combining direct spectral estimators:...
TRANSCRIPT
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Combining Direct Spectral Estimators: I
• tapering introduced as means of creating SDF estimator S(d)(·)with potentially better bias properties than periodogram S(p)(·)• both S(d)(·) & S(p)(·) inherently noisy – smoothing across fre-
quencies leads to lag window estimator S(lw)(·)• large-sample approximation to variance of S(lw)(f ) points out
price of tapering, namely, inflation of variance by Ch > 1:
var {S(lw)(f )} ≈ ChS2(f )
BWN ∆t
− Ch = 1 if and only if rectangular taper used (i.e., no real ta-pering done, leading to periodogram & use of prewhitening)
− Ch ≈ 2 for Hanning data taper
SAPA2e–371 VIII–1
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Combining Direct Spectral Estimators: II
• Q: can we compensate for increase in variance due to tapering?
• A: yes, by combining together different S(d)(·)’s formed usingsame time series
• two schemes
− multitapering: use multiple orthogonal tapers on {Xt}−Welch’s overlapped segment averaging (WOSA): use a single
taper multiple times on {Xt}• like lag window estimators, multitaper and WOSA estimators
reduce variance over that given by S(d)(·), but do so withoutextracting price similar to Ch
• will discuss multitaper estimators first and then WOSA
SAPA2e–371 VIII–2
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Multitaper Spectral Estimation: I
• tapering useful for S(·) with large dynamic range, but increases
variance of S(lw)(f ) by Ch > 1
• alternatives to S(lw)(·), i.e., smoothing S(d)(·), include prewhiten-ing (see Chapter 6), WOSA and multitapering (Thomson, 1982)
• why multitapering?
− works automatically on high dynamic range SDFs
− natural definition of resolution
− can tradeoff bias/variance/resolution easily
− for some processes, can argue
∗ superior to prewhitening (Thomson, 1990a)
∗ superior to WOSA (Bronez, 1992)
SAPA2e–375, 376, 377 VIII–3
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Multitaper Spectral Estimation: II
− produces ‘S(d)(·)’ with EDOF ν = 2K, where K is thenumber of tapers used (2 to 10 typically; recall that widthof 95% CIs shrinks considerably as ν increases from 2 to 10)
− can get internal estimate of variance (‘jackknifing’)
− can handle mixed spectra (i.e., line components)
− extends naturally to irregularly sampled processes
SAPA2e–375, 376, 377 VIII–4
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Widths of 95% CIs on dB Scale versus EDOF ν
100 101 102 103
ν
−10
0
10
20
dB
3 dB
2 dB
1 dB
SAPA2e–294 VIII–5
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Basics of Multitapering: I
• basic multitaper estimator is average of K direct spectral esti-mators:
S(mt)(f ) ≡ 1
K
K−1∑k=0
S(mt)k (f ),
where
S(mt)k (f ) ≡ ∆t
∣∣∣∣∣∣N−1∑t=0
hk,tXte−i2πft∆t
∣∣∣∣∣∣2
is called kth eigenspectrum and uses kth taper {hk,t} normal-
ized by∑t h
2k,t = 1
SAPA2e–372 VIII–6
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Basics of Multitapering: II
• generalization of interest later on: weighted multitaper estima-tor
S(wmt)(f ) ≡K−1∑k=0
dkS(mt)k (f ),
where weights dk are nonnegative and sum to unity
• basic multitaper estimator is a special case of the above: justset dk = 1/K
SAPA2e–372 VIII–7
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Basics of Multitapering: III
• spectral window for kth eigenspectrum:
Hk(f ) ≡ ∆t
∣∣∣∣∣∣N−1∑t=0
hk,te−i2πft∆t
∣∣∣∣∣∣2
• an eigenspectrum is just a direct spectral estimator, so
E{S(mt)k (f )} =
∫ fN
−fNHk(f − f ′)S(f ′) df ′
• for the basic multitaper estimator, we thus have
E{S(mt)(f )} =
∫ fN
−fNH(f−f ′)S(f ′) df ′ with H(f ) ≡ 1
K
K−1∑k=0
Hk(f )
SAPA2e–372, 373 VIII–8
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Basics of Multitapering: IV
• corresponding result for weighted multitaper estimator is
E{S(wmt)(f )} =
∫ fN
−fNH(f−f ′)S(f ′) df ′ with H(f ) ≡
K−1∑k=0
dkHk(f )
• leakage for S(mt)(·) or S(wmt)(·) OK if Hk(·)’s all have smallsidelobes
• if all K eigenspectra are approximately unbiased, then multi-taper estimators will be also approximately so
SAPA2e–372, 373 VIII–9
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Basics of Multitapering: V
• recall standard measure of effective bandwidth for direct spec-tral estimator:
BHk ≡ widtha {Hk(·)} =
(∫ fN−fNHk(f ) df
)2
∫ fN−fNH2k(f ) df
=∆t∑N−1
τ=−(N−1)(h ? hk,τ )2
• for basic and weighted multitaper estimators, measures are
BH =∆t∑N−1
τ=−(N−1)
(1K
∑K−1k=0 h ? hk,τ
)2
and
BH =∆t∑N−1
τ=−(N−1)
(∑K−1k=0 dkh ? hk,τ
)2
SAPA2e–372, 373 VIII–10
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Basics of Multitapering: VI
• rationale for averaging eigenspectra is to produce estimator ofS(f ) with variance smaller than any single eigenspectrum
• making use of Exercise [2.1e] (Problem 4), have
var {S(mt)(f )} =1
K2
K−1∑k=0
var {S(mt)k (f )} +
2
K2
∑j<k
cov {S(mt)j (f ), S
(mt)k (f )}
• assume S(·) locally constant about f
• for j 6= k, can argue cov {S(mt)j (f ), S
(mt)k (f )} ≈ 0 if
N−1∑t=0
hj,thk,t = 0, i.e., if tapers are orthogonal
• var {S(mt)k (f )} ≈ S2(f ) =⇒ var {S(mt)(f )} ≈ S2(f )/K
SAPA2e–373, 374 VIII–11
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Basics of Multitapering: VII
• corresponding result for weighted multitaper estimator is
var {S(wmt)(f )} ≈ S2(f )
K−1∑k=0
d2k,
which reduces to
var {S(mt)(f )} ≈ S2(f )
K
by setting dk = 1/K
SAPA2e–374 VIII–12
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Basics of Multitapering: VIII
• to determine approximate distribution of S(mt)(f ), note that,since each eigenspectrum is a direct spectral estimator,
S(mt)k (f )
d= S(f )χ2
2/2 for 0 < f < fNasymptotically as N →∞• factoid: if χ2
ν1& χ2
ν2are independent chi-square RVs with ν1
& ν2 DOFs, then χ2ν1
+χ2ν2
is chi-square RV with ν1 + ν2 DOF
• assuming mutual independence, implies
S(mt)(f )d= S(f )χ2
2K/2K for 0 < f < fNasymptotically as N →∞
SAPA2e–374, 375 VIII–13
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Basics of Multitapering: IX
• corresponding result for weighted multitaper estimator:
S(mt)(f )d= S(f )χ2
ν/ν for 0 < f < fNasymptotically as N →∞, where
ν =2∑K−1
k=0 d2k
(note that ν = 2K when dk = 1/K)
• two sets of orthogonal tapers in common use
− Slepian (DPSS) tapers (Thomson, 1982)
− sinusoidal tapers (Riedel and Sidorenko, 1995)
• both are termed constructive multitaper estimators (as op-posed to deconstructed weighted multitaper estimators to bediscussed later on)
SAPA2e–375 VIII–14
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Slepian Multitapers: I
• tapers minimize spectral window sidelobes
• for fixed design bandwidth 2W , measure sidelobes via
β2k(W ) ≡
∫W−W Hk(f ) df∫ fN−fNHk(f ) df
• for given W and N , {h0,t} maximizes β20(W )
• {hk,t} maximizes β2k(W ) amongst sequences orthogonal to
{h0,t}, {h1,t}, . . ., {hk−1,t}
SAPA2e–378, 95 VIII–15
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Slepian Multitapers: II
• computation of {hk,t}’s requires solution of Ahk = β2k(W )hk,
where hTk =[hk,0, · · · , hk,N−1
]and
At′,t =sin(2πW (t′ − t))
π(t′ − t)is (t′, t)th element of N ×N matrix A
• can solve using inverse iteration (stable, but slow), numericalintegration (Thomson, 1982) or tridiagonal formulation (fast!)
SAPA2e–95 VIII–16
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Slepian Multitapers: III
• number of {hk,t}’s with good leakage protection is 2NW ∆t−1or less
• strategy & considerations for picking K
− set design half-bandwidthW as multiple J of spacing 1/N ∆tbetween Fourier frequencies, where J = 2, 3, 4, . . .
−W = J/N ∆t expressed as NW∆t = J (or just NW = J)
− set K < 2NW ∆t = 2J , noting that, as K increases, vari-ance decreases, but leakage gets worse
− increasing W implies a decrease in resolution, but gain moreleakage-free tapers to work with (i.e., can increase K)
• following plots use NW = 4 (i.e, 2NW = 8), for which Kshould be no greater than 7, but eighth taper {h7,t} also shown
SAPA2e–378, 379 VIII–17
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AR(4) Series, NW = 4 Slepian {h0,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.08
0.00
0.08
Sle
pian
tape
r
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–380 VIII–18
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AR(4) Series, NW = 4 Slepian {h1,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.08
0.00
0.08
Sle
pian
tape
r
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–380 VIII–19
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AR(4) Series, NW = 4 Slepian {h6,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.08
0.00
0.08
Sle
pian
tape
r
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–380 VIII–20
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NW = 4 Slepian H0(·), S(mt)0 (·), H(·) and S(mt)(·), K = 1
−60
−40
−20
0
20
dB
k = 0
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 1
SAPA2e–381, 384 VIII–21
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NW = 4 Slepian H1(·), S(mt)1 (·), H(·) and S(mt)(·), K = 2
−60
−40
−20
0
20
dB
k = 1
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 2
SAPA2e–381, 384 VIII–22
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NW = 4 Slepian H6(·), S(mt)6 (·), H(·) and S(mt)(·), K = 7
−60
−40
−20
0
20
dB
k = 6
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 7
SAPA2e–383, 385 VIII–23
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Sinusoidal Multitapers: I
• recall notion of smoothing window bias:
bW ≡∫ fN
−fNWm(f − f ′)S(f ′) df ′ − S(f )
≈ S′′(f )
2
∫ fN
−fNφ2Wm(φ) dφ =
S′′(f )
24β2W
(part of criterion used to derive Papoulis lag window)
• recall notion of spectral window bias (Exer. [211]):
b(f ) ≡ E{S(d)(f )} − S(f )
≈ S′′(f )
2
∫ fN
−fNφ2H(φ) dφ ≡ S′′(f )
24β2H
SAPA2e–413 VIII–24
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Sinusoidal Multitapers: II
• for given N , {h0,t} minimizes β2H0
• {hk,t} minimizes β2Hk amongst sequences orthogonal to
{h0,t}, {h1,t}, . . . , {hk−1,t}
• scheme due to Riedel & Sidorenko (1995), who actually workedwith continuous parameter processes (similiar to Papoulis, 1973)
• can approximate solutions well using
hk,t =
{2
N + 1
}1/2
sin
{(k + 1)π(t + 1)
N + 1
}, t = 0, 1 . . . , N−1,
which is very easy to compute!
SAPA2e–414, 415 VIII–25
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Sinusoidal Multitapers: III
• all {hk,t}’s give moderate leakage protection (like Hanning)
• strategy & considerations for picking K
− bandwidth BH ≈ (K+1)/(N+1) increases with K (dashed& solid red lines show BH/2 & (K + 1)/(2N + 2) on plots)
− leakage relatively unchanged as K increases
− can trade off variance and resolution by increasing K, whichdecreases variance and resolution (i.e., increases bandwidth)
• sinusoidal tapers vs. Slepian tapers
− 1 parameter (K) vs. 2 parameters (2W & K)
− moderate vs. adjustable leakage protection
− juggle resolution/variance vs. leakage/resolution/variance
− simple expression vs. need software to compute
SAPA2e–423 VIII–26
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AR(4) Series, Sinusoidal {h0,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
sine
tape
r
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–417 VIII–27
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AR(4) Series, Sinusoidal {h1,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
sine
tape
r
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–417 VIII–28
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AR(4) Series, Sinusoidal {h6,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
sine
tape
r
0 512 1024t
−0.2
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tape
red
serie
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SAPA2e–417 VIII–29
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Sinusoidal H0(·), S(mt)0 (·), H(·) and S(mt)(·), K = 1
−60
−40
−20
0
20
dB
k = 0
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 1
VIII–30
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Sinusoidal H1(·), S(mt)1 (·), H(·) and S(mt)(·), K = 2
−60
−40
−20
0
20
dB
k = 1
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 2
VIII–31
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Sinusoidal H6(·), S(mt)6 (·), H(·) and S(mt)(·), K = 7
−60
−40
−20
0
20
dB
k = 6
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 7
VIII–32
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Recovery of ‘Lost Information”: I
• Slepian tapers are solutions to Ahk = β2k(W )hk
• N orthonormal solutions h0, . . . , hN−1
• can order via eigenvalues (concentration measure):
1 > β20(W ) > β2
1(W ) > · · · > β2N−1(W ) > 0
• only first K < 2NW ∆t have β2k(W ) ≈ 1
• form V =[h0, h1, . . . , hN−1
]• V TV = IN restates orthonormality, where IN is N ×N iden-
tity matrix:N−1∑t=0
hj,thk,t =
{1, j = k;
0, j 6= k
SAPA2e–388 VIII–33
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Recovery of ‘Lost Information’: II
• since V T = V −1, also have V V T = IN , yielding
N−1∑k=0
hk,thk,t′ =
{1, t = t′;
0, t 6= t′.
• thus have
N−1∑k=0
N−1∑t=0
(hk,tXt
)2=
N−1∑t=0
X2t
N−1∑k=0
h2k,t︸ ︷︷ ︸
(∗)
=
N−1∑t=0
X2t
because (∗) – the energy across tapers – is unity
• following figures shows relative influence of Xt’s by plotting∑K−1k=0 h2
k,t versus t for K = 1, . . . , 8 (here NW ∆t = 4)
SAPA2e–388 VIII–34
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Decomposition of Slepian Taper Energy for K = 1
0 256 512 768 1024
t
0
1
SAPA2e–387 VIII–35
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Decomposition of Slepian Taper Energy for K = 2
0 256 512 768 1024
t
0
1
SAPA2e–387 VIII–36
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Decomposition of Slepian Taper Energy for K = 7
0 256 512 768 1024
t
0
1
SAPA2e–387 VIII–37
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Multitapering of White Noise: I
• assume X0, . . . , XN−1 is zero mean Gaussian white noise withunknown variance s0 and SDF S(f ) = s0 (here we set ∆t tounity for convenience)
• by several criteria, best estimate of s0 is∑N−1t=0 X2
t /N = s(p)0
• implies best estimate of S(f ) is s(p)0
• can obtain best estimator from S(p)(·) via∫ 1/2
−1/2S(p)(f ) df = s
(p)0 ;
i.e., ‘smoothing’ with Wm(f ) = 1
• Equation (392) says var {s(p)0 } = 2s2
0/N
SAPA2e–392 VIII–38
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Multitapering of White Noise: II
• let S(d)(·) be direct spectral estimator based upon {h0,t}
• smoothing S(d)(·) with Wm(f ) = 1 yields∫ 1/2
−1/2S(d)(f ) df =
N−1∑t=0
h20,tX
2t = s
(d)0 .
• since var {X2t } = 2s2
0 (Isserlis – Equation (32a)), have
var {s(d)0 } =
N−1∑t=0
var {h20,tX
2t } = 2s2
0
N−1∑t=0
h40,t = 2s2
0Ch/N,
where, as before, Ch ≡ N∑N−1t=0 h4
0,t
SAPA2e–392 VIII–39
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Multitapering of White Noise: III
• use Cauchy inequality∣∣∣∣N−1∑t=0
atbt
∣∣∣∣2 ≤ N−1∑t=0
|at|2N−1∑t=0
|bt|2 ,
with at = h20,t and bt = 1 to argue
∑N−1t=0 h4
0,t ≥ 1/N ; i.e.,
Ch ≥ 1, with equality if and only if h0,t = 1/√N
• can conclude
var {s(d)0 } = 2s2
0Ch/N > 2s20/N = var {s(p)
0 }for any nonrectangular taper
SAPA2e–392 VIII–40
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Multitapering of White Noise: IV
• claim: multitapering reclaims best estimator
• let {h0,t}, {h1,t}, . . . , {hN−1,t} be orthonormal
• let V be the N ×N matrix given by
V ≡
h0,0 h1,0 . . . hN−1,0
h0,1 h1,1 . . . hN−1,1... ... . . . ...
h0,N−1 h1,N−1 . . . hN−1,N−1
• orthonormality says V T V = IN & hence V V T = IN
• kth eigenspectrum:
S(mt)k (f ) ≡
∣∣∣∣∣∣N−1∑t=0
hk,tXte−i2πft
∣∣∣∣∣∣2
SAPA2e–393 VIII–41
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Multitapering of White Noise: V
• form S(mt)(·) by averaging all S(mt)k (·)’s:
S(mt)(f ) ≡ 1
N
N−1∑k=0
S(mt)k (f )
=1
N
N−1∑k=0
N−1∑t=0
hk,tXte−i2πft
N−1∑u=0
hk,uXuei2πfu
=
1
N
N−1∑t=0
N−1∑u=0
XtXu
N−1∑k=0
hk,thk,u
︸ ︷︷ ︸
1 if t = u; 0 if t 6= u
e−i2πf (t−u)
=1
N
N−1∑t=0
X2t = s
(p)0
SAPA2e–393 VIII–42
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Multitapering of White Noise: VI
• note: holds for any set of orthonormal tapers!
• as K increases, can study rate of decay
var {S(mt)(f )} = var
1
K
K−1∑k=0
S(mt)k (f )
=
1
K2
K−1∑j=0
K−1∑k=0
cov {S(mt)j (f ), S
(mt)k (f )}
• Exercise [8.8b] indicates how to compute this for white noise
SAPA2e–394, 395 VIII–43
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Multitapering of White Noise: VII
• following plot shows example for f = 1/4 using Slepian tapers
− N = 64; NW = 4; s0 = 1; S(f ) = 1
− thick curve: var {S(mt)(1/4)} vs. K
∗K = 1: var {S(mt)(1/4)} = S2(f ) = 1
∗K = N : var {S(mt)(1/4)} = 2/N.= 0.03
∗ curve agrees with these values
− thin curve: computed assuming
cov {S(mt)j (f ), S
(mt)k (f )} = 0 when j 6= k
• dashed vertical line marks Shannon number 2NW = 8
• two curves agree closely for K ≤ 2NW
• variance decreases slowly for K > 2NW (bias then can be badfor nonwhite processes)
SAPA2e–394, 395 VIII–44
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var {S(mt)(1/4)} versus Number K of Eigenspectra
0 16 32 48 64
K
10−2
10−1
100
varia
nce
of m
ultit
aper
est
imat
e
SAPA2e–394 VIII–45
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Multitapering of White Noise: VIII
• following plot shows 2nd example, which differs from 1st onlyin that NW = 16 rather than 4
• two examples similar in some aspects, but differ in others
− NW = 4 achieves minimum only when K = 64, whereasNW = 16 achieves minimum at both K = 32 and K = 64
− NW = 4 after Shannon number has smaller values, whereasNW = 16 after Shannon has larger values (except K = 64)
• 2nd example shows that, if U0, U1, . . . are correlated RVs withcommon mean µ & common variance, variance of sample mean
µN =1
N
N−1∑n=0
Un
can actually increase as N increases!
SAPA2e–394, 395 VIII–46
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var {S(mt)(1/4)} versus Number K of Eigenspectra
0 16 32 48 64
K
10−2
10−1
100
varia
nce
of m
ultit
aper
est
imat
e
SAPA2e–394 VIII–47
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Multitapering of White Noise: IX
• following plot shows 3rd example, which differs from 1st & 2ndonly in that sinusoidal tapers are used rather than Slepian
• after accounting for numerical precision, R declares that, atall K, var {S(mt)(1/4)} is identical for Slepian tapers withNW = 16 and sinusoidal tapers!
• Q: why?
• A: hmmm . . .
SAPA2e–426 VIII–48
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var {S(mt)(1/4)} versus Number K of Eigenspectra
0 16 32 48 64
K
10−2
10−1
100
varia
nce
of m
ultit
aper
est
imat
e
SAPA2e–426 VIII–49
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Quadratic Spectral Estimators: I
• provides important motivation for multitapering
• let X0, . . . , XN−1 be portion of real-valued stationary processwith zero mean 0, SDF S(·) and ACVS {sτ}• for fixed f , define Zt ≡ Xte
i2πft∆t
• Exercise [5.13a]: {Zt} stationary with
SZ(f ′) = S(f − f ′) and sZ,τ = sτei2πfτ ∆t
(recall that S(·) is periodic with a period of 2fN )
• note: SZ(0) = S(f ), so can estimate S(f ) by estimating SZ(·)at f = 0
SAPA2e–395, 396, 177 VIII–50
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Quadratic Spectral Estimators: II
• let Z be vector with tth element Zt
• let ZH be its Hermitian transpose:
ZH ≡[Z∗0 , . . . , Z
∗N−1
]note: if A real-valued matrix, then AH = AT
• since XtXt′∆t has same units as S(f ), consider
S(q)(f ) ≡ S(q)Z (0) ≡ ∆t
N−1∑s=0
N−1∑t=0
Z∗sQs,tZt = ∆tZHQZ,
where Qs,t is (s, t)th element of weight matrix Q
• S(q)(f ) called quadratic spectral estimator
SAPA2e–396 VIII–51
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Quadratic Spectral Estimators: III
• assumptions about N ×N matrix Q:
− Qs,t is real-valued
− Q is symmetric; i.e., Qs,t = Qt,s− Qs,t does not depend on {Zt}
• if Q positive semidefinite (PSD), then S(q)(f ) ≥ 0, which isobviously a desirable property in view of S(f ) ≥ 0
• three examples of quadratic estimators
SAPA2e–396 VIII–52
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Quadratic Spectral Estimators: IV
1. lag window estimator (need not be PSD):
S(lw)(f ) ≡ ∆t
N−1∑τ=−(N−1)
wm,τ s(d)τ e−i2πfτ ∆t
= ∆t
N−1∑s=0
N−1∑t=0
wm,t−shsXshtXte−i2πf (s−t) ∆t
= ∆tN−1∑s=0
N−1∑t=0
Z∗s hswm,s−tht︸ ︷︷ ︸= Qs,t
Zt
SAPA2e–396 VIII–53
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Quadratic Spectral Estimators: V
2. direct spectral estimator (always PSD – special case of S(lw)(f )with wm,τ = 1):
S(d)(f ) ≡ ∆t
∣∣∣∣∣∣N−1∑t=0
htXte−i2πft∆t
∣∣∣∣∣∣2
= ∆t
N−1∑s=0
N−1∑t=0
Z∗s hsht︸︷︷︸= Qs,t
Zt
3. basic multitaper estimator (always PSD):
S(mt)(f ) ≡ ∆t
K
K−1∑k=0
∣∣∣∣∣∣N−1∑t=0
hk,tXte−i2πft∆t
∣∣∣∣∣∣2
= ∆t
N−1∑s=0
N−1∑t=0
Z∗s
K−1∑k=0
hk,shk,tK︸ ︷︷ ︸
= Qs,t
Zt
SAPA2e–397, 372 VIII–54
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Quadratic Spectral Estimators: VI
• goal: set Q so S(q)(·) unbiased and has small variance
• to get S(q)(f ) ≥ 0, assume Q is PSD: let K = rank of Q, andassume 1 ≤ K ≤ N (rules out K = 0, which is not of interest)
• matrix theory: there exists an N ×N orthonormal matrix HNsuch that HT
NQHN = DN , where DN is a diagonal matrixwith diagonal elements
d0 ≥ d1 ≥ · · · ≥ dK−1 > 0 and dK = · · · = dN−1 = 0
with each dk being an eigenvalue of Q
• Exercise [397a]: as a result of the above, can write
Q = HDKHT ,
where H is N×K matrix (the 1st K columns of HN ), and DKis a diagonal matrix with diagonal entries d0, d1, . . . , dK−1
SAPA2e–397 VIII–55
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Quadratic Spectral Estimators: VII
• Exercise [397b]: substituting HDKHT for Q in
S(q)(f ) = ∆tZHQZ
yields
S(q)(f ) = ∆tZHHDKHTZ = ∆t
K−1∑k=0
dk
∣∣∣∣∣∣N−1∑t=0
hk,tXte−i2πft∆t
∣∣∣∣∣∣2
,
where hk,t is the (k, t)th element of H
• reusing definition S(mt)k (f ) = ∆t
∣∣∣∑N−1t=0 hk,tXte
−i2πft∆t∣∣∣2,
can write
S(q)(f ) =
K−1∑k=0
dkS(mt)k (f ) = S(wmt)(f )
SAPA2e–397, 398 VIII–56
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Quadratic Spectral Estimators: VIII
• conclusion: can write any PSD quadratic estimator
S(q)(f ) = ∆tZHQZ
with weight matrix Q of rank K as a weighted multitaper es-timator with
− weights dk given by positive eigenvalues of Q
− tapers {hk,t} given by associated eigenvectors of Q (tapersare mutually orthonormal)
• S(q)(f ) formulated as S(wmt)(f ) termed a deconstructed mul-titaper estimator
• opposing concept is constructive multitaper estimator for whichtapers are formulated explicitly (e.g., Slepian or sinusoidal)
SAPA2e–398 VIII–57
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Deconstructed Multitaper Spectral Estimators: I
• as an example of a deconstructed multitaper spectral estimator,consider an m = 179 Parzen lag window estimator used with adirect spectral estimator employing a Hanning data taper
• following plot shows estimate for AR(4) time series of Fig-ure 36(e)
SAPA2e–435 VIII–58
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Parzen S(lw)(·) with Hanning S(d)(·) & AR(4) SDF
0.0 0.5f
−60
−40
−20
0
20
AR
(4)
spec
tra
(dB
)
m = 179
SAPA2e–327 VIII–59
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Deconstructed Multitaper Spectral Estimators: II
• weight matrix Q for m = 179 Parzen lag window estimator isof full rank so K = N , but, as shown in next plot, eigenval-ues decay rapidly to zero, implying that effective rank is muchsmaller
SAPA2e–435 VIII–60
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Eigenvalues of Q for Lag Window Estimator
●
●
●
●
●
●●
●● ● ● ● ● ● ● ●
k
eige
nval
ue
0 5 10 15
0.00
0.05
0.10
0.15
0.20
0.25
0.30
SAPA2e–435 VIII–61
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Deconstructed Multitaper Spectral Estimators: III
• following plots show first seven eigenvectors of Q – these serveas data tapers in weighted multitaper representation for lagwindow estimator
SAPA2e–435, 437, 438 VIII–62
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AR(4) Series, Deconstructed {h0,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
deco
n ta
per
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–436 VIII–63
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AR(4) Series, Deconstructed {h1,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
deco
n ta
per
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–436 VIII–64
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AR(4) Series, Deconstructed {h6,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
deco
n ta
per
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–436 VIII–65
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Deconstructed Multitaper Spectral Estimators: IV
• following plots show eigenspectra based on seven data tapers,along with weighted multitaper estimates of increasing order K
• weights dk determined by eigenvalues dk, but renormalized foreach K to sum to unity:
dk =dk∑K−1
k′=0dk′, k = 0, . . . , K − 1
(renormalization required because∑N−1k=0 dk = 1)
• plots also show spectral windows for eigenspectra and weightedmultitaper estimates, with vertical red dashed lines indicatingstandard bandwidth measures (either BHk or BH)
SAPA2e–435, 437, 438 VIII–66
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Deconstructed H0(·), S(mt)0 (·), H(·) and S(wmt)(·), K = 1
−60
−40
−20
0
20
dB
k = 0
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 1
VIII–67
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Deconstructed H1(·), S(mt)1 (·), H(·) and S(wmt)(·), K = 2
−60
−40
−20
0
20
dB
k = 1
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 2
VIII–68
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Deconstructed H6(·), S(mt)6 (·), H(·) and S(wmt)(·), K = 7
−60
−40
−20
0
20
dB
k = 6
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 7
SAPA2e–437 VIII–69
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Deconstructed Multitaper Spectral Estimators: V
• because eigenvalues decrease to zero rapidly, K = 7 multitaperestimate is a reasonable approximation to lag window estimate,as next plot shows
SAPA2e–437, 438 VIII–70
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Parzen S(lw)(·) with K = 7 S(wmt)(·) Approximation
0.0 0.5f
−60
−40
−20
0
20
AR
(4)
spec
tra
(dB
)
SAPA2e–437, 327 VIII–71
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Deconstructed Multitaper Spectral Estimators: VI
• lag window estimators are sometimes well approximated by low-order weighted multitaper estimators, suggesting that
‘lag-windowing and multiple-data-windowing are roughlyequivalent for smooth spectrum estimation’
(title of article by McCloud et al.,1999)
SAPA2e–438 VIII–72
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Quadratic Spectral Estimators: XI
• Q: what conditions on Q ensure S(q)(f ) has good bias andvariance properties?
• let’s consider line of thought leading to Slepian tapers (Bronez,1985)
SAPA2e–395 VIII–73
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First Moment of S(q)(·): I
• since S(q)(·) has a weighted multitaper representation, an ap-peal to results stated for the latter says that
E{S(q)(f )} =
∫ fN
−fNH(f − f ′)S(f ′) df ′,
where
H(f ) ≡K−1∑k=0
dkHk(f ) with Hk(f ) ≡ ∆t
∣∣∣∣∣∣N−1∑t=0
hk,te−i2πft∆t
∣∣∣∣∣∣2
SAPA2e–398 VIII–74
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First Moment of S(q)(·): II
• Exercise [8.12] gives equivalent ‘time domain’ expression:
E{S(q)(f )} = ∆t tr {QΣZ} = ∆t tr {HDKHTΣZ} = ∆t tr {DKHTΣZH},where tr = trace and ΣZ = covariance matrix for Zt’s,
− last equation above uses following result: if A and B havedimensions M ×N and N ×M , then tr {AB} = tr {BA}
• equating frequency and time domain expressions yields∫ fN
−fNH(f − f ′)S(f ′) df ′ = ∆t tr {DKHTΣZH}
SAPA2e–398 VIII–75
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First Moment of S(q)(·): III
• for general {Xt}, can get handle on first moment by incorpo-rating notion of resolution (key idea!)
• given resolution bandwidth 2W > 0, seek Q’s so
E{S(q)(f )} ≈ 1
2W
∫ f+W
f−WS(f ′) df ′ ≡ S(f ),
i.e., no longer seek E{S(q)(f )} ≈ S(f )
• rationale
− ‘regularizes’ SDF estimation problem: S(·) smooth to somedegree, whereas S(·) need not be
− incorporates filter bandwidth in filtering interpretation ofS(·) (Section 5.6)
SAPA2e–398, 399 VIII–76
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First Moment of S(q)(·): IV
• strategy
− set resolution bandwidth 2W appropriately
− optimize bias/variance within limitations imposed by choiceof 2W
• basically we are giving up finest possible resolution of 1/N ∆tto get handle on bias/variance
SAPA2e–399 VIII–77
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First Moment of S(q)(·): V
• definition for bias that incorporates notion of resolution is
b{S(q)(f )} = E{S(q)(f )} − S(f ) = E{S(q)(f )} − 1
2W
∫ f+W
f−WS(f ′) df ′
• since
E{S(q)(f )} =
∫ fN
−fNH(f − f ′)S(f ′) df ′,
insisting upon b{S(q)(f )} = 0 would require
H(f ′) =
{1/(2W ), |f ′| ≤ W ;
0, W < |f ′| ≤ fN
• H(·) is sum of functions Hk(·) that are based on Fourier trans-
forms of index-limited sequences – hence H(·) cannot be exactlyband-limited
SAPA2e–399 VIII–78
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First Moment of S(q)(·): VI
• game plan
− insist that S(q)(f ) be unbiased for white noise
− break up bias into two components (yet to be defined, butwe’ll end up calling them ‘broad-band bias’ and ‘local bias’)
− develop bounds for both components for colored noise
SAPA2e–399 VIII–79
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Unbiasedness for White Noise: I
• for white noise S(f ) = s0 ∆t for all f , so
S(f ) =1
2W
∫ f+W
f−WS(f ′) df ′ =
1
2W
∫ f+W
f−Ws0 ∆t df ′ = s0 ∆t
• since
E{S(q)(f )} =
∫ fN
−fNH(f − f ′)S(f ′) df ′ = s0 ∆t
∫ fN
−fNH(f − f ′) df ′,
have unbiasedness if ∫ fN
−fNH(f ′) df ′ = 1
SAPA2e–399 VIII–80
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Unbiasedness for White Noise: II
• noting that∫ fN
−fNH(f ′) df ′ =
∫ fN
−fN
K−1∑k=0
dkHk(f ′) df ′ =
K−1∑k=0
dk
because ∫ fN
−fNHk(f ′) df ′ =
N−1∑t=0
h2k,t = 1,
requirement for unbiasedness is
K−1∑k=0
dk = 1
SAPA2e–399, 400 VIII–81
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Unbiasedness for White Noise: III
• alternative approach: since ΣZ = s0IN for white noise, can use
E{S(q)(f )} = ∆t tr {DKHTΣZH}= s0 ∆t tr {DKHTH}= s0 ∆t tr {DK}
because columns {hk,t} of H are orthonormal
• use of the fact that
tr {DK} =
K−1∑k=0
dk
leads to the same requirement for white noise unbiasedness:K−1∑k=0
dk = 1
SAPA2e–400 VIII–82
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Broad-Band & Local Bias: I
• using new notion of bias, have
b{S(q)(f )} ≡ E{S(q)(f )} − S(f )
=
∫ fN
−fNH(f − f ′)S(f ′) df ′ − 1
2W
∫ f+W
f−WS(f ′) df ′
=
∫ f+W
f−W
[H(f − f ′)− 1
2W
]S(f ′) df ′
+
∫f ′ 6∈[f−W,f+W ]
H(f − f ′)S(f ′) df ′
≡ b(l){S(q)(f )}︸ ︷︷ ︸local bias
+ b(b){S(q)(f )}︸ ︷︷ ︸broad-band bias
• to bound bias terms, assume S(·) bounded by Smax; i.e., S(f ) ≤Smax <∞ for all f
SAPA2e–400 VIII–83
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Broad-Band and Local Bias: II
• bound on magnitude of local bias:∣∣∣b(l){S(q)(f )}∣∣∣ =
∣∣∣∣∣∫ f+W
f−W
[H(f − f ′)− 1
2W
]S(f ′) df ′
∣∣∣∣∣≤∫ f+W
f−W
∣∣∣∣H(f − f ′)− 1
2W
∣∣∣∣S(f ′) df ′
≤ Smax
∫ W
−W
∣∣∣∣H(f ′′)− 1
2W
∣∣∣∣ df ′′;integral gives useful measure of local bias
• local bias small if H(f ) ≈ 1/2W over [−W,W ]
SAPA2e–400 VIII–84
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Broad-Band and Local Bias: III
• bound on broad-band bias (must be positive!):
b(b){S(q)(f )} =
∫f ′ 6∈[f−W,f+W ]
H(f − f ′)S(f ′) df ′
≤ Smax
∫f ′ 6∈[f−W,f+W ]
H(f − f ′) df ′
= Smax
∫f 6∈[−W,W ]
H(f ′′) df ′′
= Smax
(∫ fN
−fNH(f ′′) df ′′ −
∫ W
−WH(f ′′) df ′′
)= Smax
(tr {DK} − tr {DKHTΣ(bl)H}
),
where Σ(bl) arises from the following argument
SAPA2e–401 VIII–85
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Broad-Band and Local Bias: IV
• let {Xt} be band-limited white noise; i.e., has SDF and ACVS
S(bl)(f ) ≡
{∆t, |f | ≤ W ;
0, W < |f | ≤ fN ,and s
(bl)τ ≡
{2W ∆t, τ = 0;sin (2πWτ ∆t)
πτ τ 6= 0.
• for this SDF (and letting f = 0 so Zt = Xt), have
E{S(q)(0)} =
∫ fN
−fNH(0− f ′)S(bl)(f ′) df ′
= ∆t
∫ W
−WH(f ′′) df ′′ = ∆t tr {DKHTΣ(bl)H},
where (j, k)th element of Σ(bl) is s(bl)j−k
SAPA2e–401 VIII–86
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Minimizing Broad-Band Bias Measure: I
• measure of broad-band bias (similar concept to leakage) is thus
tr {DK} − tr {DKHTΣ(bl)H}
• insisting on tr {DK} = 1 ensures unbiasedness for white noise
• to minimize broad-band bias under this restriction,
maximize tr {DKHTΣ(bl)H} subject to tr {DK} = 1
SAPA2e–401, 402 VIII–87
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Minimizing Broad-Band Bias Measure: II
• Exercsie [8.13] gives solution:
− set K = 1
−H = h0 is normalized eigenvector associated with largesteigenvalue λ0(N,W ) of Σ(bl)
− eigenvector is zeroth order Slepian sequence (technically: fi-nite subsequence of DPSS)
− broad-band bias measure = 1−λ0(N,W ), where λ0(N,W )is the concentration ratio (Exercise [402])
• solution conflicts with variance in white noise case: as K in-creases, variance decreases
• reasonable balance: use K orthonormal Slepian tapers
SAPA2e–402, 403 VIII–88
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Managing Bias and Variance
• broad-band bias: Exercise [8.14] says this can be measured by
1− 1
K
K−1∑k=0
λk(N,W );
recall that λk(N,W ) is close to unity as long as K < 2NW ∆t
• variance: previous argument says
S(mt)(f )d=S(f )
2Kχ2
2K
approximately if S(·) not rapidly varying over [f −W, f +W ];
thus have var {S(mt)(f )} ≈ S2(f )/K
• local bias: small if H(f ) ≈ 12W over [−W,W ]; as following
figures show, decreases as K increases; here NW = 4 withN = 1024 for which 1
2W = N8 = 128
.= 21 dB
SAPA2e–403 VIII–89
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NW = 4 Slepian H0(·), S(mt)0 (·), H(·) and S(mt)(·), K = 1
−60
−40
−20
0
20
dB
k = 0
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 1
SAPA2e–381, 384 VIII–90
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NW = 4 Slepian H1(·), S(mt)1 (·), H(·) and S(mt)(·), K = 2
−60
−40
−20
0
20
dB
k = 1
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 2
SAPA2e–381, 384 VIII–91
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NW = 4 Slepian H6(·), S(mt)6 (·), H(·) and S(mt)(·), K = 7
−60
−40
−20
0
20
dB
k = 6
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 7
SAPA2e–383, 385 VIII–92
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Adaptive Multitaper Estimation: I
• Section 8.5 discusses data-adaptive refinement to basic multi-tapering (developed for Slepian tapers)
• idea: weight eigenspectra adaptively according to need for leak-age suppression at each f
− if S(f ) relatively large, leakage not a concern, so can makeK large
− if S(f ) relatively small, leakage might be a concern, so shouldmake K small
SAPA2e–412 VIII–93
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Adaptive Multitaper Estimation: II
• adaptive multitaper estimator given by
S(amt)(f ) ≡∑K−1k=0 b2k(f )λkS
(mt)k (f )∑K−1
k=0 b2k(f )λk
where λk ≈ 1− 1/10j (with j ↓ as k ↑) &
bk(f ) =1
λk + (1− λk)s0 ∆tS(f )
≈ 1
1 + s0 ∆t10jS(f )
− λk’s downweight higher eigenspectra (slightly)
− s0 ∆t = average value of S(·)− bk(f ) small if 10jS(f )� s0 ∆t & large if 10jS(f )� s0 ∆t
• determine bk(f ) using preliminary estimate of S(·); can iterateto refine bk(f )’s if desired
SAPA2e–412 VIII–94
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Adaptive Multitaper Estimation: III
• assume
− S(mt)k (f )
d= S(f )χ2
2/2 for each eigenspectrum
− S(mt)k (f )’s are pairwise uncorrelated
• as before, assume S(amt)(f )d= aχ2
ν
• EDOF argument similar to S(lw)(·) and S (WOSA)(·) yields
ν =2(E{S(amt)(f )}
)2
var {S(amt)(f )}≈
2(∑K−1
k=0 b2k(f )λk
)2
∑K−1k=0 b4k(f )λ2
k
SAPA2e–412 VIII–95
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Motivation for WOSA: I
• letX0, . . . , XN−1 be sample of zero mean Gaussian white noise
• partition into NB blocks of size NS = N/NB:
X0, . . . , XNS−1; XNS, . . . , X2NS−1; . . . ; X(NB−1)NS, . . . , XN−1
b = 0 b = 1 b = 2 b = 3
0 256 512 768 1024
t
−4
−2
0
2
4
whi
te n
oise
SAPA2e–412 VIII–96
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Motivation for WOSA: II
• form periodograms for blocks b = 0, . . . , NB − 1:
S(p)b (fk) ≡ ∆t
NS
∣∣∣∣∣∣NS−1∑t=0
XbNS+te−i2πfkt∆t
∣∣∣∣∣∣2
0.0 0.5f
0
2
4
6
8
perio
dogr
am
b = 0
0.0 0.5f
b = 1
0.0 0.5f
b = 2
0.0 0.5f
b = 3
SAPA2e–412 VIII–97
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Motivation for WOSA: III
• form average of NB periodograms:
S(p)(fk) ≡ 1
NB
NB−1∑b=0
S(p)b (fk)
0.0 0.5f
0
2
4
6
8av
erag
ed p
erio
dogr
am
SAPA2e–412 VIII–98
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Motivation for WOSA: IV
• S(p)b (fk)
d= S(fk)χ2
2/2 for 0 < fk < fN
• S(p)b (fk) independent of S
(p)b′ (fk) for b′ 6= b
• χ2ν = Y 2
0 + Y 21 + · · · + Y 2
ν−1 for IID N(0, 1) RVs Yj implies
χ2ν1
+ χ2ν2
= χ2ν1+ν2
if χ2ν1
and χ2ν2
are independent
• with ν ≡ 2NB, have
NB−1∑b=0
S(p)b (fk)
d=S(fk)
2χ2
2NB=⇒ S(p)(fk) =
1
NB
NB−1∑b=0
S(p)b (fk)
d=S(fk)
νχ2ν
• note similarity to S(lw)(f )d= S(f )χ2
ν/ν
SAPA2e–412 VIII–99
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Definition of WOSA: I
•Welch’s overlapped segment averaging (WOSA)
− generalization applicable to other {Xt}’s− break time series into NB blocks
∗ each block has NS points
∗ blocks now allowed to overlap
− apply data taper {h0, . . . , hNS−1} to each block
− form direct spectral estimator for each block
− average NB estimators together
SAPA2e–427, 428 VIII–100
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Definition of WOSA: II
• for 0 ≤ l ≤ N −NS, let
S(d)l (f ) ≡ ∆t
∣∣∣∣∣∣NS−1∑t=0
htXt+le−i2πft∆t
∣∣∣∣∣∣2
•WOSA spectral estimator given by
S (WOSA)(f ) ≡ 1
NB
NB−1∑j=0
S(d)jn (f ),
where n is integer such that n(NB − 1) = N −NS• overlapping recovers ‘information’ lost in tapering (not obvi-
ously useful if ht = 1/√NS, i.e., S
(d)l (f ) = S
(p)l (f ))
• plots show example N = 100, NS = 32, NB = 5 and n = 17
SAPA2e–427, 428 VIII–101
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Block j = 4 Before and After Tapering
0 50 100t
SAPA2e–428 VIII–102
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First Moment Properties of WOSA: I
• first moment:
E{S (WOSA)(f )} =1
NB
NB−1∑j=0
E{S(d)jn (f )};
however,
E{S(d)jn (f )} =
∫ fN
−fNH(f − f ′)S(f ′) df ′
is the same for all j, where H(f ) ≡ |H(f )|2 is the spectralwindow associated with
{h0, . . . , hNS−1} ←→ H(·)
SAPA2e–428, 429 VIII–103
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First Moment Properties of WOSA: II
• thus have
E{S (WOSA)(f )} =
∫ fN
−fNH(f − f ′)S(f ′) df ′
• note that this expected value
− depends on just NS and data taper
− does not depend on N , NB or n
− can be biased if we make NS too small
SAPA2e–428, 429 VIII–104
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Variance of WOSA Estimator: I
• assume E{S (WOSA)(f )} ≈ S(f )
• variance is given by
var {S (WOSA)(f )} = cov
1
NB
NB−1∑j=0
S(d)jn (f ),
1
NB
NB−1∑k=0
S(d)kn (f ),
=
1
N2B
NB−1∑j=0
var {S(d)jn (f )}
+2
N2B
∑j<k
cov {S(d)jn (f ), S
(d)kn (f )}
• for 0 < f < fN can use var {S(d)jn (f )} ≈ S2(f )
SAPA2e–429 VIII–105
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Variance of WOSA Estimator: II
• assume:
− S(·) is locally constant
− f is not too close to 0 or fN− ht = 0 for t ≥ NS
• can argue (Exercise [8.25]):
cov {S(d)jn (f ), S
(d)kn (f )} ≈ S2(f )
∣∣∣∣∣∣NS−1∑t=0
htht+|k−j|n
∣∣∣∣∣∣2
;
i.e., depends on autocorrelation of {ht}
SAPA2e–429 VIII–106
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Variance of WOSA Estimator: III
• leads to the useful expression
var {S (WOSA)(f )} ≈ S2(f )
NB
1 +2
NB
∑j<k
∣∣∣∣∣∣NS−1∑t=0
htht+|k−j|n
∣∣∣∣∣∣2
=S2(f )
NB
1 + 2
NB−1∑m=1
(1− m
NB
) ∣∣∣∣∣∣NS−1∑t=0
htht+mn
∣∣∣∣∣∣2
via a ‘diagonalization’ argument
SAPA2e–429 VIII–107
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Distribution of WOSA Estimator
• assuming S (WOSA)(f )d= aχ2
ν, usual EDOF argument yields
ν =2(E{S (WOSA)(f )}
)2
var {S (WOSA)(f )}≈ 2NB
1 + 2∑NB−1m=1
(1− m
NB
) ∣∣∣∑NS−1t=0 htht+mn
∣∣∣2• specialize to Hanning data taper
− following plot shows EDOF versus percentage overlap
− 50% overlap gets close to maximum EDOF
− for Hanning + 50% overlap (i.e., n = NS/2):
ν ≈ 2NB
1 + 2 (1− 1/NB)∣∣∣∑NS/2−1
t=0 htht+NS/2
∣∣∣2 ≈36N2
B
19NB − 1≈ 3.79N
NS
• need 65% overlap for Slepian with NW = 4
SAPA2e–429, 430 VIII–108
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DOFs ν versus % of Block Overlap for N = 1024
0 50 100percentage of overlap
0
10
20
30
40
50
60
70
ν
NS = 256
NS = 128
NS = 64
SAPA2e–430 VIII–109
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S (WOSA)(·) with Hanning & 50% Overlap & AR(2) SDF
0.0 0.5f
−20
−10
0
10
20
AR
(2)
spec
tra
(dB
)
NS = 4
SAPA2e–431 VIII–110
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S (WOSA)(·) with Hanning & 50% Overlap & AR(2) SDF
0.0 0.5f
−20
−10
0
10
20
AR
(2)
spec
tra
(dB
)
NS = 16
SAPA2e–431 VIII–111
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S (WOSA)(·) with Hanning & 50% Overlap & AR(2) SDF
0.0 0.5f
−20
−10
0
10
20
AR
(2)
spec
tra
(dB
)
NS = 32
SAPA2e–431 VIII–112
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S (WOSA)(·) with Hanning & 50% Overlap & AR(2) SDF
0.0 0.5f
−20
−10
0
10
20
AR
(2)
spec
tra
(dB
)
NS = 64
SAPA2e–431 VIII–113
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S (WOSA)(·) with Hanning & 50% Overlap & AR(4) SDF
0.0 0.5f
−60
−40
−20
0
20
AR
(4)
spec
tra
(dB
)
NS = 64
SAPA2e–432 VIII–114
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S (WOSA)(·) with Hanning & 50% Overlap & AR(4) SDF
0.0 0.5f
−60
−40
−20
0
20
AR
(4)
spec
tra
(dB
)
NS = 128
SAPA2e–432 VIII–115
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S (WOSA)(·) with Hanning & 50% Overlap & AR(4) SDF
0.0 0.5f
−60
−40
−20
0
20
AR
(4)
spec
tra
(dB
)
NS = 256
SAPA2e–432 VIII–116
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S (WOSA)(·) with Hanning & 50% Overlap & AR(4) SDF
0.0 0.5f
−60
−40
−20
0
20
AR
(4)
spec
tra
(dB
)
NS = 512
SAPA2e–432 VIII–117
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Advantages/Disadvantages of WOSA
• widely used in spectrum analyzers
• advantages
− computationally efficient
− can handle large N
− can handle ‘locally stationary’ processes
− can be ‘robustified’ (Chave et al., 1987)
• disadvantages
− leakage if NS too small
− loss of resolution if NS too small
SAPA2e–434 VIII–118
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Deconstructed Multitaper Spectral Estimators: VII
• as 2nd example of deconstructed multitaper estimator, considerWOSA estimator using a Hanning data taper for N = 1024with block size NS = 256 and 50% overlap, yielding NB = 7blocks
• following plot redisplays estimate for AR(4) time series of Fig-ure 36(e)
SAPA2e–435, 438 VIII–119
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S (WOSA)(·) with Hanning & 50% Overlap & AR(4) SDF
0.0 0.5f
−60
−40
−20
0
20
AR
(4)
spec
tra
(dB
)
NS = 256
SAPA2e–432 VIII–120
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Deconstructed Multitaper Spectral Estimators: VIII
• weight matrix Q for WOSA estimator has rank K = NB = 7,as following plot demonstrates because there are only 7 nonzeroeigenvalues
SAPA2e–435, 438 VIII–121
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Eigenvalues of Q for WOSA Estimator
++
++
++
+
+ + + + + + + + +
k
eige
nval
ue
0 5 10 15
0.00
0.05
0.10
0.15
SAPA2e–435 VIII–122
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Deconstructed Multitaper Spectral Estimators: IX
• following plots show eigenvectors associated with 7 nonzeroeigenvalues – these serve as data tapers in weighted multita-per representation for WOSA estimator
SAPA2e–435, 438 VIII–123
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AR(4) Series, Deconstructed {h0,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
deco
n ta
per
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–436 VIII–124
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AR(4) Series, Deconstructed {h1,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
deco
n ta
per
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–436 VIII–125
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AR(4) Series, Deconstructed {h6,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
deco
n ta
per
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–436 VIII–126
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Deconstructed Multitaper Spectral Estimators: X
• following plots show eigenspectra based on seven data tapers,along with weighted multitaper estimates of increasing order K
• because Q is of rank 7, weighted multitaper estimate of orderK = 7 is exactly the same as WOSA estimate
• weights dk determined by eigenvalues dk, but renormalized forK < 7 to sum to unity:
dk =dk∑K−1
k′=0dk′, k = 0, . . . , K − 1
(renormalization required because∑6k=0 dk = 1)
• plots also show spectral windows for eigenspectra and weightedmultitaper estimates, with vertical red dashed lines indicatingstandard bandwidth measures (either BHk or BH)
SAPA2e–435, 437, 438 VIII–127
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Deconstructed H0(·), S(mt)0 (·), H(·) and S(wmt)(·), K = 1
−60
−40
−20
0
20
dB
k = 0
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 1
VIII–128
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Deconstructed H1(·), S(mt)1 (·), H(·) and S(wmt)(·), K = 2
−60
−40
−20
0
20
dB
k = 1
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 2
VIII–129
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Deconstructed H6(·), S(mt)6 (·), H(·) and S(wmt)(·), K = 7
−60
−40
−20
0
20
dB
k = 6
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 7
VIII–130
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Ocean Wave Data: I
• sea level time series for which N = 1024, ∆t = 1/4 second andfN = 2 cycles per second
0 50 100 150 200 250
−10
000
500
1000
t (seconds)
rela
tive
heig
ht
SAPA2e–248 VIII–131
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Ocean Wave Data: II
• arguably best SDF estimate seen so far is Gaussian S(lw)(·)based on S(d)(·) using NW = 2 Slepian data taper
• effective bandwidth of S(lw)(·) given by BU.= 0.135 Hz
• following plot shows basic multitaper estimate S(mt)(·)− set NW = 4 (resolution not main concern)
− maximum of 7 possible reasonable tapers, but S(mt)6 (·) poor
at high frequencies
− set K = 6, yielding ν = 12 EDOF
SAPA2e–439, 441, 443 VIII–132
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S(mt)(·), NW = 4, K = 6 Slepian Tapers
0.0 0.5 1.0 1.5 2.0f
−40
−20
0
20
40
60
80
ocea
n w
ave
spec
tra
(dB
)
SAPA2e–442 VIII–133
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Ocean Wave Data: III
• next plots compare another S(mt)(·) estimate with Parzen lag
window estimate S(lw)(·)− set NW = 6 and K = 10 so ν = 20
− bandwidth of S(lw)(·) is 0.049 Hz ≈ 2W.= 0.047 Hz
− good agreement between S(mt)(·) and S(lw)(·)
SAPA2e–443 VIII–134
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S(mt)(·), NW = 6, K = 10 Slepian Tapers
0.0 0.5 1.0 1.5 2.0f
−40
−20
0
20
40
60
80
ocea
n w
ave
spec
tra
(dB
)
SAPA2e–442 VIII–135
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Parzen S(lw)(·), m = 150, using NW = 2 Slepian S(d)(·)
0.0 0.5 1.0 1.5 2.0f
−40
−20
0
20
40
60
80
ocea
n w
ave
spec
tra
(dB
)
SAPA2e–341 VIII–136
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S(mt)(·), NW = 6, K = 10 Slepian Tapers
0.0 0.5 1.0 1.5 2.0f
−40
−20
0
20
40
60
80
ocea
n w
ave
spec
tra
(dB
)
SAPA2e–442 VIII–137
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Ocean Wave Data: IV
• next two plots show
1. adaptive multitaper estimate S(amt)(·) with NW = 4 andK = 7, along with 95% confidence intervals (CIs)
2. EDOFs ν versus f
• wider CIs for f > 1 Hz due to decreasing degrees of freedom
• final plot compares S(amt)(·) with Parzen S(lw)(·) – good over-all agreement between estimates
SAPA2e–443 VIII–138
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S(amt)(·), NW = 4, K = 7 Slepian Tapers
0.0 0.5 1.0 1.5 2.0f
−40
−20
0
20
40
60
80
ocea
n w
ave
spec
tra
(dB
)
SAPA2e–442 VIII–139
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Equivalent Degrees of Freedom ν for S(amt)(·)
0.0 0.5 1.0 1.5 2.0f
0
4
8
12
16
degr
ees
of fr
eedo
m
SAPA2e–442 VIII–140
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S(amt)(·) and Parzen S(lw)(·)
0.0 0.5 1.0 1.5 2.0f
−40
−20
0
20
40
60
80
ocea
n w
ave
spec
tra
(dB
)
VIII–141
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Ocean Wave Data: V
• recall that effective bandwidth of Gaussian-based S(lw)(·) isBU
.= 0.135 Hz
• for WOSA estimator, effective bandwidth is BH, which de-pends upon selected data taper {ht} and block size NS
• following plot shows BH versus NS for Hanning data taper,along with dashed line showing BU for Gaussian lag windowestimate
SAPA2e–440 VIII–142
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BH versus NS for Hanning Data Taper (∆t = 0.25 sec)
40 60 80 100 120block size
0.10
0.15
0.20
0.25
band
wid
th
SAPA2e–440 VIII–143
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Ocean Wave Data: VI
• NS = 61 yields BH.= 0.134 Hz (closest to BU
.= 0.135 Hz)
• setting n = 30 gives blocks that overlap by ≈ 50%
• let X ′t = Xt −X , i.e., centered time series
− block 1: X ′0, X′1, . . . , X
′60
− block 2: X ′30, X′31, . . . , X
′90
− block 3: X ′60, X′91, . . . , X
′120
− ...
− block 33: X ′960, X′961, . . . , X
′1020
• redefining last block to be X ′963, X′964, . . . , X
′1023 allows use of
entire time series
SAPA2e–440 VIII–144
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Hanning-based S (WOSA)(·) and Gaussian S(lw)(·)
0.0 0.5 1.0 1.5 2.0f
−40
−20
0
20
40
60
80
ocea
n w
ave
spec
tra
(dB
)
SAPA2e–441 VIII–145
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Ocean Wave Data: VII
• S (WOSA)(f ) has EDOF ν.= 62.6, whereas S(lw)(f ) has ν
.= 34.4
• recall of that variance is inversely proportional to ν
• 95% CIs based on S (WOSA)(f ) are tighter
• greater variability on explanation for S(lw)(·) being somewhatmore wobbly in appearance than S (WOSA)(f )
SAPA2e–440, 441 VIII–146