Download - The Empirical FT
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The Empirical FT.
)()0( Note
real ),(}exp{}exp{)(:
}...{0
1 0
1
TNd
tdNtiidEFT
TData
T
N
N T
j
T
N
N
What is the large sample distribution of the EFT?
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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
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Theorem. Suppose p.p. N 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
NN
C
lLLTT
lNN
C
L
T
N
T
N
NN
C
NNN
T
N
TfIN
TrrrT
rd
Tr
diii
TfIN
ddii
TfN
TfTpN
di
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Evaluate first and second-order cumulants
Bound higher cumulants
Normal is determined by its moments
Proof. Write
)()()( N
TT
N dZd
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Consider
)(2)()(
saw We
)()(2~
)()()(
)()()()()}(),(cov{
TTT
NN
T
NN
TT
NN
TTT
N
T
N
d
f
df
ddfdd
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Comments.
Already used to study rate estimate
Tapering makes
dfHd NN
TT
N )(|)(|~)(var
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 fNN( )
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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
NNNN
NNNN
NN
NN
NN
C
T
N
T
NN
ff
ffE
f
f
TfNT
dT
I
An estimate whose limit is a random variable
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Some moments.
2222
0
22
|)(|)(|)(||)(|
)(}exp{)(
|| var||
r.v.Complex
T
NNN
TT
N
T T
NN
T
N
pdfdE
so
pdtptiEd
EE
The estimate is asymptotically unbiased
Final term drops out if = 2r/T 0
Best to correct for mean, work with
)()( TT
N
T
N pd
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Periodogram values are asymptotically independent since dT values are -
independent exponentials
Use to form estimates
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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
NNLNNNNLNN
LNN
T
NN
ll
TT
NN
l
L
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Approximate marginal confidence intervals
LVT
L
ff
fT
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{log2
2
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More on choice of L
biased constant,not is If
radians /2
(.) of width affects L of Choice
)()(W
)2/)/(()2/)(sin2
11
/2/2 },/)({)}({
Consider
T
2
T
T
T
lll
lll
l
T
ff
TL
W
df
dfTTL
TrTrLIEfE
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Approximation to bias
0)( symmetricFor W
...2/)()(")()(')()(
...]2/)(")(')()[(
)()(
)()(
)()( Suppose
22
22
11
dW
dWfBdWfBdWf
dfBfBfW
BfW
dfW
BWBW
TTT
TT
T
T
TT
T
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Indirect estimate
duuquwuip T
NN
TT
N )()(}exp{21
2
Could approximate p.p. by 0-1 time series and use t.s. estimate
choice of cells?
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Estimation of finite dimensional .
approximate likelihood (assuming IT values independent exponentials)
)}/2(/)/2(exp{)/2()(
in ),;( spectrum
1 TrfTrITrfL
f
r
T
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Bivariate case.
)( )(
)( )(
matrixdensity spectral
)()()}(),(cov{
)(
)(/]1}[exp{
)(
)(
NNNM
MNMM
MNNM
N
M
ff
ff
dfdZdZ
dZ
dZitit
tN
tM
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Crossperiodogram.
T
NN
T
NM
T
MN
T
MMT
T
N
T
M
T
MN
II
II
ddT
I
)( formmatrix
)()(2
1)(
I
Smoothed periodogram.
LlTrLTr ll
l
TT
NN ,...,1 ,/2 ,/)/2()( If
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Complex Wishart
ΣW
XXWΣ
0XX
nE
nW
IN
n T
jj
C
r
rn
squared-chi diagonals
~),(
),(~,...,
1
1
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Predicting N via M
)()(1}exp{
)(
}exp{)()2()(
)()(/)()( :coherency
)(]|)(|1[ :MSE
phase :)(arggain |:)(|
functionfer trans,)()()(
|)(-)(|min
)(by )( predicting
1
2
1
2
M
NNMMNM
NN
MMNM
MNA
MN
dZAiit
tN
diuAua
fffR
fR
AA
ffA
AdZdZE
dZdZ
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Plug in estimates.
LRE
R
LL
RRLLFR
R
fffR
fR
AA
ffA
T
TL
T
NNMMNM
T
T
NN
T
TT
T
MM
T
NM
T
/1||
)-(1-1
point %100approx 0|| If
)1()1()(
)||||;1;,()||1(
|| ofDensity
)()(/)()(
)(]|)(|1[ :MSE
)(arg |)(|
)()()(
2
1)-1/(L
2
22
122
2
2
1
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Large sample distributions.
var log|AT| [|R|-2 -1]/L
var argAT [|R|-2 -1]/L
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Networks.
partial spectra - trivariate (M,N,O)
Remove linear time invariant effect of M from N and O and examine coherency of residuals,
)(
)()()()()(
M of effects
invariant lineartime removed having O and N of spectrum-cross partial
),()()()(
),()()()(
|
1
|
1
|
1
|
T
MNO
MNMMNMNOMNO
MMOMMOMO
MMNMMNMN
R
fffff
ffBdZBdZdZ
ffAdZAdZdZ
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Point process system.
An operation carrying a point process, M, into another, N
N = S[M]
S = {input,mapping, output}
Pr{dN(t)=1|M} = { + a(t-u)dM(u)}dt
System identification: determining the charcteristics of a system from inputs and corresponding outputs
{M(t),N(t);0 t<T}
Like regression vs. bivariate
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Realizable case.
a(u) = 0 u<0
A() is of special form
e.g. N(t) = M(t-)
arg A( ) = -
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Advantages of frequency domain approach.
techniques formany stationary processes look the same
approximate i.i.d sample values
assessing models (character of departure)
time varying variant
...
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Published by AAAS
R. Fuentes et al., Science 323, 1578 -1582 (2009)
Fig. 2. DCS restores locomotion and desynchronizes corticostriatal activity
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Muscle spindle example