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Structural Break Detection in Time Series ModelsStructural Break Detection in Time Series Models

Richard A. Davis

Thomas Lee

Gabriel Rodriguez-Yam

Colorado State University

(http://www.stat.colostate.edu/~rdavis/lectures)

This research supported in part by:• National Science Foundation• IBM faculty award

• EPA funded project entitled: Space-Time Aquatic Resources Modeling and Analysis Program (STARMAPSTARMAP))

The work reported here was developed under STAR Research Assistance Agreements CR-829095 awarded by the U.S. Environmental Protection Agency (EPA) to Colorado State University. This presentation has not been formally reviewed by EPA. EPA does not endorse any products or commercial services mentioned in this presentation.

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Illustrative Example

time

0 100 200 300 400

-6-4

-20

24

6

How many segments do you see?

1 = 51 2 = 151 3 = 251

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Illustrative Example

time

0 100 200 300 400

-6-4

-20

24

6

1 = 51 2 = 157 3 = 259

Auto-PARM=Auto-Piecewise AutoRegressive Modeling

4 pieces, 2.58 seconds.

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Introduction Setup Examples

AR GARCH Stochastic volatility State space model

Model Selection Using Minimum Description Length (MDL) General principles Application to AR models with breaks

Optimization using Genetic Algorithms Basics Implementation for structural break estimation

Simulation results

Applications

Simulation results for GARCH and SSM

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Examples

1. Piecewise AR model:

where 01m-1mn + 1, and {t} is

IID(0,1).

Goal: Estimate

m = number of segments

j = location of jth break point

j = level in jth epoch

pj = order of AR process in jth epoch

= AR coefficients in jth epoch

j = scale in jth epoch

, if , 111 jj-tjptjptjjt tYYYjj

),,( 1 jjpj

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Piecewise AR models (cont)

Structural breaks:

Kitagawa and Akaike (1978)

• fitting locally stationary autoregressive models using AIC

• computations facilitated by the use of the Householder transformation

Davis, Huang, and Yao (1995)

• likelihood ratio test for testing a change in the parameters and/or order of an AR process.

Kitagawa, Takanami, and Matsumoto (2001)

• signal extraction in seismology-estimate the arrival time of a seismic signal.

Ombao, Raz, von Sachs, and Malow (2001)

• orthogonal complex-valued transforms that are localized in time and frequency- smooth localized complex exponential (SLEX) transform.

• applications to EEG time series and speech data.

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Motivation for using piecewise AR models:

Piecewise AR is a special case of a piecewise stationary process (see Adak 1998),

where , j = 1, . . . , m is a sequence of stationary processes. It is

argued in Ombao et al. (2001), that if {Yt,n} is a locally stationary

process (in the sense of Dahlhaus), then there exists a piecewise

stationary process with

that approximates {Yt,n} (in average mean square).

Roughly speaking:{Yt,n} is a locally stationary process if it has a time-

varying spectrum that is approximately |A(t/n,|where A(u,is a

continuous function in u.

,)/(~

1),[, 1

m

j

jtnt ntIYY

jj

}{ jtY

, as ,0/ with nnmm nn

}~

{ ,ntY

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Examples (cont)

2. Segmented GARCH model:

where 01m-1mn + 1, and {t} is

IID(0,1).

3. Segmented stochastic volatility model:

4. Segmented state-space model (SVM a special case):

, if ,

,

122

1122

112

jj-qtjqtjptjptjjt

ttt

tYY

Y

jjjj

. if ,loglog log

,

122

112

jj-tjptjptjjt

ttt

t

Y

jj

. if ,

specified is )|(),...,,,...,|(

111

111

jj-tjptjptjjt

ttttt

t

ypyyyp

jj

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Model Selection Using Minimum Description Length

Basics of MDL:

Choose the model which maximizes the compression of the data

or, equivalently, select the model that minimizes the code length of

the data (i.e., amount of memory required to encode the data).

M = class of operating models for y = (y1, . . . , yn)

LF (y) = = code length of y relative to F MTypically, this term can be decomposed into two pieces (two-part code),

where

= code length of the fitted model for F

= code length of the residuals based on the fitted model

,ˆ|ˆ( ˆ()( )eL|y)LyL FFF

|y)L F(

)|eL Fˆ(

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Model Selection Using Minimum Description Length (cont)

Applied to the segmented AR model:

First term :

, if , 111 jj-tjptjptjjt tYYYjj

|y)L F(

m

jj

jm

jj

mmm

np

pnmm

yLyLppLLL(m)|y)L

12

1222

111

log2

2logloglog

)|ˆ()|ˆ(),,(),,(ˆ( F

Encoding: integer I : log2 I bits (if I unbounded)

log2 IU bits (if I bounded by IU)

MLE : ½ log2N bits (where N = number of observations used to

compute ; Rissanen (1989))

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Second term : Using Shannon’s classical results on

information theory, Rissanen demonstrates that the code length of

can be approximated by the negative of the log-likelihood of the

fitted model, i.e., by

)eL F|ˆ(

m

j

yL)eL1

12 )|ˆ(logˆ|ˆ( F

For fixed values of m, (1,p1),. . . , (m,pm), we define the MDL as

m

j

m

jjj

jm

jj

mm

yLnp

pnmm

ppmMDL

1 122

1222

11

)|ˆ(loglog2

2logloglog

)),(,),,(,(

The strategy is to find the best segmentation that minimizes

MDL(m,1,p1,…, m,pm). To speed things up for AR processes, we use

Y-W estimates of AR parameters and we replace

with jj n )ˆ2(log 22

)|ˆ(log2 yL j

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Optimization Using Genetic Algorithms

Basics of GA:

Class of optimization algorithms that mimic natural evolution.

• Start with an initial set of chromosomes, or population, of

possible solutions to the optimization problem.

• Parent chromosomes are randomly selected (proportional to

the rank of their objective function values), and produce

offspring using crossover or mutation operations.

• After a sufficient number of offspring are produced to form a

second generation, the process then restarts to produce a third

generation.

• Based on Darwin’s theory of natural selection, the process

should produce future generations that give a smaller (or

larger) objective function.

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Map the break points with a chromosome c via

where

For example,

c = (2, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, 3, -1, -1, -1, -1,-1) t: 1 6 11 15

would correspond to a process as follows:

AR(2), t=1:5; AR(0), t=6:10; AR(0), t=11:14; AR(3), t=15:20

Application to Structural Breaks—(cont)

Genetic Algorithm: Chromosome consists of n genes, each taking

the value of 1 (no break) or p (order of AR process). Use natural

selection to find a near optimal solution.

),,,( )),(,),(,( n111 cppm mm

. isorder AR and at timepoint break if ,

,at point break no if ,1

1 jjjt ptp

t

),,,( )),(,),(,( n111 cppm mm

. isorder AR and at timepoint break if ,

,at point break no if ,1

1 jjjt ptp

t

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Simulation Examples-based on Ombao et al. (2001) test cases

1. Piecewise stationary with dyadic structure: Consider a time

series following the model,

where {t} ~ IID N(0,1).

,1024769 if ,81.32.1

,769513 if ,81.69.1

,5131 if ,9.

21

21

1

tYY

tYY

tY

Y

ttt

ttt

tt

t

Time

1 200 400 600 800 1000

-10

-50

510

,1024769 if ,81.32.1

,769513 if ,81.69.1

,5131 if ,9.

21

21

1

tYY

tYY

tY

Y

ttt

ttt

tt

t

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1. Piecewise stat (cont)

GA results: 3 pieces breaks at 1=513; 2=769. Total run time 16.31 secs

Time

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

Time

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

True Model Fitted Model

Fitted model:

1- 512: .857 .9945

513- 768: 1.68 -0.801 1.1134

769-1024: 1.36 -0.801 1.1300

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1 200 400 600 800 1000

-6-4

-20

24

Simulation Examples (cont)

2. Slowly varying AR(2) model:

where and {t} ~ IID N(0,1).

10241 if 81. 21 tYYaY ttttt

)],1024/cos(5.01[8. tat

0 200 400 600 800 1000

time

0.4

0.6

0.8

1.0

1.2

a_t

10241 if 81. 21 tYYaY ttttt

)],1024/cos(5.01[8. tat

20Corvallis 9/05Time

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

2. Slowly varying AR(2) (cont)

GA results: 3 pieces, breaks at 1=293, 2=615. Total run time 27.45 secs

True Model Fitted Model

Fitted model:

1- 292: .365 -0.753 1.149

293- 614: .821 -0.790 1.176

615-1024: 1.084 -0.760 0.960

Time

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

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2. Slowly varying AR(2) (cont)

True Model Average Model

In the graph below right, we average the spectogram over the GA fitted models generated from each of the 200 simulated realizations.

Time

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

Time

Freq

uenc

y

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

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Examples

Speech signal: GREASY

Time

1 1000 2000 3000 4000 5000

-150

0-5

000

500

1000

1500

G R EA S Y

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Time

1 1000 2000 3000 4000 5000

-150

0-5

000

500

1000

1500

G R EA S Y

Time

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

Speech signal: GREASY

n = 5762 observations

m = 15 break points

Run time = 18.02 secs

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Application to GARCH (cont)

Garch(1,1) model:

.1000501 if ,6.1.4.

,5011 if ,5.1.4.2

121

21

212

tY

tY

tt

ttt

. if ,

IID(0,1)~}{ ,

12

121

2jj-tjtjjt

tttt

tY

Y

# of CPs GA%

AG%

0 80.4 72.0

1 19.2 24.0

2 0.4 0.4

AG = Andreou and Ghysels (2002)

.1000501 if ,6.1.4.

,5011 if ,5.1.4.2

121

21

212

tY

tY

tt

ttt

Time

1 200 400 600 800 1000

-4-2

02

CP estimate = 506

. if ,

IID(0,1)~}{ ,

12

121

2jj-tjtjjt

tttt

tY

Y

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time

y

1 100 200 300 400 500

05

1015

Count Data Example

Model: Yt | t Pois(exp{t}), t = t-1+ t , {t}~IID N(0,

Breaking PointM

DL

1 100 200 300 400 500

1002

1004

1006

1008

1010

1012

1014

True model:

Yt | t ~ Pois(exp{.7 + t }), t = .5t-1+ t , {t}~IID N(0, .3), t < 250

Yt | t ~ Pois(exp{.7 + t }), t = -.5t-1+ t , {t}~IID N(0, .3), t > 250.

GA estimate 251, time 267secs

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time

y

1 500 1000 1500 2000 2500 3000

-2-1

01

23

Model: Yt | t N(0,exp{t}), t = t-1+ t , {t}~IID N(0, )

SV Process Example

True model:

Yt | t ~ N(0, exp{t}), t = -.05 + .975t-1+ t , {t}~IID N(0, .05), t 750

Yt | t ~ N(0, exp{t }), t = -.25 +.900t-1+ t , {t}~IID N(0, .25), t > 750.

GA estimate 754, time 1053 secs

Breaking Point

MD

L

1 500 1000 1500 2000 2500 3000

1295

1300

1305

1310

1315

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time

y

1 100 200 300 400 500

-0.5

0.0

0.5

Model: Yt | t N(0,exp{t}), t = t-1+ t , {t}~IID N(0, )

SV Process Example

True model:

Yt | t ~ N(0, exp{t}), t = -.175 + .977t-1+ t , {t}~IID N(0, .1810), t 250

Yt | t ~ N(0, exp{t }), t = -.010 +.996t-1+ t , {t}~IID N(0, .0089), t > 250.

GA estimate 251, time 269s

Breaking Point

MD

L

1 100 200 300 400 500

-530

-525

-520

-515

-510

-505

-500

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SV Process Example-(cont)

time

y

1 100 200 300 400 500

-0.5

0.0

0.5

Fitted model based on no structural break:

Yt | t N(0, exp{t}), t = -.0645 + t-1+ t , {t}~IID N(0,

True model:

Yt | t ~ N(0, exp{at}), t = -.175 + .977t-1+ et , {t}~IID N(0, .1810), t 250

Yt | t N(0, exp{t}), t = -.010 t-1+ t , {t}~IID N(0, t > 250.

time

y

1 100 200 300 400 500

-0.5

0.0

0.5

1.0

simulated seriesoriginal series

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SV Process Example-(cont)

Fitted model based on no structural break:

Yt | t N(0, exp{t}), t = -.0645 + t-1+ t , {t}~IID N(0,

time

y

1 100 200 300 400 500

-0.5

0.0

0.5

1.0 simulated series

Breaking Point

MD

L

1 100 200 300 400 500

-478

-476

-474

-472

-470

-468

-466

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Summary Remarks

1. MDL appears to be a good criterion for detecting structural breaks.

2. Optimization using a genetic algorithm is well suited to find a near optimal value of MDL.

3. This procedure extends easily to multivariate problems.

4. While estimating structural breaks for nonlinear time series models is more challenging, this paradigm of using MDL together GA holds promise for break detection in parameter-driven models and other nonlinear models.