Post on 09-Sep-2020
Time series power spectral density.
frequency-side, , vs. time-side, t
X(t), t= 0, ±1, ±2,… Suppose stationary
cXX(u) = cov{X(t+u),X(t)} u = 0, ±1, ±2, … lag
f XX()= (1/2π) Σ exp {-iu} cXX(u) - < ≤
period 2π non-negative
/2 : frequency (cycles/unit time)
cXX(u) = ∫ -pi pi exp{iuλ} f XX() d
The Empirical FT.
Data X(0), …, X(T-1)
dXT()= Σ exp {-iu} X(u)
dXT(0)= Σ X(u)
What is the large sample distribution of the EFT?
The complex normal.
2var
)1,0( ...
onentialexp2/2/)(|)1,0(|
2/)()2/1,0()1,0(
||var
:Notes
)2/,(Im ),2/,(Ret independen are V and Uwhere
V i U Y
form theof variatea is ),,( normal,complex The
22
22
1
2
2
2
2
2
2
1
2
21
22
22
2
E
INZZZ
ZZN
iZZINN
YEY
EY
NN
N
j
C
C
C
Theorem. Suppose X is stationary mixing, then
))(2,0( asymp are
~/2 with 0 integersdistinct ,..., ),2
(),...,2
( ).
)(2,0(
asymp are 0 anddistinct ,..., ),(),...,( ).
0 )),(2,0(
0 )),0(2,(
allyasymptotic is )( ).
11
L11
XX
C
lLLTT
lXX
C
L
T
X
T
X
XX
C
XXX
T
X
TfIN
TrrrT
rd
T
rdiii
TfIN
ddii
TfN
TfTcN
di
Evaluate first and second-order cumulants
Bound higher cumulants
Normal is determined by its moments
Proof. Write
)()()( X
TT
XdZd
Consider
)(2)()(
have We
)()(2~
)()()(
)()()()()}(),(cov{
TTT
XX
T
XX
TT
XX
TTT
X
T
X
d
f
df
ddfdd
Comments.
Already used to study mean estimate
Tapering, h(t)X(t). makes
dfHdXX
TT
X)(|)(|~)(var 2
Get asymp independence for different frequencies
The frequencies 2r/T are special, e.g. T(2r/T)=0, r 0
Also get asymp independence if consider separate stretches
p-vector version involves p by p spectral density matrix fXX( )
Estimation of the (power) spectrum.
22
2
2
2
2
2
2
2
)(}2/)(var{ and
)(}2/)({ notebut
0)( unlessnt inconsiste appears Estimate
lexponentia ,2/)(~
|))(2,0(|2
1~
|)(|2
1)(
m,periodogra heconsider t ,0For
XXXX
XXXX
XX
XX
XX
C
T
X
T
XX
ff
ffE
f
f
TfNT
dT
I
An estimate whose limit is a random variable
Some moments.
2222
0
22
|)(|)(|)(||)(|
)(}exp{)()(
|| var||
T
NXX
TT
X
T T
X
T
X
cdfdE
so
ctitXEd
EE
The estimate is asymptotically unbiased
Final term drops out if = 2r/T 0
Best to correct for mean, work with
)()( TT
X
T
Xcd
Periodogram values are asymptotically independent since dT
values are -
independent exponentials
Use to form estimates
sL' severalTry variance.controlCan
/)(}2/)(var{ )(}2/)({
Now
ondistributiin 2/)()(
gives CLT
/)/2()(
Estimate
near /2
0 integersdistinct ,...,Consider .
22
2
2
2
2
2
1
LfLffLfE
Lff
LTrIf
Tr
rrmperiodograSmoothed
XXLNNXXLXX
LXX
T
XX
l l
TT
XX
l
L
Approximate marginal confidence intervals
LVT
L
ff
f
T
T
data,split might
FT or taperedmean, weightedmight take
estimate consistentfor might take
ondistributi valueextreme viaband ussimultaneo
levelmean about CIset
logby stabilized variance
Notes.
)}2/(/log)(log)(log
)2/1(/log)(Pr{log
2
2
More on choice of L
biased constant,not is If
radians /2
(.) of width affects L of Choice
)()(W
)2/)/(sin()2/)(sin2
11
/2/2 },/)({)}({
Consider
T
2
T
T
T
ll
l
ll
l
l
T
ff
TL
W
df
dfTTL
TrTrLIEfE
Approximation to bias
0)( symmetricFor W
...2/)()(")()(')()(
...]2/)(")(')()[(
)()(
)()(
)()( Suppose
22
22
11
dW
dWfBdWfBdWf
dfBfBfW
BfW
dfW
BWBW
TTT
TT
T
T
TT
T
Indirect estimate
duucuwui T
XX
T )()(}exp{2
1
Estimation of finite dimensional .
approximate likelihood (assuming IT values
independent exponentials)
)}/2(/)/2(exp{)/2()(
in ),;( spectrum
1 TrfTrITrfL
f
r
T
Bivariate case.
)( )(
)( )(
matrixdensity spectral
)()()}(),(cov{
)(
)(}exp{
)(
)(
YYYX
XYXX
XYYX
Y
X
ff
ff
dfdZdZ
dZ
dZit
tY
tX
Crossperiodogram.
T
YY
T
YX
T
XY
T
XXT
T
Y
T
X
T
XY
II
II
ddT
I
)( formmatrix
)()(2
1)(
I
Smoothed periodogram.
LlTrLTrl
ll
TT
NN,...,1 ,/2 ,/)/2()( If
Complex Wishart
ΣW
XXWΣ
0XX
nE
nW
IN
n T
jj
C
r
rn
squared-chi diagonals
~),(
),(~,...,
1
1
Predicting Y via X
)()(}exp{)(
}exp{)()2()(
)()(/)()( :coherency
)(]|)(|1[ :MSE
phase :)(arggain |:)(|
functionfer trans,)()()(
|)(-)(|min
)(by )( predicting
1
2
1
2
X
YYXXYX
YY
XXYX
XYA
XY
dZAtitY
diuAua
fffR
fR
AA
ffA
AdZdZE
dZdZ
Plug in estimates.
LRE
R
L
LRRLLFR
R
fffR
fR
AA
ffA
T
TL
T
T
YY
T
XX
T
YX
T
T
YY
T
TT
T
XX
T
YX
T
/1||
)-(1-1
point %100approx 0|| If
)1()1(
)()||||;1;,()||1(
|| ofDensity
)()(/)()(
)(]|)(|1[ :MSE
)(arg |)(|
)()()(
2
1)-1/(L
2
22
1
2
2
2
2
1
Large sample distributions.
var log|AT| [|R|-2 -1]/L
var argAT [|R|-2 -1]/L
Berlin and
Vienna
monthly
temperatures
Recife
SOI
Furnace data
RXZ|Y =
(R XZ – R XZ R ZY )/[(1- |R XZ|2 )(1- |RZY | 2 )]
Partial coherence/coherency. Mississipi dams
Advantages of frequency domain approach.
techniques for many stationary processes look the same
approximate i.i.d sample values
assessing models (character of departure)
time varying variant...
Cleveland, RB, Cleveland, WS, McRae, JE & Terpenning, I (1990),
‘STL: a seasonal-trend decomposition procedure
based on loess’, Journal of Official Statistics
Y(t) = S(t) + T(t) + E(t)
Seasonal, trend. error
London water usage
Dynamic spectrum, spectrogram: IT (t,). London water
Earthquake? Explosion?
Lucilia cuprina
nobs = length(EXP6) # number of observations
wsize = 256 # window size
overlap = 128 # overlap
ovr = wsize-overlap
nseg = floor(nobs/ovr)-1; # number of segments
krnl = kernel("daniell", c(1,1)) # kernel
ex.spec = matrix(0, wsize/2, nseg)
for (k in 1:nseg)
{ a = ovr*(k-1)+1
b = wsize+ovr*(k-1)
ex.spec[,k] = mvspec(EXP6[a:b], krnl, taper=.5, plot=FALSE)$spec }
x = seq(0, 10, len = nrow(ex.spec)/2)
y = seq(0, ovr*nseg, len = ncol(ex.spec))
z = ex.spec[1:(nrow(ex.spec)/2),] # below is text version
filled.contour(x,y,log(z),ylab="time",xlab="frequency (Hz)",nlevels=12
,col=gray(11:0/11),main="Explosion")
dev.new() # a nicer version with color
filled.contour(x, y, log(z), ylab="time", xlab="frequency(Hz)", main=
"Explosion") dev.new() # below not shown in text
persp(x,y,z,zlab="Power",xlab="frequency(Hz)",ylab="time",ticktype="det
ailed",theta=25,d=2,main="Explosion")