
K∑k=1
∫E (Xc(s, t)Xc(u, v)) fk(t)fk(v)gj(s)gj(u)dsdtdudv
+P∑
j=1
K∑k=1
∫E (Xc(s, t)Xc(u, v)) fk(t)fk(v)gj(s)gj(u)dsdtdudv
=E‖Xc‖2 −P∑
j=1
K∑k=1
∫E (Xc(s, t)Xc(u, v)) fk(t)fk(v)gj(s)gj(u)dsdtdudv.
(35)
33
Page 35
Let fk(t) and gj(s) denote the optimal basis to achieve the minimum reconstruction
error Q∗, and define
(I) =P∑
j=1
K∑k=1
∫E (Xc(s, t)Xc(u, v))φk(t)φk(v)ψj(s)ψj(u)dsdtdudv,
and
(II) =P∑
j=1
K∑k=1
∫E (Xc(s, t)Xc(u, v)) fk(t)fk(v)gj(s)gj(u)dsdtdudv.
By Eq. (35), to prove the theorem, we only need to show that (II)− (I) < aE‖Xc‖2.
We further define
(III) =P∑
j=1
∫S×S
∫TE (Xc(s, t)Xc(u, t)) dtgj(s)gj(u)dsdu,
and
(IV ) =P∑
j=1
∫S×S
∫TE (Xc(s, t)Xc(u, t)) dtψj(s)ψj(u)dsdu.
We will prove that (II) < (III) < (IV ) and (IV )− (I) < aE‖Xc‖2, which implies that
(II)− (I) < aE‖Xc‖2.
By definition, the ψj are the leading eigenfunctions of the marginal kernel GS(s, u) so
that (III) < (IV ).
To show (II) < (III), let ξj(t) = 〈Xc(s, t), gj(s)〉, we observe
(II) =P∑
j=1
K∑k=1
∫E (Xc(s, t)Xc(u, v)) fk(t)fk(v)gj(s)gj(u)dsdtdudv
=P∑
j=1
K∑k=1
∫E(ξj(t)ξj(v)
)fk(t)fk(v)dtdv
<P∑
j=1
∫E(ξj(t)ξj(t)
)dt
=P∑
j=1
∫E
(∫Xc(s, t)gj(s)ds
∫Xc(u, t)gj(u)du
)dt
=(III). (36)
34
Page 36
Finally, we prove (IV )− (I) < aE‖Xc‖2 = max(aS , aT )E‖Xc‖2. Recall that τj and ϑk
are the eigenvalues of GT (s, u) and GS(t, v). Then
τj =
∫GT (s, u)ψj(s)ψj(u)dsdu
=
∫ ∫E (Xc(s, t)Xc(u, t)) dtψj(s)ψj(u)dsdu
=
∫ ∫ ∑j′
∑k
∑l
∑h
E(χj′kχlh)φk(t)φh(t)dtψj′(s)ψl(u)ψj(s)ψj(u)dsdu
=∞∑k=1
var(χjk),
and by symmetry, we obtain ϑk =∑∞
j=1 var(χjk). Then,
(IV )− (I) =P∑
j=1
τj −P∑
j=1
K∑k=1
var(χjk) =P∑
j=1
∞∑k=K+1
var(χjk)
<∞∑
k=K+1
ϑk = aTE‖Xc‖2,
By symmetry, we also have (IV )− (I) < aSE‖Xc‖2.
35
Page 37
Supplement B: Additional Tables and Figures for the analysis of the
fertility data and simulations
Additional materials on the fertility data, which were downloaded from the human fertility
database on March 18, 2013, are provided in Table 4 and Figures 5-11. These complement
the results presented in sections 5 and 6 of the main part of the paper.
Table 4: The abbreviations and names of the 17 countries (or territories), whose data are
used in the fertility application (those with available data for the period 1951-2006). The
colors used for representing each country in Figures 4 and 6 are also shown.
Color Abbreviation Country name First year Last year
�� SWE Sweden 1891 2010�� CAN Canada 1921 2007�� ESP Spain 1922 2006�� CHE Switzerland 1932 2009�� USA U.S. 1933 2010�� GBRTENW U.K., England and Wales 1938 2009�� FIN Finland 1939 2009�� PRT Portugal 1940 2009�� GBR SCO U.K., Scotland 1945 2009�� FRA France 1946 2010�� BGR Bulgaria 1947 2009�� JPN Japan 1947 2009�� CZE Czech Republic 1950 2011�� HUN Hungary 1950 2009�� NLD Netherlands 1950 2009�� SVK Slovakia 1950 2009�� AUT Austria 1951 2010
36
Page 38
1960 1970 1980 1990 2000
2030
4050
AUT
YearA
ge
1960 1970 1980 1990 2000
2030
4050
BGR
Year
Age
1960 1970 1980 1990 2000
2030
4050
CAN
Year
Age
1960 1970 1980 1990 2000
2030
4050
CZE
Year
Age
1960 1970 1980 1990 2000
2030
4050
FIN
Year
Age
1960 1970 1980 1990 2000
2030
4050
FRA
Year
Age
1960 1970 1980 1990 2000
2030
4050
HUN
Year
Age
1960 1970 1980 1990 2000
2030
4050
JPN
Year
Age
1960 1970 1980 1990 2000
2030
4050
NLD
Year
Age
1960 1970 1980 1990 2000
2030
4050
PRT
Year
Age
1960 1970 1980 1990 2000
2030
4050
SVK
Year
Age
1960 1970 1980 1990 2000
2030
4050
SWE
Year
Age
1960 1970 1980 1990 2000
2030
4050
CHE
Year
Age
1960 1970 1980 1990 2000
2030
4050
GBRTENW
Year
Age
1960 1970 1980 1990 2000
2030
4050
GBR_SCO
Year
Age
1960 1970 1980 1990 2000
2030
4050
USA
Year
Age
1960 1970 1980 1990 2000
2030
4050
ESP
Year
Age
0.00
0.05
0.10
0.15
0.20
0.25
Figure 5: Age-specific fertility rates (ASFR) for 17 countries, red colors correspond to low
values and yellow colors to high values.
37
Page 39
ASFR. Country−year data
Age
AS
FR
20 30 40 50
−0.
10−
0.05
0.00
0.05
0.10
Figure 6: All available functional fertility data as functions of age for 952 combinations of
17 countries and 56 calendar years, centered around the mean. Functions corresponding to
the same country are in the same color.
38
Page 40
−0.2 −0.1 0.0 0.1 0.2
−0.
2−
0.1
0.0
0.1
0.2
Scores at eigenfunctions 2 vs. 1
Scores at eigenfunction 1
Sco
res
at e
igen
func
tion
2
1951.AUT
1966.AUT
1981.AUT
1996.AUT
2006.AUT
1951.BGR
1966.BGR
1981.BGR
1996.BGR
2006.BGR
1951.CAN
1966.CAN
1981.CAN
1996.CAN2006.CAN 1951.CZE
1966.CZE
1981.CZE
1996.CZE2006.CZE
1951.FIN
1966.FIN1981.FIN
1996.FIN2006.FIN
1951.FRA
1966.FRA
1981.FRA1996.FRA
2006.FRA
1951.HUN
1966.HUN
1981.HUN1996.HUN
2006.HUN
1951.JPN
1966.JPN
1981.JPN1996.JPN
2006.JPN
1951.NLD
1966.NLD
1981.NLD
1996.NLD
2006.NLD
1951.PRT
1966.PRT
1981.PRT
1996.PRT
2006.PRT
1951.SVK
1966.SVK
1981.SVK
1996.SVK
2006.SVK
1951.SWE
1966.SWE1981.SWE
1996.SWE
2006.SWE
1951.CHE
1966.CHE
1981.CHE
1996.CHE
2006.CHE
1951.GBRTENW
1966.GBRTENW
1981.GBRTENW
1996.GBRTENW2006.GBRTENW
1951.GBR_SCO
1966.GBR_SCO
1981.GBR_SCO1996.GBR_SCO2006.GBR_SCO
1951.USA
1966.USA
1981.USA
1996.USA
2006.USA
1951.ESP
1966.ESP
1981.ESP
1996.ESP
2006.ESP
1951.CZE
1966.CZE
1981.CZE
1996.CZE2006.CZE
1951.PRT
1966.PRT
1981.PRT
1996.PRT
2006.PRT
1951.USA
1966.USA
1981.USA
1996.USA
2006.USA
1951.ESP
1966.ESP
1981.ESP
1996.ESP
2006.ESP
−0.2 −0.1 0.0 0.1 0.2
−0.
15−
0.10
−0.
050.
000.
05
Scores at eigenfunctions 3 vs. 1
Scores at eigenfunction 1
Sco
res
at e
igen
func
tion
3
1951.AUT
1966.AUT
1981.AUT
1996.AUT2006.AUT
1951.BGR
1966.BGR
1981.BGR
1996.BGR
2006.BGR
1951.CAN
1966.CAN
1981.CAN1996.CAN2006.CAN
1951.CZE
1966.CZE
1981.CZE
1996.CZE2006.CZE
1951.FIN1966.FIN1981.FIN
1996.FIN2006.FIN
1951.FRA1966.FRA
1981.FRA1996.FRA2006.FRA
1951.HUN
1966.HUN1981.HUN
1996.HUN
2006.HUN
1951.JPN
1966.JPN
1981.JPN
1996.JPN 2006.JPN
1951.NLD
1966.NLD
1981.NLD
1996.NLD
2006.NLD
1951.PRT
1966.PRT
1981.PRT
1996.PRT
2006.PRT
1951.SVK1966.SVK
1981.SVK
1996.SVK2006.SVK
1951.SWE
1966.SWE1981.SWE
1996.SWE
2006.SWE
1951.CHE
1966.CHE
1981.CHE
1996.CHE
2006.CHE
1951.GBRTENW1966.GBRTENW
1981.GBRTENW
1996.GBRTENW2006.GBRTENW
1951.GBR_SCO
1966.GBR_SCO
1981.GBR_SCO
1996.GBR_SCO2006.GBR_SCO
1951.USA
1966.USA
1981.USA
1996.USA
2006.USA
1951.ESP
1966.ESP 1981.ESP
1996.ESP
2006.ESP
1951.JPN
1966.JPN
1981.JPN
1996.JPN 2006.JPN
1951.NLD
1966.NLD
1981.NLD
1996.NLD
2006.NLD1951.ESP
1966.ESP 1981.ESP
1996.ESP
2006.ESP
Figure 7: Track-plots corresponding to the implicitly parametrized planar curves
{(ξi,1(t), ξi,2(t)), t = 1951, . . . , 2006}, parametrized by calendar time t, where ξi,j(t) is the
j-th score function for country i as in (4).
39
Page 41
1950 1960 1970 1980 1990 2000
0.06
0.10
0.14
Eigenfunction 1 (FVE: 88.83%)
Year
Eig
enfu
nctio
n 1
(3rd
ste
p)
1950 1960 1970 1980 1990 2000
−0.
050.
050.
150.
25
Eigenfunction 1 (FVE: 47.03%)
Year
Eig
enfu
nctio
n 1
(3rd
ste
p)
1950 1960 1970 1980 1990 2000
−0.
20−
0.10
0.00
Eigenfunction 1 (FVE: 60.87%)
Year
Eig
enfu
nctio
n 1
(3rd
ste
p)
1950 1960 1970 1980 1990 2000
−0.
20.
00.
1
Eigenfunction 2 (FVE: 7.55%)
Year
Eig
enfu
nctio
n 2
(3rd
ste
p)
1950 1960 1970 1980 1990 2000
−0.
10.
10.
20.
3
Eigenfunction 2 (FVE: 24.83%)
Year
Eig
enfu
nctio
n 2
(3rd
ste
p)1950 1960 1970 1980 1990 2000
−0.
10.
00.
10.
2
Eigenfunction 2 (FVE: 26.23%)
Year
Eig
enfu
nctio
n 2
(3rd
ste
p)
1950 1960 1970 1980 1990 2000
−0.
20.
00.
10.
2
Eigenfunction 3 (FVE: 2.24%)
Year
Eig
enfu
nctio
n 3
(3rd
ste
p)
1950 1960 1970 1980 1990 2000
−0.
20.
00.
2
Eigenfunction 3 (FVE: 15.87%)
Year
Eig
enfu
nctio
n 3
(3rd
ste
p)
1950 1960 1970 1980 1990 2000
−0.
20.
00.
1
Eigenfunction 3 (FVE: 7.61%)
Year
Eig
enfu
nctio
n 3
(3rd
ste
p)
Figure 8: Estimated eigenfunctions φjk(t) of the random scores ξj(t). These quantities are
as defined in (4).
40
Page 42
1960 1970 1980 1990 2000
2030
4050
j=1, k=1 (54.33% of var.)
Year
Age
1960 1970 1980 1990 2000
2030
4050
j=2, k=1 (13.04% of var.)
Year
Age
1960 1970 1980 1990 2000
2030
4050
j=2, k=2 (6.88% of var.)
Year
Age
1960 1970 1980 1990 2000
2030
4050
j=1, k=2 (4.62% of var.)
YearA
ge
1960 1970 1980 1990 2000
2030
4050
j=2, k=3 (4.4% of var.)
Year
Age
1960 1970 1980 1990 2000
2030
4050
j=3, k=1 (4.22% of var.)
Year
Age
Figure 9: Product functions φjk(t)ψj(s) corresponding to the six terms with higher FVE
in the marginal FPCA representation (4) of ASFR(s, t).
41
Page 43
10 20 30 40 50 60−0.4
−0.2
0
0.2
0.4
sψ1(s)
10 20 30 40 50 60−0.1
0
0.1
0.2
0.3
s
ψ2(s)
10 20 30 40 50 60−0.4
−0.2
0
0.2
0.4
s
ψ3(s)
1950 1960 1970 1980 1990 2000 20100.05
0.1
0.15
0.2
0.25
t
φ11(t)
1950 1960 1970 1980 1990 2000 2010−0.4
−0.2
0
0.2
0.4
t
φ12(t)
1950 1960 1970 1980 1990 2000 2010−0.4
−0.2
0
0.2
0.4
t
φ13(t)
1950 1960 1970 1980 1990 2000 2010−0.1
0
0.1
0.2
0.3
t
φ21(t)
1950 1960 1970 1980 1990 2000 2010−0.2
−0.1
0
0.1
0.2
t
φ22(t)
1950 1960 1970 1980 1990 2000 2010−0.4
−0.2
0
0.2
0.4
t
φ23(t)
1950 1960 1970 1980 1990 2000 2010
−0.2
−0.1
0
0.1
t
φ31(t)
1950 1960 1970 1980 1990 2000 2010−0.2
−0.1
0
0.1
0.2
t
φ32(t)
1950 1960 1970 1980 1990 2000 2010−0.4
−0.2
0
0.2
0.4
t
φ33(t)
Figure 10: True (red-solid) and estimated (blue-dashed) eigenfunctions ψj(s) and φjk(t) as
in model (4) for j = 1, 2, 3 and k = 1, 2, 3, for one run of simulation 1 with sample size
n = 50.
10 15 20 25 30 35 40 45 50 55
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
s
ψ1(s)
10 15 20 25 30 35 40 45 50 55−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
s
ψ2(s)
10 15 20 25 30 35 40 45 50 55−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
s
ψ3(s)
1950 1960 1970 1980 1990 2000 20100.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
t
φ 1(t)
1950 1960 1970 1980 1990 2000 2010−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
t
φ 2(t)
1950 1960 1970 1980 1990 2000 2010−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
t
φ 3(t)
Figure 11: True (red-solid) and estimated (blue-dashed) eigenfunctions ψj(s) and φk(t) in
model (6) for j = 1, 2, 3 and k = 1, 2, 3, as obtained in one run of simulation 2 with sample
size n = 50.
42
Page 44
Supplement C: Standard two-dimensional FPCA and product FPCA for
fertility data
Here we present the standard FPCA analysis of the ASFR data with the Karhunen-
Loeve representation, considering the data as random functions in two arguments. We
performed FPCA for this type of functional data following Yao et al. (2005) as implemented
in the PACE package (http://www.stat.ucdavis.edu/PACE). First, we rearrange the n = 17
matrices with dimension L×M = 44× 56, containing the observed functional data, into a
big data matrix with dimension n× (M · L). Then we perform FPCA on this big matrix.
Finally we rearrange the estimated eigenfunctions (stored at this point as arrays of length
M · L) into matrices of dimension M × L. Figure 12 graphically summarizes the main
results of this standard FPCA.
The first 4 eigenfunctions (which are eigensurfaces in this unconstrained approach) have
a FVE of 89.73%. The first one (with FVE equal to 58.93%) is almost constant in calendar
year and corresponds to a contrast between young fertility (women aged between 18 and 25
years) and fertility in mature years (mothers being from 25 to 40 years old). Countries with
larger positive coefficients in this eigenfunction are Bulgaria, Czech Republic, Slovakia,
Hungary and U.S., while the Netherlands, Japan, Spain and Switzerland have negative
coefficients.
The second eigenfunction (or eigensurface) reflects the specificity of the baby-boom
around 1960 in Canada and U.S. (both have high positive coefficients in this eigenfunction).
Countries with negative scores (such as Japan, Spain, Bulgaria, Hungary or Czech Republic)
do not show a drop in fertility rates at the end of the 1960s. The third eigenfunction appears
to correspond to a sudden drop at the end of the 1970s in fertility for women aged between
30 and 40 years. This could be associated with women’s decision on reducing the number
of children, as the high fertility rates for ages in the interval [30,40] before 1977 are mainly
associated with large families or, in more technical terms, with high parities, parity being
defined as the cumulative number of a woman’s live births; see Preston et al. (2001)). This
43
Page 45
5 10 15
010
2030
4050
60
Eigenfunction
FV
E in
%
FVE by each eigenfunction
58.93
13.7111.04
6.05
−1.0 −0.5 0.0 0.5 1.0 1.5
−0.
40.
00.
20.
40.
60.
8
1st eigenfunction
2nd
eige
nfun
ctio
n
Scores at 2nd vs 1st eigenfunctions
AUT
BGR
CAN
CZE
FINFRA
HUNJPN
NLD
PRT
SVK
SWECHE
GBRTENWGBR_SCO
USA
ESP
−1.0 −0.5 0.0 0.5 1.0 1.5
−0.
40.
00.
20.
40.
6
1st eigenfunction
3rd
eige
nfun
ctio
n
Scores at 3rd vs 1st eigenfunctions
AUT
BGRCAN CZE
FIN
FRAHUN
JPN
NLD
PRT
SVK
SWE
CHE GBRTENW
GBR_SCO
USA
ESP
−1.0 −0.5 0.0 0.5 1.0 1.5−
0.4
0.0
0.2
0.4
0.6
1st eigenfunction
4th
eige
nfun
ctio
n
Scores at 4th vs 1st eigenfunctions
AUT
BGR
CAN CZE
FIN
FRAHUN
JPN
NLD
PRT
SVK
SWECHE GBRTENWGBR_SCO
USA
ESP
1960 1970 1980 1990 2000
2030
4050
Ppal. Funct. 1 (FVE: 58.93%)
Year
Age
1960 1970 1980 1990 2000
2030
4050
Ppal. Funct. 2 (FVE: 13.71%)
Year
Age
1960 1970 1980 1990 2000
2030
4050
Ppal. Funct. 3 (FVE: 11.04%)
Year
Age
1960 1970 1980 1990 2000
2030
4050
Ppal. Funct. 4 (FVE: 6.05%)
Year
Age
Figure 12: Standard FPCA of the fertility data ASFRi(s, t), i = 1, . . . , n = 17, where the
lower four panels display the first four eigensurfaces.
44
Page 46
drop may be related to advances in birth control. These social changes arrived with a
certain lag in countries with positive scores (Portugal, Spain, Slovakia) while they were
adopted much earlier in countries with negative scores (Sweden, Finland, Switzerland).
Other characteristics of this third eigenfunction are less intuitive.
Regarding the fourth eigenfunction, the score map in the panel in column 2, row 2, of
Figure 12 indicates that Japan strongly weighs in this eigenfunction. Meanwhile the heat
map (panel in column 2, row 4) shows a contrast between fertility concentrated around the
age of 25 years (this strongly applies for Japan, with an outstanding positive score in this
eigenfunction) and spread out fertility between the ages of 18 to 40, mainly between 1955
and 1980. Moreover, this heat map also shows an anomalous behavior (that appears as a
discontinuity) at the year 1966. This fact corroborates that the fourth eigenfunction is a
Japan specific function. We refer to the discussion in Section 5 for the anomaly in Japanese
fertility in 1966.
Fitting the product FPC model to the fertility data resulted in estimates for the first
four eigenfunctions φk of the operator GT (t, v) as shown in Figure 13. The first of these
time trend functions particularly weighs the pre-1990 fertility, while the others are contrasts
between different calendar time periods. These estimated eigenfunctions are then multiplied
with the age eigenfunction estimates ψj(s) of Figure 3 to obtain the product functions
φk(t)ψj(s) that appear in the product FPC model representation (6) of ASFR(s, t). Figure
18 displays these product functions corresponding to the seven terms with larger FVEs
among those with j ≤ 4 and k ≤ 4, which together explain 87.38% of the total variability;
see also Table 1.
The product functions φk(t)ψj(s) in Figure 18 match well with the corresponding prod-
ucts φjk(t)ψj(s) in Figure 9 that result from the more general marginal approach (see
Appendix B). These functions can thus be similarly interpreted as previously described in
Section 5.1. The simplified product FPCA provides representations that are thus slightly
less flexible and therefore explain somewhat less of the variance when compared with those
45
Page 47
obtained from marginal FPCA, but have equally good, if not better, interpretability.
1950 1960 1970 1980 1990 2000
0.06
0.10
0.14
0.18
Eigenfunction 1 (FVE: 67.58%)
Year
Eig
enfu
nctio
n 1
1950 1960 1970 1980 1990 2000
−0.
2−
0.1
0.0
0.1
0.2
Eigenfunction 2 (FVE: 13.53%)
Year
Eig
enfu
nctio
n 2
1950 1960 1970 1980 1990 2000
−0.
2−
0.1
0.0
0.1
0.2
Eigenfunction 3 (FVE: 10.18%)
Year
Eig
enfu
nctio
n 3
1950 1960 1970 1980 1990 2000
−0.
2−
0.1
0.0
0.1
0.2
0.3
Eigenfunction 4 (FVE: 4.7%)
Year
Eig
enfu
nctio
n 4
Figure 13: Estimated eigenfunctions φk(t), k = 1, 2, 3, 4, in the product FPC model (6) for
the fertility data.
46
Page 48
1960 1970 1980 1990 2000
2030
4050
j=1, k=1 (FVE: 53.69%)
Year
Age
1960 1970 1980 1990 2000
2030
4050
j=2, k=2 (FVE: 8.1%)
Year
Age
1960 1970 1980 1990 2000
2030
4050
j=2, k=1 (FVE: 8.08%)
Year
Age
1960 1970 1980 1990 2000
2030
4050
j=2, k=3 (FVE: 5.51%)
Year
Age
1960 1970 1980 1990 2000
2030
4050
j=1, k=2 (FVE: 4.47%)
Year
Age
1960 1970 1980 1990 2000
2030
4050
j=2, k=4 (FVE: 3.85%)
Year
Age
1960 1970 1980 1990 2000
2030
4050
j=3, k=1 (FVE: 3.68%)
Year
Age
Figure 14: Product functions φk(t)ψj(s) corresponding to the seven terms with higher FVE
in the product FPC model.
When applying product FPCA, one needs 7 terms to explain 87.38% of variance,
while for the marginal FPCA it is sufficient to include 6 terms to explain 87.49% of
the variance. Of course product FPCA has the big advantage that the final repre-
sentation in general involves fewer functions ψj(s) and φk(t) than the number of func-
tions needed for the marginal FPCA representation and therefore is much simpler. For
instance, the analysis of the fertility data with marginal FPCA involves 9 functions
(ψ1(s), ψ2(s), ψ3(s), φ11(t), φ12(t), φ21(t), φ22(t), φ23(t), φ31(t)), while only 7 functions are in-
volved in the product FPC model (ψ1(s), ψ2(s), ψ3(s), φ1(t), φ2(t), φ3(t), φ4(t)).
47
Page 49
Supplement D: Male mortality rates as an additional example
Mortality rates (or death rates) are defined as a ratio of the death count for a given age-
time interval divided by an estimate of the population exposed to the risk of death during
some age-time interval (Preston et al. 2001). The Human Mortality Database (http:
//www.mortality.org/) provides detailed information on mortality rates for 37 countries
or areas with precision of one year in both age and calendar time. Such rich information
can be provided only by countries with well developed official statistics agencies. This is
the reason why only 37 countries are covered by this database.
An alternative database including a much larger number of countries can be accessed
through the Population Division of the Department of Economic and Social Affairs of
the United Nations (WPP 2012). This database contains information for more than 200
countries on deaths grouped into five-year age intervals, from 1950 to 2010 (every 5 years).
The price to be paid for including a much larger number of countries is a loss in precision,
i.e., aggregation over 5 year intervals, both in terms of age and calendar time. As definition
of the mortality rate for a given country during a period of consecutive years and an interval
of ages, we consider the ratio between the number of deaths reported for a specific country
over the selected 5 year calendar period for people with age at death in the selected 5
year age interval, divided by the number of people that at the beginning of the calendar
time interval were in the age interval. As male and female mortality rates are different, we
consider here male data that were downloaded (on the 14th of January 2015) from
http://esa.un.org/wpp/Excel-Data/EXCEL_FILES/3_Mortality/
WPP2012_MORT_F04_2_DEATHS_BY_AGE_MALE.XLS
http://esa.un.org/wpp/Excel-Data/EXCEL_FILES/1_Population/
WPP2012_POP_F15_2_ANNUAL_POPULATION_BY_AGE_MALE.XLS
We work with log-Mortality Rates, for which we use log(mortality rate+1), considered
as functions of men’s age grouped into intervals of 5 years (s) and repeatedly measured for
48
Page 50
every 5 calendar years t for various countries. The aggregated log-mortalities constitute
the functional data X(s, t) = log-mortality rate(s, t).
In WPP (2012), data are provided for ages s in the year intervals {[0, 5), [5, 10), . . . ,
[90, 95), [95,∞)}. The interval of calendar years with available data are {[1950, 1955), . . . ,
[2005, 2010)}. The variability of mortality rates increases with age and decreases with
population. So we limit ourselves to ages lower than 80. We also excluded countries with
a 0 value for population size at any year or age. Then our database finally consisted of 166
countries, with 12 periods of five years each (which we labeled with the first year of the
respective interval: 1950 to 2005) and 16 five years age intervals (labeled from 0 to 75).
0.0
0.1
0.2
0.3
0.4
0.5
0.6
1950 1960 1970 1980 1990 2000
0
10
20
30
40
50
60
70
Male log−mortatlity rate sample mean
Year
Age
Year
1950
1960
1970
1980
1990
2000
Age
0
20
40
60
Male log−m
ortatlity rate
0.1
0.2
0.3
0.4
Figure 15: Sample mean of the 166 male log-mortality rate functions by calendar year.
The sample mean of the male log-mortality rate functions for 166 countries displayed
in Figure 15 shows that mortality rates are, on average, highest for children under 5, and
for men aged more than 60; and that they are decreasing with increasing calendar year.
The male log-mortality rate data include one log-mortality rate curve over age per
calendar year and per country and are observed on a regular grid spaced in years across both
coordinates age s and calendar year t, which means that the empirical estimators described
in Section 2 can be applied to these data. Figure 16 displays the nM = 1992 centered
functional data male log-mortality rates Xci (sl, tm) = Xi(sl, tm) for l = 1, . . . , L = 16,
49
Page 51
Centered male log mortality rates. Country−year data
Age
Cen
tere
d lg
_mr
0 20 40 60
−0.
4−
0.2
0.0
0.2
0.4
Cambod.1975
Iran.1980Equato.1980
Botswa.2005Zimbab.2000
Figure 16: All available functional male log-mortality data as functions of age for 1992
combinations of 166 countries and 12 calendar years, centered around the mean. Functions
corresponding to the same country are in the same color.
m = 1, . . . ,M = 12 and i = 1, . . . , n = 166, demonstrating that there is substantial
variation across countries and calendar years. Several outliers in the centered log mortality
profiles have been highlighted in the figure. Some of these reflect periods of war, e.g. Iran
1980-1985 or genocides, e.g. Cambodia 1975-1980. Others correspond to high mortality
rates due to the HIV/AIDS pandemic. The bloody reign of the Macias Nguema dictatorship
in Equatorial Guinea also left its mark in this country’s mortality profile.
We fitted the marginal FPC model and found that the φjk(t) are similar for j = 1 and
2. This is an indication that the product FPCA is appropriate for these data, and we
directly applied it. Fitting the product FPC model to the male log-mortality data resulted
in estimates for ψj and φk as shown in Figure 17. The shape of the first eigenfunction
ψ1(s) (that takes positive values for all ages s) is similar to that of the mean function for a
fixed year t (see the right panel of Figure 15). Therefore ψ1(s) can be interpreted as a size
component: Country-years with positive score in the direction of this eigenfunction have
higher male log-mortality ratios than the mean function for all ages, with larger differences
50
Page 52
for larger values of the average log-mortality rates. The second eigenfunction ψ2(s) repre-
sents a contrast between infant mortality and older age mortality. The third eigenfunction
ψ3(s) appears to point to difficulties in obtaining accurate estimates of mortality rates for
the last age interval.
0 20 40 60
0.1
0.3
0.5
Eigenfunction 1 (FVE: 67.35%)
Age
Eig
enfu
nctio
n 1
0 20 40 60
−0.
50.
00.
5
Eigenfunction 2 (FVE: 13.83%)
Age
Eig
enfu
nctio
n 2
0 20 40 60
−0.
40.
00.
4
Eigenfunction 3 (FVE: 8.35%)
Age
Eig
enfu
nctio
n 3
1950 1960 1970 1980 1990 2000
0.25
0.30
0.35
Eigenfunction 1 (FVE: 69.91%)
Year
Eig
enfu
nctio
n 1
1950 1960 1970 1980 1990 2000
−0.
40.
00.
20.
4
Eigenfunction 2 (FVE: 10.64%)
Year
Eig
enfu
nctio
n 2
1950 1960 1970 1980 1990 2000
−0.
20.
00.
20.
4
Eigenfunction 3 (FVE: 6.57%)
YearE
igen
func
tion
3
Figure 17: Estimated eigenfunctions ψj(s) (first row) and φk(t) (second row), in the product
FPC model for the log-mortality data.
The first calendar year trend function φ1(t) shows a continuous reduction in male log-
mortality rates, with a pattern similar to the average evolution of male log-mortality over
time (see Figure 15, right panel). So positive scores associated with this eigenfunction
indicate larger reductions than the average (the opposite for negative scores). The second
and third trend functions are contrasts between different calendar time periods. Positive
(resp., negative) scores in the second trend function φ2(s) indicate higher (resp., lower) than
average mortality at the beginning of the overall calendar period, and lower than average
mortality for the final years of the calendar period, i.e., a faster decline in mortality as
compared to the average decline. The third eigenfunction is associated with differences
in changes in log-mortality rates over calendar time between the middle period and the
early/late periods.
51
Page 53
The product functions φk(t)ψj(s) are shown in Figure 18. These functions can be easily
interpreted by taking into account that they are the product of a function ψj(s) and a
function φk(t), as represented in Figure 17. When applying product FPCA, one needs 4
terms to achieve a FVE of 71.06%, and 6 terms to achieve a FVE of 75%.
1950 1960 1970 1980 1990 2000
020
4060
j=1, k=1 (FVE: 55.81%)
Year
Age
1950 1960 1970 1980 1990 2000
020
4060
j=2, k=1 (FVE: 6.57%)
YearA
ge
1950 1960 1970 1980 1990 2000
020
4060
j=1, k=2 (FVE: 5.15%)
Year
Age
1950 1960 1970 1980 1990 2000
020
4060
j=3, k=1 (FVE: 3.53%)
Year
Age
1950 1960 1970 1980 1990 2000
020
4060
j=1, k=3 (FVE: 2.84%)
Year
Age
1950 1960 1970 1980 1990 2000
020
4060
j=2, k=2 (FVE: 2.79%)
Year
Age
Figure 18: Product functions φk(t)ψj(s) corresponding to the six terms with higher FVE
in the product FPC model representation for the log mortality data.
52
Page 54
The first product of estimated eigenfunctions (with FVE equal to 55.81%) is φ1(t)ψ1(s),
which is the product of the function φ1(t) that is similar to the average evolution of log-
mortality rate over calendar years, and the function ψ1(s) that has a shape similar to
the average log-mortality rate pattern. As a consequence, the product function is always
positive and very similar to the mean function (see Figure 15). So countries with positive
random coefficients χ11 at this product function φ1(t)ψ1(s) have larger male log-mortality
rates than the average for all ages and all calendar years, with larger differences for larger
values of the average log-mortality rates, and vice versa for the countries with a negative
coefficient. We refer to Table 5 for a list of countries with most extreme (positive or
negative) coefficients at this first product component.
The second product of eigenfunctions is φ1(t)ψ2(s) (FVE: 6.57%). It represents a con-
trast between infant mortality and old age mortality, due to the shape of ψ2(s), which is
more marked at the beginning than at the end of the period (because of φ1(t)). Countries
with negative scores (see Table 5) have lower than average infant log-mortality rates and
higher than average old age log-mortality rates. The opposite applies to countries with
positive scores at this product. The third product of eigenfunctions is φ2(t)ψ1(s) and it
separates countries with faster than average reduction in male log-mortality rates (positive
coefficients) from those with slower than average reduction (negative coefficients). This is
the main effect of φ2(t). This effect is more marked for extreme ages, due to the shape of
ψ1(s). The countries with extreme coefficients as listed in Table 5 are extremes in a certain
shape direction and deserve further study.
Alternatively, one can also apply marginal FPCA to quantify the observed variability
across countries. The results for marginal FPCA are summarized in Figures 19 and 20 for
the first three eigenfunctions, ψj(s), j = 1, 2, 3, resulting in a FVE of 89.52%. The first
row of Figure 19 displays the estimated eigenfunctions ψj(s), which are identical to the
functions ψj(s) used in the product FPCA. The second row of panels in Figure 19 shows
the score functions ξi,j(t), t ∈ T , which are country-specific functions of calendar year.
53
Page 55
Table 5: Countries with the most extreme estimates of the random coefficients χjk ob-
tained by fitting the product FPCA model (6) for the six terms with higher FVE in the
representation of male log-mortality rates as linear combinations of the product functions
φk(t)ψj(s).
φ1(t)ψ1(s) (FVE: 55.81%)
Most − Iceland, Channel Islands, Sweden, Norway, Puerto Rico, Barbados
Most + Sierra Leone, Mali, Eritrea, Equatorial Guinea, Timor-Leste, Liberia
φ1(t)ψ2(s) (FVE: 6.57%)
Most − Fiji, Suriname, Martinique, Mauritius, Dem People’s Republic of Korea, Guyana
Most + Reunion, Central African Republic, El Salvador, Honduras, Pakistan, Angola
φ2(t)ψ1(s) (FVE: 5.15%)
Most − Channel Islands, Barbados, Iceland, Belarus, Rwanda, Sierra Leone
Most + China, Oman, Tunisia, Singapore, Hong Kong SAR, Japan
φ1(t)ψ3(s) (FVE: 3.53%)
Most − Channel Islands, Iceland, Martinique, Guinea-Bissau, Timor-Leste, Oman
Most + Reunion, Papua New Guinea, Eritrea, South Africa,
Dem People’s Republic of Korea, Guadeloupe
φ3(t)ψ1(s) (FVE: 2.84%)
Most − Cape Verde, Tajikistan, Kazakhstan, Azerbaijan, Belarus, Kyrgyzstan
Most + Cambodia, Barbados, Channel Islands, Reunion, Guadeloupe, Martinique
φ2(t)ψ2(s) (FVE: 2.79%)
Most − Martinique, Japan, Fiji, Malta, Guyana, Botswana
Most + Reunion, Barbados, Channel Islands, Iceland, Yemen, Eritrea
54
Page 56
Their evolution over calendar time can be visualized by the track plot in Figure 20, showing
the planar curves for the pairs (ξi,1(t), ξi,2(t)), t ∈ T . In this example, the track plot is
particularly useful to detect country-years with extreme scores in some eigenfunctions. For
instance, Cambodia 1975-1980 and Rwanda 1990-1995 have extremely positive high scores
in the first eigenfunction. This corresponds to periods in the history of these two countries
during which they experienced a very high mortality rate: the Cambodian Genocide from
1975 to 1979, and the Rwandan Genocide in 1994.
The third step of the marginal FPCA (performing a separate standard FPCA for the
estimated score functions ξi,j(t), i = 1, . . . , n, for j = 1, 2, 3) yields estimated eigenfunc-
tions φjk. For k = 1, 2, 3 these estimates are shown in Figure 19 (three lower rows). It can
be seen that results are similar (up to sign changes) for the first and second sets of score
functions.
To conclude this second example, we present the standard FPCA of the log-mortality
data with the Karhunen-Loeve representation, considering the data as random functions in
two arguments. Figure 21 graphically summarizes the main results of this standard FPCA.
The first four eigenfunctions have a FVE of 78.55%. There are similarities between these
eigenfunctions and, respectively, the 1st, 2nd, 3rd and 5th eigenfunction products repre-
sented in Figure 18 (in the two last cases, up to a sign change). Therefore the interpretation
we have made above for these eigenfunctions products are valid for the eigenfunctions ob-
tained by standard FPCA. Nevertheless, to arrive at these interpretations is much more
difficult if the starting point is Figure 21, without the benefit of the functions represented
in Figure 18 for the product FPCA and their decomposition as products of functions in
Figure 17.
55
Page 57
0 20 40 60
0.1
0.3
0.5
Eigenfunction 1 (FVE: 67.35%)
Age
Eig
enfu
nctio
n 1
0 20 40 60
−0.
50.
00.
5
Eigenfunction 2 (FVE: 13.83%)
Age
Eig
enfu
nctio
n 2
0 20 40 60
−0.
40.
00.
4
Eigenfunction 3 (FVE: 8.35%)
Age
Eig
enfu
nctio
n 3
1950 1960 1970 1980 1990 2000
−0.
6−
0.2
0.2
0.6
Functional scores at eigenfunction 1
Year
Sco
res
at e
igen
func
tion
1
Burund BurundEritre EritreEthiopEthiopKenya Kenya
MadagaMadaga
Malawi MalawiMauritMaurit
MozambMozamb
Reunio
Reunio
RwandaRwanda
SomaliSomali
South South
Uganda UgandaUnited
United
ZambiaZambia
ZimbabZimbab
AngolaAngola
Camero CameroCentra CentraChad Chad
CongoCongoDemocr
DemocrEquato
EquatoGabon GabonAlgeri
AlgeriEgypt EgyptLibya
LibyaMorocc
MoroccSudan SudanTunisi
Tunisi
BotswaBotswa
Lesoth
Lesoth
Namibi
NamibiBenin Benin
Burkin
BurkinCape V
Cape V
Cote dCote d
Ghana GhanaGuinea
Guinea
Liberi
Liberi
Mali
MaliNiger NigerNigeri NigeriSenega
Senega
SierraSierra
Togo TogoChina
ChinaChina,
China,
Dem PeDem Pe
JapanJapan
MongolMongol
Republ RepublOther
Other
Kazakh
KazakhKyrgyz KyrgyzTajiki
TajikiTurkme TurkmeUzbeki Uzbeki
Afghan
Afghan
Bangla Bangla
IndiaIndiaIran (
Iran (
Nepal
NepalPakist PakistSri La
Sri La
Cambod
Cambod
IndoneIndone
Lao Pe Lao Pe
MalaysMalays
Myanma
Myanma
Philip
PhilipSingap
SingapThaila Thaila
Timor−
Timor−Viet NViet N
ArmeniArmeniAzerbaAzerba
Cyprus
CyprusGeorgi Georgi
IraqIraq
Israel Israel
JordanJordan
Lebano Lebano
Oman
Oman
Saudi
Saudi State State Syrian Syrian
Turkey
Turkey
Yemen
Yemen
Belaru
Belaru
Bulgar
Bulgar
Czech Czech
Hungar
HungarPoland PolandRomani
RomaniRussia
Russia
Slovak
SlovakUkrain
Ukrain
Channe
ChanneDenmar
DenmarEstoni
EstoniFinlan Finlan
Icelan
IcelanIrelan IrelanLatvia
Latvia
Lithua
Lithua
Norway
NorwaySweden
SwedenAlbani
AlbaniBosnia BosniaCroati CroatiGreece GreeceItaly ItalyMalta
MaltaMonten
MontenPortug PortugSerbia
SerbiaSloven SlovenSpain SpainTFYR M
TFYR M
Austri AustriBelgiu BelgiuFrance FranceGerman GermanLuxemb
Luxemb
NetherNetherSwitze Switze
Barbad
Barbad
Cuba Cuba
DominiDomini
Guadel
Guadel
Haiti Haiti
Jamaic Jamaic
Martin
MartinPuertoPuertoTrinid
Trinid
Costa Costa
El SalEl Sal
GuatemGuatem
Hondur
HondurMexicoMexico
Nicara
NicaraPanama PanamaArgent Argent
BoliviBolivi
Brazil BrazilChileChile
Colomb ColombEcuado
Ecuado
Guyana Guyana
Paragu Paragu
PeruPeru
Surina
Surina
UruguaUrugua
VenezuVenezu
Canada CanadaAustra AustraNew Ze New Ze
FijiFiji
Papua Papua
1950 1960 1970 1980 1990 2000−
0.3
−0.
10.
10.
20.
3
Functional scores at eigenfunction 2
Year
Sco
res
at e
igen
func
tion
2
Burund BurundEritre
Eritre
EthiopEthiop
Kenya KenyaMadagaMadagaMalawiMalawi
Maurit
MauritMozamb Mozamb
Reunio
ReunioRwanda RwandaSomali
SomaliSouth
South
Uganda UgandaUnited
UnitedZambia ZambiaZimbab
ZimbabAngola
AngolaCamero
CameroCentra CentraChad ChadCongo CongoDemocr
DemocrEquato EquatoGabon
GabonAlgeri
Algeri
Egypt
Egypt
Libya
LibyaMorocc
MoroccSudan SudanTunisi
TunisiBotswa
BotswaLesothLesoth
Namibi
Namibi
Benin
BeninBurkin Burkin
Cape V
Cape V
Cote d
Cote dGhana Ghana
GuineaGuineaLiberi
Liberi
MaliMali
Niger
NigerNigeriNigeri
SenegaSenega
Sierra
Sierra
TogoTogo
ChinaChinaChina,
China,Dem Pe
Dem Pe
Japan
Japan
Mongol MongolRepubl RepublOther
Other Kazakh
Kazakh
KyrgyzKyrgyz
Tajiki
TajikiTurkme TurkmeUzbeki
Uzbeki
Afghan
Afghan
BanglaBangla
India
India
Iran (
Iran (Nepal Nepal
Pakist
PakistSri La Sri LaCambod
CambodIndone
Indone
Lao Pe
Lao PeMalays
Malays
Myanma
MyanmaPhilipPhilipSingap
SingapThaila ThailaTimor−Timor−Viet NViet NArmeni ArmeniAzerba
AzerbaCyprus CyprusGeorgi
Georgi
Iraq
IraqIsrael
IsraelJordan
JordanLebanoLebanoOman Oman
Saudi
Saudi State State
SyrianSyrian
Turkey
Turkey
Yemen
YemenBelaru
BelaruBulgar BulgarCzech
Czech Hungar
HungarPoland
Poland
Romani
RomaniRussia RussiaSlovak SlovakUkrain Ukrain
Channe
ChanneDenmar
DenmarEstoni Estoni
Finlan
Finlan
Icelan
IcelanIrelan
IrelanLatviaLatvia
Lithua
LithuaNorway
Norway
Sweden
SwedenAlbani
AlbaniBosnia
BosniaCroati
CroatiGreece
Greece
Italy
Italy
Malta
MaltaMonten
MontenPortug
PortugSerbia SerbiaSloven
SlovenSpain
SpainTFYR M
TFYR MAustri
Austri
Belgiu
Belgiu
France
France
German
GermanLuxemb
LuxembNether
Nether
Switze
Switze
Barbad
BarbadCubaCubaDomini Domini
Guadel
Guadel
Haiti
HaitiJamaic
Jamaic
Martin
MartinPuertoPuerto
Trinid TrinidCosta Costa El Sal
El SalGuatem GuatemHondur HondurMexico Mexico
NicaraNicaraPanama
Panama
ArgentArgent
BoliviBolivi
Brazil BrazilChile
ChileColomb ColombEcuadoEcuado
Guyana
Guyana
Paragu
ParaguPeru Peru
Surina
SurinaUrugua
UruguaVenezu
Venezu
Canada
Canada
Austra
Austra
New Ze
New Ze
Fiji
Fiji
Papua
Papua
1950 1960 1970 1980 1990 2000
−0.
4−
0.2
0.0
0.2
0.4
Functional scores at eigenfunction 3
Year
Sco
res
at e
igen
func
tion
3
Burund Burund
Eritre Eritre
EthiopEthiopKenya Kenya
Madaga MadagaMalawi MalawiMaurit
MauritMozamb Mozamb
Reunio Reunio
RwandaRwanda
Somali SomaliSouth
South
Uganda UgandaUnitedUnited
Zambia ZambiaZimbab ZimbabAngola AngolaCamero CameroCentra CentraChad ChadCongo
CongoDemocr DemocrEquato EquatoGabon GabonAlgeri AlgeriEgypt
EgyptLibya LibyaMorocc MoroccSudan SudanTunisi
TunisiBotswa
BotswaLesoth Lesoth
NamibiNamibi
Benin BeninBurkin
BurkinCape V Cape VCote d
Cote dGhana GhanaGuinea Guinea
Liberi
Liberi
Mali
Mali
NigerNiger
Nigeri NigeriSenegaSenegaSierra
SierraTogo Togo
China
China
China,
China,
Dem Pe
Dem Pe
Japan
Japan
MongolMongol
RepublRepubl
Other Other
KazakhKazakh
KyrgyzKyrgyz
Tajiki
TajikiTurkme
TurkmeUzbeki
UzbekiAfghan Afghan
BanglaBangla
India IndiaIran (
Iran (
Nepal
Nepal
PakistPakistSri La Sri La
Cambod
CambodIndone Indone
Lao PeLao Pe
MalaysMalays
Myanma MyanmaPhilip
PhilipSingap
SingapThaila ThailaTimor−
Timor−Viet N
Viet NArmeni ArmeniAzerba
AzerbaCyprus CyprusGeorgi GeorgiIraq Iraq
Israel Israel
JordanJordan
Lebano LebanoOman OmanSaudi Saudi State State Syrian SyrianTurkey
Turkey
Yemen Yemen
Belaru
Belaru
BulgarBulgarCzech Czech Hungar
HungarPoland PolandRomani RomaniRussia
Russia
SlovakSlovak
Ukrain
Ukrain
ChanneChanne
Denmar DenmarEstoni EstoniFinlan
Finlan
Icelan
IcelanIrelan IrelanLatvia
Latvia
Lithua
Lithua
Norway NorwaySweden SwedenAlbaniAlbani
Bosnia BosniaCroati CroatiGreece
GreeceItaly ItalyMalta MaltaMonten MontenPortug PortugSerbia SerbiaSloven
SlovenSpainSpain
TFYR M TFYR MAustri AustriBelgiu BelgiuFranceFranceGerman GermanLuxemb LuxembNether NetherSwitzeSwitze
Barbad
Barbad
Cuba CubaDomini Domini
Guadel
GuadelHaiti Haiti
JamaicJamaic
Martin
MartinPuerto PuertoTrinid
Trinid
Costa Costa El Sal El SalGuatem GuatemHondur
HondurMexico MexicoNicara NicaraPanama
PanamaArgent ArgentBolivi BoliviBrazil BrazilChile ChileColomb ColombEcuado
Ecuado
Guyana GuyanaParagu ParaguPeru Peru
Surina
SurinaUrugua UruguaVenezu VenezuCanada CanadaAustra
AustraNew Ze New ZeFiji
FijiPapua
Papua
1950 1960 1970 1980 1990 2000
0.26
0.30
0.34
0.38
Eigenfunction 1 (FVE: 82.38%)
Year
Eig
enfu
nctio
n 1
(3rd
ste
p)
1950 1960 1970 1980 1990 2000
0.10
0.20
0.30
0.40
Eigenfunction 1 (FVE: 52.55%)
Year
Eig
enfu
nctio
n 1
(3rd
ste
p)
1950 1960 1970 1980 1990 2000
−0.
40−
0.30
Eigenfunction 1 (FVE: 57.54%)
Year
Eig
enfu
nctio
n 1
(3rd
ste
p)
1950 1960 1970 1980 1990 2000
−0.
40.
00.
20.
4
Eigenfunction 2 (FVE: 8.27%)
Year
Eig
enfu
nctio
n 2
(3rd
ste
p)
1950 1960 1970 1980 1990 2000
−0.
40.
00.
2
Eigenfunction 2 (FVE: 20.11%)
Year
Eig
enfu
nctio
n 2
(3rd
ste
p)
1950 1960 1970 1980 1990 2000
−0.
20.
00.
20.
4Eigenfunction 2 (FVE: 13.81%)
Year
Eig
enfu
nctio
n 2
(3rd
ste
p)
1950 1960 1970 1980 1990 2000
−0.
40.
00.
4
Eigenfunction 3 (FVE: 4.41%)
Year
Eig
enfu
nctio
n 3
(3rd
ste
p)
1950 1960 1970 1980 1990 2000
−0.
40.
00.
20.
4
Eigenfunction 3 (FVE: 11.09%)
Year
Eig
enfu
nctio
n 3
(3rd
ste
p)
1950 1960 1970 1980 1990 2000
−0.
40.
00.
4
Eigenfunction 3 (FVE: 9.7%)
Year
Eig
enfu
nctio
n 3
(3rd
ste
p)
Figure 19: Results of the marginal FPCA for the male log-mortality rate data. Columns 1,
2 and 3 correspond to eigenfunctions 1, 2 and 3 in the second step of the marginal FPCA,
respectively. First row: Estimated eigenfunctions ψj(s), where s is age. Second row: Score
functions ξi,j(t), where t is calendar year. Rows 3, 4 and 5: Estimated eigenfunctions
φjk(t), k = 1, 2, 3, in the third step.
56
Page 58
−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6
−0.
3−
0.2
−0.
10.
00.
10.
20.
3
Scores at eigenfunctions 2 vs. 1
Scores at eigenfunction 1
Sco
res
at e
igen
func
tion
2
1950.Burund1955.Burund1960.Burund
1965.Burund
1970.Burund
1975.Burund1980.Burund1985.Burund1990.Burund1995.Burund
2000.Burund2005.Burund2005.Burund
1950.Eritre
1955.Eritre
1960.Eritre
1965.Eritre1970.Eritre1975.Eritre
1980.Eritre1985.Eritre
1990.Eritre1995.Eritre
2000.Eritre
2005.Eritre2005.Eritre
1950.Ethiop1955.Ethiop1960.Ethiop1965.Ethiop1970.Ethiop
1975.Ethiop
1980.Ethiop
1985.Ethiop1990.Ethiop
1995.Ethiop
2000.Ethiop
2005.Ethiop2005.Ethiop
1950.Kenya1955.Kenya1960.Kenya1965.Kenya1970.Kenya
1975.Kenya1980.Kenya1985.Kenya1990.Kenya
1995.Kenya2000.Kenya2005.Kenya2005.Kenya1950.Madaga1955.Madaga
1960.Madaga1965.Madaga1970.Madaga
1975.Madaga1980.Madaga
1985.Madaga1990.Madaga1995.Madaga
2000.Madaga2005.Madaga2005.Madaga1950.Malawi
1955.Malawi1960.Malawi1965.Malawi
1970.Malawi
1975.Malawi1980.Malawi
1985.Malawi1990.Malawi
1995.Malawi2000.Malawi
2005.Malawi2005.Malawi
1950.Maurit
1955.Maurit
1960.Maurit1965.Maurit1970.Maurit
1975.Maurit
1980.Maurit
1985.Maurit1990.Maurit
1995.Maurit2000.Maurit2005.Maurit
1950.Maurit
1955.Maurit1960.Maurit
1965.Maurit
1970.Maurit
1975.Maurit
1980.Maurit1985.Maurit
1990.Maurit
1995.Maurit
2000.Maurit
2005.Maurit2005.Maurit
1950.Mozamb1955.Mozamb1960.Mozamb
1965.Mozamb1970.Mozamb1975.Mozamb
1980.Mozamb1985.Mozamb1990.Mozamb1995.Mozamb2000.Mozamb2005.Mozamb2005.Mozamb
1950.Reunio1955.Reunio
1960.Reunio
1965.Reunio1970.Reunio
1975.Reunio1980.Reunio
1985.Reunio1990.Reunio
1995.Reunio2000.Reunio2005.Reunio2005.Reunio
1950.Rwanda
1955.Rwanda
1960.Rwanda1965.Rwanda
1970.Rwanda1975.Rwanda1980.Rwanda1985.Rwanda
1990.Rwanda
1995.Rwanda
2000.Rwanda
2005.Rwanda2005.Rwanda 1950.Somali
1955.Somali1960.Somali
1965.Somali1970.Somali1975.Somali
1980.Somali1985.Somali1990.Somali1995.Somali
2000.Somali2005.Somali2005.Somali
1950.South 1955.South
1960.South 1965.South 1970.South 1975.South 1980.South
1985.South 1990.South 1995.South 2000.South 2005.South
1950.South 1955.South 1960.South
1965.South 1970.South 1975.South 1980.South 1985.South 1990.South
1995.South 2000.South 2005.South 2005.South
1950.Uganda1955.Uganda1960.Uganda1965.Uganda1970.Uganda1975.Uganda1980.Uganda1985.Uganda
1990.Uganda1995.Uganda
2000.Uganda2005.Uganda2005.Uganda
1950.United
1955.United1960.United
1965.United1970.United1975.United1980.United1985.United
1990.United1995.United
2000.United
2005.United
1950.United1955.United1960.United
1965.United1970.United1975.United1980.United1985.United
1990.United1995.United
2000.United2005.United
1950.United1955.United1960.United
1965.United1970.United1975.United1980.United1985.United
1990.United1995.United2000.United2005.United2005.United 1950.Zambia1955.Zambia1960.Zambia
1965.Zambia
1970.Zambia1975.Zambia1980.Zambia
1985.Zambia
1990.Zambia
1995.Zambia
2000.Zambia
2005.Zambia2005.Zambia
1950.Zimbab1955.Zimbab1960.Zimbab1965.Zimbab
1970.Zimbab1975.Zimbab1980.Zimbab1985.Zimbab
1990.Zimbab
1995.Zimbab2000.Zimbab
2005.Zimbab2005.Zimbab
1950.Angola1955.Angola
1960.Angola1965.Angola
1970.Angola1975.Angola
1980.Angola1985.Angola1990.Angola1995.Angola2000.Angola
2005.Angola2005.Angola1950.Camero
1955.Camero1960.Camero1965.Camero
1970.Camero1975.Camero1980.Camero1985.Camero1990.Camero1995.Camero2000.Camero2005.Camero2005.Camero1950.Centra
1955.Centra
1960.Centra1965.Centra1970.Centra
1975.Centra1980.Centra1985.Centra1990.Centra1995.Centra2000.Centra
2005.Centra2005.Centra1950.Chad1955.Chad1960.Chad1965.Chad1970.Chad1975.Chad1980.Chad
1985.Chad1990.Chad1995.Chad
2000.Chad2005.Chad2005.Chad
1950.Congo1955.Congo
1960.Congo1965.Congo
1970.Congo
1975.Congo1980.Congo
1985.Congo1990.Congo
1995.Congo
2000.Congo2005.Congo2005.Congo
1950.Democr1955.Democr1960.Democr
1965.Democr1970.Democr1975.Democr1980.Democr1985.Democr1990.Democr1995.Democr2000.Democr2005.Democr2005.Democr
1950.Equato1955.Equato1960.Equato
1965.Equato
1970.Equato
1975.Equato
1980.Equato
1985.Equato1990.Equato1995.Equato
2000.Equato
2005.Equato2005.Equato1950.Gabon
1955.Gabon
1960.Gabon1965.Gabon
1970.Gabon1975.Gabon
1980.Gabon1985.Gabon1990.Gabon
1995.Gabon2000.Gabon
2005.Gabon2005.Gabon1950.Algeri1955.Algeri1960.Algeri1965.Algeri1970.Algeri1975.Algeri
1980.Algeri1985.Algeri1990.Algeri1995.Algeri
2000.Algeri2005.Algeri2005.Algeri
1950.Egypt1955.Egypt1960.Egypt1965.Egypt1970.Egypt1975.Egypt1980.Egypt
1985.Egypt1990.Egypt1995.Egypt2000.Egypt2005.Egypt2005.Egypt
1950.Libya1955.Libya
1960.Libya1965.Libya1970.Libya
1975.Libya1980.Libya
1985.Libya1990.Libya1995.Libya2000.Libya2005.Libya2005.Libya
1950.Morocc
1955.Morocc
1960.Morocc1965.Morocc
1970.Morocc1975.Morocc1980.Morocc1985.Morocc
1990.Morocc1995.Morocc2000.Morocc2005.Morocc2005.Morocc
1950.Sudan1955.Sudan1960.Sudan1965.Sudan1970.Sudan
1975.Sudan1980.Sudan1985.Sudan1990.Sudan1995.Sudan2000.Sudan2005.Sudan2005.Sudan
1950.Tunisi
1955.Tunisi1960.Tunisi
1965.Tunisi
1970.Tunisi1975.Tunisi1980.Tunisi1985.Tunisi
1990.Tunisi
1995.Tunisi2000.Tunisi2005.Tunisi2005.Tunisi 1950.Botswa
1955.Botswa1960.Botswa1965.Botswa1970.Botswa1975.Botswa
1980.Botswa1985.Botswa
1990.Botswa
1995.Botswa2000.Botswa
2005.Botswa2005.Botswa1950.Lesoth1955.Lesoth1960.Lesoth
1965.Lesoth
1970.Lesoth
1975.Lesoth
1980.Lesoth
1985.Lesoth1990.Lesoth
1995.Lesoth2000.Lesoth
2005.Lesoth2005.Lesoth
1950.Namibi1955.Namibi1960.Namibi1965.Namibi1970.Namibi1975.Namibi
1980.Namibi
1985.Namibi1990.Namibi
1995.Namibi2000.Namibi
2005.Namibi2005.Namibi
1950.Benin1955.Benin
1960.Benin1965.Benin1970.Benin1975.Benin1980.Benin1985.Benin1990.Benin1995.Benin
2000.Benin2005.Benin2005.Benin
1950.Burkin1955.Burkin1960.Burkin1965.Burkin
1970.Burkin1975.Burkin
1980.Burkin1985.Burkin1990.Burkin1995.Burkin2000.Burkin2005.Burkin2005.Burkin
1950.Cape V
1955.Cape V
1960.Cape V1965.Cape V
1970.Cape V
1975.Cape V
1980.Cape V
1985.Cape V1990.Cape V1995.Cape V2000.Cape V2005.Cape V2005.Cape V
1950.Cote d1955.Cote d1960.Cote d
1965.Cote d1970.Cote d
1975.Cote d1980.Cote d1985.Cote d1990.Cote d1995.Cote d2000.Cote d2005.Cote d2005.Cote d
1950.Ghana
1955.Ghana1960.Ghana1965.Ghana1970.Ghana1975.Ghana1980.Ghana1985.Ghana
1990.Ghana1995.Ghana
2000.Ghana2005.Ghana2005.Ghana
1950.Guinea1955.Guinea1960.Guinea1965.Guinea1970.Guinea
1975.Guinea1980.Guinea
1985.Guinea1990.Guinea1995.Guinea
2000.Guinea2005.Guinea
1950.Guinea
1955.Guinea1960.Guinea1965.Guinea
1970.Guinea1975.Guinea
1980.Guinea
1985.Guinea
1990.Guinea1995.Guinea2000.Guinea2005.Guinea2005.Guinea 1950.Liberi
1955.Liberi1960.Liberi
1965.Liberi1970.Liberi
1975.Liberi
1980.Liberi
1985.Liberi1990.Liberi
1995.Liberi
2000.Liberi
2005.Liberi2005.Liberi
1950.Mali1955.Mali1960.Mali1965.Mali1970.Mali
1975.Mali1980.Mali1985.Mali1990.Mali1995.Mali2000.Mali2005.Mali2005.Mali
1950.Niger
1955.Niger
1960.Niger
1965.Niger1970.Niger
1975.Niger1980.Niger
1985.Niger1990.Niger
1995.Niger2000.Niger2005.Niger2005.Niger1950.Nigeri1955.Nigeri1960.Nigeri
1965.Nigeri1970.Nigeri1975.Nigeri
1980.Nigeri1985.Nigeri1990.Nigeri1995.Nigeri2000.Nigeri2005.Nigeri2005.Nigeri
1950.Senega1955.Senega1960.Senega1965.Senega1970.Senega1975.Senega
1980.Senega1985.Senega1990.Senega
1995.Senega2000.Senega
2005.Senega2005.Senega
1950.Sierra1955.Sierra1960.Sierra1965.Sierra
1970.Sierra
1975.Sierra1980.Sierra1985.Sierra
1990.Sierra1995.Sierra2000.Sierra
2005.Sierra2005.Sierra
1950.Togo1955.Togo1960.Togo
1965.Togo
1970.Togo
1975.Togo
1980.Togo1985.Togo1990.Togo
1995.Togo2000.Togo2005.Togo2005.Togo
1950.China1955.China
1960.China
1965.China
1970.China1975.China1980.China1985.China1990.China1995.China2000.China2005.China2005.China
1950.China,
1955.China,1960.China,1965.China,1970.China,
1975.China,
1980.China,
1985.China,1990.China,1995.China,
2000.China,2005.China,2005.China,
1950.Dem Pe
1955.Dem Pe
1960.Dem Pe1965.Dem Pe1970.Dem Pe1975.Dem Pe1980.Dem Pe1985.Dem Pe
1990.Dem Pe1995.Dem Pe
2000.Dem Pe
2005.Dem Pe2005.Dem Pe
1950.Japan1955.Japan1960.Japan1965.Japan
1970.Japan1975.Japan
1980.Japan1985.Japan1990.Japan1995.Japan2000.Japan2005.Japan2005.Japan
1950.Mongol1955.Mongol
1960.Mongol
1965.Mongol
1970.Mongol1975.Mongol1980.Mongol1985.Mongol
1990.Mongol1995.Mongol
2000.Mongol
2005.Mongol2005.Mongol1950.Republ1955.Republ
1960.Republ
1965.Republ
1970.Republ1975.Republ1980.Republ
1985.Republ1990.Republ
1995.Republ2000.Republ2005.Republ
1950.Republ1955.Republ1960.Republ
1965.Republ1970.Republ1975.Republ
1980.Republ1985.Republ1990.Republ
1995.Republ2000.Republ2005.Republ2005.Republ
1950.Other
1955.Other 1960.Other 1965.Other 1970.Other 1975.Other 1980.Other 1985.Other
1990.Other 1995.Other 2000.Other 2005.Other 2005.Other
1950.Kazakh1955.Kazakh1960.Kazakh
1965.Kazakh1970.Kazakh1975.Kazakh1980.Kazakh1985.Kazakh
1990.Kazakh
1995.Kazakh
2000.Kazakh
2005.Kazakh2005.Kazakh
1950.Kyrgyz1955.Kyrgyz1960.Kyrgyz
1965.Kyrgyz1970.Kyrgyz1975.Kyrgyz1980.Kyrgyz
1985.Kyrgyz1990.Kyrgyz
1995.Kyrgyz2000.Kyrgyz
2005.Kyrgyz2005.Kyrgyz
1950.Tajiki
1955.Tajiki1960.Tajiki
1965.Tajiki1970.Tajiki1975.Tajiki
1980.Tajiki
1985.Tajiki
1990.Tajiki
1995.Tajiki2000.Tajiki
2005.Tajiki2005.Tajiki1950.Turkme
1955.Turkme
1960.Turkme1965.Turkme1970.Turkme1975.Turkme
1980.Turkme
1985.Turkme1990.Turkme1995.Turkme
2000.Turkme
2005.Turkme2005.Turkme
1950.Uzbeki1955.Uzbeki1960.Uzbeki
1965.Uzbeki1970.Uzbeki1975.Uzbeki1980.Uzbeki1985.Uzbeki
1990.Uzbeki
1995.Uzbeki2000.Uzbeki
2005.Uzbeki2005.Uzbeki
1950.Afghan1955.Afghan1960.Afghan1965.Afghan1970.Afghan1975.Afghan1980.Afghan
1985.Afghan
1990.Afghan
1995.Afghan2000.Afghan2005.Afghan2005.Afghan
1950.Bangla1955.Bangla1960.Bangla1965.Bangla
1970.Bangla
1975.Bangla1980.Bangla1985.Bangla1990.Bangla1995.Bangla
2000.Bangla2005.Bangla2005.Bangla
1950.India1955.India
1960.India1965.India
1970.India1975.India1980.India1985.India1990.India1995.India2000.India2005.India2005.India
1950.Iran (1955.Iran (1960.Iran (1965.Iran (
1970.Iran (
1975.Iran (
1980.Iran (
1985.Iran (
1990.Iran (1995.Iran (2000.Iran (2005.Iran (2005.Iran (
1950.Nepal
1955.Nepal
1960.Nepal1965.Nepal1970.Nepal1975.Nepal1980.Nepal
1985.Nepal1990.Nepal1995.Nepal
2000.Nepal2005.Nepal2005.Nepal
1950.Pakist1955.Pakist
1960.Pakist1965.Pakist1970.Pakist1975.Pakist1980.Pakist1985.Pakist1990.Pakist1995.Pakist
2000.Pakist2005.Pakist2005.Pakist1950.Sri La
1955.Sri La
1960.Sri La1965.Sri La
1970.Sri La1975.Sri La1980.Sri La1985.Sri La1990.Sri La1995.Sri La2000.Sri La
2005.Sri La2005.Sri La
1950.Cambod
1955.Cambod1960.Cambod
1965.Cambod1970.Cambod
1975.Cambod
1980.Cambod1985.Cambod1990.Cambod
1995.Cambod2000.Cambod2005.Cambod2005.Cambod 1950.Indone1955.Indone
1960.Indone1965.Indone1970.Indone1975.Indone1980.Indone1985.Indone1990.Indone1995.Indone2000.Indone
2005.Indone2005.Indone
1950.Lao Pe1955.Lao Pe
1960.Lao Pe
1965.Lao Pe1970.Lao Pe
1975.Lao Pe1980.Lao Pe1985.Lao Pe1990.Lao Pe
1995.Lao Pe2000.Lao Pe2005.Lao Pe2005.Lao Pe
1950.Malays1955.Malays1960.Malays1965.Malays1970.Malays1975.Malays
1980.Malays1985.Malays1990.Malays1995.Malays
2000.Malays2005.Malays2005.Malays
1950.Myanma
1955.Myanma1960.Myanma1965.Myanma1970.Myanma1975.Myanma1980.Myanma1985.Myanma1990.Myanma1995.Myanma
2000.Myanma2005.Myanma2005.Myanma1950.Philip1955.Philip1960.Philip1965.Philip1970.Philip1975.Philip1980.Philip1985.Philip1990.Philip1995.Philip
2000.Philip2005.Philip2005.Philip
1950.Singap
1955.Singap
1960.Singap1965.Singap
1970.Singap
1975.Singap
1980.Singap
1985.Singap1990.Singap1995.Singap2000.Singap
2005.Singap2005.Singap1950.Thaila1955.Thaila1960.Thaila1965.Thaila1970.Thaila
1975.Thaila1980.Thaila1985.Thaila1990.Thaila1995.Thaila2000.Thaila2005.Thaila2005.Thaila
1950.Timor−1955.Timor−
1960.Timor−
1965.Timor−
1970.Timor−
1975.Timor−
1980.Timor−
1985.Timor−
1990.Timor−1995.Timor−
2000.Timor−
2005.Timor−2005.Timor−1950.Viet N1955.Viet N1960.Viet N1965.Viet N1970.Viet N
1975.Viet N1980.Viet N1985.Viet N1990.Viet N1995.Viet N2000.Viet N2005.Viet N2005.Viet N
1950.Armeni1955.Armeni1960.Armeni1965.Armeni
1970.Armeni1975.Armeni1980.Armeni
1985.Armeni
1990.Armeni
1995.Armeni2000.Armeni
2005.Armeni2005.Armeni1950.Azerba1955.Azerba1960.Azerba
1965.Azerba1970.Azerba1975.Azerba1980.Azerba
1985.Azerba1990.Azerba
1995.Azerba
2000.Azerba
2005.Azerba2005.Azerba
1950.Cyprus
1955.Cyprus1960.Cyprus
1965.Cyprus
1970.Cyprus1975.Cyprus1980.Cyprus1985.Cyprus
1990.Cyprus1995.Cyprus2000.Cyprus2005.Cyprus2005.Cyprus
1950.Georgi1955.Georgi1960.Georgi
1965.Georgi1970.Georgi1975.Georgi1980.Georgi1985.Georgi1990.Georgi
1995.Georgi2000.Georgi
2005.Georgi2005.Georgi
1950.Iraq
1955.Iraq1960.Iraq
1965.Iraq1970.Iraq1975.Iraq
1980.Iraq
1985.Iraq1990.Iraq1995.Iraq
2000.Iraq2005.Iraq2005.Iraq1950.Israel
1955.Israel1960.Israel1965.Israel1970.Israel
1975.Israel1980.Israel
1985.Israel1990.Israel1995.Israel2000.Israel2005.Israel2005.Israel
1950.Jordan
1955.Jordan1960.Jordan1965.Jordan
1970.Jordan1975.Jordan
1980.Jordan
1985.Jordan1990.Jordan
1995.Jordan2000.Jordan
2005.Jordan2005.Jordan1950.Lebano1955.Lebano1960.Lebano1965.Lebano1970.Lebano1975.Lebano1980.Lebano1985.Lebano
1990.Lebano1995.Lebano
2000.Lebano2005.Lebano2005.Lebano1950.Oman1955.Oman
1960.Oman1965.Oman
1970.Oman1975.Oman
1980.Oman
1985.Oman1990.Oman
1995.Oman
2000.Oman
2005.Oman2005.Oman
1950.Saudi 1955.Saudi 1960.Saudi 1965.Saudi 1970.Saudi 1975.Saudi
1980.Saudi
1985.Saudi
1990.Saudi
1995.Saudi
2000.Saudi 2005.Saudi 2005.Saudi 1950.State
1955.State 1960.State 1965.State
1970.State
1975.State 1980.State 1985.State
1990.State
1995.State 2000.State
2005.State 2005.State
1950.Syrian1955.Syrian1960.Syrian1965.Syrian1970.Syrian1975.Syrian1980.Syrian1985.Syrian1990.Syrian
1995.Syrian2000.Syrian2005.Syrian2005.Syrian
1950.Turkey1955.Turkey
1960.Turkey
1965.Turkey
1970.Turkey1975.Turkey1980.Turkey
1985.Turkey1990.Turkey
1995.Turkey2000.Turkey2005.Turkey2005.Turkey
1950.Yemen1955.Yemen
1960.Yemen1965.Yemen
1970.Yemen1975.Yemen
1980.Yemen
1985.Yemen
1990.Yemen
1995.Yemen2000.Yemen2005.Yemen2005.Yemen1950.Belaru
1955.Belaru1960.Belaru1965.Belaru1970.Belaru1975.Belaru1980.Belaru
1985.Belaru
1990.Belaru
1995.Belaru2000.Belaru
2005.Belaru2005.Belaru
1950.Bulgar1955.Bulgar
1960.Bulgar1965.Bulgar1970.Bulgar1975.Bulgar1980.Bulgar1985.Bulgar1990.Bulgar
1995.Bulgar
2000.Bulgar2005.Bulgar2005.Bulgar
1950.Czech 1955.Czech 1960.Czech 1965.Czech 1970.Czech 1975.Czech 1980.Czech 1985.Czech 1990.Czech
1995.Czech
2000.Czech 2005.Czech 2005.Czech
1950.Hungar1955.Hungar1960.Hungar1965.Hungar1970.Hungar1975.Hungar1980.Hungar
1985.Hungar1990.Hungar
1995.Hungar
2000.Hungar2005.Hungar2005.Hungar
1950.Poland1955.Poland1960.Poland1965.Poland1970.Poland1975.Poland
1980.Poland1985.Poland1990.Poland
1995.Poland
2000.Poland2005.Poland2005.Poland
1950.Romani1955.Romani
1960.Romani1965.Romani
1970.Romani1975.Romani
1980.Romani1985.Romani1990.Romani
1995.Romani
2000.Romani2005.Romani2005.Romani1950.Russia
1955.Russia1960.Russia1965.Russia1970.Russia1975.Russia1980.Russia1985.Russia1990.Russia
1995.Russia2000.Russia
2005.Russia2005.Russia1950.Slovak1955.Slovak1960.Slovak1965.Slovak1970.Slovak
1975.Slovak1980.Slovak1985.Slovak1990.Slovak
1995.Slovak
2000.Slovak2005.Slovak2005.Slovak1950.Ukrain
1955.Ukrain1960.Ukrain1965.Ukrain1970.Ukrain
1975.Ukrain1980.Ukrain
1985.Ukrain1990.Ukrain
1995.Ukrain2000.Ukrain2005.Ukrain2005.Ukrain
1950.Channe1955.Channe1960.Channe
1965.Channe
1970.Channe1975.Channe1980.Channe
1985.Channe1990.Channe1995.Channe
2000.Channe2005.Channe2005.Channe
1950.Denmar1955.Denmar1960.Denmar1965.Denmar1970.Denmar1975.Denmar1980.Denmar
1985.Denmar1990.Denmar1995.Denmar2000.Denmar2005.Denmar2005.Denmar
1950.Estoni1955.Estoni
1960.Estoni
1965.Estoni1970.Estoni
1975.Estoni1980.Estoni1985.Estoni1990.Estoni1995.Estoni2000.Estoni2005.Estoni2005.Estoni
1950.Finlan1955.Finlan1960.Finlan1965.Finlan1970.Finlan1975.Finlan
1980.Finlan1985.Finlan1990.Finlan1995.Finlan
2000.Finlan2005.Finlan2005.Finlan
1950.Icelan1955.Icelan1960.Icelan
1965.Icelan1970.Icelan
1975.Icelan
1980.Icelan1985.Icelan1990.Icelan
1995.Icelan2000.Icelan2005.Icelan2005.Icelan
1950.Irelan1955.Irelan1960.Irelan1965.Irelan1970.Irelan1975.Irelan
1980.Irelan1985.Irelan1990.Irelan1995.Irelan
2000.Irelan2005.Irelan2005.Irelan1950.Latvia
1955.Latvia1960.Latvia1965.Latvia1970.Latvia
1975.Latvia
1980.Latvia1985.Latvia
1990.Latvia
1995.Latvia2000.Latvia2005.Latvia2005.Latvia
1950.Lithua1955.Lithua
1960.Lithua1965.Lithua1970.Lithua1975.Lithua
1980.Lithua1985.Lithua1990.Lithua
1995.Lithua2000.Lithua2005.Lithua2005.Lithua1950.Norway
1955.Norway1960.Norway1965.Norway
1970.Norway1975.Norway1980.Norway1985.Norway1990.Norway1995.Norway2000.Norway2005.Norway2005.Norway
1950.Sweden1955.Sweden1960.Sweden
1965.Sweden1970.Sweden1975.Sweden1980.Sweden1985.Sweden1990.Sweden1995.Sweden2000.Sweden2005.Sweden2005.Sweden1950.Albani1955.Albani1960.Albani1965.Albani
1970.Albani1975.Albani
1980.Albani1985.Albani1990.Albani
1995.Albani
2000.Albani2005.Albani2005.Albani
1950.Bosnia1955.Bosnia
1960.Bosnia1965.Bosnia1970.Bosnia
1975.Bosnia1980.Bosnia1985.Bosnia
1990.Bosnia
1995.Bosnia2000.Bosnia2005.Bosnia2005.Bosnia
1950.Croati1955.Croati1960.Croati
1965.Croati1970.Croati
1975.Croati1980.Croati1985.Croati
1990.Croati
1995.Croati2000.Croati
2005.Croati2005.Croati
1950.Greece1955.Greece
1960.Greece1965.Greece1970.Greece1975.Greece
1980.Greece1985.Greece
1990.Greece1995.Greece2000.Greece2005.Greece2005.Greece
1950.Italy1955.Italy1960.Italy1965.Italy1970.Italy
1975.Italy1980.Italy1985.Italy
1990.Italy1995.Italy
2000.Italy2005.Italy2005.Italy
1950.Malta
1955.Malta
1960.Malta1965.Malta1970.Malta1975.Malta
1980.Malta1985.Malta1990.Malta1995.Malta2000.Malta
2005.Malta2005.Malta1950.Monten
1955.Monten
1960.Monten1965.Monten1970.Monten
1975.Monten
1980.Monten1985.Monten1990.Monten
1995.Monten
2000.Monten
2005.Monten2005.Monten
1950.Portug1955.Portug1960.Portug1965.Portug1970.Portug1975.Portug
1980.Portug1985.Portug1990.Portug1995.Portug2000.Portug2005.Portug2005.Portug
1950.Serbia1955.Serbia1960.Serbia1965.Serbia1970.Serbia1975.Serbia1980.Serbia1985.Serbia
1990.Serbia
1995.Serbia2000.Serbia
2005.Serbia2005.Serbia
1950.Sloven1955.Sloven
1960.Sloven1965.Sloven1970.Sloven
1975.Sloven1980.Sloven1985.Sloven1990.Sloven
1995.Sloven
2000.Sloven2005.Sloven2005.Sloven
1950.Spain1955.Spain1960.Spain1965.Spain1970.Spain1975.Spain
1980.Spain1985.Spain1990.Spain1995.Spain2000.Spain2005.Spain2005.Spain
1950.TFYR M1955.TFYR M1960.TFYR M1965.TFYR M
1970.TFYR M1975.TFYR M1980.TFYR M1985.TFYR M
1990.TFYR M
1995.TFYR M
2000.TFYR M2005.TFYR M2005.TFYR M
1950.Austri1955.Austri1960.Austri1965.Austri1970.Austri
1975.Austri
1980.Austri1985.Austri1990.Austri
1995.Austri
2000.Austri2005.Austri2005.Austri
1950.Belgiu1955.Belgiu1960.Belgiu1965.Belgiu1970.Belgiu1975.Belgiu
1980.Belgiu1985.Belgiu1990.Belgiu1995.Belgiu
2000.Belgiu2005.Belgiu2005.Belgiu
1950.France1955.France1960.France
1965.France1970.France1975.France
1980.France1985.France1990.France1995.France
2000.France2005.France2005.France
1950.German1955.German1960.German1965.German1970.German1975.German
1980.German1985.German1990.German1995.German
2000.German2005.German2005.German
1950.Luxemb
1955.Luxemb1960.Luxemb
1965.Luxemb1970.Luxemb
1975.Luxemb
1980.Luxemb1985.Luxemb
1990.Luxemb1995.Luxemb
2000.Luxemb
2005.Luxemb2005.Luxemb
1950.Nether1955.Nether1960.Nether
1965.Nether1970.Nether1975.Nether1980.Nether1985.Nether1990.Nether1995.Nether2000.Nether2005.Nether2005.Nether
1950.Switze1955.Switze1960.Switze1965.Switze
1970.Switze1975.Switze1980.Switze1985.Switze1990.Switze1995.Switze2000.Switze2005.Switze2005.Switze
1950.Barbad1955.Barbad
1960.Barbad
1965.Barbad
1970.Barbad
1975.Barbad
1980.Barbad1985.Barbad1990.Barbad1995.Barbad
2000.Barbad2005.Barbad2005.Barbad1950.Cuba1955.Cuba1960.Cuba1965.Cuba1970.Cuba
1975.Cuba1980.Cuba1985.Cuba1990.Cuba
1995.Cuba2000.Cuba2005.Cuba2005.Cuba1950.Domini
1955.Domini1960.Domini1965.Domini
1970.Domini1975.Domini1980.Domini1985.Domini1990.Domini1995.Domini2000.Domini2005.Domini2005.Domini
1950.Guadel
1955.Guadel
1960.Guadel
1965.Guadel
1970.Guadel1975.Guadel
1980.Guadel
1985.Guadel
1990.Guadel1995.Guadel
2000.Guadel2005.Guadel2005.Guadel
1950.Haiti1955.Haiti
1960.Haiti1965.Haiti
1970.Haiti
1975.Haiti1980.Haiti
1985.Haiti1990.Haiti1995.Haiti2000.Haiti2005.Haiti2005.Haiti
1950.Jamaic
1955.Jamaic
1960.Jamaic
1965.Jamaic1970.Jamaic
1975.Jamaic
1980.Jamaic1985.Jamaic1990.Jamaic1995.Jamaic2000.Jamaic
2005.Jamaic2005.Jamaic
1950.Martin
1955.Martin1960.Martin
1965.Martin
1970.Martin1975.Martin
1980.Martin1985.Martin
1990.Martin1995.Martin2000.Martin2005.Martin2005.Martin1950.Puerto
1955.Puerto1960.Puerto
1965.Puerto
1970.Puerto
1975.Puerto1980.Puerto1985.Puerto1990.Puerto1995.Puerto2000.Puerto2005.Puerto2005.Puerto
1950.Trinid1955.Trinid
1960.Trinid
1965.Trinid1970.Trinid
1975.Trinid1980.Trinid1985.Trinid
1990.Trinid1995.Trinid2000.Trinid2005.Trinid2005.Trinid
1950.Costa 1955.Costa
1960.Costa
1965.Costa
1970.Costa 1975.Costa 1980.Costa
1985.Costa 1990.Costa
1995.Costa 2000.Costa 2005.Costa 2005.Costa
1950.El Sal
1955.El Sal1960.El Sal1965.El Sal1970.El Sal1975.El Sal1980.El Sal
1985.El Sal1990.El Sal1995.El Sal2000.El Sal2005.El Sal2005.El Sal
1950.Guatem1955.Guatem1960.Guatem
1965.Guatem1970.Guatem1975.Guatem1980.Guatem
1985.Guatem1990.Guatem
1995.Guatem2000.Guatem2005.Guatem2005.Guatem1950.Hondur
1955.Hondur
1960.Hondur
1965.Hondur1970.Hondur
1975.Hondur
1980.Hondur1985.Hondur1990.Hondur
1995.Hondur2000.Hondur2005.Hondur2005.Hondur1950.Mexico
1955.Mexico1960.Mexico1965.Mexico1970.Mexico1975.Mexico1980.Mexico1985.Mexico1990.Mexico1995.Mexico2000.Mexico2005.Mexico2005.Mexico
1950.Nicara
1955.Nicara1960.Nicara1965.Nicara1970.Nicara
1975.Nicara1980.Nicara1985.Nicara1990.Nicara1995.Nicara2000.Nicara2005.Nicara2005.Nicara1950.Panama1955.Panama
1960.Panama1965.Panama
1970.Panama1975.Panama
1980.Panama1985.Panama
1990.Panama1995.Panama2000.Panama2005.Panama2005.Panama
1950.Argent1955.Argent1960.Argent
1965.Argent1970.Argent1975.Argent1980.Argent1985.Argent1990.Argent1995.Argent2000.Argent2005.Argent2005.Argent
1950.Bolivi1955.Bolivi1960.Bolivi
1965.Bolivi1970.Bolivi1975.Bolivi1980.Bolivi
1985.Bolivi1990.Bolivi1995.Bolivi2000.Bolivi2005.Bolivi2005.Bolivi
1950.Brazil1955.Brazil1960.Brazil1965.Brazil
1970.Brazil1975.Brazil
1980.Brazil1985.Brazil1990.Brazil1995.Brazil2000.Brazil2005.Brazil2005.Brazil
1950.Chile1955.Chile1960.Chile1965.Chile1970.Chile
1975.Chile1980.Chile1985.Chile1990.Chile
1995.Chile2000.Chile2005.Chile2005.Chile1950.Colomb
1955.Colomb1960.Colomb1965.Colomb1970.Colomb1975.Colomb1980.Colomb1985.Colomb1990.Colomb1995.Colomb2000.Colomb2005.Colomb2005.Colomb
1950.Ecuado1955.Ecuado1960.Ecuado
1965.Ecuado1970.Ecuado1975.Ecuado1980.Ecuado1985.Ecuado1990.Ecuado
1995.Ecuado2000.Ecuado2005.Ecuado2005.Ecuado
1950.Guyana1955.Guyana1960.Guyana
1965.Guyana1970.Guyana
1975.Guyana
1980.Guyana1985.Guyana
1990.Guyana1995.Guyana
2000.Guyana
2005.Guyana2005.Guyana
1950.Paragu
1955.Paragu1960.Paragu1965.Paragu
1970.Paragu1975.Paragu1980.Paragu
1985.Paragu1990.Paragu1995.Paragu2000.Paragu2005.Paragu2005.Paragu
1950.Peru1955.Peru1960.Peru
1965.Peru1970.Peru1975.Peru1980.Peru1985.Peru1990.Peru1995.Peru
2000.Peru2005.Peru2005.Peru
1950.Surina
1955.Surina
1960.Surina1965.Surina
1970.Surina
1975.Surina
1980.Surina1985.Surina
1990.Surina
1995.Surina
2000.Surina2005.Surina2005.Surina
1950.Urugua1955.Urugua1960.Urugua
1965.Urugua1970.Urugua1975.Urugua1980.Urugua1985.Urugua
1990.Urugua1995.Urugua2000.Urugua2005.Urugua2005.Urugua
1950.Venezu1955.Venezu1960.Venezu1965.Venezu
1970.Venezu1975.Venezu1980.Venezu1985.Venezu1990.Venezu
1995.Venezu2000.Venezu2005.Venezu2005.Venezu
1950.Canada1955.Canada1960.Canada1965.Canada1970.Canada1975.Canada1980.Canada1985.Canada
1990.Canada1995.Canada2000.Canada2005.Canada2005.Canada
1950.Austra1955.Austra1960.Austra
1965.Austra1970.Austra1975.Austra1980.Austra1985.Austra
1990.Austra1995.Austra
2000.Austra2005.Austra2005.Austra
1950.New Ze1955.New Ze1960.New Ze
1965.New Ze1970.New Ze1975.New Ze1980.New Ze1985.New Ze
1990.New Ze1995.New Ze2000.New Ze
2005.New Ze2005.New Ze
1950.Fiji
1955.Fiji
1960.Fiji
1965.Fiji
1970.Fiji1975.Fiji1980.Fiji
1985.Fiji1990.Fiji1995.Fiji2000.Fiji
2005.Fiji2005.Fiji
1950.Papua 1955.Papua
1960.Papua 1965.Papua 1970.Papua
1975.Papua
1980.Papua 1985.Papua 1990.Papua
1995.Papua
2000.Papua 2005.Papua 2005.Papua
Figure 20: Track-plot corresponding to the implicitly parametrized planar curves
{(ξi,1(t), ξi,2(t)) : t = 1950, 1955, . . . , 2005}, parametrized by calendar time t, where ξi,j(t)
is the j-th score function for country i.
57
Page 59
5 10 15 20
010
2030
4050
60
Eigenfunction
FV
E in
%
FVE by each eigenfunction
59.41
7.516.844.79
−1.0 −0.5 0.0 0.5 1.0 1.5
−0.
40.
00.
20.
40.
6
1st eigenfunction
2nd
eige
nfun
ctio
n
Scores at 2nd vs 1st eigenfunctions
Burund
Eritre
EthiopKenya
MadagaMalawi
Maurit
MozambReunio Rwanda
SomaliSouth UgandaUnitedZambiaZimbab AngolaCameroCentra
ChadCongo
Democr
Equato
Gabon
Algeri
EgyptLibyaMorocc
Sudan
Tunisi
Botswa
Lesoth
Namibi
South
Benin
Burkin
Cape V
Cote d
Ghana
GuineaGuinea
LiberiMali
Maurit
Niger
NigeriSenega
SierraTogo
ChinaChina,
Dem Pe
JapanMongol
RepublOther
Kazakh
Kyrgyz
Tajiki
Turkme
UzbekiAfghan
Bangla
India
Iran (
Nepal
Pakist
Sri La CambodIndoneLao PeMalays Myanma
Philip
Singap
Thaila
Timor−Viet N
ArmeniAzerbaCyprus
GeorgiIraqIsrael JordanLebano OmanSaudi State Syrian Turkey
Yemen
BelaruBulgar
Czech HungarPoland
Republ
Romani
RussiaSlovakUkrain
Channe
Denmar
EstoniFinlan
Icelan
IrelanLatvia
LithuaNorwaySweden
United
Albani
Bosnia
Croati
GreeceItaly
Malta
Monten
PortugSerbia
Sloven
SpainTFYR M
AustriBelgiuFrance
GermanLuxemb
NetherSwitze
BarbadCuba Domini
Guadel
HaitiJamaic
Martin
Puerto
Trinid
Costa
El SalGuatemHondurMexico
NicaraPanama
Argent
BoliviBrazil
ChileColomb
Ecuado
Guyana
ParaguPeru
Surina
UruguaVenezu
CanadaUnitedAustraNew Ze
Fiji
Papua
−1.0 −0.5 0.0 0.5 1.0 1.5
−0.
40.
00.
20.
40.
60.
8
1st eigenfunction
3rd
eige
nfun
ctio
n
Scores at 3rd vs 1st eigenfunctions
Burund
Eritre
EthiopKenya MadagaMalawi
MauritMozamb
Reunio
Rwanda
SomaliSouth UgandaUnited
Zambia
ZimbabAngola
CameroCentraChadCongo
Democr Equato
GabonAlgeriEgypt
LibyaMorocc
Sudan
TunisiBotswa
LesothNamibi South
Benin
Burkin
Cape VCote dGhana
GuineaGuineaLiberi
MaliMaurit
NigerNigeri
Senega
Sierra
Togo
ChinaChina,
Dem Pe
Japan
MongolRepubl
Other
Kazakh
KyrgyzTajikiTurkmeUzbeki
AfghanBanglaIndia
Iran (Nepal
PakistSri La
Cambod
IndoneLao PeMalays
Myanma
Philip
Singap
ThailaTimor−
Viet N
ArmeniAzerba
Cyprus GeorgiIraq
Israel
JordanLebano
Oman
Saudi State Syrian
TurkeyYemen
Belaru
BulgarCzech HungarPoland RepublRomani
Russia
Slovak
Ukrain
Channe
Denmar
Estoni
Finlan
Icelan
Irelan
LatviaLithua
NorwaySwedenUnited
AlbaniBosniaCroati
GreeceItaly
Malta
Monten
Portug
Serbia
Sloven
Spain
TFYR M
AustriBelgiu
FranceGermanLuxembNether
Switze
Barbad
Cuba Domini
Guadel
Haiti
Jamaic
Martin
Puerto
Trinid
Costa El SalGuatemHondurMexico
NicaraPanama
Argent
BoliviBrazil
ChileColombEcuado
GuyanaParagu
Peru
Surina
Urugua
VenezuCanadaUnited
AustraNew Ze
Fiji
Papua
−1.0 −0.5 0.0 0.5 1.0 1.5
−0.
6−
0.2
0.2
0.4
1st eigenfunction
4th
eige
nfun
ctio
n
Scores at 4th vs 1st eigenfunctions
Burund
Eritre
Ethiop
KenyaMadaga
MalawiMaurit Mozamb
Reunio
Rwanda
SomaliSouth
UgandaUnitedZambiaZimbab
AngolaCameroCentra
ChadCongo
Democr
EquatoGabonAlgeri
EgyptLibyaMoroccSudanTunisi
BotswaLesoth
NamibiSouth
BeninBurkin
Cape V
Cote dGhana
Guinea
GuineaLiberi
Mali
Maurit
Niger
Nigeri
Senega
SierraTogoChina
China,Dem PeJapan
Mongol
RepublOther
KazakhKyrgyzTajikiTurkmeUzbeki
AfghanBangla
India
Iran (
NepalPakistSri La
Cambod
Indone
Lao Pe
Malays MyanmaPhilip
Singap
Thaila
Timor−
Viet N
ArmeniAzerba
Cyprus
Georgi
IraqIsrael
JordanLebano OmanSaudi
State Syrian
Turkey Yemen
Belaru
Bulgar
Czech
HungarPoland
RepublRomani
Russia
Slovak
Ukrain
Channe
DenmarEstoni
Finlan
Icelan
Irelan
LatviaLithua
NorwaySwedenUnitedAlbaniBosnia
CroatiGreeceItaly
MaltaMonten
Portug
SerbiaSlovenSpain TFYR M
AustriBelgiuFranceGerman
Luxemb
NetherSwitze
Barbad
CubaDomini
Guadel
Haiti
Jamaic
Martin
Puerto
TrinidCosta El Sal
GuatemHondurMexico
NicaraPanamaArgent
Bolivi
BrazilChileColombEcuado
GuyanaParaguPeru
Surina
UruguaVenezuCanadaUnited
AustraNew ZeFiji
Papua
1950 1960 1970 1980 1990 2000
020
4060
Ppal. Funct. 1 (FVE: 59.41%)
Year
Age
1950 1960 1970 1980 1990 2000
020
4060
Ppal. Funct. 2 (FVE: 7.51%)
Year
Age
1950 1960 1970 1980 1990 2000
020
4060
Ppal. Funct. 3 (FVE: 6.84%)
Year
Age
1950 1960 1970 1980 1990 2000
020
4060
Ppal. Funct. 4 (FVE: 4.79%)
Year
Age
Figure 21: Standard FPCA of the male log-mortality data.
58
![References - Springer978-3-0348-8641-3/1.pdf · (1989) Haus Stapel, D-4409 Havixbeck, Germany. [12] Barchielli, A, Lupieri, G.: Quantum stochastic calculus, operation-valued stochastic](https://static.documents.pub/doc/80x56/5e0eb45e46561740f64d12b6/references-springer-978-3-0348-8641-31pdf-1989-haus-stapel-d-4409-havixbeck.jpg)
