BUDI NURANI RUCHJANA

DEPARTMENT OF MATHEMATICS

FACULTY OF MATHEMATICS AND NATURAL SCIENCES

UNIVERSITAS PADJADJARAN

Indonesia

budi.nurabi@unpad.ac.id

Paper is presented at SEAMS School

Spatio Temporal Data Mining and Optmization Modeling

UTC-Bandung, 9-19 August 2016

Development of Sciences and its Applications using a natural resources and environmental for welfare of society

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Research Roadmap of Faculty of Math and Natural Sciences

Universitas Padjadjaran

PIPUNPAD

Research on Basic Sciences

Model SAR, SAR-Kriging, Cross

Variogram, Co-Kriging

Model Clustering GSTAR, GSTARI-SUR,

GSTARIMAXDisease mapping

Model Clustering GSTAR-Kriging menggunakan

KDD

Pengembangan Model

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Temporal Data Mining

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dalam prediksi panas bumi

Kestabilan lereng

Open source R, Matlab

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Aplikasi user friendly

Hak Merk Model Clustering

GSTAR-Kriging

Statistica Neerlandica,

Journal Data Mining, SOLA

IJAMAS, JIMS

ITS, IPB, RuG, LEAD

Leiden, IndoMS

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Pusair, Balai Pangan

PERTAMINA

PIP Unpad

CG Jabar

Multivariat time series

untuk bidang keuangan,

pendidikan dan lingkungan

Metode penaksiran:

SUR, Bayesian

Algoritma integrasi clustering

spatio temporal dan

SDM ethnomathematics

http://www.jims-a.org/

2010 Mathematics Subject Classification :62M10 (TIME SERIES0, 63M30 (SPATIAL PROCESSES), 62H11 (SPATIAL STATISTICS)

A STOCHASTIC PROCESSES IS A FAMILY OF RANDOM VARIABLES X(t) PARAMETRIZED BY

WHERE . (OSAKI, 1980)

WHEN {T=1, 2, 3, ...}, WE SAY X(t) is a STOCHASTIC PROCESS IN DISCRETE TIME

WHEN T IS AN INTERVAL IN R (TYPICALLY

, WE SAY X(t) is a STOCHASTIC PROCESS IN CONTIONOUS TIME

Tt RT

),0[ T

Random Variable a realization of

TttX ),(

)( 1tX

Set of all realizations of

is a state space of S

( )X t

S discrete S continous

Proces is called a chain

Proces

T

Parameter Indexes

T discrete

T continous

Stochastic Processes

)( 1tX

Stoc. Pro. in discrete time

Stoc. Pro. in continous time

THE AIMS

•To develop study on a space time models, especially on the Space Time Autoregressive (STAR) and Gene-ralized (GSTAR) model.

•To inform the research activities on application of the space time models.

TO INTRODUCE THE SPACE TIME MODELS BASEDON TIME SERIES MODELS FROM BOX-JENKINS (1976AS PART OF MULTIVARIATE TIME SERIES ANALYSIS)

TO EXPLAIN THE SPACE TIME MODELS, FOCUS ONTHE STAR AND GSTAR MODELS

TO EXTEND THE GSTAR-ARCH, GSTAR-SUR ANDGSTAR-KRIGING MODELS

TO APPLY THE STAR, GSTAR AND GSTAR-KRIGINGMODELS TO REAL PHENOMENA SUCH AS OILPRODUCTION DATA AT VOLCANIC LAYER INJATIBARANG, CLIMATE PHENOMENA IN INDONESIA

FURTHER RESEARCH : CLUSTERING SPATIAL OF GSTAR-KRIGING

GSTARIMA-X MODEL

DEVELOPMENT STUDY ON THE SPACE TIME MODELS

GSTAR-ARCHGSTAR-SUR

CLUSTERING GSTAR-KRIGINGGSTARIMA-X

GSTAR-OLSSTAR

VAR M-ARCH

AR, MA, ARMA, ARIMA

ARCH, GARCH,ARMA-ARCH

TIME SERIES MODEL

HOMOSCEDASTICITY HETEROSCEDASTICITY:

WEIGHT MATRICES W

3

DATA

IDENTIFICATION

ESTIMATION

PARAMETER MODEL

CHECKING DIAGNOSTIC

Box-Jenkins’s Procedure in Analysis of Time Series

-MODEL

-PREDICTION

Space-Time Model

Checking on stationary

Identification Process

Checking diagnostic

Fitted Space-Time model

Prediction

Y Y

Y

YN

N

N N

12

Parameters Estimation

The random weight matrix W

Fitting theoretical model

to experimental semivariogram

Semivariogram Estimation

Characterization of weights(independent of time)

Parameters estimation

and checking diagnostic

Identification process

Study on Stationary

Univariate data from each location

Space-Time Phenomenon

Procedure in Space-Time Analysis

Source: Box-Jenkins, p. 19

time Bandung (Kota) Bandung Bandung Barat Banjar (Kota) Bekasi Bekasi (Kota) Bogor Bogor (Kota)

Jan-81 25.2 23.8 25.2 26.3 27.0 27.0 27.0 27.0

Feb-81 24.9 23.5 25.0 26.0 26.9 26.9 26.9 26.9

Mar-81 25.5 23.8 25.6 26.5 27.3 27.4 27.4 27.4

Apr-81 26.1 24.5 26.2 27.2 28.1 28.0 28.0 28.0

May-81 26.3 24.6 26.4 27.2 28.2 28.3 28.3 28.3

Jun-81 25.4 23.6 25.5 26.2 27.4 27.4 27.4 27.4

Jul-81 25.1 23.4 25.2 26.0 27.0 27.1 27.1 27.1

Aug-81 26.1 24.3 26.2 27.0 28.0 28.0 28.0 28.0

Sep-81 25.8 24.2 25.9 26.8 27.8 27.8 27.8 27.8

Oct-81 26.5 24.8 26.5 27.4 28.4 28.5 28.5 28.5

Nov-81 25.3 23.6 25.4 26.1 27.3 27.4 27.4 27.4

Dec-81 25.2 23.7 25.3 26.2 27.1 27.2 27.2 27.2

Jan-82 25.0 23.6 25.1 26.1 26.8 26.8 26.8 26.8

Feb-82 25.3 23.8 25.4 26.4 27.2 27.2 27.2 27.2

Mar-82 25.5 23.9 25.6 26.5 27.4 27.4 27.4 27.4

Apr-82 26.1 24.5 26.2 27.2 28.1 28.0 28.0 28.0

May-82 26.0 24.3 26.1 27.0 27.9 28.0 28.0 28.0

Jun-82 25.2 23.4 25.3 26.0 27.2 27.2 27.2 27.2

Jul-82 25.7 24.0 25.8 26.6 27.6 27.7 27.7 27.7

Aug-82 25.8 24.0 25.9 26.8 27.7 27.7 27.7 27.7

Sep-82 25.5 23.8 25.6 26.5 27.5 27.5 27.5 27.5

Oct-82 26.1 24.4 26.2 27.0 28.0 28.1 28.1 28.1

Nov-82 26.3 24.6 26.4 27.2 28.2 28.3 28.3 28.3

Dec-82 26.1 24.6 26.2 27.0 28.0 28.1 28.1 28.1

Jan-83 25.7 24.3 25.8 26.8 27.5 27.6 27.6 27.6

Feb-83 26.3 24.9 26.4 27.4 28.2 28.2 28.2 28.2

Mar-83 25.7 24.1 25.8 26.5 27.6 27.7 27.7 27.7

Apr-83 26.8 25.2 26.9 27.8 28.8 28.8 28.8 28.8

May-83 26.7 25.0 26.8 27.7 28.6 28.7 28.7 28.7

Jun-83 26.0 24.3 26.1 27.0 28.0 28.0 28.0 28.0

Jul-83 25.3 23.6 25.4 26.2 27.2 27.3 27.3 27.3

Aug-83 25.8 24.0 25.9 26.6 27.7 27.8 27.8 27.8

Sep-83 25.7 24.0 25.8 26.6 27.6 27.6 27.6 27.6

Oct-83 26.2 24.5 26.3 27.0 28.2 28.3 28.3 28.3

Nov-83 25.8 24.2 25.9 26.7 27.7 27.8 27.8 27.8

Dec-83 24.9 23.4 25.0 25.8 26.8 26.9 26.9 26.9

Jan-84 25.0 23.6 25.0 26.1 26.7 26.8 26.8 26.8

Feb-84 24.7 23.3 24.8 25.8 26.6 26.6 26.6 26.6

Mar-84 25.1 23.5 25.2 26.2 26.9 27.0 27.0 27.0

Apr-84 25.7 24.1 25.8 26.7 27.6 27.6 27.6 27.6

May-84 26.0 24.3 26.1 27.0 27.8 27.9 27.9 27.9

Jun-84 25.5 23.8 25.6 26.3 27.4 27.5 27.5 27.5

Jul-84 24.6 22.9 24.8 25.5 26.6 26.7 26.7 26.7

Aug-84 25.1 23.3 25.2 25.9 27.1 27.2 27.2 27.2

Sep-84 24.8 23.1 25.0 25.6 26.9 27.0 27.0 27.0

Oct-84 25.8 24.2 25.9 26.7 27.8 27.9 27.9 27.9

Nov-84 25.6 23.9 25.8 26.4 27.6 27.7 27.7 27.7

Dec-84 25.0 23.5 25.1 25.8 26.9 27.0 27.0 27.0

Jan-85 25.1 23.7 25.2 26.1 26.9 27.0 27.0 27.0

TEMPERATURE DATA AT WEST JAVA

Univariate Time Series

AR(1) Model with assumption E[Z(t)=0]

Z(t)= Z(t-1)+e(t)

e(t)iid

~ N(0,2) dan t N

1)(0.41)(0.1)(ˆ

)1(3.0)1(5.0)(ˆ

122

211

tztztz

tztztz

Bivariate Autoregressive,

Vector Autoregressive (VAR)

SPATIAL DEPENDENCE

Apart from time dependence most ecological variables also fit the First Law of Geography:

“everything is related to everything else, but near things are more related than distant things” (Tobler in Cliff-Ord, 1975)

The spatial autoregressive (SAR) model is a spatiallag model or mixed regression as an extension oflinear regression model with spatial dependence.

Cressie (1993) explain a spatial processes, for adata location at dimension-d and is a randomvariable at location s, random process is a spatialprocesses as a stochastic processes.

In real phenomena a spatial processes has aneratics aspect, it means a variability is big. So aspatial processes need a stationary assumption.

Purely spatial models do not capture timedependencies

VARMA(p,q) models have too manyparameters

Spatial dependence is often pre-determinedby external factors and sometimes”known”. We need a weight matrix W.

Combination of spatial and time seriesmodel is called the Spatial TimeAutoregressive (STAR) model

STAR Model of Cliff-Ord (1975)

Prod . Welli (t) = 00 Prod. Welli (t) +

11 wij Prod.Wellj (t-1) + Error Welli (t)

STAR Model of Pfeifer (1979)

Prod . Welli (t) = 01 Prod. Welli (t-1) +

11 wij Prod.Wellj (t-1) + Error Welli ( t)

The weight matrix to describe the

characteristic of locations

To estimate the parameters model

To determine the least squares

properties of estimators

LACK OF THE STAR MODEL JUST ONLY APPLICABLE FOR HOMOGENOUS LOCATIONS

Source: Siregar [20]

Dual Porosity from Matrix and Fracturer

0 50 100 150

50

01

00

01

50

02

00

02

50

03

00

0

Data Produksi JTB68

Semivariogram

The semivariogram is a graphical device for

modelling spatial continuity. The spatial

process is assumed to hold intrinsic hypothesis:

for h a vector in R2, we have:

E[Y(s+h)-Y(s)]=0

E[Y(s+h)-Y(s)]2= 2(h)

The quantity 2(h) which is a function only the

increment h is known as the variogram [(h)

has been called semivariogram] and is the

crucial parameter of spatial processes

Estimation of Semivariogram

Under the constant mean assumption, E[Y(s)]=m, so a natural estimation of the

semivariogram is defined (Cressie, 1993):

for h >0 and N(h) is number of pairs of locations (si,sj) with d(si-sj) h and d is

distance.

2

),(d:),(

)]()([)(N2

1)(ˆ

jiji

hssss

ssh

h ji YY

Theoretical Model of Semivariogram

The spherical model probably the most commonly used model. It is presented as:

where c is sill or variance of observations and a is range.

ah,c

ah,/a)0.5(h)/a1.5(hc)(hγ

ij

ij3

ijijij

][ˆ

The Spatial Weights using the Semivariogram

N)h(w

i,1hw

tan,

,

N

1i

N

1j

ijss

N

1j

ijss

)(

weightspatiallydardizedsthe

ji,0

ji,A

a

)(hw

aA

weightspatiallyoriginalthe

ji,0

ji,

)](hγ[1

1

a

ij

ij

ijss

N

ij

ijij

ij

^

ij

Estimation of Semivariogram

Under the constant mean assumption, E[Y(s)]=m, so a natural estimation of the

semivariogram is defined (Cressie, 1993):

for h >0 and N(h) is number of pairs of locations (si,sj) with d(si-sj) h and d is

distance.

2

),(d:),(

)]()([)(N2

1)(ˆ

jiji

hssss

ssh

h ji YY

Theoretical Model of Semivariogram

The spherical model probably the most commonly used model. It is presented as:

where c is sill or variance of observations and a is range.

ah,c

ah,/a)0.5(h)/a1.5(hc)(hγ

ij

ij3

ijijij

][ˆ

The Spatial Weights using the Semivariogram

N)h(w

i,1hw

tan,

,

N

1i

N

1j

ijss

N

1j

ijss

)(

weightspatiallydardizedsthe

ji,0

ji,A

a

)(hw

aA

weightspatiallyoriginalthe

ji,0

ji,

)](hγ[1

1

a

ij

ij

ijss

N

ij

ijij

ij

^

ij

Ruchjana (2002)

Based on the oil production phenomenon at volcanic layer at West Java are heterogeneous

Prod . Welli (t) = (i)01 Prod. Welli (t-1)

+ (i)11 wij Prod.Wellj (t-1) + Error Welli (t)

the autoregressive and space time parameters are different for each well

Generalized Spatial Time AutoRegressive order one Model,

GSTAR(1;1)

)()1(),,()1(),,(

)()1()1()(

)1()(

11

)1(

11

)(

01

)1(

01

)1()1(

)1()(

)1(

)(11)1()(01)1(

ttdiagtdiag

tttt

NN

NxNxNxNNxNNxNxNNx

ezWz

ezWzz

01 : diagonal matrices containing the autoregressive parameter at time lag 1

11 : diagonal matrices containing the spatial-time parameter at spatial lag 1

and time lag 1

z(t) : the random vector of observation at time t

W(1) : the weight matrix at spatial lag 1

e(t) ~ iid (0, 2IN)

GSTAR(1;1) Model

For locatons i =1,2,…,N at time t :

)(

)(

)1(

)1(

)1(0

0

)1(

)1(

0

0

)(

)( 11

)1(

1

)1(

1

)1(

11

)(

11

)1(

111

)(

01

)1(

011

te

te

tz

tz

ww

ww

tz

tz

tz

tz

NNNNN

N

N

N

N

N

GSTAR(1;1) model can be written as VAR(1):

)()1()( ttt ezz

][ )1(

1101 W

We can use the least square method to estimate the parameter GSTAR(1;1)

N

j

jNj

N

j

jj

N

j

jj

N

N

tzw

tzw

tzw

ttt

G

1

)1(

1

)1(

2

1

)1(

1

)(

11

)1(

11

)(

01

)1(

01

)1(00

00

0)1(0

00)1(

where

)(diag)1(diag)(

2

V

eVzz

X

(V.1)

The space time software still not yet provide

We built the GSTAR Script S-Plus to estimate the parameters using least squares method, for order 1, 2, …, p in time and order 1, 2, …, p in space.

The GSTAR(2;1) using binary weights gave the better result than STAR(1;1) or GSTAR(1;1) with binary and uniform weights, it has AIC minimum

Mesosphere

Ionosphere/

Thermosphee

Stratosphere

Troposphere

Collaboration with the

Center for Application at

Atmospheric Science and

Climate of National Institute

of Aeronautics and Space

(LAPAN), Bandung

Equatorial Atmospheric Radar (Padang)

Application of STAR(1;1) and

GSTAR(1;1) Models to Oil Production

The Jatibarang Field is located near

Cirebon, West Java-Indonesia. The

reservoir formation is a fractured volcanic

rock.

We use the monthly oil production data

from 3 wells in 80 observations: JTB68,

JTB72 and Jtb120.

Application STAR and GSTAR model in Petroleum Field, Volcanic Layer-West Java

Collaboration between Unpad, ITB, TU Delft, Pertamina since 1999-now, and Chevron Indonesia Company

Production Correlation Data

between 2 wells (185 months)

Well JTB68 JTB72 JTB120

JTB68 1

JTB72 0.414 1

JTB120 0.547 0.456 1

-10000 10002000

05

1015

20

oil68b[1:60]

JTB68b (60obs)

oil68b[1:59]

oil68b[2:60]

-5000500100015002000

-500050

01500

z(t-1) vs z(t) Jtb68b

bulan

z(t) (m3/hari)

0102030405060

-500050

01500

JTB68b (60obs)

Lag

Partial ACF

051015202530

-0.20.0

0.20.4

0.6

Series : oil3p[, 1]

-50005001000150020002500

05

1015

20

oil3p[, 2]

JTB72b (60obs)

oil3p[1:59, 2]

oil3p[2:60, 2]

-5000500100015002000

-5000

5001500

z(t-1) vs z(t) Jtb72b

bulan

z(t) (m3/hari)

0102030405060

-500050

01500

JTB72b (60obs)

Lag

Partial ACF

051015202530

-0.20.0

0.20.4

0.6

Series : oil3p[, 2]

18

The uniform weight matrix Wu , the spatial weight matrix Ws

using the thickness data of 83 wells

úúú

û

ù

êêê

ë

é

úúú

û

ù

êêê

ë

é

úúú

û

ù

êêê

ë

é

0462.0538.0

452.00548.0

490.0510.00

0000657.0000797.0

000657.00000766.0

000797.0000766.00

05.05.0

5.005.0

5.05.00

s

s

u

W

W

W

Wells Well1 Well2 Well3

Well1 71,50 1,80 9,10

Well2 10,20 71,50 8,40

Well3 10,00 8,60 71,50

715.0086.0100.0

084.0715.0102.0

091.0018.0715.0

ˆ

Wells Well1 Well2 Well3

Well1 62,20 15,70 15,10

Well2 12,22 63,80 10,08

Well3 0,02 0,01 83,90

MonthlyP

rod

uctio

n0 50 100 150

01

00

02

00

03

00

0Monthly

Pro

du

ctio

n0 50 100 150

01

00

02

00

03

00

0Monthly

Pro

du

ctio

n0 50 100 150

01

00

02

00

03

00

0

z1(t)z1hat(t)r1(t)

Fitted GSTAR Well1

Least Squares Estimator of GSTAR(1;1) Ws

Well )(10ˆ i )(

11ˆ i SSE Total

SSE

MAPE Total

MAPE

JTB68

b

0.623

(0.108)

0.308

(0.114)

6,361

x 106

1,048 %

JTB72

b

0.638

(0.094)

0.223

(0.101)

9,504

x 106

21,360

x 106

1,494 %

3,223 %

JTB12

0b

0.839

(0.084)

0.028

(0.115)

5,495

x 106

0,681 %

)3(11

)2(11

)1(11

11)3(

10

)2(10

)1(10

10

1110

ˆ00

0ˆ0

00ˆ

ˆ,

ˆ00

0ˆ0

00ˆ

ˆ,)()(~

)1(~ˆ)1(~ˆ)(~̂

or

zzz

zWzz

tt

ttt s

)49(ˆ

)49(ˆ

)49(ˆ

0462,0538,0

452,00548,0

490,0510,00

028,000

0223,00

00308,0

)49(ˆ

)49(ˆ

)49(ˆ

839,000

0638,00

00622,0

)50(ˆ

)50(ˆ

)50(ˆ

50

3

2

1

3

2

1

3

2

1

z

z

z

z

z

z

z

z

z

t

Sufficient Conditions for the GSTAR(1;1) stationary

If for all locations, i =1, 2 ,…,N, theparameters and satisfies)(

10i

)(11

i

1,1 )(

11

)(

10

)(

11

)(

10 iiii

then the GSTAR(1;1) model is

stationary process.

GSTAR KRIGING

The coordinate’s position of ten excluding sample locations between two oil wells (JTB68 and JTB72)

Using the (i)10 and (i)

11 parameters, we can simulate the

GSTAR(1;1) for ten excluding sample locations, and we have the model for prediction of oil production for each new oil

well.

Well's coordinate

0

100

200

300

400

14000 14050 14100 14150 14200 14250 14300 14350 14400

X

Y

)(

11is

322430003224320032243400322436003224380032244000322442003224440032244600

21950000 21952000 21954000

Y

X

Koordinat Sumur Minyak

J. Bruining

H. P. Lopuhaä

Budi Nurani Ruchjana

Asep K. Permadi

Sutawanir Darwis

*) Research is funded by KNAW the Netherlands

through Mobility Programme 2007 and General

Manager Deepwater Project, Chevron Indonesia

Company, Chevron IndoAsia Business

Oil Well Placement Design through Geostatistical Analysis using the Generalized Space Time Autoregressive (GSTAR) Kriging Model *)

GSTAR(1;1) model

1. (t) has zero mean and constant of variance-

covariance matrix

2. (t) doesn’t have a correlation with error in the last

time

3. There is no autocorrelation between error in

location at time t : i(t) dan j(t) , ij = 0

4. (t) is identically independent normal

GSTAR(1,1)-ARCH(1) MODEL (NAINGGOLAN, 2011)

• New error assumption: heteroscedasticity of variance covariance matrix

10 11( ) ( 1) ( 1) ( )t t t t Z Z W Z ε

2( ( ) ~ (0, )iid

Nt N ε I

2

1 12 1

2

21 2 2

1

2

1 2

( ) ( ) ( )

( ) ( ) ( )( ( ) )

( ) ( ) ( )

N

N

t t

N N N

t t t

t t tVar t F

t t t

ε Σ

2

1 0 0

0 1 0( ( ))

0 0 1

Var t

ε

( ) ( )

01 11( ) ( 1) ( ) ( )i i

i i i i it t t t z μ Φ z Φ V ε

( ) ( ) ( )

01 11

z (1) z (0) (0) ε (1)1

z (2) z (1) (1) ε (2)1, , , , ,

z ( ) z ( 1) ( 1) ε ( )1

i i i i

i i i ii i i

i i i i

i i i iT T T T

é ù é ù é ùê ú ê ú ê úê ú ê ú ê ú ê ú ê ú ê úê ú ê ú ê ú

ë û ë û ë û

V

VX ε

V

y

)()1( tt

where

iVzX

autocorrelation

error ≠ 0

11 12 13 14 15

21 22 23 24 25

31 32 33 33 35

41 42 43 44 45

51 52 53 54 55

( )Var

ε

Error Variance

between location is

heteroscedasticity

2

2

2 2

2

2

0 0 0 0

0 0 0 0

( ) 0 0 0 0

0 0 0 0

0 0 0 0

Var

ε I

GSTAR-SUR in Linear Model

y = Xβ+ε

where

1

2

1011 11

11 1

201 21 1 1 21

2 2 2

22 2

1 10

; ;

M

KT T

KT T

M M M

M MM

MT MMK

y

y

y

y

y

y

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This paper introduces a novel model - GeneralizedSpace-Time Autoregressive Integrated MovingAverage (GSTARIMA) methodology - into the field ofshort-term traffic flow forecasting in urban network.Compared to traditional STARIMA, GSTARIMA is amore flexible model class where parameters aredesigned to vary per location.

GSTARIMA-X model and its applications?

Sahid, Ph.D student at Graduate Program ITS

The space time models, especially STAR, and GSTAR models are informed.

The application of the space time models are shown.

The continuation research on space time models are introduced

Atje Setiawan Abdullah. (2009). Spatial Data Mining using the Spatial Autoregressive-Kriging (SAR-Kriging) Model to Predict a Quality of Education in Indonesia.Unpublished Ph.D Dissertation. Computer Science Department. Universitas GadjahMada, Yogyakarta.

Budi N. Ruchjana. (2002). The Generalized Space Time Autoregressive and Its Application to Oil Production. Unpublished Ph.D Dissertation. Mathematics Department. Institut Teknologi Bandung.

G.E.P Box, and W.L. Jenkins. (1974). The analysis of time series: Prognosis and Control. Issue 1, Moscow, Mir.

G.E.P. Box and W.L. Jenkins. (1976). Time Series Analysis, Forecasting and Control. Holden-Day. Inc., San Fransisco.

Kartlos J. Kachiashvili . (1990). Simulation of real multidimensional stationary Gaussian Markov series with given depth of coherence. Multidimensional statistical analysis and probabilistic simulation of real processes. Academic notations in statistics, Vol. 54, Moscow, Nauka, 254-256.

Noel Cressie (1993). Statistics for Spatial Data. New York: Wiley & Sons.

P. E. Pfeifer. (1979). Spatial Dynamic Modeling. Unpublished Ph.D Dissertation. Georgia Institute of Technology. Georgia.

Svetlana A. Borovkova, H.P. Lopuhaa. and Budi N. Ruchjana. (2004). Prediction of Oil Production using the Generalized Space-Time Autoregressive Model. Poster Session. The European Conference on Mathematical Industry. TU Eindhoven, the Netherlands

Svetlana A. Borovkova, H.P. Lopuhaä, H. P. and Budi N. Ruchjana. (2008). Consistency and asymptotic normality of least squares estimator in generalized STAR model. Statistica Neerlandica. vol. 6, nr. 4, p. 482-508.

Sutawanir Darwis, Budi N. Ruchjana, Asep K. Permadi (2009). Robust Decline Curve

Analysis. Journal of the Indonesian Mathematical Society, vol. 15, nr. 2, p. 105-111.

ASSMS-Lahore-Pakistan 2008