307 Spectral Analysis of Qualitative Time Series
where {b
j
} is an absolutely summable p×1 lter, is such that the mean square approximation
error
∗
E X
t
− X
t
X
t
− X
t
(13.19)
is minimized.
Let b(ω) and c(ω) be the transforms of {b
j
} and {c
j
}, respectively. For example,
∞
c(ω) =
c
j
exp(−2πijω), (13.20)
j=−∞
and, consequently,
1
/2
c
j
= c(ω) exp(2πijω)dω. (13.21)
−1/2
Brillinger (1981, Theorem 9.3.1) shows the solution to the problem is to choose c(ω) to
satisfy (13.15) and to set b(ω) =
c(ω). This is precisely the previous problem, with the
solution given by (13.16). That is, we choose c(ω) = e
1
(ω) and b(ω) = e
1
(ω);thelter
values can be obtained via the inversion formula given by (13.21). Using these results, in
view of (13.17), we may form the rst principal component series, say Y
t1
.
This technique may be extended by requesting another series, say, Y
t2
, for approximating
X
t
with respect to minimum mean square error, but where the coherency between Y
t2
and
Y
t1
is zero. In this case, we choose c(ω) = e
2
(ω). Continuing this way, we can obtain the
rst q ≤ p principal component series, say, Y
t
= (Y
t1
, ..., Y
tq
)
, having spectral density
f
yy
(ω) = diag{λ
1
(ω), ..., λ
q
(ω)}. The series Y
tk
is the kth principal component series.
13.8 R Code
The following R script can be used to calculate the spectral envelope and optimal scalings.
The script can also be used to perform dynamic analysis in a piecewise fashion. The scripts
are used to produce Figures 13.4 and 13.5 and Table 13.3. The scripts require the use of the R
package astsa, which must be downloaded and installed prior to running the script. The
data are included in the package as bnrf1ebv, which lists the bp using the code A = 1,
C = 2, G = 3, T = 4.
require(astsa)
u= factor(bnrf1ebv)
# first, input the data as factors and then
x= model.matrix(~u-1)[,1:3]
# make an indicator matrix
Var = var(x)
# var-cov matrix
xspec = mvspec(x, spans=c(7,7), plot=FALSE)
# spectral matrices are an array
# called fxx
fxxr = Re(xspec$fxx)
# fxxr is real(fxx)
308 Handbook of Discrete-Valued Time Series
#--- compute Q = Var^-1
/
2 ---#
ev = eigen(Var)
Q =ev$vectors%*%diag(1/sqrt(ev$values))%*%t(ev$vectors)
#--- compute spectral envelope and scale vectors ---#
num = xspec$n.used
# effective sample size
nfreq = length(xspec$freq)
# number of frequencies
specenv = matrix(0,nfreq,1)
# initialize the spectral envelope
beta = matrix(0,nfreq,3)
# initialize the scale vectors
for (k in 1:nfreq){
ev = eigen(2*Q%*%fxxr[,,k]%*%Q/num)
# get evalues of normalized spectral
# matrix at freq k
/
n
specenv[k] = ev$values[1]
# spec env at freq k
/
n is max evalue
b= Q%*%ev$vectors[,1]
# beta at freq k
/
n
beta[k,] = b/sqrt(sum(b^2))
# helps to normalize beta
}
#--- output and graphics ---#
dev.new(height=3)
par(mar=c(3,3,2,1), mgp=c(1.6,.6,0))
frequency = (0:(nfreq-1))/num
plot(frequency, 100*specenv, type="l", ylab="Spectral Envelope (%)")
title("Epstein-Barr BNRF1")
## add significance threshold to plot ##
m = xspec$kernel$m
nuinv=sqrt(sum(xspec$kernel[-m:m]^2))
thresh=100*(2/num)*exp(qnorm(.9999)*nuinv)*matrix(1,nfreq,1)
lines(frequency, thresh, lty="dashed", col="blue")
#-- details --#
output = cbind(frequency, specenv, beta)
# results
colnames(output)=c("freq","specenv","A", "C", "G")
##--- dynamic part ---##
z= matrix(0,250,8)
output2 = array(0, dim=c(250,5,8))
colnames(output2) = c("freq","specenv","A", "C", "G")
for (j in 1:8){
ind = (500*(j-1)+1):(500*j)
if (j==8) ind=3501:length(bnrf1ebv)
xx = x[ind,]
# select subsequence -- the rest of the this part is the same
# as above
Var = var(xx)
xspec = mvspec(xx, spans=c(3,3), plot=FALSE)
fxxr = Re(xspec$fxx)
ev = eigen(Var)
Q =ev$vectors%*%diag(1/sqrt(ev$values))%*%t(ev$vectors)
num = xspec$n.used
nfreq = length(xspec$freq)
frequency = (0:(nfreq-1))/num
specenv = matrix(0, nfreq, 1)
beta = matrix(0, nfreq, 3)
for (k in 1:nfreq){
ev = eigen(2*Q%*%fxxr[,,k]%*%Q/num)
specenv[k] = ev$values[1]
b= Q%*%ev$vectors[,1]
beta[k,] = b/sqrt(sum(b^2))
}
309 Spectral Analysis of Qualitative Time Series
if(j<8) { z[,j] = specenv; output2[,,j] = cbind(frequency, specenv, beta)}
if(j==8) { z[1:240,8] = specenv; output2[1:240,,j] = cbind(frequency, specenv,
beta)}
}
#--- output and graphics (results in output2)---#
zz = 100*t(z)
# zz is 8x250
rowss = rep(1:8, each=2)
zz = zz[rowss,]
# now it’s 16x250
# threshold
m = xspec$kernel$m
nuinv = sqrt(sum(xspec$kernel[-m:m]^2))
thresh=100*(2/num)*exp(qnorm(.995)*nuinv)
dev.new(height = 5)
par(mar=c(3,3,3,1), mgp=c(1.6,.6,0))
xa = 0:249/500
ya1 = 1736+500*0:8
rowss = c(1, rep(2:8, each=2), 9)
ya = ya1[rowss]
ya[seq(2,16,by=2)] = ya[seq(2,16,by=2)]-.5
levs = thresh*seq(0, 4.5, by=.5)
colr = gray(c(10,9,5,4.5,4,3,2,1,0)/10)
contour(ya, xa, zz, xlab="base pair", ylab="frequency", levels=levs, col=colr,
main="Epstein-Barr BNRF1", lwd=2, drawlabels=FALSE)
Acknowledgment
This work was supported, in part, by a grant from the U.S. National Science Foundation.
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