Hierarchical Bayesian of ARMA Models Using Simulated Annealing Algorithm - Repository Universitas Ahmad Dahlan

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TELKOMNIKA

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Editor-in_Chief Tole Sutikno

Universitas _yogyakarta, Ahmad Dahlan lndonesia email: thsutikno@ieee.org

Co-Editor_in-Chief

Signal processin(] Arianna Mencattirii

O:?-1nl"i, _{of Etectronic Engineering}

untversity of Rome .Tor _V6rgata,. Roma, ltajV

email; mencattini@ing.uniroma2.it Telecommu n ications E ngi neeri ng

Sunardi

Dept. of Ejectrjcal Engineering Universitas Ahmad Dahlan

Yogyakarta, Indonesia email; sunargm@ieee.org

Editors

Control Engineerino

.. Omar Lengerke

, ,_ Mechatronics Engineering Faculty unrversidad Autonoma de BrJcaramanga

Bucaramanga, Colombia olengerke@unab.edu.co Com put ng a nd I nformatics

Wanquan Liu

Dept of of Comoutino

Curtin _{University ot iecnriotogy} perth _{WA, Australia}

w.liu@curtin.edu.au

E I e ctr-on ic s E n g i neeri ng

. Mark S. Hooper ,__Analog/mixed signal/RFIC Desjgn

rEEE Consultants Network of Silico; Viley California, USA

m. hooper@ieee.org

power Engineerino

Auzani Jidin

Dept of Power _{Electronics and D.ive}

Universiti Teknjkal Malaysia lvlelaka Melaka, Maiaysia auzani@ utem. edu. my

Ahmad Saudi Samosir

Universjtas Lampung

Lampunq, Indonesja saudi@ieee.org

Deris Stiawan

Sriwrjaya Universitv Palembang, lndoneiia

deris@ieee.org

Ahmet Teke

Qukurova Universjty

Aoana, I urkev ahmetteke@cu.edu.tr

Faygat Djeffat

Unrversity of Batna

_ Batna, Algeria taycaldzdz@hotmai j. _com

Jumril yunas

Universiti Kebangsaan Malavsia Kuala Lumpur, Malaysia

iumrilyunas@ukm. my

Mochammad Facta

Ur ivers jtas D iponegoro

semarang, Indonesia facta@ieee.org

Balza Achmad

u"i"*"it"" _{C-"lt"i']v"0"} D' Jude Hemanth

Yogyakarta. _lndonesia Karunya university

uur."i'nog"dju;;r"o"iJo, Tamil Nadu lndia

jude-hemanth@rediffmail.com

lrwan _{Prasetya Gunawan}

Brk,j" _Lj;;;;it;"'" Jacek stando

.r"ourtu,lnOl,nJsL ,"chnicat University of Lodz

r*"n.s;;u;;;6lriii5."".,o

Lodz, Porand Jacek.stando@p. lodz. pl

Jyoteesh Malhotra Guru _{Nanak Dev Universitv}

Punlab, lndia jyoteesh@gmail.com

Lunchakorn Wuttisittikulkii

Chulalongkorn Universitv Banqkok, Thailancj '

Luncha korn. W@chu la. ac. th

Yin Liu

Symantec Research Labs, Core Symantec Corporation Mountain View, CA. USA

liuy@cs.rpi.edu

Leo p. Ligthart

Delft U_niversity of Technology Delft, Netherlands

L. P.Ligthart@tudetft . nl

..MunawarARiyadi

Ultversitas _Diponegoro Jemarang, lndonesia munawar.riyadi@ieee.org

Srinivasan _Alavandar

Caledonian Univ. ol Engineering Sib, Oman seenu. phd@gmail.com

l_inawati

Universitas Udavana

Bali, lndonesia linawati@u n ud. ac. id

Nidhal Bouaynaya

unrv. of Arkansas at Litfle Rock Little Rock, Arkansas, _USA

nxbouaynaya@ualr.edu Supavadee Aramvith Chulalongkorn Universitv

Bangkok, Thailand supavadee. a @ch u la. ac. th

Nik Rumzi Nik ldris

u n iuu,iitl i"r,ioj",'ni",""^

n,

",""t# I'jl#,:${rT" u.,"

Johor..Malaysia -

Johor, Maiavsia

nikrumzi@ieee.ors

_{.#;;;6ffi.:i;1_y}

- , .surinder singh Tarek Bouktir sant _{Longowal,lnsr of Eng &}

L_arbi _{e"n Vntiitjniu"r.ttv}

Tech

Puniab, lndia oum El-Bouaqhi Alqeria

surinoer-soJ'ni@i"iiiruir

"o.

tarek bouktir@estg'ouf,s _org

Tutut Herawan

u"i"".J;;;;;;"?un"ns

_{Universrtyorlils#ls.iun"uuno}

pahang, _Malavsia

t,r,t@;;;ffi;y

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Yutthapong Tuppaduno

Provincial _{Electriciti Autho;itv} Bangkok, Thajland ytuppadung@gmail. com

Youssef Said

Ecole Nationale d,lngenieurs de Tunis

Tunis, Tunisia y.said@ttnet.tn

Bimat K. Bose (USA)

Hamid A. Totiyar (USA1

Abdut Fadtil

Adhi Susanto

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A h m a d D a h r a n ( u A D ₎

TelD. +62 c

p

_oz zr+ latan prof. Soepomo ianturan-, yogyakafta, 37e4tB, _{s63sls, s11B2e, s1-i'3o,}

3;ii;;;;i

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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

Regular

Papers

The Architecture of lndonesian Publication lndex: A Major lndonesian

Academic

1

Database

lmam Much lbnu Subroto, Tole Sutikno, Deris Stiawan

A Review on Favourable Maximum Power Point Tracking Systems in Solar

Energy

6

Application

Awang Jusoh, Tole Sutikno, Tan Kar Guan, Saad Mekhilef

The Analysis and Application on a Fraetional-Order Chaotic

System

23

Ye Qian, liang Ling Liu

Cost and Benefit Analysis of Desulfurization System in Power

Plant

33

Zhang Caiqing, Liu Meilong

lnvestigation of Wave Propagation to PV-Solar Panel Due to Lightning

lnduced

47 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

Signal

53

Comparison Method

Hamzah Eteruddin, Abdullah Asuhaimi Mohd Zin, Belyamin Belyamin

lmproved Ambiguity-Resolving for Virtual

Baseline

63 Hailiang Song, Yongqing Fu, Xue Liu

lntegrated Vehicle Accident Detection and Location

System

73 Md. SyedulAmin, Mamun Bin lbne Reaz, Salwa Sheikh Nasir

Neural Network Adaptive Control for X-Y Position Platform with

Uncertainty

79

Ye Xiaoping, Zhang Wenhui, Fang Yamin

Hierarchical Bayesian of ARMA Models Using Simulated Annealing

Algorithm

87

Suparman, Michel Doisy

Water Pump Mechanical Faults Display at Various Frequency

Resolutions

97 Linggo Sumarno, Tjendro, Wwien Widyastuti, R.B. Dwiseno \Mhadi

Endocardial Border Detection Using Radial Search and Domain

Knowledge

107

Yong Chen, DongC Liu

New Modelling of Modified Two Dimensional Fisherface Based Feature

Extraction

115

Arif Muntasa

Emergency PrenatalTelemonitoring System in Wireless Mesh

Network

123 Muhammad HaikalSatria, Jasmy bin Yunus, Eko Supriyanto

A New ControlCurve Method for lmage

Deformation

135

Hong-an Li, Jie Zhang, LeiZhang, Baosheng Kang

TELKOMNIKA

ISSN

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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

LTE Coverage Network Planning and Comparison with Different Propagation Models

RekaM Sabir Hassan, T. A. Rahman, A. Y' Abdulrahman

A Threshold Based Triggering Scheme for Cellular-to-WLAN Handovers Liming Chen, Qing Guo, Zhenyu Na, Kaiyuan Jiang

Model and Optimal Solution of Single Link Pricing Scheme Multiservice Network

lrmeilyana, lndrawati, Fitri Maya Puspita, Juniwati

An Analytical Expression for k-connectivity of Wireless Ad Hoc Networks Nagesh kallollu Narayanaswamy, Satyanarcyana D, M'N Giri Prasad

Two Text Classifiers in Online Discussion: Support Vector Machine vs Back-Propagation

Neural Network

Erlin, Rahmiati, Unang Rio

Research on Traffic ldentification Based on Multi Layer Perceptron Dingding Zhou, Wei Liu, Wengang Zhou, Shi Dong

plagiarism _{Detection through lnlernet} using Hybrid Artificial Neural Network and Support Vectors Machine

lmam Much lbnu Subroto, AliSelamat

Reliability Analysis of Components Life Based on Copula Model Han Wen Qin, Zhou Jin Yu

Cobb-Douglass Utility Function in Optimizing the lnternet Pricing Scheme Model

lndrawati, lrmeilyana, Fitri Maya Puspita, Meiza Putri Lestari

CApBLAT: An lnnovative Computer-Assisted Assessment for Problem-Based Learning Approach

Muhammad Qomaruddin, Azizah Abdul Rahman, Noorminshah A. lahad

publications Repository Based on oAl-PMH 2.0 Using Google App Engine Hendra, Jimmy

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Gede Ngurah Pasek Pusia Putra, Ketut Gede Darma Putra, Putu \Mra Buana

143

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153

163

173

179

189

201

209

227

241

251

TELKOMNIKA, Vol.12, No.1, March 2014,pp.87

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96

ISSN: 1693€930, accredited A by DtKTt, Decree No: 58/DtKit/Kepl2O13

DOl: 1 0. 1 2928ffELK0MNlKA,v 12i1.fl _A6

f87

Hierarchical

Bayesian of ARMA

Models

Using

Simulated Annealing

Algorithm

1 _{Deparrment or Marhematicar}

rclcX?i3l:tXH:',jl$""?flr3:fil:r,rn,

_Jr pror. _{Dr soepomo, sH} ,

_

Yogyakarta, lndonesia

'ENSEEIHT/TESA, 2 Rue Camichel, Toulouse, France

"Conesponding author, email: e_mail: suparmancict@yahoo.co.id

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

Suppose

xpx2t"

',xo

is a time series data. Time series

xpx22...,xn

is said to be an

ARMA(p,q)

modet, if

x,,X2,...,Xn

satisff following stochastic _{equation [3]} :

xt

:

zt

*

I;_,

0,2,_i

*ff=,0,r,_,,

t

=I,2,...,n

where

z,

is the random error at the time

t,

0t

(i

=

1,2,...,

p) and

0:

₀

=

1,2,

..., q) is the coefficient-coefficient. Here

z,

is assumed

to

have the normal distribution with mean

0

and

variance

o'

. ARMA model

(*,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

(",),."

is called invertibte if and onty if the

equality of polynomial

t +

Fl

_|HJ=L olq)bj

:

o

J

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.

(1)

BBI

ISSN: 1693-6930

While for the order

(p,q)

'.

not known, the order identification and the parameters

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 orOer

(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

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, _{xp*q +2,.} . .

)*,

)

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 :

z(rln,r,otn',0(n),o')=

(#)7.*o-#I*,.,*,e'(t,p,e,oin),s{o)

(2)

where

g(t,p,q,0(0r,6{o))=

*,

*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)

tel

6.gro)

= (efor,e!o),...,efrr)e In F+ p(o)

=(p,pr,...,pn). Ft,t)-)

the ARMA model

(*,)*

stationery

if

and only

i1

.(n)

=(rr,rr,"

',.0).(-t,t)n)

t2l

ano tne ARMA

model

(",),.,

invertible

if

and only

if

o(t)

-

b,pr,"',Po).(-t't)t)

[1]. Then the

likelihood function can be rewritten as:

n-p

/(rlp, e, ,(') , o{a) , o2

)=

(#)T

"*p-

*I,=n.'.,

g' (t, p, q, F-l(o(n) ;,

6-r

lgta'))

tol

The prior distribution determination for parameters are as follows:

a)

Order p have Binomial distribution with parameters

l":

,,blr)=

[o'-)uu

-

]")r*-n

b)

Order q have Binomial distribution with parameters p:'

TELKOMNIKA ISSN: 1693-6930

r89

,,(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 uniform distribution on the interval

(-

r,r)n .

e)

Variance

o'

have inverse gamma distribution with parameter

I

anO 2

rq');

,,("'F,

u):

:*

{"'

I('+) .*p-

#

'lz

)

q

2'

Here,

the

parameters

l"

and

p

are assumed

to

have uniform distribution

on

the

interval (0,1), the value

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

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 H1 and Hz can be expressed as:

"(u,,

H,lr)*

z(';n,11n,,

H,

)

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,}

(6)

ISSN: 1693-6930

90 I

used to simulate the distribution a(Hr

lUr,.)

. Tn" reversible jump MCMC algorithm is generally

of the Metropolis-Hastings algorithm [6]' [8]'

2.2.1 Slmulatlon of the

dlstrlbution n(H,lH,

_' s)

The conditional distribution of Hz given (Hr,s), writen

,r(HrlH,,

r),

..n

be expressed as

,,(urlu,,s).. il(1-1;n*-n

po(l-p)q@-q

[i)t

**# i

This distribution is inversion gamma distribution with parameters

A

ana

L'

so the

Gibbs

algorithm is used to simulate it'

2.2.2 Simulation of the

distributlon

n(HtlH,s)

lf the conditionat distribution of H1 given (He, s), writen

n(H,lUr,

r) ,

i.

lnt"gtated with respect

to

02, then we get

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

*o-#uo'=#

ln.

(o'ft'."

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'

$

So that

we

can

express

the

distribution

n(H,pfr,t)

.as

a

result

of

multiplication

of

the distribution

n(p,q,rtnl,p(u)lH,s)

ano tne oistrioution, n(ouln,qor(o)op(o)'H"s) '

TELKOMNIKA ISSN: 1693-6930

r91

,r(u,lur, ,)

=

n(p, q, r(o), p,n,lHr,

r)

r

"("'ln,

q, r(p), p(n),

Hr,

,)

Next to simulate the distribution

r(U,lH,s),

we use

a

hybrid algorithm that consists of two

phases:

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)

,epresent actual values for the order, we will write:

nfl

tne probability

tojumpfromthepto

p+1,6fl

theprobabilitytojumpfromthepto

p-1,

Ei"

theprobabitity

to jump from p to

p,

nf

the probability to jump from q

to

q

*

1,

6yo

the probabitity to jump from the q

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

Under this restriction, the probability

will

be

nfl

=.*r,'{t,g+}

and

sfl

=.rnir,{r,-"(p).-}

t

n(p)

)

'

['n(p+l)J

with constant c, as much as possible so that

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

-l<u<0

0<u<l

[u+1"

s(u)=1r_",

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ISSN: 1693-6930

We complete the vector

rb)

random variables with u. So the new coefficient vector is

proposed

,(r1)

-

(.,,

rr....,

rp,

u)

Note that this transformation will change the total value of all. lt is clearly seen that the

Jacobian of the transformation of value is 1.

lnstead,

to

jump

from

p +

1

to

p

is

done

by

removing

the last

coefficients in r(p+l)

-

(tr,rr....,rn,ro*r).

So the

new

coefficient

vectors

that

is

proposed become r.(n) ₌ (r,, 12... . , fp _).

fn"

probability of acceptance / rejection respectively is

crN

=min[,rnu]

and

oo

=mint,r"t]

where

rN

Finally, we get We have

l'

q(p

*

1,1(n*t);P,r'o' =

sffi)

to(p,

tto', p + 1, r(o*t) =

nf*e(ro.'

))

[si'tr,

-

frl,sin(r,

+

frr]

-

-

w(F,p+l,q,r(o*t),p(d)-u(")

p.o

-p r f n[

'"

-;6,p,q,r0Lptrrfvtc')

P+1 1-)'

2

sf

g(ro*r)

2.2.5 Changes in coefficients

Suflpose now that the AR part is selected to jump from p to p without a order change, but only the coefficient is changed .

lf

r(p) =

(r,,rr....,rn)

is the coefficient vector, we modify

the

coefficient

vector.

Consider

that

t1,t2,...,tp

is

courant

point

and

supposing that

tJlert2t ..'

up

new point' we define the point ui in the following way:

ui=sin(ri+s)

s

is taken with the uniform distribution on the interval

[-+,+-l.

Then

u,

is selected with

L

lo',lo.l

I

the distribution

f(*,1r,)=+

\ rrrl

n.r/t-uf

in the interval

q(p + 1, r(n+') ' _o, r(n)

q(p, r(o); p + 1, r(n*t)

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r93

fc=

because

;; ,.(r)

=

(ri,r;.'..,i,,ri,ri*r,...,r0)

and

7.(n)

-

(+i,t;....,r,1,,ui,ri*l,...,r0),

th"n the probability acceptance / rejection can be written by

crc =

min[,rc]

Where

n(p, q, r-(o), p(n),

H,ls)

n(;r, q, ?.(o), _{Otrl, H, ls) e}

w(F, p, q, r*(o), prol )-"(")

*(F,

p, q, r(o), _otu)

fv(')

I

(

t+r, 1-4')t

l.t.r\

t-r,l

So

fc=

2.3 Simulated Annealing Algorithm

Simulated Annealing algorithm

[7]

is

obtained

by

adding

a

line

in

the

temperature

T1,T2,.'.,

T.

at the top of the MCMC method. Next simulated annealing algorithm will produce

a

Markov

chain

M(T,),M(Tr),...,M(T-)

which

is

no

longer

homogeneous.

Wth

a

hypothetical on

a

certain

TpT2,...,T.

_[14]

will

be convergent

to

maximize the value

of

a posteriori distribution n(H,, ff

,lr).

3. Results and Analysis

ln As an illustration, we will apply this method to identify the order and estimate the

parameter synthesis ARMA data and real ARMA data. Simulation studies are done to confirm

that the performance of simulated annealing algorithm is able to work well. V/hile case studies

are given to exemplify the application of research in solving problems in everyday life.

For

both synthesis ARMA data

and

real ARMA data,

we will

use

the

simulated

annealing algorithm to identify order and estimate the parameters of the ARMA model. For this

purpose, the simulated annealing algorithm is implemented for 70000 iterations with a value of

initial temperature To ₌ 10 and the temperature is derived with the temperature factor 0.995 up

to the end temperature Tlase ₌ 0.01 . Value of order p and q is limited to a maximum of 10. So

that

p.o=

e-*=

10.

3.1 Synthetic ARMA data

Figure 1 shows a synthetic ARMA data. The data are made according to the equation

(1) above, with the nurnber of data

n =

250, order

p

= 2, order

q = 1,

_O(t)

=(-t.lO,O.Z),

g(r)

-

_{(O.Z), anO}

oz

=I

, r*(l).pr 7*(r) , ?*(P). p, 1'(P)

94r

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Figure 1. ARMA Synthetic Data

Based on the synthetic data in Figure 1, next order p, _{q order, ARMA model parameter}

and variance

o'

are estimated by using the SA algorithm. The order p, q order, ARMA model

parameter and variance 62 produced by the simulated annealing algorithm are p

=2,

Q=I,

$tzr

-

(0.+t,o.zs),

6ttr

-

(o.lz)

and 62 = 1.06. \Mren we compare between the actuatvatue

and the estimator value, it shows that simulated annealing algorithm can work well.

3.2 ReaIARMA Data

The real data in Figure 4 below is a passenger service charge (PSC) at the Adisutjipto

lnternational Airport in Yogyakarta lndonesia for the period 55 from January 2001 to July 2005.

@

Figure 2. First distinction of PSC data at the Adisutjipto lnternational Airport Yogyakarta.

Clearly visible in Figure 4, the data are not stationary very day. To get stationary data the

first

distinction is made and the results shown in Figure 3.

x 1or

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r95

-2t

0 _{50 60}

Figure 3. Second distinction of PSC data at the Adisutjipto lnternationat Airport.

Based on the data

in

Figure

3,

next order p,

q

order, ARTMA model parameter and

variance are estimated

by

using the simulated anneaiing algorithm. The results

are

p

-

1,

4 =

0,

6,',

= (O.lS) oan

6'

= 5.75x 107 .

4. Conclusion

The description above is a study of the theory

of

simulated annealing algorithms and its

application in the identification of order p and q, _{coefficient vectors estimation}

0,',

"nd

e,n), and variance estimation

o'

from the ARMA model. The results _{of the simulation show that the}

simulated annealing algorithm can estimate the parameters _{well. Simulated annealing algorithm}

can also be implemented with good results on Synthetic Aperture Radar image segmentation

[13].

As _{the implementation, the simulated annealing algorithm is applied to the pSC data at} the Adisutjipto lnternational Airport.

lts

result is that the PSC data can be modeled with the ARIMA model (1,0). The model can be used to predict the number of PSC at the Adisutjipto

lnternational Airport in the future.

References

t1l

Barndorff-Nielsen O, and Schou G. On the parametrization of autoregressive models by partial

autocorrelation. J. Multivar. Anal. 1973;3: 408-419.

l2l

Bhansali RJ. The inverse partial correlation function of a time series and its applications. J. Multivar. Anal. 1983: 13: 310-327.

13] Box GEP, Jenkins GM, Reinsel GC. Time Sen'es Analysr.s: Forecasting and Control. New Jersey: Prentice Hall. 2013

t4l

BrockwellPJ, Davis RA. Ilrnes Sen'es.'Theory and Methods. Springer: NewYork.200g.

l5l

Green

PJ.

Reversible _{Jump Markov Chain Monte Carlo} Computation and Bayesian model determination . Biometrika. 1 995; 82: 7 1 1-732.

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Hastings WK. Monte Carlo _{sampling methods using Markov chains and their applications. Biometrika.} 1970:57:97-109.

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Kirpatrick S. Optimization by Simulated Annealing: Quantitative Studies. Joumal of Statistical physics

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t81 Metropolis N, Rosenbluth AW, Teller AH, and Teller E. Equations of state calculations by fast computing machines. Joumal Chemical Physics. 19S3; 21: 1087-1091.

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[10] Robert CP. The Bayesian Choice. A Dlecision-Thearetic Matiiation. Springer Texts in Statistics. 1994. [1 1] Shaarawy S, Broemeling L. Bayesian inferences and forecasts with moving averages processes.

Commun. Sfafisf. - Theory M eth. 1 984; 1 3(1 5): 1 871 -1 888.

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[12] Suparman, Soejoeti Z. Bayesian estimation of ARMA Time Series Models. WKSI Joumal. 1999; 2(3):

91-98.

[13] Suparman. Segmentasi Bayesian Hierarki dalam Gitra SAR dengan Menggunakan Algoritma SA. Jumal Pakar. 2005; 6: 227-235.

[14]Winkler G. lmage Analysis, Random ,Fields and Dynamic Monte Calo Methods : A Mathematical

Intrcduction. New York: Springer Yerlag.2012.

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