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MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
Spectral Analysis ofNonstationary Time Series
DAVID S. STOFFERDepartment of StatisticsUniversity of Pittsburgh
Ori Rosen Sally WoodUTEP U.Melbourne
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
IN THE BEGINNING ...
Original problem was to do local spectral analysis for a multivariate pointprocess, X t = e j ; for j = 1; : : : ; p, where e j = (0; : : : ; 0; 1; 0; : : : ; 0)0.
It ain’t easy.
Series of papers:
Rosen, O. & Stoffer, D.S. (2007). Automatic Estimation of MultivariateSpectra via Smoothing Splines. Biometrika, 335 – 345.
Rosen, O., Stoffer, D.S. & Wood, S. (2009). Local Spectral Analysis viaa Bayesian Mixture of Smoothing Splines. JASA, 104, 249 – 262.
Rosen, O., Wood, S. & Stoffer, D.S. (2012). AdaptSPEC: AdaptiveSpectral Estimation for Nonstationary Time Series. JASA, 107,1575–1589.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
IN THE BEGINNING ...
Original problem was to do local spectral analysis for a multivariate pointprocess, X t = e j ; for j = 1; : : : ; p, where e j = (0; : : : ; 0; 1; 0; : : : ; 0)0.
It ain’t easy.
Series of papers:
Rosen, O. & Stoffer, D.S. (2007). Automatic Estimation of MultivariateSpectra via Smoothing Splines. Biometrika, 335 – 345.
Rosen, O., Stoffer, D.S. & Wood, S. (2009). Local Spectral Analysis viaa Bayesian Mixture of Smoothing Splines. JASA, 104, 249 – 262.
Rosen, O., Wood, S. & Stoffer, D.S. (2012). AdaptSPEC: AdaptiveSpectral Estimation for Nonstationary Time Series. JASA, 107,1575–1589.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
IN THE BEGINNING ...
Original problem was to do local spectral analysis for a multivariate pointprocess, X t = e j ; for j = 1; : : : ; p, where e j = (0; : : : ; 0; 1; 0; : : : ; 0)0.
It ain’t easy.
Series of papers:
Rosen, O. & Stoffer, D.S. (2007). Automatic Estimation of MultivariateSpectra via Smoothing Splines. Biometrika, 335 – 345.
Rosen, O., Stoffer, D.S. & Wood, S. (2009). Local Spectral Analysis viaa Bayesian Mixture of Smoothing Splines. JASA, 104, 249 – 262.
Rosen, O., Wood, S. & Stoffer, D.S. (2012). AdaptSPEC: AdaptiveSpectral Estimation for Nonstationary Time Series. JASA, 107,1575–1589.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
El Niño – Southern Oscillation
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
SPECTRAL ESTIMATION – STATIONARY CASE
For a stationary time series with bounded spectral density f (�), givenmean-zero data fX1; : : : ;Xng, let Pn(�k ) denote the periodogram. Forlarge n , approximately
Pn(�k ) = f (�k )Uk
where Ukiid� Gamma(1; 1).
Taking logs, we have a GLM
y(�k ) = g(�k ) + �k
where y(�k ) = logPn(�k ) , g(�k ) = log f (�k ) and �k are iid log(�22=2)s.
Want to fit the model with the constraint that g(�) is smooth. Wahba (1980)suggested smoothing splines. This can be done in a Bayesian framework.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
SPECTRAL ESTIMATION – STATIONARY CASE
For a stationary time series with bounded spectral density f (�), givenmean-zero data fX1; : : : ;Xng, let Pn(�k ) denote the periodogram. Forlarge n , approximately
Pn(�k ) = f (�k )Uk
where Ukiid� Gamma(1; 1).
Taking logs, we have a GLM
y(�k ) = g(�k ) + �k
where y(�k ) = logPn(�k ) , g(�k ) = log f (�k ) and �k are iid log(�22=2)s.
Want to fit the model with the constraint that g(�) is smooth. Wahba (1980)suggested smoothing splines. This can be done in a Bayesian framework.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
SPECTRAL ESTIMATION – STATIONARY CASE
For a stationary time series with bounded spectral density f (�), givenmean-zero data fX1; : : : ;Xng, let Pn(�k ) denote the periodogram. Forlarge n , approximately
Pn(�k ) = f (�k )Uk
where Ukiid� Gamma(1; 1).
Taking logs, we have a GLM
y(�k ) = g(�k ) + �k
where y(�k ) = logPn(�k ) , g(�k ) = log f (�k ) and �k are iid log(�22=2)s.
Want to fit the model with the constraint that g(�) is smooth. Wahba (1980)suggested smoothing splines. This can be done in a Bayesian framework.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
HOW IT’S DONE:(1) Need a Likelihood
Whittle LikelihoodSpectral density f (�), and mean-zero data x = fX1; : : : ;Xng, for large n ,
logL( f �� x ) � �mX
k=1
�log f (�k ) +
Pn(�k )
f (�k )
�;
where Pn(�k ) is the periodogram, �k = k=n , and k = 1; : : : ;m = bn�12c.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
HOW IT’S DONE:(2) Need Priors
Place a linear smoothing spline prior on log f (�):recall yn(�k ) = logPn(�k ) and g(�k ) = log f (�k )
g(�k ) = �0 + �1�k + h(�k ) linear [�] + nonlinear [h(�)]h(�) = �
R�
0W (u)du )W (u) is a GP with kernel depending on �.
�0 � N (0; �2�), �1 � 0, since (@g(�)=@�)j�=0 = 0.
h = Dβ, is a linear combination of basis functions whereh = (h(�1); : : : ; h(�m))
0, and the j th column of D isp2 cos(j�ν),
ν = (�1; : : : ; �m)0.
β � N (0; � 2I )
� 2 � U (0; c�2) is the smoothing parameter.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
HOW IT’S DONE:(2) Need Priors
Place a linear smoothing spline prior on log f (�):recall yn(�k ) = logPn(�k ) and g(�k ) = log f (�k )
g(�k ) = �0 + �1�k + h(�k ) linear [�] + nonlinear [h(�)]h(�) = �
R�
0W (u)du )W (u) is a GP with kernel depending on �.
�0 � N (0; �2�), �1 � 0, since (@g(�)=@�)j�=0 = 0.
h = Dβ, is a linear combination of basis functions whereh = (h(�1); : : : ; h(�m))
0, and the j th column of D isp2 cos(j�ν),
ν = (�1; : : : ; �m)0.
β � N (0; � 2I )
� 2 � U (0; c�2) is the smoothing parameter.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
HOW IT’S DONE:(2) Need Priors
Place a linear smoothing spline prior on log f (�):recall yn(�k ) = logPn(�k ) and g(�k ) = log f (�k )
g(�k ) = �0 + �1�k + h(�k ) linear [�] + nonlinear [h(�)]h(�) = �
R�
0W (u)du )W (u) is a GP with kernel depending on �.
�0 � N (0; �2�), �1 � 0, since (@g(�)=@�)j�=0 = 0.
h = Dβ, is a linear combination of basis functions whereh = (h(�1); : : : ; h(�m))
0, and the j th column of D isp2 cos(j�ν),
ν = (�1; : : : ; �m)0.
β � N (0; � 2I )
� 2 � U (0; c�2) is the smoothing parameter.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
HOW IT’S DONE:(2) Need Priors
Place a linear smoothing spline prior on log f (�):recall yn(�k ) = logPn(�k ) and g(�k ) = log f (�k )
g(�k ) = �0 + �1�k + h(�k ) linear [�] + nonlinear [h(�)]h(�) = �
R�
0W (u)du )W (u) is a GP with kernel depending on �.
�0 � N (0; �2�), �1 � 0, since (@g(�)=@�)j�=0 = 0.
h = Dβ, is a linear combination of basis functions whereh = (h(�1); : : : ; h(�m))
0, and the j th column of D isp2 cos(j�ν),
ν = (�1; : : : ; �m)0.
β � N (0; � 2I )
� 2 � U (0; c�2) is the smoothing parameter.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
HOW IT’S DONE:(2) Need Priors
Place a linear smoothing spline prior on log f (�):recall yn(�k ) = logPn(�k ) and g(�k ) = log f (�k )
g(�k ) = �0 + �1�k + h(�k ) linear [�] + nonlinear [h(�)]h(�) = �
R�
0W (u)du )W (u) is a GP with kernel depending on �.
�0 � N (0; �2�), �1 � 0, since (@g(�)=@�)j�=0 = 0.
h = Dβ, is a linear combination of basis functions whereh = (h(�1); : : : ; h(�m))
0, and the j th column of D isp2 cos(j�ν),
ν = (�1; : : : ; �m)0.
β � N (0; � 2I )
� 2 � U (0; c�2) is the smoothing parameter.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
HOW IT’S DONE:(2) Need Priors
Place a linear smoothing spline prior on log f (�):recall yn(�k ) = logPn(�k ) and g(�k ) = log f (�k )
g(�k ) = �0 + �1�k + h(�k ) linear [�] + nonlinear [h(�)]h(�) = �
R�
0W (u)du )W (u) is a GP with kernel depending on �.
�0 � N (0; �2�), �1 � 0, since (@g(�)=@�)j�=0 = 0.
h = Dβ, is a linear combination of basis functions whereh = (h(�1); : : : ; h(�m))
0, and the j th column of D isp2 cos(j�ν),
ν = (�1; : : : ; �m)0.
β � N (0; � 2I )
� 2 � U (0; c�2) is the smoothing parameter.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
SAMPLING SCHEMEThe parameters �0, β and �
2 are drawn from the posterior distributionp(�0;β; �
2�� y), where y = (yn(�1); : : : ; yn(�m))
0, using MCMC:
�0 and β are sampled jointly via an M-H step from
log p(�0;� � 2;y) /
�
mXk=1
h�0 + d
0kβ + exp
�yn(�k )� �0 � d
0kβ�i�
�2
0
2�2��
1
2� 2β0β;
where d 0k is the k th row of D .
�2 is sampled from the truncated inverse gamma distribution,
p(� 2�� β) / (� 2)�m=2 exp
��
1
2� 2β0β
�; �
22 (0; c�2 ]:
A consistency result can be established for the sampled posterior under restrictiveassumptions: The time series is short-memory Gaussian and � is known + otherminor details. Choudhuri et al. (2004). Bayesian Estimation of the Spectral Densityof a Time Series, JASA.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
SAMPLING SCHEMEThe parameters �0, β and �
2 are drawn from the posterior distributionp(�0;β; �
2�� y), where y = (yn(�1); : : : ; yn(�m))
0, using MCMC:
�0 and β are sampled jointly via an M-H step from
log p(�0;� � 2;y) /
�
mXk=1
h�0 + d
0kβ + exp
�yn(�k )� �0 � d
0kβ�i�
�2
0
2�2��
1
2� 2β0β;
where d 0k is the k th row of D .
�2 is sampled from the truncated inverse gamma distribution,
p(� 2�� β) / (� 2)�m=2 exp
��
1
2� 2β0β
�; �
22 (0; c�2 ]:
A consistency result can be established for the sampled posterior under restrictiveassumptions: The time series is short-memory Gaussian and � is known + otherminor details. Choudhuri et al. (2004). Bayesian Estimation of the Spectral Densityof a Time Series, JASA.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
SAMPLING SCHEMEThe parameters �0, β and �
2 are drawn from the posterior distributionp(�0;β; �
2�� y), where y = (yn(�1); : : : ; yn(�m))
0, using MCMC:
�0 and β are sampled jointly via an M-H step from
log p(�0;� � 2;y) /
�
mXk=1
h�0 + d
0kβ + exp
�yn(�k )� �0 � d
0kβ�i�
�2
0
2�2��
1
2� 2β0β;
where d 0k is the k th row of D .
�2 is sampled from the truncated inverse gamma distribution,
p(� 2�� β) / (� 2)�m=2 exp
��
1
2� 2β0β
�; �
22 (0; c�2 ]:
A consistency result can be established for the sampled posterior under restrictiveassumptions: The time series is short-memory Gaussian and � is known + otherminor details. Choudhuri et al. (2004). Bayesian Estimation of the Spectral Densityof a Time Series, JASA.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
AR(2) Example
� logPn — true g = log f - - - 95% credible intervals
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−10
−5
0
5
ν
log(f11
)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−6
−4
−2
0
2
4
ν
log(f22
)
frequency
0.0 0.2 0.4
12
34
frequency
spec
trum
from specified model
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
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ADAPTSPEC – AN ADAPTIVE METHODPiecewise Stationary
Locally Stationary Time Series
. . . . . .ξ0,m ξ1,m ξj−1,m ξj,m ξm,m
n1,m nj−1,m nj,m nm,m
1 j − 1 j m
1
Suppose xxx = (x1, . . . , xn)′ is a time series with an unknown number oflocally stationary segments.
m unknown number of segments
nj,m number of observations in the jth segment, nj,m ≥ tmin.
ξj,m location of the end of the jth segment, j = 0, . . . ,m, ξ0,m ≡ 0and ξm,m ≡ n.
Suppose x = fX1; : : : ;Xng is a time series with an unknown number ofstationary segments.
m : unknown number of segments (m = 1 means stationary)
nj ;m : number of observations in the j th segment, nj ;m � tmin.
�j ;m : location of the end of the j th segment, j = 0; : : : ;m , �0;m � 0 and�m;m � n .
fj ;m : spectral densities
Pnj ;m : periodograms at �kj = kj =nj ;m , 0 � kj � nj ;m � 1.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
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PRIOR DISTRIBUTIONS
Priors on gj ;m(�) = log fj ;m(�), j = 1; : : : ;m , as before.
Pr(�j ;m = t�� m) = 1=pjm ; for j = 1; : : : ;m � 1;
where pjm is the number of available locations for split point �j ;m .
The prior on the number of segments
Pr(m = k) = 1=M ; for k = 1; : : : ;M :
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
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SAMPLING SCHEMELIFE GOES ON WITHIN MOVES AND WITHOUT MOVES
Within-model moves: (location of end points)
Given m , �k�;m is proposed to be relocated.
The corresponding and βs are updated (absorb �0s into βs).
These two steps are jointly accepted or rejected in a M-H step.
The � 2s are then updated in a Gibbs step.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
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Between-model moves: (number of segments)
mp = mc + 1y
Select a segment to split
Select a new split point in this segment.
Two new �2s are formed from the current � 2
Two new βs are drawn.
mp = mc � 1
Select a split point to be removed.
A single �2 is then formed from the current � 2s
A new β is proposed.
Accept or Reject in a RJ-MCMC to move between models.
yc=current, p=proposedDAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
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EXAMPLE – the benefits of model averaging
Consider two tvAR(1) models Xt = atXt�1 +Wt for t = 1; : : : ; 500(blue) at = t=500� :5 There is no optimal segmentation in this case.
(green) at = :5 sign(t � 250)
Time
Xt
0 100 200 300 400 500
−2
02
4
Time
a t
0 100 200 300 400 500
−0.
40.
00.
4
Time
Xt
0 100 200 300 400 500
−3
−1
13
Time
a t
0 100 200 300 400 500
−0.
40.
00.
4
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
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In each case, m = 2 is the modal value [posteriors in paper] on thenumber of partitions. Plotted below are Pr(�1;2 = t
�� data) andPr(�1;2 < t
�� data), where �1;2 is the change point when m = 2.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
MOTIVATE THIS IT’S ALL THE SAME ADAPTSPEC WE’RE SO SORRY, UNCLE ENSO WE KNOW WHERE YOU ARE
0
0.2
0.4
0.6
0.8
1
0
0.1
0.2
0.3
0.4
0.5
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Time (rescaled)Frequency
Log(
Pow
er)
change-point!
time-varying
00.2
0.40.6
0.81
0
0.1
0.2
0.3
0.4
0.5
−1.5
−1
−0.5
0
0.5
1
1.5
2
Time (rescaled)Frequency
Log(
Pow
er)
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
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ENSO
1951 1970 1990 2010−4
−3
−2
−1
0
1
2
3
4
Year
(c)
1950 1970 1990 2010−3
−2
−1
0
1
2
3
Year
(b)
1880 1920 1960 2000−50
−40
−30
−20
−10
0
10
20
30
40
Year
(a)
Plots of (a) SOI from 1876–2011; (b) Niño3.4 index from 1950-2011;(c) DSLPA from 1951–2010.
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
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The Posteriors – Pr(#segments = k��� x )
k SOI Niño3.4 DSLPA1 0.95 0.93 0.992 0.05 0.07 0.013 0.00 0.00 0.00
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
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Simulation
xt =
8>>>>><>>>>>:
P6
k=1 �k1xt�k + �1�t for 1 � t � 200P6
k=1 �k2xt�k + �2�t for 201 � t � 1000P6
k=1 �k3xt�k + �3�t for 1001 � t � 1300P6
k=1 �k4xt�k + �4�t for 1301 � t � 1600P6
k=1 �k5xt�k + �5�t for 1601 � t � 2000;
500 1000 1500 2000
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Time
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
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0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Number of Segments
Po
ste
rio
r P
rob
ab
ility
(b)
0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Number of Segments
Po
ste
rio
r P
rob
ab
ility
(a)
0 10240
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Den
sity
ofE(ξ
j,3|X
)
(a)
0 20000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Den
sity
ofE(ξ
j,5|X
)
(b)
ξ1,3 ξ2,5 ξ3,5 ξ4,5ξ1,5ξ2,3
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE
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THANKS TO THE ORGANIZERS
ALL YOU NEED IS LOVE – WITH BREAKS
DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE