297 Spectral Analysis of Qualitative Time Series
Epstein−Barr BNRF1
0.1
0.3
0.5
0.7
Spectral envelope (%)
0.0 0.1 0.2 0.3 0.4 0.5
Frequency
FIGURE 13.4
Smoothed sample spectral envelope of the BNRF1 gene from the Epstein–Barr virus.
Epstein−Barr BNRF1
0.0
0.1
0.2
Frequency
0.3
0.4
0.5
2000
3000 4000 5000
Base pair
FIGURE 13.5
Dynamic spectral envelope estimates for the BNRF1 gene (bp 1736–5689) of the Epstein–Barr virus (EBV). The
vertical dashed lines indicate the blocks, and values over the approximate 0.005 null signicance threshold are
indicated by darker regions.
evident from the gure that even within small segments of the gene, there is heterogeneity.
There is, however, a basic cyclic pattern that exists through most of the gene as evidenced
by the peak at ω = 1/3, except at the end of the gene. Table 13.3 shows the optimal scalings
at the one-third frequency and we note that the corresponding alphabets are somewhat
consistent in the “signicant" blocks, with each block in the beginning of the sequence
indicating a weak–strong bonding alphabet (A = T, C = G). This alphabet starts to break
down at block number ve (bp 3736–4235). As previously indicated, there is a substantial
difference in the nal 1000 bp (blocks 7 and 8) of the gene.
298 Handbook of Discrete-Valued Time Series
TABLE 13.3
1
Blockwise (500 bp) Optimal Scaling, β
3
, for the Epstein–Barr BNRF1
Gene Example
Block (bp) A C G T
1. 1736–2235 0.26 0.69 0.68 0
2. 2236–2735 0.23 0.71 0.67 0
3. 2736–3235 0.16 0.56 0.82 0
4. 3236–3735 0.15 0.61 0.78 0
5. 3736–4235 0.30 0.35 0.89 0
6. 4236–4735 0.22 0.61 0.76 0
7. 4736–5235
a
0.41 0.56 0.72 0
8. 5236–5689
a
0.90 −0.43 −0.07 0
a
λ
1
3
is not signicant in this block.
13.6 Enveloping Spectral Surfaces
Motivated by problems in Sleep Medicine and Circadian Biology, the author and colleagues
developed a method for the analysis of cross-sectional categorical time series collected from
multiple subjects where the effect of static continuous-valued covariates is of interest; see
Krafty et al. (2012). In particular, the spectral envelope was extended for the analysis of
cross-sectional categorical processes that are possibly covariate dependent. This extension
introduces an enveloping spectral surface for describing the association between the fre-
quency domain properties of qualitative time series and covariates. The resulting surface
offers an intuitively interpretable measure of association between covariates and a qualita-
tive time series by nding the maximum possible conditional power at a given frequency
from scalings of the qualitative time series conditional on the covariates. The optimal scal-
ings that maximize the power provide scientic insight by identifying the aspects of the
qualitative series, which have the most pronounced periodic features at a given frequency
conditional on the value of the covariates. The approach is entirely nonparametric, and we
summarize the technique in this section.
In this section, we suppose we observe qualitative time series {X
jt
; t = 0, ±1, ±2, ...}
with nite state-space C ={c
1
, c
2
, ..., c
p+1
} and a covariate vector S
j
= (S
j1
, ..., S
jq
)
∈
S ⊂ R
q
for j = 1, ..., N independent subjects. We assume that {X
jt
; t = 0, ±1, ±2, ...} is
stationary conditional on S
j
such that
s∈S,k=
inf
1,...,p+1
Pr{X
jt
= c
k
| S
j
= s} > 0, (13.10)
so that there are no absorbing states. The covariates S
j
are assumed to be independent
and identically distributed second-order random variables with density function φ
s
. Aside
from making some smoothness assumptions about the conditional spectral distribution of
X
jt
given S
j
to aid estimation, we will only assume a very general nonparametric model for
X
jt
and S
j
.
Analogous to the discussion in Section 13.3, we consider the quantitative time series
X
jt
(β) obtained from scaling X
jt
such that X
jt
(β) = β
when X
jt
= c
,for 1 ≤ ≤ p,
299 Spectral Analysis of Qualitative Time Series
and X
jt
(β) = 0 when X
jt
= c
p+1
. Further, we suppose that the p-dimensional random
vector process Y
jt
, which has one in the th element if X
jt
= c
for = 1, ..., p and zeros
elsewhere, has a spectral density conditional on the value of the covariate. In this case,
dene the conditional spectral density and variance of Y
jt
as
∞
f (ω, s) =
Cov
Y
jt
, Y
jt+τ
| S
j
= s
e
−2πiωτ
(13.11)
τ=−∞
V(s) = Var
Y
jt
| S
j
= s
. (13.12)
We will assume that f (ω, s) and V(s) are nonsingular for all frequencies ω and s ∈ S.
Under this assumption, we have the existence of the spectral density of X
jt
(β) conditional
on S
j
= s for all β ∈ R
p
0
p
. Thus, dene
∞
f
x
(ω, s; β) =
Cov
X
jt
(β), X
jt+τ
(β) | S
j
= s
e
−2πiωτ
.
τ=−∞
As an extension of the spectral envelope, for every frequency ω and covariate s ∈ S,we
dene the enveloping spectral surface, λ(ω, s), to be the maximal normalized power among
all possible scalings at frequency ω, conditional on the covariate value S
j
= s.
Letting V
x
(s; β) = Var
X
jt
(β) | S
j
= s
be the conditional variance of the scaled time
series, we formally dened the enveloping spectral surface as
f
x
(ω, s; β)
λ(ω, s) = max
. (13.13)
β=0
p
V
x
(s; β)
In addition to the maximum value λ(ω, s), the scalings where this maximum is achieved
can provide important information by locating the scalings of the qualitative time series for
which cycles at a given frequency are most prominent conditional on the covariate vector.
Equivalently, note that λ(ω, s) is the largest eigenvector associated with
g(ω, s) = V
−1/2
(s)f (ω, s)V
−1/2
(s) ,
and the optimal scaling is linearly related to the eigenvector associated with the largest
eigenvalue.
An aspect of the enveloping spectral surface that can be of scientic interest are fre-
quencies where the enveloping spectral surface changes based on covariate values. An
interpretable measure of the dynamics of the enveloping spectral surface with respect to
the covariates depends on the form of the covariates. Here, we consider the case where the
covariates are continuous random variables such that the spectrum g
re
(ω, S) is smooth.
For a metric space D,let C
d
(D) be the space of real-valued functions over D such that
all dth-order partial derivatives exist and are continuous. We need the following two
smoothness assumptions:
A(i): The support of the density function φ
s
is S and φ
s
∈ C
2
(
S
)
.
A(ii): Each element of the p × p spectral density matrix g
re
is an element of the space
C
4
[R × S].
300 Handbook of Discrete-Valued Time Series
Under Assumptions A(i) and A(ii), the continuity of the eigenvalues of a matrix-valued
function implies that the enveloping spectral surface λ is continuous in both the frequency
and covariates. In this setting, the rst-order partial derivatives of the enveloping spec-
tral surface in the direction of the covariates provide an assessment of the dependence of
the enveloping spectral surface on the covariates. Let Dλ(ω, s) be the q-dimensional vec-
tor of partial derivatives of λ(ω, s) with respect to s which has jth element D
j
λ(ω, s) =
∂λ(ω, s)/∂s
j
. For a frequency ω ∈ R,ifDλ(ω, s) = 0 for all s ∈ S, then there is no association
between the the maximal amount of normalized power at frequency ω and the covariates.
The following result, which follows from Magnus and Neudecker (1988, Chapter 8, Theo-
rem 7) provides a computationally useful form for D
j
λ in terms of the derivatives of the real
part of the spectral density of the indicator variables. In particular, under Assumptions A(i)
and A(ii), if λ(ω, s) is a unique eigenvalue of g
re
(ω, s), then
re
D
j
λ(ω, s) = γ(ω, s)
V
1/2
(s)D
j
g (ω, s)V
1/2
(s)γ(ω, s) (13.14)
where D
j
g
re
(ω, s) is the p ×p matrix with mth element D
j
g
re
m
(ω, s) =∂g
re
m
(ω, s)/∂s
j
,and
γ(ω, s) is the corresponding eigenvector. Equation (13.14) provides a useful tool for devel-
oping an estimation procedure for the derivatives of the enveloping spectral surface
directly from estimates of the eigenvectors and derivatives of g
re
.
It is assumed that epochs of the qualitative time series of length T, X
jt
; t = 1, ..., T ,
are observed for j = 1, ..., N subjects. Asymptotic properties are established as both the
number of time points and the number of subjects are large so that T, N →∞. Since the
spectral density is Hermitian and periodic with period 1, we restrict our attention to ω ∈
[0, 1/2].
Estimation procedures based on multivariate local quadratic regression are proposed.
Let K
q
and K
q+1
be spherically symmetric q and q + 1-dimensional compactly supported
density functions that possess eighth-order marginal moments over S and [0, 1/2]×S,
respectively. For k = q or k = (q + 1), the bandwidth will be parameterized by assuming
that there exists a positive denite k ×k symmetric real matrix H
∗
and a scaling bandwidth
h > 0 such that the bandwidth is parameterized as H = hH
∗
and the corresponding weight
functions are |H|
−1
K
k
[H
−1
s]. Asymptotic properties will be established when h → 0as
N, T →∞.
Local quadratic estimation will be used to estimate V, g
re
,and D
k
g
re
. A comprehensive
review of local polynomial regression is given in Fan and Gijbels (1996). Although V, g
re
,
and its derivatives can be of scientic interest in their own right, we are concerned with the
estimation of these quantities exclusively for use in estimating and performing inference
on the enveloping spectral surface and its derivatives.
To estimate g
re
and its derivatives, rst note that the conditional spectral density of
V
−1/2
(S
j
)Y
jt
is g(ω, S
j
). Dene the normalized periodogram I
jk
for j = 1, ..., N and
k =1, ..., T/2as I
jk
=
V
−1/2
(S
j
)
Y
jk
Y
jk
∗
V
−1/2
(S
j
) where
Y
jk
= T
−1/2
t
T
=1
Y
jt
− Y
j
e
−2πiω
k
t
∗
is the nite Fourier transform of Y
jt
at frequency ω
k
= k/T and Y
jk
is the conjugate transpose
of Y
jk
. The components I
jk
re
m
provide asymptotically unbiased but inconsistent estimates
of g
re
m
(ω
k
, S
j
) and we apply a local quadratic regression to these real components of the
periodograms to obtain a consistent estimate of g
re
(ω, s). For ease of notation, dene the
(q + 1)-dimensional vector
301 Spectral Analysis of Qualitative Time Series
ω
k
− ω
ξ
jk
=
S
j
− s
.
Then, for ω ∈[0, 1/2] and s ∈ S, dene the p × p matrices
g
re
(ω, s) and
D
k
g
re
(ω, s) for
D
k
g
re
k = 1, ... , q with respective mth elements
g
re
(ω, s) =
α
0
and
(ω, s) =
α
k+1
,
m m
α
0
N
T/2
2
α
=
argmin
I
jk
re
m
− α
0
− α
ξ
jk
− ξ
jk
Qξ
jk
vec(
Q
)
α
0
∈R,α∈R
q+1
,Q∈Q
q+1
j=1 k=0
× K
q+1
H
−1
ξ
jk
/ |H
g
|,
g
=
α
1
, ... ,
1
)
,and H
g
is a symmetric positive denite (q + 1) × (q + 1) real matrix.
α (
α
q+
The following theorem provides the asymptotic consistency of
V, D
k
g
re
, which
g
re
,and
allows for the consistent estimation of the enveloping spectral surface and its derivatives.
Theorem 13.1 Let H
v
= h
v
H
v
∗
and H
g
= h
g
H
g
∗
for positive denite symmetric matrices H
v
∗
, H
g
∗
and positive real numbers h
v
, h
g
. Under Assumptions A(i) and A(ii), the rst-order optimal con-
ditional mean squared error of
V(s),
D
k
g
re
(ω, s) for ω ∈[0, 1/2] and s ∈ S are
g
re
(ω, s), and
achieved when h
v
∼
(
NT
)
−1/(q+6)
and h
g
∼
(
NT
)
−1/(q+7)
as N, T →∞.Ifh
v
∼
(
NT
)
−1/(q+6)
,
h
g
∼
(
NT
)
−1/(q+7)
, and N ∼ T
(q/6)−
for some ∈ (0, q/6),then
V
(s) = V(s) + O
p
(NT)
−3/(q+6)
,
g
re
(ω, s) = g
re
(ω, s) + O
p
(NT)
−3/(q+7)
,
D
k
g
re
(NT)
−2/(q+7)
,
D
k
g
re
(ω, s) = (ω, s) + O
p
conditional on S
1
, ... , S
N
.
We can now estimate the enveloping spectral surface and optimal scalings. Dene
λ(ω, s)
as the largest eigenvalue of
g
re
(ω, s).Let
γ(ω, s) =
V
−1/2
(s)
ψ
(ω, s) where
ψ(ω, s) is the
eigenvector of
g
re
(ω, s) associated with
λ(ω, s) such that
γ(ω, s)
V(s)
γ(ω, s) = 1andthe
rst nonzero element of
γ(ω, s) is positive. The next theorem establishes the consistency
and asymptotic distribution of
λ(ω, s) and the consistency of
γ(ω, s).
Theorem 13.2 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
λ(ω, s) = λ(ω, s) + O
p
(NT)
−3/(q+7)
γ(ω, s) = γ(ω, s) + O
p
(NT)
−3/(q+7)
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