Detection of Structural Breaks and Outliers
in Time Series
Richard A. Davis
Columbia University
Collaborators:
Implementation of AutoPARM/AutoPARMO:
Thomas Lee, UC Davis
Gabriel Rodriguez-Yam, Universida Autónoma Chapingo
Theory behind AutoPARM:
Stacey Hancock, UC Irvine (to be).
Yi-Ching Yao, Academia Sinica
General theory:
Chun Yip Yau, Chinese University of Hong Kong
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1
Illustrative Example
10
Can you spot any unusual observations?
-5
0
Y
5
IO
0
100
200
300
400
500
600
time
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2
Illustrative Example
10
Auto-PARMO=Auto-Piecewise AutoRegressive Modeling Outlier
Misclassified as
AO
-5
0
Y
5
5 AOs
0 IOs
0
100
200
300
400
500
600
time
1 = 51
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2 = 151
3 = 252
4
Illustrative Example
-5
0
Y
5
10
The series w/o any outlier effects—structural breaks easier to identify
0
100
200
300
400
500
600
time
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5
A Second Example
Weekly data: January 2004--December 2011.
0
5
Y
10
15
Any breaks? Outliers?
2004
2006
2008
time
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2010
2012
6
Game Plan
 Introduction
 Piecewise Models (AR, GARCH, SV, State-space)
 Model selection using Minimum Description Length (MDL)
 General principles
 Application to AR models with breaks
 Optimization using a Genetic Algorithm
 Basics
 Simulation Examples
 Applications
 Comments about theory for AutoPARM
 Consistency (FLIL)
 AutoPARM with Outliers
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7
Piecewise Models—examples
1. Piecewise AR model:
Yt   j   j1Yt 1    jp jYt  p j   j t , if  j-1  t   j ,
where 0 1 m m+1 n, and {t} is IID(0,1).
Goal: Estimate
m = number of change-points
j = location of jth break point
j = level in jth segment
pj = order of AR process in jth segment
( j1,,  jp j ) = AR coefficients in jth segment
j = scale in jth segment
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8
Examples (cont)
2. Segmented GARCH model:
Yt  t t ,
t2   j   j1Yt 21    jp jYt 2 p j   j1t21    jq j t2q j ,
if  j-1  t   j ,
where 0 1 m m+1 n + 1, and {t} is IID(0,1).
3. Segmented stochastic volatility model:
Yt  t t ,
logt2   j   j1 logt21    jp j logt2 p j   j t ,
if  j-1  t   j .
4. Segmented state-space model (SVM a special case):
p( yt | t ,...,1, yt 1,..., y1 )  p( yt | t ) is specified
t   j   j1t 1    jp j t  p j   j t ,
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if  j-1  t   j .
9
Model Selection Using Minimum Description Length
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  M
Typically, this term can be decomposed into two pieces (two-part code),
LF ( y)  L(Fˆ|y)  L(eˆ | Fˆ),
where
L(F̂|y) = code length of the fitted model for F
L(eˆ|F̂) = code length of the residuals based on the fitted model
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10
Model Selection Using Minimum Description Length (cont)
Code length
Using results from information theory and
likelihood
for τ1,…, τm
approximations, the minimum description length for the
Code length
, pm length
Penalty
Code
segmented AR model
with parameters (m, 1 ,  , m , p1 , 
1)
for m
for p1,…, pm+1
is Code length for Term
AR parameters
m 1
ψ1,…, ψ
mm+1
MDL(
, 1 ,  , m , p1 , , pm 1 )  log m  m log n  log p j

j 1
m 1
pj  2
j 1
2

m 1
nj
j 1
2
log n j  
log(2ˆ 2j ),
-log(like)
th
Code
length
where nj is the number of observations in the j segment, and
residuals
ˆ 2 is the Yule-Walker estimate of the process for
variance
in the jth
j
segment.
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11
Minimizing the MDL

Select the best-fitting model for the data as the model that minimizes
the minimum description length with respect to the number of
change-points m, the change-point locations, 1, m, and the AR
orders p1,…, pm+1.

The dependence of the minimum description length on the
autoregressive coefficient parameter estimates is only through the
white noise estimates ˆ 2j , j = 1,…, m + 1.

Numerical minimization carried out using a genetic algorithm, which
mimics natural evolution (see Davis et al., 2006).
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12
Optimization Using Genetic Algorithm
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.
Map the break points with a chromosome c via
(m, (1, p1 ), (m , pm )) 
 c  (1,, n ),
where
if no
no break
break point
point at
at tt,,
11,, if
tt  
if break
break point
point at
at timett  jj11 and
and AR
AR order is ppjj..
 ppjj,, if
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
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15
Implementation of Genetic Algorithm—(cont)
Generation 0: Start with L (200) randomly generated chromosomes,
c1, . . . ,cL with associated MDL values, MDL(c1), . . . , MDL(cL).
Generation 1: A new child in the next generation is formed from the
chromosomes c1, . . . , cL of the previous generation as follows:
 with probability c, crossover occurs.
 two parent chromosomes ci and cj are selected at random with
probabilities proportional to the ranks of MDL(ci).
 kth gene of child is k = ik w.p. ½ and jk w.p. ½
 with probability  c, mutation occurs.
 a parent chromosome ci is selected
 kth gene of child is k = ik w.p. 1 ; 1 w.p. 2;and p w.p.  12.
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16
Implementation of Genetic Algorithm—(cont)
Execution of GA: Run GA until convergence or until a maximum
number of generations has been reached. .
Various Strategies:
 include the top ten chromosomes from last generation in next
generation.
 use multiple islands, in which populations run independently,
and then allow migration after a fixed number of generations.
This implementation is amenable to parallel computing.
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17
Simulation Examples (cont)
2. Slowly varying AR(2) model:
Yt  atYt 1  .81Yt 2  t if 1  t  1024
1024
a_t
0.4
-6
-4
0.6
-2
0.8
0
2
1.0
4
1.2
where at  .8[1  0.5 cos(t / 1024)], and {t} ~ IID N(0,1).
1
200
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400
600
Time
800
1000
0
200
400
600
time
800
1000
21
2. Slowly varying AR(2) (cont)
GA results: 3 pieces, breaks at 1=293, 2=615. Total run time 27.45 secs



1- 292: .365 -0.753 1.149
293- 614: .821 -0.790 1.176
615-1024: 1.084 -0.760 0.960
Fitted model:
Fitted Model
0.0
0.0
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
True Model
0.0
0.2
0.4
0.6
Time
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0.8
1.0
0.0
0.2
0.4
0.6
Time
0.8
1.0
22
2. Slowly varying AR(2) (cont)
In the graph below right, we average the spectogram over the GA
fitted models generated from each of the 200 simulated realizations.
Average Model
0.3
0.2
0.0
0.0
0.1
0.1
0.2
Frequency
0.3
0.4
0.4
0.5
0.5
True Model
0.0
0.2
0.4
0.6
Time
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0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Time
23
Simulation Examples (cont)
3. Piecewise ARMA:
  .9Ytt11  tt  .7tt11, if 1  t  513,

Ytt   .9Ytt11  tt,
if 513  t  769,
   .7 ,
if 769  t  1024.
tt
tt11

-4
-2
0
2
4
where {t} ~ IID N(0,1).
1
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200
400
600
Time
800
1000
24
3. Piecewise ARMA (cont)
GA results: 3 pieces, breaks at 1=513, 2=769. Total run time 1.53 secs
Fitted model: AR orders 4, 1, 2
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
Fitted Model
0.5
True Model
0.0
0.2
0.4
0.6
Time
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0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Time
25
Example--Monthly Deaths & Serious Injuries, UK
1600
1200
1400
Counts
1800
2000
2200
Data: yt = number of monthly deaths and serious injuries in UK, Jan
`75 – Dec `84, (t = 1,…, 120)
Remark: Seat belt legislation introduced in Feb `83 (t = 99).
1976
1978
1980
1982
1984
Year
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26
Example -- Monthly Deaths & Serious Injuries, UK (cont)
Data: xt = number of monthly deaths and serious injuries in UK,
differenced at lag 12; Jan `75 – Dec `84, (t = 13,…, 120)
Remark: Seat belt legislation introduced in Feb `83 (t = 99).
-200
0
bf ((tt)) W
Wt,,
YYtt  aa  bf
t
0, if
if 11 tt  98
98,,
0,
f(t)

f(t)  1, if 98  t  120.
1, if 98  t  120.
XXtt YYtt YYtt1212
bg((tt)) N
Ntt
 bg
-600
-400
Differenced Counts
200
Traditional regression
analysis:
1976
1978
1980
1982
1984
99  tt 110
110,,
11,, ifif 99
g(t)  
g(t)
0, otherwise
otherwise..
0,
Year
Model: b373.4, {Nt}~AR(13).
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27
Example: Monthly Deaths & Serious Injuries, UK
0
-200
-600
-400
Differenced Counts
200
Data: Yt = number of monthly deaths and serious injuries in UK, Jan
`75 – Dec `84, (t = 1,…, 120)
Remark: Seat belt legislation introduced in Feb `83 (t = 99).
1976
1978
1980
1982
1984
Year
Results from GA: 3 pieces; time = 4.4secs
Piece 1: t=1,…, 98IID; Piece 2: (t=99,…108) IID; Piece 3: t=109,…,120 AR(1)
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28
Application to GARCH
Garch(1,1) model: Yt  t t ,
t
t t
{tt} ~ IID(0,1)
t2t2   jj   jjYtt2211   jjt2t211,
if  j-j-11  t   jj.
 .4  .1Yt21  .5t21, if 1  t  501,
 
2
2
.4  .1Yt1  .6t1, if 501 t  1000.
-4
-2
0
2
2
t
1
200
400
600
800
1000
# of CPs
GA
%
AG
%
0
80.4
72.0
1
19.2
24.0
2
0.4
0.4
Time
CP estimate = 506
AG = Andreou and Ghysels (2002)
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33
Application to GARCH (cont)
More simulation results for Garch(1,1) : Yt  t t ,
{t } ~ IID(0,1)
 .05  .4Yt 21  .3t21, if 1  t  1,
 
2
2



1
.
00
.
3
Y
.
2
if 1  t  1000.
t 1
t 1,

2
t

Mean
SE
Med
Freq
50
GA
Berkes
52.62
71.40
11.70
12.40
50
71
.98
250
GA
Berkes
251.18
272.30
4.50
18.10
250
271
.99
500
GA
Berkes
501.22
516.40
4.76
54.70
502
538
.98
Berkes = Berkes, Gombay, Horvath, and Kokoszka (2004).
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35
Application to Parameter-Driven SS Models
State Space Model Setup:
Observation equation:
p(yt | t) exp{t yt  b(t) + c(yt)}.
State equation: {t} follows the piecewise AR(1) model given by
t = k  kt-1  kt , if k-1  t k ,
where  0  1   m n, and {t } ~ IID N(0,).
Parameters:
m = number of break points
k = location of break points
k = level in kth epoch
k = AR coefficients kth epoch
k = scale in kth epoch
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38
Application to Structural Breaks—(cont)
Estimation: For (m, 1, . . . , m fixed, calculate the approximate
likelihood evaluated at the “MLE”, i.e.,
| Gn |1/ 2
*
*
*
T *
T
T
ˆ ; yn ) 
exp{
y


1
{
b
(

)

c
(
y
)}

(



)
G
(

 ) / 2},
La (
n
n
n
1/ 2
K  Gn 
where 
ˆ  ( ˆ1,, ˆ m , ˆ 1,, ˆ m , ˆ 12 ,, ˆ 2m ) is the MLE.
Remark: The exact likelihood is given by the following formula
where
L(; y n )  La (; y n ) Era (),
Era ()   exp{R( n ; *)} pa ( n | y n ; ) dn .
It turns out that log( Era ( )) is nearly linear and can be approximated
by a linear function via importance sampling,
e( ) ~ e(ˆ AL )  e(ˆ AL )(  ˆ AL )
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39
SV Process Example
1305
1295
-2
-1
1300
0
y
MDL
1
1310
2
3
1315
Model: Yt | t N(0,exp{t}), t = t-1+ t , {t}~IID N(0, )
1
500
1000
1500
2000
2500
time
3000
1
500
1000
1500
2000
2500
3000
Breaking Point
True model:
 Yt | t ~ N(0, exp{t}), t = -.05 + .975t-1+ t , {t}~IID N(0, .05), t  750
 Yt | t ~ N(0, exp{t }), t = -.25 +.900t-1+ t , {t}~IID N(0, .25), t > 750.
 GA estimate 754, time 1053 secs
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41
SV Process Example
-530
-0.5
-525
-520
-515
MDL
0.0
y
-510
0.5
-505
-500
Model: Yt | t N(0,exp{t}), t = t-1+ t , {t}~IID N(0, )
1
100
200
300
time
400
500
1
100
200
300
400
500
Breaking Point
True model:
 Yt | t ~ N(0, exp{t}), t = -.175 + .977t-1+ t , {t}~IID N(0, .1810), t  250
 Yt | t ~ N(0, exp{t }), t = -.010 +.996t-1+ t , {t}~IID N(0, .0089), t > 250.
 GA estimate 251, time 269s
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42
SV Process Example-(cont)
True model:
 Yt | t ~ N(0, exp{at}), t = -.175 + .977t-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.
Fitted model based on no structural break:
1.0
 Yt | t N(0, exp{t}), t = -.0645 + t-1+ t , {t}~IID N(0, 
0.5
simulated series
0.0
y
0.0
-0.5
-0.5
y
0.5
original series
1
100
200
300
time
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400
500
1
100
200
300
400
500
time
43
SV Process Example-(cont)
Fitted model based on no structural break:
-468
-466
simulated series
-472
-478
-0.5
-476
-474
0.0
y
MDL
-470
0.5
1.0
 Yt | t N(0, exp{t}), t = -.0645 + t-1+ t , {t}~IID N(0, 
1
100
200
300
time
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400
500
1
100
200
300
400
500
Breaking Point
44
Comments on the theory for AutoPARM
Consistency of
:
Suppose the true number of change points is m and let
1=1/n, . . . , m=m/n
be the relative (true) change-points. Then AutoPARM estimates
of
and
satisfy:
→
,
→
.
Remarks: The proof is a “cool” application of Strassen’s function
law of the iterated logarithm.
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45
Comments on the theory for AutoPARM (cont)
Extension: If the process in the jth segment is stationary with ACVF
j(h) and with j(h)  j-1(h) for some lag h{0,1,…,p*}. Then the
estimator which minimizes MDL by fitting AR(p*) models in each
segment produces weakly consistent estimators of m = true number of
change-points.
Remark: This result can be applied to detecting change-points in
piecewise GARCH models. Here one can take p*=0 so that we are
only talking about changes in variance.
Further extensions: This result has been extended to include a more
general framework by Davis and Yau (2012). Base models include
ARMA and GARCH—results address some identifiability issues
(Andrews and Cheng (2011)).
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46
Comments on the theory for AutoPARM (cont)
Theorem: Suppose
is the AR(1) process
,
1, and the
where
is IID(0,1) with finite 5th moment and pdf
satisfying
0 for all
•
•
•
liminf
→
e
0, for some 0.
Then, MDL is NOT consistent using Yule-Walker estimation. That
is, there is a positive probability that a change-point model will be
selected as
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→ ∞.
47
Modeling Framework for Outlier Detection
,…,
Data:
Model parameters
Segments:
m = number of change-points
j = start of jth segment (0 1 m m+1 n)
,
Intra-segment parameters (jth segment
,
Integer valued:
•
•
,
∗
,
,…,
∗
∗
,
,
,
,…,
,
locationofadditiveoutliers
,…,
∗
∗
,
locationofinnovationaloutliers
,
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,
order of AR process
∗
,
•
∗
,…,
,
55
Modeling Framework (cont)
,…,
Data:
,
Intra-segment parameters (jth segment
,
Real valued:
,
•
∗
•
,
≫
,…,
,
,
,…,
,
,
,
∗
,
AR model parameters
,
is AO variance
≫ 1is proportional increase in innovation variance.
•
Intra-segment AR process with IO:
,
where
,
,
⋯
,
,
,
is an independent sequence with
Var
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,
,
,if ∈ ,
1, if ∉ .
56
Modeling Framework (cont)
Data:
,…,
Intra-segment AR process with IO:
,
where
,
,
⋯
,
,
,
,
is an independent sequence with
Var
,
,if ∈ ,
1, if ∉ .
Intra-segment AR process with IO and AO:
,
,
,
where the
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,
,
,if ∈
ift ∉
are independent and N(0,
∗
,
,
distributed.
57
Modeling Framework (cont)
Remarks:
•
If the
,
and the
,
are Gaussian, then model (for fixed
integer parameter ) can be expressed as a Gaussian linear
state-space model.
•
Take advantage of state-space model and Kalman recursions
for likelihood calculation.
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58
Model Selection Using Minimum Description Length
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  M
Typically, this term can be decomposed into two pieces (two-part code of
Rissanen),
LF ( y)  L(Fˆ|y)  L(eˆ | Fˆ),
where
L(F̂|y) = code length of the fitted model for F
L(eˆ|F̂) = code length of the residuals based on the fitted model
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59
Model Selection Using Minimum Description Length
Take
M = class of operating models for y = (y1, . . . , yn) that
includes piecewise AR models with AO’s and IO’s.
Let
,…,
and
,…,
valued parameter vectors, where
, ∗ , , ∗, , … ,
,
Then
L
∑
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log
log
,
,…,
,
be the integer- and real∗
,
∗
,
,
,
∗
,
,…,
,
.
,
1 log
log
∗
log
∗
log
1/2
3 log
60
Model Selection Using Minimum Description Length
Moreover, using results from information theory, Rissanen argues
L ̂|
where
∑
~
;
log
;
) is the likelihood evaluated at the MLE of the jth
segment; so that
, , ,
,
∗
, ,
1 log
log
∗
log
log
∗
log
m
log
1/2
3 log
log 2 (2
ˆ; 2j )  n j )
 (log
j 1
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61
Model Selection Using Minimum Description Length
Computing the likelihood
in a segment given
;
: Start by computing likelihood
and locations of additive and innovational
outliers.
;
,
; ,
,
,
,
,
|
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62
Model Selection Using Minimum Description Length
;
Computing the likelihood
;
•
~
:
,
;
|
,
multivariate normal
•
,…,
; ,
2
⋯
exp
∉
2
exp
⋯
∈
Remark: Use Kalman recursions to compute likelihood.
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63
Optimization Using Genetic Algorithm
Genetic Algorithm: Chromosome consists of n genes, each taking
the value of -3 (AO), -2 (AI), 1 (no break), p (order of AR process).
Use natural selection to find a near optimal solution.
Map the break points with a chromosome c via
, , , ,
↔
,…,
where
3, ifAOoccursat ,
2, ifIOoccursat ,
1, ifnobreakoccursat ,
, ifbreakoccursattime andARmodelis .
For example,
c = (2, -1, -3, -1, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, 3, -1, -1, -2, -1,-1)
t: 1
6
11
15
would correspond to a process with AO at t=3, IO at t=18.
AR(2), t=1:5; AR(0), t=6:10; AR(0), t=11:14; AR(3), t=15:20
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64
Robust AutoPARM-IO contaminated AR model
Single AR model with
single IO at ?
MDL’s w/ top 5
residuals indicated
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66
Robust AutoPARM-AO contaminated AR model
Single AR model with
single AO at 101
MDL’s w/ top 5
residuals indicated
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67
Accelerating Robust AutoPARM
Remark: Observe that large reductions in MDL due to outliers seem
to occur at locations of largest residuals. Use this info to facilitate the
search optimizer.
Search modification: for simplicity, assume one AR(p) signal
,…,
1. Select locations
2. Let
and set
(resp
be an AR(p) model with AO(IO) at time t
,
3. Define
4. Set
argmin
, ∈
;
∈
, ,
argmin
5. Continue to obtain
arg min
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of k largest absolute residuals.
,
∈
,…,
location of outlier
, ∈
;
location of outlier
and define
0
68
Accelerating Robust AutoPARM
6. Only allow time locations , … , for outliers in the current
generation of the genetic algorithm.
7. After producing the next generation, select largest residuals again
and repeat.
Remark: The outlier selection step has elements that are similar in
spirit to one proposed by Bianco et al. (2001) and Sánchez and Peña
(2003). Our method is not as likely to stop prematurely.
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69
Accelerating Robust AutoPARM
Example: AR(1) model
.8
, with AO’s at {16,32,48,80} and no IOs.
~
0,1
Residuals selected by
minimizing MDL
Correctly
One
wrong
identified
classification
locations
Minimum
MDL
10 largest
residuals
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70
Robust AutoPARM-sample results
DGP1: one-piece AR model; no outliers
DGP2: one-piece AR model; 5 innovation outliers
DGP3: one-piece AR model; 4 additive outliers
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71
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72
Robust AutoPARM-sample results
Three-piece AR model:
.8
1.69
1.32
3 , if1
.81
, if128
.81
3 , if192
128,
t 192,
t 256.
DGP4: three-piece AR model; no outliers
DGP5: three-piece AR model; 3 innovation outliers at 16, 32, 48.
DGP6: three-piece AR model; 3 additive outliers at 16, 32, 48.
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73
Robust AutoPARM-sample results
-5
0
5
Y
10
15
20
DGP5: three-piece AR model; 3 innovation outliers at 16, 32, 48.
0
50
Results from AutoPARMO:
100
150
200
250
time
Breaks: 124 and 194; 3 IA outliers: 16, 32, 48
Results from AutoPARM(no outliers): Breaks at 52, 126, 294
arima.rob: AO at 16, 17,32, 48, 49
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74
Robust AutoPARM-sample results
DGP4: three-piece AR model; no outliers
DGP5: three-piece AR model; 3 innovation outliers
DGP6: three-piece AR model; 3 additive outliers
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75
Robust AutoPARM-sample results
DGP5: two-piece AR model; 3 innovation outliers
DGP6: two-piece AR model; 3 additive outliers
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76
Application to Google Trends Example: Tea Party
Weekly data: January 2004--December 2011.
15
Taxes due
9-12 taxpayer
march on
Washington DC
Y
10
3 weeks after
Obama takes
office
5
Midterm
elections
0
Texas Tea
Party straw
poll
2004
2006
Results from AutoPARMO:
2008
2010
2012
time
Ron Paul
raised
$4.4M on
internet in
1 day
Break: Feb 15, 2009 (3 weeks after Obama took office)
AO outliers: 11-5-06(election week); 9-2-07; 12-16-07;
4-12-09 (taxes due); 4-11-10 (taxes); 9-12-10; 10-31-10 (election)
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79
Application to Google Trends Example: Tea Party
Refinement: January 2004--December 2011.
15
Taxes due
9-12 taxpayer
march on
Washington DC
10
3 weeks after
Obama takes
office
Midterm
elections
0
Texas Tea
Party straw
poll
5
Y
Tea Party
protests
2004
2006
2008
2010
time
Results from AutoPARMO on the two segments:
2012
Ron Paul
raised
$4.4M on
internet in
1 day
AO outliers (7): 149, 192, 207, 276, 287, 328, 357
IO outliers (2): 325, 350
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80
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. While estimating structural breaks for nonlinear time series models is
more challenging, this paradigm of using MDL together with GA holds
promise for break detection in parameter-driven models and other
nonlinear models.
4. Extensions to outlier (both innovation and additive) detection are
currently under study. Results look promising—new implementation is
almost ready.
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81
Running the Program AutoPARM
• Download the zip file AutoPARM (only runs on PC) from
www.stat.columbia.edu/~rdavis/AutoPARM/AutoPARM.zip
• Extract the zip file (folder called AutoPARM) will be created.
• Open a command prompt window
(In run, type the command cmd)
• Navigate to the AutoPARM folder
• AutoPARM needs an input file (first several lines of example file
eqn5.in are:
• DATA:
eqn5.dat
[name of file containing input data]
• OUTPUT:
eqn5.out
[name of file to export results]
• Execute program by typing on command line
AutoPARM eqn5.in
• Results are written to file named eqn5.out
• More details about running the program can be found in AutoPARM.pdf
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