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Spectral Analysis ofNonstationary Time Series

DAVID S. STOFFERDepartment of StatisticsUniversity of Pittsburgh

Ori Rosen Sally WoodUTEP U.Melbourne

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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

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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

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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.

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El Niño – Southern Oscillation

DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE

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DAVID STOFFER SPECTRAL ANALYSIS OF THE FUTURE

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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

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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

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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.

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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.

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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.

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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.

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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.

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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

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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

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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.

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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.

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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

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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.

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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

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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.

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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 :

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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.

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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.

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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

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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.

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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)

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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.

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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

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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

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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

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THANKS TO THE ORGANIZERS

ALL YOU NEED IS LOVE – WITH BREAKS

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