302 Handbook of Discrete-Valued Time Series
conditional on S
1
, ..., S
N
. In addition,
λ(ω, s) is conditionally asymptotically normal with large
sample variance
λ
2
(ω, s)ρ
Var
λ(ω, s) | S
1
, ..., S
N
≈
NT|H
g
|φ
s
(s)
,
where
ρ =
(
μ
14
− μ
12
)
−2
2
×
μ
20
μ
14
+ μ
2
− 2(q + 1)μ
12
μ
22
μ
14
+ qμ
2
+ (q + 1)μ
2
μ
24
+ qμ
2
12 12 12 22
u
m
q+1
(u)du.
for μ
m
=
u∈R×S
1
K
Equation (13.14) can be used to obtain a consistent estimate of Dλ(ω, s) from the estimates
in Theorem 13.1. Dene
Dλ(ω, s) as the q-vector with kth element
D
k
λ(ω, s) where
V
1/2
D
k
g
re
(ω, s)
V
1/2
D
k
λ(ω, s) =
γ(ω, s)
(s)
(s)
γ(ω, s).
The consistency of
Dλ(ω, s) and the large sample distribution of its elements are established
in the following theorem.
Theorem 13.3 If Assumptions A(i) and A(ii) hold, h
v
∼
(
NT
)
−1/(q+6)
,h
g
∼
(
NT
)
−1/(q+7)
and
N ∼ T
(q/6)−
for some ∈ (0, q/6) as N, T →∞, then for ω ∈[0, 1/2] and s ∈ S
D
λ(ω, s) = Dλ(ω, s) + O
p
(NT)
−2/(q+7)
conditional on S
1
, ..., S
N
. In addition,
D
k
λ(ω, s) is conditionally asymptotically normal with large
sample variance
e
k
H
g
−2
e
k
μ
22
Var
D
k
λ(ω, s) | S
1
, ..., S
N
≈
NT|H
g
|φ
s
(s)μ
2
(ω, S)
12
where e
k
is the p-dimensional vector of zeros except for a one in k-th element, μ
m
=
u∈R×S
u
m
1
K
q
+1
(u)du, and (ω, s) is uniquely determined by g(ω, s).
The explicit form for (ω, s) is given in Krafty et al. (2012, Appendix). Note that (ω, s)
depends on both g
re
(ω, s) and g
im
(ω, s). Subsequently, large sample condence intervals
for D
k
λ(ω, s) require consistent estimates of both g
re
(ω, s) and g
im
(ω, s). An estimate for
g
re
(ω, s) has been obtained through
g
re
(ω, s). The local quadratic smoother applied to the
real part of the periodograms can be applied to the imaginary part to obtain an estimate of
g
im
(ω, s). The ensuing pointwise condence intervals can be used to assess if, for a given
303 Spectral Analysis of Qualitative Time Series
frequency, the rst derivatives in the direction of the covariates are different from zero and
that the enveloping spectral surface at that frequency depends on the covariate vector.
As an example of this extension to the spectral envelope, we consider data from the
AgeWise study conducted at the University of Pittsburgh. The goal of the AgeWise study
is to understand the connections between sleep and health, functioning, and well-being in
older adults. In the study, data are collected from N = 98 adults between 60 and 89 years
of age.
Each subject was monitored during a night of in-home sleep through ambulatory
poly-somnography. Polysomnography is a comprehensive recording of the biophysiolog-
ical changes that occur during sleep and includes the collection of electrocardiograph
(), electroencephalographic (), electro-oculography (), and electromyography
() signals. The polysomnographic signals were used by a trained technician to score
20 seconds (s) epochs into stages of , on-, or wakefulness. Our analysis considers
6 h immediately following the onset of sleep. The resulting qualitative time series for each
subject consists of sleep stage as one of the p = 3 categories of , on-, and wake-
fulness for each of the T = 1080 20 s intervals during the rst 6 h after sleep onset. Sleep
stages for two subjects are plotted in Figure 13.6.
In addition to polysomnography, each patient completed the questionnaire from
which a score was computed. The , which was introduced by Buysse et al. (1989),
is a common instrument for measuring self-reported sleep quality. The questionnaire
collects information about sleep quality and disturbance over a 1-month period. scores
can take values between 0 and 21 with larger numbers representing poorer sleep quality. A
score of 6 or larger has been shown to be clinically associated with poor sleep quality in a
variety of settings. The observed scores in our sample of N = 98 subjects range from
1 to 16 with a mean of 8.88 and a standard deviation of 3.64. The goal of our investigation is
to determine the association between the frequency domain properties of sleep stages and
the single, that is, q = 1, quantitative variable of the score.
We choose on- sleep as the referent group so that sleep stages for each subject are
represented through indicator variables for wakefulness and for sleep. Local quadratic
0
Non-REM
REM
Awake
Non-REM
REM
Awake
2 4 6 0 2 4 6
Hours Hours
FIGURE 13.6
Sleep stages from two subjects for 6 h following sleep onset. The plot on the left is from a subject with a score
of 1 while the plot on the right is from a subject with a score of 15.
304 Handbook of Discrete-Valued Time Series
λ(ω, PSQI)
16
14
12
10
PSQI
8
6
4
2
0
1
2
3
4
5
6
0
0.5 1 1.5
2 2.5 3
Cycles per hour
FIGURE 13.7
The estimated enveloping spectral surface for sleep staging conditional on score.
regression is implemented for the estimation of
V using the univariate Epanechnikov kernel
and for the estimation of g
re
using the spherical bivariate Epanechnikov kernel. The
scores in our sample are approximately unimodal and symmetric; we estimate φ
s
as the
truncated normal distribution with support between 1 and 16, mean 8.88, and standard
deviation 3.64.
The estimated enveloping spectral surface
λ(ω, ) displayed in Figure 13.7 indicates
that power exists primarily within two frequency bands: a band of low frequencies between
0.17 and 0.23 cycles per hour, or between one cycle every 4 h and 20 min and one cycle every
6 h, and a band of higher frequencies between 0.33 and 0.75 cycles per hour, or between
one cycle every 1 h and 20 min and one cycle every 3 h. The estimated scalings
γ where
the power in the enveloping spectral surface is achieved are approximately equivalent to
an indicator variable for wakefulness for all frequencies in the low-frequency band and
for all and approximately equivalent to an indicator variable for sleep for all fre-
quencies in the higher-frequency band and for all . For example, the estimated optimal
scaling at ω = 0.2 cycles per hour, or 1 cycle every 5 h, and = 2 assigns the value 0.42
for awake, −0.09 for REM, and 0 for on-. The estimated optimal scaling at ω = 0.5
cycles per hour, or 1 cycle every 2 h, and = 6 assigns the value 0.01 for awake, 0.37
for , and 0 for on-. To illustrate the scalings within these two bands, Figure 13.6
displays the estimated conditional spectra for wakefulness, or with the scaling that assigns
1 to wakefulness and 0 to both and on- sleep, and for , or with the scaling that
assigns 1 to , sleep and 0 to both wakefulness and on- sleep. The low-frequency
band indicates that for all subjects, regardless of score, 6 h following sleep onset con-
sists primarily of time segments in which the subject is mostly asleep with the presence of
epochs of up to 1 h and 40 min of mostly wakefulness. The higher-frequency band indi-
cates that most subjects experience sleep approximately every 2 h. It should be noted
that the subjects in our study are older adults and are subsequently expected to have more
disturbed sleep than younger adults and children. The presented inference is specic to
older adults and is not generalizable to other populations.
To investigate if the power of the enveloping spectral surface within the frequency bands
for wakefulness and for sleep changes with score, we compute the estimated
305 Spectral Analysis of Qualitative Time Series
Derivative at Derivative at
1 cycle every 5 h 1 cycle every 2 h
0.05
10
PSQI
Dλ(ω = 0.5, PSQI)
0.05
0
0
15
D
λ(ω = 0.2, PSQI)
–0.05 –0.05
–0.1
–0.1
–0.15
–0.15
–0.2
–0.2
5 15
510
PSQI
FIGURE 13.8
The estimated partial derivatives and pointwise 95% condence intervals of the enveloping spectral surface for
sleep staging with respect to the score for frequencies of one cycle every 5 h and one cycle every 2 h.
partial derivative of the enveloping spectral surface with respect to score at frequen-
cies ω = 0.2 cycles per hour, or one cycle every 5 h, and at ω = 0.5, or one cycle every
2 h. Figure 13.8 displays Dλ(ω = 0.2, ) and Dλ(ω = 0.5, ) along with estimated
pointwise 95% condence intervals. The estimated derivative at one cycle every 5 h is neg-
ative for values less than 6 and not signicantly different from zero for larger
values. This indicates that, although signicant power exists at periods wakefulness every
5 h for all values, this power decreases with an increase in for subjects with a
less than 6 and is approximately the same for subjects with a greater than 6.
Recall that the enveloping spectral surface is dened as the maximal normalized condi-
tional power. From this result, we hypothesize that subjects experience approximately the
same frequency of long periods of wakefulness regardless of their score but that sub-
jects with a above 6 experience more short epochs of wakefulness. Consequently, the
normalized conditional power for wakefulness once every 5 h is larger in subjects with
smaller scores. The derivative of the enveloping spectral surface at the frequency of
one cycle every 2 h is not signicantly different from zero and we conclude that the amount
of conditional power of sleep at one cycle every 2 h does not depend on score.
13.7 Principal Components
It may have been noticed that the theory associated with the spectral envelope is seemingly
related to principal component analysis for time series. In this section, we summarize the
technique so that the connection between the concepts is evident.
For the case of principal component analysis for time series, suppose we have a zero
mean, p × 1, stationary vector process X
t
that has a p × p spectral density matrix given
by f
xx
(ω). Recall f
xx
(ω) is a complex-valued, nonnegative denite, Hermitian matrix.
306 Handbook of Discrete-Valued Time Series
Using the analogy of classical principal components, suppose for a xed value of ω,we
want to nd a complex-valued univariate process Y
t
(ω) = c(ω)
∗
X
t
, where c(ω) is com-
plex, such that the spectral density of Y
t
(ω) is maximized at frequency ω,and c(ω) is of
unit length, c(ω)
∗
c(ω) = 1. Because, at frequency ω, the spectral density of Y
t
(ω) is f
y
(ω) =
c(ω)
∗
f
xx
(ω)c(ω), the problem can be restated as, nd complex vector c(ω) such that
c(ω)
∗
f
xx
(ω)c(ω)
max
. (13.15)
c(ω)=0
c(ω)
∗
c(ω)
Let {(λ
1
(ω), e
1
(ω)) ,…, (λ
p
(ω), e
p
(ω))} denote the eigenvalue–eigenvector pairs of f
xx
(ω),
where λ
1
(ω) ≥ λ
2
(ω) ≥ ··· ≥ λ
p
(ω) ≥ 0, and the eigenvectors are of unit length. We
note that the eigenvalues of a Hermitian matrix are real. The solution to (13.15) is to choose
c(ω) = e
1
(ω), in which case the desired linear combination is Y
t
(ω) = e
1
(ω)
∗
X
t
. For this
choice,
c(ω)
∗
f
xx
(ω)c(ω)
e
1
(ω)
∗
f
xx
(ω)e
1
(ω)
max
= = λ
1
(ω). (13.16)
c(ω)=0
c(ω)
∗
c(ω) e
1
(ω)
∗
e
1
(ω)
This process may be repeated for any frequency ω, and the complex-valued process,
Y
t1
(ω) = e
1
(ω)
∗
X
t
, is called the rst principal component at frequency ω.The kth prin-
cipal component at frequency ω,for k = 1, 2, ... , p, is the complex-valued time series
Y
tk
(ω) = e
k
(ω)
∗
X
t
, in analogy to the classical case. In this case, the spectral density of
Y
tk
(ω) at frequency ω is f
y
k
(ω) = e
k
(ω)
∗
f
xx
(ω)e
k
(ω) = λ
k
(ω).
The previous development of spectral domain principal components is related to the
spectral envelope methodology as discussed in equation (13.4). In particular, the spectral
envelope is a principal component analysis on the real part of f
xx
(ω). Hence, the difference
between spectral domain principal component analysis and the spectral envelope is that,
for the spectral envelope, the c(ω) are restricted to be real. If, in the development of the
spectral envelope, we allowed for complex scalings, the two methods would be identical.
Another way to motivate the use of principal components in the frequency domain was
given in Brillinger (1981, Chapter 9). Although the technique appears to be different, it
leads to the same analysis. In this case, we suppose we have a stationary, p-dimensional,
vector-valued process X
t
and we are only able to keep a univariate process Y
t
such that,
when needed, we may reconstruct the vector-valued process, X
t
, according to an optimal-
ity criterion. Specically, we suppose we want to approximate a mean-zero, stationary,
vector-valued time series, X
t
, with spectral matrix f
xx
(ω), by a univariate process Y
t
dened by
∞
∗
Y
t
= c
t−j
X
j
, (13.17)
j=−∞
where {c
j
} is a p × 1 vector-valued lter, such that {c
j
} is absolutely summable; that is,
∞
j=−∞
|c
j
| < ∞. The approximation is accomplished so the reconstruction of X
t
from
y
t
,say,
∞
X
t
= b
t−j
Y
j
, (13.18)
j=−∞
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