Hierarchical Bayesian of ARMA Models Using Simulated Annealing Algorithm - Repository Universitas Ahmad Dahlan
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Accredited,,A,,byDGHE(DtKl),r"", j"""*.1#?^?rt;Ur7,?r?
TELKOMNIKA
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Editor-in_Chief
Tole Sutikno
Universitas Ahmad Dahlan
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E I e ctr-on ic
s E n g i neering
Mark S. Hooper
.
Signal processin(]
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signal/RFIC Desjgn
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Consultants Network of Silico;
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of Etectronic Engineering
untversity of Rome .Tor V6rgata,.
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Universiti Kebangsaan Malavsia
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TELKOMNIKA
Vof, 12, No. 1, March
2Al4
ISSN ,1693-6930
Accredited "A" by DGHE (DlKTl), Decree No.58/DlKTl/Kep/2013
Table of Contents
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Hamzah Eteruddin, Abdullah Asuhaimi Mohd Zin, Belyamin Belyamin
Baseline
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Hailiang Song, Yongqing Fu, Xue Liu
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lntegrated Vehicle Accident Detection and Location
Md. SyedulAmin, Mamun Bin lbne Reaz, Salwa Sheikh Nasir
Neural Network Adaptive Control for X-Y Position Platform with
Ye Xiaoping, Zhang Wenhui, Fang Yamin
73
Uncertainty
Hierarchical Bayesian of ARMA Models Using Simulated Annealing
Suparman, Michel Doisy
Algorithm
Resolutions
Water Pump Mechanical Faults Display at Various Frequency
Linggo Sumarno, Tjendro, Wwien Widyastuti, R.B. Dwiseno \Mhadi
Endocardial Border Detection Using Radial Search and Domain
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79
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Yong Chen, DongC Liu
New Modelling of Modified Two Dimensional Fisherface Based Feature
Arif Muntasa
Extraction
Network
Emergency PrenatalTelemonitoring System in Wireless Mesh
Muhammad HaikalSatria, Jasmy bin Yunus, Eko Supriyanto
Deformation
A New ControlCurve Method for lmage
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I
115
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TELKOMNIKA
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Vol. 12, No. 1, March 2OL4
Accredited 'rA" by DGHE (DlKTl), Decree No.58/DlKTl/Kep/2013
Orthogonal Frequency-Division Multiplexing-Based Cooperative Spectrum Sensing for
Cognitive Radio Networks
Arief Marwanto, Sharifah Kamilah Syed Yusof, M. Haikal Satria
143
LTE Coverage Network Planning and Comparison with Different Propagation Models
RekaM Sabir Hassan, T. A. Rahman, A. Y' Abdulrahman
153
A Threshold Based Triggering Scheme for Cellular-to-WLAN Handovers
Liming Chen, Qing Guo, Zhenyu Na, Kaiyuan Jiang
163
Model and Optimal Solution of Single Link Pricing Scheme Multiservice Network
lrmeilyana, lndrawati, Fitri Maya Puspita, Juniwati
173
An Analytical Expression for k-connectivity of Wireless Ad Hoc Networks
Nagesh kallollu Narayanaswamy, Satyanarcyana D, M'N Giri Prasad
179
Two Text Classifiers in Online Discussion: Support Vector Machine vs Back-Propagation
Neural Network
Erlin, Rahmiati, Unang Rio
189
Research on Traffic ldentification Based on Multi Layer Perceptron
Dingding Zhou, Wei Liu, Wengang Zhou, Shi Dong
201
plagiarism Detection through lnlernet using Hybrid Artificial Neural Network and Support
Vectors Machine
lmam Much lbnu Subroto, AliSelamat
209
Reliability Analysis of Components Life Based on Copula Model
Han Wen Qin, Zhou Jin Yu
219
Cobb-Douglass Utility Function in Optimizing the lnternet Pricing Scheme Model
lndrawati, lrmeilyana, Fitri Maya Puspita, Meiza Putri Lestari
227
CApBLAT: An lnnovative Computer-Assisted Assessment for Problem-Based Learning
241
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Muhammad Qomaruddin, Azizah Abdul Rahman, Noorminshah A. lahad
publications Repository Based on oAl-PMH 2.0 Using Google App Engine
251
Hendra, Jimmy
Android Based Palmprint Recognition System
Gede Ngurah Pasek Pusia Putra, Ketut Gede Darma Putra, Putu \Mra Buana
263
\
,
TELKOMNIKA, Vol.12, No.1, March 2014,pp.87 - 96
ISSN: 1693€930, accredited A by DtKTt, Decree No: 58/DtKit/Kepl2O13
DOl:
1
0.
1
f87
2928ffELK0MNlKA,v 12i1.fl A6
Hierarchical Bayesian of ARMA Models Using
Simulated Annealing Algorithm
1
Deparrment or Marhematicar
_
rclcX?i3l:tXH:',jl$""?flr3:fil:r,rn, Jr pror. Dr
Yogyakarta, lndonesia
'ENSEEIHT/TESA, 2 Rue Camichel, Toulouse, France
"Conesponding author, email: e_mail: suparmancict@yahoo.co.id
soepomo, sH
,
Abstract
When the Autorcgressive Moving Average (ARMA) model is fifted with rcal data, the actual
value
of the model order and the model parameter are often unknown. The goal of this paper is to frnd an
estimator for the model order and the model parameter based on the dati. ln this paper, ihe model
order
identification and model parameter estimation
is given in a hierarchicat Bayeiiai fimiiorx. tn this
framework, the model order and model panmeter are assurned to have pior distibution, which
summarizes all the infomation available about the p/ocess. All the information about the characteristics
of
the model order and the model panmeter are expressed in the posteior distibution. prcbability
determination of the model order and the model panmeter requirei the integration of the posteior
djstnbution resulting. lt is an operation which is very ditricuft to be-solved analytiiatty. 6ere ine Simuated
Annealing Reversible Jump Markov Chain Monte Cailo (MCMC) aQofthm *as aeretopei to iompute
the
required integration over the posterior distibution simulation. Methods developed are evaluated in
simulation sfudies in a number of set of synthetic data and real data.
Keywords: Simulated Annealing, ARMA model, oiler identification, parameter estimation
l.lntroduction
xpx2t" ',xo is a time series data. Time series xpx22...,xn
ARMA(p,q) modet, if x,,X2,...,Xn satisff following stochastic equation [3]
Suppose
is said to be an
:
xt
where
z,
: zt * I;_, 0,2,_i *ff=,0,r,_,,
is the random error at the time t,
coefficient-coefficient. Here
variance
o'
.
ARMA model
z,
t =I,2,...,n
(1)
0t (i = 1,2,..., p) and 0: 0 = 1,2, ..., q) is the
is assumed to have the normal distribution with mean 0 and
(*,I*
is called stationary if and onty if the equality of potynomial
l+I:=,Sf')bi:o
must lie outside the unit circle. NextARMA model
equality of polynomial
t+
Fl
|HJ=L
olq)bj
J
:
(",),."
is called invertibte if and onty if the
o
must lie outside the unit circle [4].
The order (p,q)'" assumed to be known, parameter estimation of the ARMA model
has been examined by several researchers, for example t3l, t4l and [12]. ln practice when we
match the ARMA modelto the data, in generatorOer (p,q) is not known.
Received september 19,2013; Revlsed December 18,2013;Accepted January 12, 2014
BBI
ISSN: 1693-6930
not known, the order identification and the parameters
While for the order (p,q)
'.
estimation are done in two stages. The first stage is to estimate the coefficients and error
variance with the assumption of order (p,q) i. known. Based on the parameter estimation in
the first stage, the second stage is to identify the order (p,q) A criterium used to determine
(p,q) has been
proposed by many researchers, among others are the Akaike lnformation
Criteria Criterion (AlC), Bayesian lnformation Criterion (BlC), and the Final Prediction Error
Criterion (FPE).
Based on the data x, , t =\,2,,.., n , this research proposes a method to estimate the
orOer
value p,q,Q(n),6(o), and o,2 simultaneously. To do that, we will use a hierarchical Bayesian
approach [10], which will be described below.
2. Research Method
2.1 Hierarchical Bayesian
t-\
Let s = (xp*q*r,
)*, ) o" a realization of the ARMA (p, q) model. lf the value
,o =(*,,*r,...,Xp+q) i, knorn, then the likelihood function of s can bewritten approximately
as follows
xp*q +2,. . .
:
(#)7.*o-#I*,.,*,e'(t,p,e,oin),s{o)
z(rln,r,otn',0(n),o')=
where g(t,p,q,0(0r,6{o))= *,
(2)
*fi,0fo'*,-, -I]=,ejttt,-, for t =p+q+1,p+ q+2,"',n
with initial value x, =xz=...=Xo*n =0 [11]. Suppose Snand In are respectively stationary
region and invertible region. By using the transformation
F:otnr = (O{",0y),"',0[n').
Sn
F] ro) =Qr,'r,"
','n)'(-t,t)n)
6.gro) = (efor,e!o),...,efrr)e In F+ p(o) =(p,pr,...,pn).
the ARMA model
ARMA model
(*,)*
(",),.,
stationery if and only
invertible
, o{a) ,
o2
)=
.(n)
if and only if o(t) -
likelihood function can be rewritten as:
/(rlp, e, ,(')
i1
b,pr,"',Po).(-t't)t)
b)
[1]. Then the
n-p
(#)T
"*p-
*I,=n.'.,
a)
[o'-)uu
Ft,t)-)
=(rr,rr," ',.0).(-t,t)n) t2l ano tne
g' (t, p, q, F-l(o(n) ;, 6-r lgta')) tol
The prior distribution determination for parameters are as follows:
Order p have Binomial distribution with parameters l":
,,blr)=
tel
- ]")r*-n
Order q have Binomial distribution with parameters p:'
TELKOMNIKA Vol. 12, No. 1, March 2014:87
-96
TELKOMNIKA
r89
ISSN: 1693-6930
,,(qlr)=
[o;
)u',t
- p)q*-q
Order p is known, the coefficient vectors r(o) have uniform distribution on the interval
(- lr)r
Order q is known, the coefficient vectors p(n) have
(- r,r)n
uniform distribution on the interval
.
e)
Variance
o'
have inverse gamma distribution with parameter
rq');
,,("'F, u):
:*
'lz
{"' I('+) .*p)
Here, the parameters
l" and p
I2
anO
q
2'
#
are assumed to have uniform distribution on the
of cr is 2 and the parameters B is assumed have Jeffrey distribution,
So the prior distribution for the parameters H, =(p,q,r(o),Otc),rz) and H, =(f.,8) can be
interval (0,1), the value
expressed as:
n(H,, u, ) = *(plr)"(.t" lp) n(qlp)n(ot" lo) ,,("'1",p)n(i)"G)
(5)
According to Bayes theorems, then the a posteriori distribution for the parameters
and Hz can be expressed as:
"(u,,
H,lr)* z(';n,11n,,
H, )
H1
(6)
A posteriori distribution is a combination of the likelihood function and prior distribution.
The prior distribution is determined before the data is taken. The likelihood function is objective
while this priordistribution is subjective. ln this case, the a posteriori distribution n(U,,Hrlr)
has the form of a very complex, so it can not be solved analytically. To handle this problem,
reversible jump MCMC method is proposed.
2.2 Revercible Jump MCMC Method
Suppose 111=(Hr,Hr). tn general, the MCMC method is a method of sampting, as
how to create a homogeneous Markov chain that meet aperiodic nature and irreducible (tgl)
M1,M2,"',M. to be eonsidered such as a random variable following the distribution
n(U,,Hrlt). Thus M1,M2,"',M- it can be used to estimate the parameter M. To realize
this, the Gibbs Hybrid algorithm (tgl) is adopted, which consists of two phases:
1. Simulation of the distribution rr(H,lH,, s)
2. Simulation of the distribution
rr(H,lHr,r)
Gibbs algorithm is used to simulate the distri6ution n(UrlH,,s) anO the hybrid
algorithm, which combine the reversible jump MCMC algorithm [5] and the Gibbs atgorithm,
Hierarchical Bayesian of ARMA Models Using Simulated Annealing Algorithm (Suparman)
ISSN: 1693-6930
90 I
generally
. Tn" reversible jump MCMC algorithm is
used to simulate the distribution a(Hr lUr,.)
of the Metropolis-Hastings algorithm [6]' [8]'
2.2.1 Slmulatlon of the
dlstrlbution n(H,lH, ' s)
(Hr,s), writen ,r(HrlH,,
The conditional distribution of Hz given
,,(urlu,,s).. il(1-1;n*-n po(l-p)q@-q
[i)t
r), ..n
be expressed as
**# i
with parameters
This distribution is inversion gamma distribution
A
ana
L'
so the Gibbs
algorithm is used to simulate it'
2.2.2 Simulation of the
distributlon n(HtlH,s)
given (He, s), writen
lf the conditionat distribution of H1
n(H,lUr, r) , i. lnt"gtated with
to 02, then we get
respect
nb,q,rrnr, pto)lHr,r) = L. n(H,lur,t)do'
Let v
-g+-yand
w =9-*|X=n._., g'(t,p,q,F-1(r(p))c-(pt')
and we use
ln.
(o'ft'." *o-#uo'=#
then we get
rq)'
n(p, q, rrnr,
p(a)
lHz, r)
*
[on-)^,
u
- ],)n*-n
[-, ]'
(1
- n)o*-r (;)-'
tr
;P
On the other hand, we have also
n(o'ln,t,rto), p(o),Hr,r)" (o'f('*t)
t*P'
$
a result of multiplication of the
So that we can express the distribution n(H,pfr,t) .as
tne oistrioution, n(ouln,qor(o)op(o)'H"s) '
distribution n(p,q,rtnl,p(u)lH,s) ano
ffiMNtKA
Vol.
D,
No' 1,
March2ol4:87-96
TELKOMNIKA
r91
ISSN: 1693-6930
,r(u,lur, ,) =
n(p, q, r(o), p,n,lHr,
Next to simulate the distribution
phases:
r) r
r(U,lH,s),
"("'ln,
q, r(p), p(n),
Hr, ,)
we use a hybrid algorithm that consists of two
t
. phase 1: Simulate the distribution or n(o'?fp,g,r(o),p(n),Hr,.)
. phase 2: Simulate the distribution n(p, q, r(n), p(o)
lH, s)
The Gibbs algorithm is used to simulate the distribu,'on ,r(o'ln,Q,r(o),p,o,,Hr,r).
conversely, the distribution n(p,q,r(o),p(n)lHr,r)
n"r a complex form. The reversible jump
MCMC is used to simulate it.
\A/hen the order (p,q)'. determined, we can use the Metropolis Hastings atgorithm.
Therefore, in the case that this order is not known, Markov chain must jump from the order
(p,q) with parameters (rrrr,otol) to tn" order (p-,g-) with the parameter (rro'r,0,")). ro
solve this problem, we use the Reversible Jump MCMC algorithm.
2.2.3 Type of jump selection
Suppose
(p,q)
tojumpfromthepto
p+1,6fl
to jump from p to
nf
from the q
p,
nfl
tne probability
p-1, Ei"
theprobabitity
,epresent actual values for the order, we will write:
theprobabilitytojumpfromthepto
the probability to jump from q to q * 1, 6yo the probabitity to jump
to q - 1 and (p tne probability to jump from q to q. For each component, we wiil
choose the uniform distribution on the possible jump. As an example for the AR, this distribution
depends on p and satisfy
ni**51"*(fl=t
We set
6fl
=
(f* = 0 and Sf: = 0
nfl =.*r,'{t,g+}
n(p)
t
)
and
Under this restriction, the probability
will
be
sfl =.rnir,{r,-"(p).-}
'
with constant c, as much as possible so that
['n(p+l)J
nfl * SfR < 0.9 for p = 0,1,...,p-*.
The goal
is to have
nf"n(P)= 6ffin(P+1)
2.2.4Bifth / Death of Order
As for the AR example, suppose that p is the actual value for the order of the ARMA
model, ,trl = (q,rr....,ro) is the coefficient value. Consider that we want to jump from p to
p+
l.
We take the random variable u according to the triangular distribution with mean 0
[u+1"
s(u)=1r_",
-l
F* * * *c'vt trBT ffi s=
F
E
*: *€$ qs* u
#*re
**
*€$
m
g
=
ffi
$
***n* * * s m * # ** * *r*
*
#
-&-
. :: ::.-;::::!,,..::€
:r,':.:i:rl'.i1-+.+i$
ffiw
w
$
Accredited,,A,,byDGHE(DtKl),r"", j"""*.1#?^?rt;Ur7,?r?
TELKOMNIKA
TELKoMNTKA (Telecommunication,,computing,
Erectronics and controt)
lhe journal is first published
peer reviewed internationar journar.
on..December zoog. i-n'"1,r
ot the journar i, to orr,,l 3
aspects of the latest outst6ntins--jg;"r;;;";.r'li
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,nu ,,",0 of erecrricar unl',t-l1'in-oi;td;i;;l#;:dicat;d to;ii
:,'J.iff[i'JJ,Jlffi',"';;;;"", ;;;;i?['J::', [,':;,1[f .*;;TT;'1J.,?[?J:l'":jt;,.,"Jf
;:."#H;:ff j,l;
Editor-in_Chief
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Universitas Ahmad Dahlan
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.
Signal processin(]
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signal/RFIC Desjgn
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rEEE
Consultants Network of Silico;
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California, USA
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of Etectronic Engineering
untversity of Rome .Tor V6rgata,.
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Universiti Kebangsaan Malavsia
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Unrversity of Batna
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ent o
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ors
TELKOMNIKA
Vof, 12, No. 1, March
2Al4
ISSN ,1693-6930
Accredited "A" by DGHE (DlKTl), Decree No.58/DlKTl/Kep/2013
Table of Contents
Regular Papers
The Architecture of lndonesian Publication lndex: A Major lndonesian
Database
lmam Much lbnu Subroto, Tole Sutikno, Deris Stiawan
Academic
A Review on Favourable Maximum Power Point Tracking Systems in Solar
Application
Awang Jusoh, Tole Sutikno, Tan Kar Guan, Saad Mekhilef
The Analysis and Application on a Fraetional-Order Chaotic
Ye Qian, liang Ling Liu
Energy
1
6
System
23
Plant
33
Cost and Benefit Analysis of Desulfurization System in Power
Zhang Caiqing, Liu Meilong
lnduced
47
Signal
53
lnvestigation of Wave Propagation to PV-Solar Panel Due to Lightning
Overvoltage
Nur Hidayu Abdul Rahim, ZikriAbadi Baharudin, Md Nazri Othman, Zahriladha
Takaria, Mohd Shahril Ahmad Khiar, Nur Zawani Safiaruddin, Azlinda Ahmad
Line Differential Protection Modeling with Composite Current and Voltage
Comparison Method
Hamzah Eteruddin, Abdullah Asuhaimi Mohd Zin, Belyamin Belyamin
Baseline
lmproved Ambiguity-Resolving for Virtual
Hailiang Song, Yongqing Fu, Xue Liu
63
System
lntegrated Vehicle Accident Detection and Location
Md. SyedulAmin, Mamun Bin lbne Reaz, Salwa Sheikh Nasir
Neural Network Adaptive Control for X-Y Position Platform with
Ye Xiaoping, Zhang Wenhui, Fang Yamin
73
Uncertainty
Hierarchical Bayesian of ARMA Models Using Simulated Annealing
Suparman, Michel Doisy
Algorithm
Resolutions
Water Pump Mechanical Faults Display at Various Frequency
Linggo Sumarno, Tjendro, Wwien Widyastuti, R.B. Dwiseno \Mhadi
Endocardial Border Detection Using Radial Search and Domain
Knowledge
79
87
97
107
Yong Chen, DongC Liu
New Modelling of Modified Two Dimensional Fisherface Based Feature
Arif Muntasa
Extraction
Network
Emergency PrenatalTelemonitoring System in Wireless Mesh
Muhammad HaikalSatria, Jasmy bin Yunus, Eko Supriyanto
Deformation
A New ControlCurve Method for lmage
Hong-an Li, Jie Zhang, LeiZhang, Baosheng Kang
I
115
123
135
TELKOMNIKA
ISSN
1693.6930
Vol. 12, No. 1, March 2OL4
Accredited 'rA" by DGHE (DlKTl), Decree No.58/DlKTl/Kep/2013
Orthogonal Frequency-Division Multiplexing-Based Cooperative Spectrum Sensing for
Cognitive Radio Networks
Arief Marwanto, Sharifah Kamilah Syed Yusof, M. Haikal Satria
143
LTE Coverage Network Planning and Comparison with Different Propagation Models
RekaM Sabir Hassan, T. A. Rahman, A. Y' Abdulrahman
153
A Threshold Based Triggering Scheme for Cellular-to-WLAN Handovers
Liming Chen, Qing Guo, Zhenyu Na, Kaiyuan Jiang
163
Model and Optimal Solution of Single Link Pricing Scheme Multiservice Network
lrmeilyana, lndrawati, Fitri Maya Puspita, Juniwati
173
An Analytical Expression for k-connectivity of Wireless Ad Hoc Networks
Nagesh kallollu Narayanaswamy, Satyanarcyana D, M'N Giri Prasad
179
Two Text Classifiers in Online Discussion: Support Vector Machine vs Back-Propagation
Neural Network
Erlin, Rahmiati, Unang Rio
189
Research on Traffic ldentification Based on Multi Layer Perceptron
Dingding Zhou, Wei Liu, Wengang Zhou, Shi Dong
201
plagiarism Detection through lnlernet using Hybrid Artificial Neural Network and Support
Vectors Machine
lmam Much lbnu Subroto, AliSelamat
209
Reliability Analysis of Components Life Based on Copula Model
Han Wen Qin, Zhou Jin Yu
219
Cobb-Douglass Utility Function in Optimizing the lnternet Pricing Scheme Model
lndrawati, lrmeilyana, Fitri Maya Puspita, Meiza Putri Lestari
227
CApBLAT: An lnnovative Computer-Assisted Assessment for Problem-Based Learning
241
Approach
Muhammad Qomaruddin, Azizah Abdul Rahman, Noorminshah A. lahad
publications Repository Based on oAl-PMH 2.0 Using Google App Engine
251
Hendra, Jimmy
Android Based Palmprint Recognition System
Gede Ngurah Pasek Pusia Putra, Ketut Gede Darma Putra, Putu \Mra Buana
263
\
,
TELKOMNIKA, Vol.12, No.1, March 2014,pp.87 - 96
ISSN: 1693€930, accredited A by DtKTt, Decree No: 58/DtKit/Kepl2O13
DOl:
1
0.
1
f87
2928ffELK0MNlKA,v 12i1.fl A6
Hierarchical Bayesian of ARMA Models Using
Simulated Annealing Algorithm
1
Deparrment or Marhematicar
_
rclcX?i3l:tXH:',jl$""?flr3:fil:r,rn, Jr pror. Dr
Yogyakarta, lndonesia
'ENSEEIHT/TESA, 2 Rue Camichel, Toulouse, France
"Conesponding author, email: e_mail: suparmancict@yahoo.co.id
soepomo, sH
,
Abstract
When the Autorcgressive Moving Average (ARMA) model is fifted with rcal data, the actual
value
of the model order and the model parameter are often unknown. The goal of this paper is to frnd an
estimator for the model order and the model parameter based on the dati. ln this paper, ihe model
order
identification and model parameter estimation
is given in a hierarchicat Bayeiiai fimiiorx. tn this
framework, the model order and model panmeter are assurned to have pior distibution, which
summarizes all the infomation available about the p/ocess. All the information about the characteristics
of
the model order and the model panmeter are expressed in the posteior distibution. prcbability
determination of the model order and the model panmeter requirei the integration of the posteior
djstnbution resulting. lt is an operation which is very ditricuft to be-solved analytiiatty. 6ere ine Simuated
Annealing Reversible Jump Markov Chain Monte Cailo (MCMC) aQofthm *as aeretopei to iompute
the
required integration over the posterior distibution simulation. Methods developed are evaluated in
simulation sfudies in a number of set of synthetic data and real data.
Keywords: Simulated Annealing, ARMA model, oiler identification, parameter estimation
l.lntroduction
xpx2t" ',xo is a time series data. Time series xpx22...,xn
ARMA(p,q) modet, if x,,X2,...,Xn satisff following stochastic equation [3]
Suppose
is said to be an
:
xt
where
z,
: zt * I;_, 0,2,_i *ff=,0,r,_,,
is the random error at the time t,
coefficient-coefficient. Here
variance
o'
.
ARMA model
z,
t =I,2,...,n
(1)
0t (i = 1,2,..., p) and 0: 0 = 1,2, ..., q) is the
is assumed to have the normal distribution with mean 0 and
(*,I*
is called stationary if and onty if the equality of potynomial
l+I:=,Sf')bi:o
must lie outside the unit circle. NextARMA model
equality of polynomial
t+
Fl
|HJ=L
olq)bj
J
:
(",),."
is called invertibte if and onty if the
o
must lie outside the unit circle [4].
The order (p,q)'" assumed to be known, parameter estimation of the ARMA model
has been examined by several researchers, for example t3l, t4l and [12]. ln practice when we
match the ARMA modelto the data, in generatorOer (p,q) is not known.
Received september 19,2013; Revlsed December 18,2013;Accepted January 12, 2014
BBI
ISSN: 1693-6930
not known, the order identification and the parameters
While for the order (p,q)
'.
estimation are done in two stages. The first stage is to estimate the coefficients and error
variance with the assumption of order (p,q) i. known. Based on the parameter estimation in
the first stage, the second stage is to identify the order (p,q) A criterium used to determine
(p,q) has been
proposed by many researchers, among others are the Akaike lnformation
Criteria Criterion (AlC), Bayesian lnformation Criterion (BlC), and the Final Prediction Error
Criterion (FPE).
Based on the data x, , t =\,2,,.., n , this research proposes a method to estimate the
orOer
value p,q,Q(n),6(o), and o,2 simultaneously. To do that, we will use a hierarchical Bayesian
approach [10], which will be described below.
2. Research Method
2.1 Hierarchical Bayesian
t-\
Let s = (xp*q*r,
)*, ) o" a realization of the ARMA (p, q) model. lf the value
,o =(*,,*r,...,Xp+q) i, knorn, then the likelihood function of s can bewritten approximately
as follows
xp*q +2,. . .
:
(#)7.*o-#I*,.,*,e'(t,p,e,oin),s{o)
z(rln,r,otn',0(n),o')=
where g(t,p,q,0(0r,6{o))= *,
(2)
*fi,0fo'*,-, -I]=,ejttt,-, for t =p+q+1,p+ q+2,"',n
with initial value x, =xz=...=Xo*n =0 [11]. Suppose Snand In are respectively stationary
region and invertible region. By using the transformation
F:otnr = (O{",0y),"',0[n').
Sn
F] ro) =Qr,'r,"
','n)'(-t,t)n)
6.gro) = (efor,e!o),...,efrr)e In F+ p(o) =(p,pr,...,pn).
the ARMA model
ARMA model
(*,)*
(",),.,
stationery if and only
invertible
, o{a) ,
o2
)=
.(n)
if and only if o(t) -
likelihood function can be rewritten as:
/(rlp, e, ,(')
i1
b,pr,"',Po).(-t't)t)
b)
[1]. Then the
n-p
(#)T
"*p-
*I,=n.'.,
a)
[o'-)uu
Ft,t)-)
=(rr,rr," ',.0).(-t,t)n) t2l ano tne
g' (t, p, q, F-l(o(n) ;, 6-r lgta')) tol
The prior distribution determination for parameters are as follows:
Order p have Binomial distribution with parameters l":
,,blr)=
tel
- ]")r*-n
Order q have Binomial distribution with parameters p:'
TELKOMNIKA Vol. 12, No. 1, March 2014:87
-96
TELKOMNIKA
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ISSN: 1693-6930
,,(qlr)=
[o;
)u',t
- p)q*-q
Order p is known, the coefficient vectors r(o) have uniform distribution on the interval
(- lr)r
Order q is known, the coefficient vectors p(n) have
(- r,r)n
uniform distribution on the interval
.
e)
Variance
o'
have inverse gamma distribution with parameter
rq');
,,("'F, u):
:*
'lz
{"' I('+) .*p)
Here, the parameters
l" and p
I2
anO
q
2'
#
are assumed to have uniform distribution on the
of cr is 2 and the parameters B is assumed have Jeffrey distribution,
So the prior distribution for the parameters H, =(p,q,r(o),Otc),rz) and H, =(f.,8) can be
interval (0,1), the value
expressed as:
n(H,, u, ) = *(plr)"(.t" lp) n(qlp)n(ot" lo) ,,("'1",p)n(i)"G)
(5)
According to Bayes theorems, then the a posteriori distribution for the parameters
and Hz can be expressed as:
"(u,,
H,lr)* z(';n,11n,,
H, )
H1
(6)
A posteriori distribution is a combination of the likelihood function and prior distribution.
The prior distribution is determined before the data is taken. The likelihood function is objective
while this priordistribution is subjective. ln this case, the a posteriori distribution n(U,,Hrlr)
has the form of a very complex, so it can not be solved analytically. To handle this problem,
reversible jump MCMC method is proposed.
2.2 Revercible Jump MCMC Method
Suppose 111=(Hr,Hr). tn general, the MCMC method is a method of sampting, as
how to create a homogeneous Markov chain that meet aperiodic nature and irreducible (tgl)
M1,M2,"',M. to be eonsidered such as a random variable following the distribution
n(U,,Hrlt). Thus M1,M2,"',M- it can be used to estimate the parameter M. To realize
this, the Gibbs Hybrid algorithm (tgl) is adopted, which consists of two phases:
1. Simulation of the distribution rr(H,lH,, s)
2. Simulation of the distribution
rr(H,lHr,r)
Gibbs algorithm is used to simulate the distri6ution n(UrlH,,s) anO the hybrid
algorithm, which combine the reversible jump MCMC algorithm [5] and the Gibbs atgorithm,
Hierarchical Bayesian of ARMA Models Using Simulated Annealing Algorithm (Suparman)
ISSN: 1693-6930
90 I
generally
. Tn" reversible jump MCMC algorithm is
used to simulate the distribution a(Hr lUr,.)
of the Metropolis-Hastings algorithm [6]' [8]'
2.2.1 Slmulatlon of the
dlstrlbution n(H,lH, ' s)
(Hr,s), writen ,r(HrlH,,
The conditional distribution of Hz given
,,(urlu,,s).. il(1-1;n*-n po(l-p)q@-q
[i)t
r), ..n
be expressed as
**# i
with parameters
This distribution is inversion gamma distribution
A
ana
L'
so the Gibbs
algorithm is used to simulate it'
2.2.2 Simulation of the
distributlon n(HtlH,s)
given (He, s), writen
lf the conditionat distribution of H1
n(H,lUr, r) , i. lnt"gtated with
to 02, then we get
respect
nb,q,rrnr, pto)lHr,r) = L. n(H,lur,t)do'
Let v
-g+-yand
w =9-*|X=n._., g'(t,p,q,F-1(r(p))c-(pt')
and we use
ln.
(o'ft'." *o-#uo'=#
then we get
rq)'
n(p, q, rrnr,
p(a)
lHz, r)
*
[on-)^,
u
- ],)n*-n
[-, ]'
(1
- n)o*-r (;)-'
tr
;P
On the other hand, we have also
n(o'ln,t,rto), p(o),Hr,r)" (o'f('*t)
t*P'
$
a result of multiplication of the
So that we can express the distribution n(H,pfr,t) .as
tne oistrioution, n(ouln,qor(o)op(o)'H"s) '
distribution n(p,q,rtnl,p(u)lH,s) ano
ffiMNtKA
Vol.
D,
No' 1,
March2ol4:87-96
TELKOMNIKA
r91
ISSN: 1693-6930
,r(u,lur, ,) =
n(p, q, r(o), p,n,lHr,
Next to simulate the distribution
phases:
r) r
r(U,lH,s),
"("'ln,
q, r(p), p(n),
Hr, ,)
we use a hybrid algorithm that consists of two
t
. phase 1: Simulate the distribution or n(o'?fp,g,r(o),p(n),Hr,.)
. phase 2: Simulate the distribution n(p, q, r(n), p(o)
lH, s)
The Gibbs algorithm is used to simulate the distribu,'on ,r(o'ln,Q,r(o),p,o,,Hr,r).
conversely, the distribution n(p,q,r(o),p(n)lHr,r)
n"r a complex form. The reversible jump
MCMC is used to simulate it.
\A/hen the order (p,q)'. determined, we can use the Metropolis Hastings atgorithm.
Therefore, in the case that this order is not known, Markov chain must jump from the order
(p,q) with parameters (rrrr,otol) to tn" order (p-,g-) with the parameter (rro'r,0,")). ro
solve this problem, we use the Reversible Jump MCMC algorithm.
2.2.3 Type of jump selection
Suppose
(p,q)
tojumpfromthepto
p+1,6fl
to jump from p to
nf
from the q
p,
nfl
tne probability
p-1, Ei"
theprobabitity
,epresent actual values for the order, we will write:
theprobabilitytojumpfromthepto
the probability to jump from q to q * 1, 6yo the probabitity to jump
to q - 1 and (p tne probability to jump from q to q. For each component, we wiil
choose the uniform distribution on the possible jump. As an example for the AR, this distribution
depends on p and satisfy
ni**51"*(fl=t
We set
6fl
=
(f* = 0 and Sf: = 0
nfl =.*r,'{t,g+}
n(p)
t
)
and
Under this restriction, the probability
will
be
sfl =.rnir,{r,-"(p).-}
'
with constant c, as much as possible so that
['n(p+l)J
nfl * SfR < 0.9 for p = 0,1,...,p-*.
The goal
is to have
nf"n(P)= 6ffin(P+1)
2.2.4Bifth / Death of Order
As for the AR example, suppose that p is the actual value for the order of the ARMA
model, ,trl = (q,rr....,ro) is the coefficient value. Consider that we want to jump from p to
p+
l.
We take the random variable u according to the triangular distribution with mean 0
[u+1"
s(u)=1r_",
-l