The Empirical FT.
)()0( Note
1 0 real }exp{)()(:
1)}-TX(1),...X( {X(0),
tXTXd
N TtitXTXdEFT
Data
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 UwhereV i U Y
form theof variatea is ),,( normal,complex The
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21
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E
INZZZ
ZZNiZZINN
YEYEY
NN
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j
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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
XXC
lLLTT
lXXC
LTX
TX
XXC
XXX
TX
TfIN
TrrrTrd
Trdiii
TfIN
ddii
TfN
TfTcNdi
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
fdf
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.
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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
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C
T
X
T
XX
ffffE
f
f
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dT
I
An estimate whose limit is a random variable
Some moments.
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22
|)(|)(|)(||)(|
)(}exp{)()( || var||
T
NXX
TT
X
T T
X
T
X
cdfdEso
ctitXEdEE
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 .
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1
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Lff
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TrrrmperiodograSmoothed
XXLNNXXLXX
LXX
T
XX
l l
TT
XX
l
L
Approximate marginal confidence intervals
LVT
L
fff
T
T
data,split might FT or taperedmean, weightedmight take
estimate consistentfor might takeondistributi 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
lll
lll
lT
ff
TLW
df
dfTTL
TrTrLIEfE
Approximation to bias
0)( symmetricFor W
...2/)()(")()(')()(
...]2/)(")(')()[(
)()(
)()(
)()( Suppose
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dW
dWfBdWfBdWf
dfBfBfW
BfW
dfW
BWBW
TTT
TT
T
T
TT
T
Indirect estimate
duucuwui T
XX
T )()(}exp{21
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
ffff
dfdZdZ
dZdZ
ittYtX
Crossperiodogram.
T
YY
T
YX
T
XY
T
XXT
T
Y
T
X
T
XY
IIII
ddT
I
)( formmatrix
)()(2
1)(
I
Smoothed periodogram.
LlTrLTr ll
lTT
NN ,...,1 ,/2 ,/)/2()( If
Complex Wishart
ΣW
XXWΣ
0XX
nE
nW
IN
n Tjj
Cr
rn
squared-chi diagonals
~),(
),(~,...,
1
1
Predicting Y via X
)()(}exp{)(
}exp{)()2()()()(/)()( :coherency
)(]|)(|1[ :MSEphase :)(arggain |:)(|
functionfer trans,)()()(
|)(-)(|min )(by )( predicting
1
2
1
2
X
YYXXYX
YY
XXYX
XYA
XY
dZAtitY
diuAuafffR
fRAA
ffA
AdZdZEdZdZ
Plug in estimates.
LRE
R
LLRRLLFR
R
fffR
fRAA
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
12
2
2
2
1
Large sample distributions.
var log|AT| [|R|-2 -1]/L
var argAT [|R|-2 -1]/L
Berlin andVienna monthlytemperatures
RecifeSOI
Furnace data
RXZ|Y =
(R XZ – R XZ R ZY )/[(1- |R XZ|2 )(1- |RZY | 2 )]
Partial coherence/coherency. Mississipi dams
Cleveland, RB, Cleveland, WS, McRae, JE & Terpenning, I (1990), ‘STL: a seasonal-trend decomposition procedurebased 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="detailed",theta=25,d=2,main="Explosion")
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...