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A First-Order Spatial Integer-Valued AutoregressiveSINAR(1, 1) ModelAlireza Ghodsi a , Mahendran Shitan a b & Hassan S. Bakouch ca Department of Mathematics , University Putra Malaysia , Selangor , Malaysiab Laboratory of Computational Statistics and Operations Research , Institute forMathematical Research, University Putra Malaysia , Selangor , Malaysiac Department of Statistics , King Abdulaziz University , Jeddah , Saudi ArabiaPublished online: 13 Jun 2012.

To cite this article: Alireza Ghodsi , Mahendran Shitan & Hassan S. Bakouch (2012) A First-Order Spatial Integer-Valued Autoregressive SINAR(1, 1) Model, Communications in Statistics - Theory and Methods, 41:15, 2773-2787, DOI:10.1080/03610926.2011.560739

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Communications in Statistics—Theory and Methods, 41: 2773–2787, 2012Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610926.2011.560739

A First-Order Spatial Integer-ValuedAutoregressive SINAR�1� 1�Model

ALIREZA GHODSI1, MAHENDRAN SHITAN1�2,AND HASSAN S. BAKOUCH3

1Department of Mathematics, University Putra Malaysia,Selangor, Malaysia2Laboratory of Computational Statistics and Operations Research,Institute for Mathematical Research, University Putra Malaysia,Selangor, Malaysia3Department of Statistics, King Abdulaziz University, Jeddah, SaudiArabia

Binomial thinning operator has a major role in modeling one-dimensional integer-valued autoregressive time series models. The purpose of this article is to extendthe use of such operator to define a new stationary first-order spatial non negative,integer-valued autoregressive SINAR�1� 1� model. We study some properties of thismodel like the mean, variance and autocorrelation function. Yule-Walker estimatorof the model parameters is also obtained. Some numerical results of the model arepresented and, moreover, this model is applied to a real data set.

Keywords Binomial thinning operator; Estimation; Modeling; SINAR�1� 1�model; Spatial integer-valued autoregressive.

Mathematics Subject Classification 62M30.

1. Introduction

Recently, there has been growing interest in modeling non negative integer-valuedtime series. Counts of accidents, patients, crime victimizations, transmittedmessages, detected errors, and so on, are examples of these type of time series.A popular approach for modeling such time series is the use of the binomial thinningoperator “�”, which was introduced by Steutel and van Harn (1979).

Let X be a non negative integer-valued random variable, then for any � ∈ �0� 1�the binomial thinning operator is defined as follows:

� � X =X∑i=1

Yi�

Received October 12, 2010; Accepted January 25, 2011Address correspondence to Mahendran Shitan, Department of Mathematics, University

Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia; E-mail: sarasmahen@gmail.com

2773

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2774 Ghodsi et al.

where �Yi� is a sequence of i.i.d.random variables with Bernoulli distribution suchthat P�Yi = 1� = 1− P�Yi = 0� = � and is independent of X.

The first time series models based on this operator was proposed by McKenzie(1985). An example of the time series model using the binomial operation is theinteger-valued autoregressive (INAR) model which has been studied by Al-Oshand Alzaid (1987). They discussed several methods for estimating the parametersof the model. Alzaid and Al-Osh (1990), who introduced the INAR(p) model,obtained its probability generating function, correlation properties and derived thestate space representation of the model. Du and Li (1991) discussed the existenceand ergodic property of the INAR(p) models and obtained the correlation structureof the model. Silva and Oliveira (2004, 2005) considered the higher-order momentsand cumulants of the INAR(1) and INAR(p) models . They also used Whittlecriterion to estimate the parameters of the model. Silva and Silva (2006) derived theasymptotic distribution of the Yule-Walker estimator of the INAR(p) model. Weiß(2008) reviewed a broad variety of thinning operations and showed how they aresuccessfully applied to define integer-valued ARMA models. Recently, Ristic et al.(2009) and Bakouch (2010) worked on an INAR model using the negative binomialthinning operator “∗”.

In the area of spatial modeling, the standard first-order spatial autoregressive(AR�1� 1�) model (Pickard, 1980; Basu and Reinsel, 1993; Martin, 1996) is defined as

Xij = �1Xi−1�j + �2Xi�j−1 + �3Xi−1�j−1 + ij� (1)

where ��1� < 1, ��2� < 1, ��1 + �2� < 1− �3, ��1 − �2� < 1+ �3, and �ij� is atwo-dimensional white noise process with mean zero and variance 2

.Spatial models are of interest in many fields such as environmental science,

geology, geography, agriculture and image processing. Two-dimensional data fromsuch fields are usually analyzed by Gaussian spatial processes (Basu and Reinsel,1993; Martin, 1996; Yao and Brockwell, 2006). However, in the spatial case datamay also have non negative integer values and so there is a need to introduce nonGaussian integer-valued spatial model. In particular, we introduce the first-orderspatial non negative integer-valued autoregressive SINAR�1� 1� model with discretemarginal in Sec. 2. In the rest of Sec. 2, some properties of this model (mean,variance and autocorrelation functions) are established. The Yule-Walker estimatorsof the parameters of the model are introduced in Sec. 3. Numerical results includingsome theoretical values of the autocorrelation function, a simulation study for themodel and estimation of its parameters are discussed in Sec. 4. Finally, in Sec. 5 theconclusion is presented.

2. The SINAR�1� 1� Model and Some of Its Properties

In this section, we introduce the SINAR�1� 1� model and discuss some of itsproperties.

2.1. The SINAR�1� 1� Model

Let

�1 � Xi−1�j =Xi−1�j∑k=1

Y1k� �Y1k� ∼ i�i�d Bernoulli��1�

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First-Order SINAR Model 2775

�2 � Xi�j−1 =Xi�j−1∑k=1

Y2k� �Y2k� ∼ i�i�d Bernoulli��2�

�3 � Xi−1�j−1 =Xi−1�j−1∑k=1

Y3k� �Y3k� ∼ i�i�d Bernoulli��3��

where �Y1k�� �Y2k�, and �Y3k� are mutually independent, E�Yik� = �i, Var�Yik� =�i�1− �i� and �i ∈ �0� 1� for i = 1� 2� 3.

We define the first-order spatial integer-valued autoregressive (SINAR�1� 1�)model as

Xij = �1 � Xi−1�j + �2 � Xi�j−1 + �3 � Xi−1�j−1 + ij� (2)

where �Xij� i� j ∈ �� with mean �X and finite variance is a spatial non negativeinteger-valued process on a two-dimensional regular lattice; �1� �2� �3 ∈ �0� 1� arethe parameters of the model where �1 + �2 + �3 < 1, �ij� is a sequence of i.i.d.non negative integer-valued random variables having mean � and finite variance2, independent of �Y1k�� �Y2k�, and �Y3k�, and independent of Xi−k�j−l for all k ≥ 1

or l ≥ 1. Note that �1� �2� �3 ∈ �0� 1� and therefore �1 + �2 + �3 ≥ 0. Further, therestriction �1 + �2 + �3 < 1 is necessary because the quantity �X must be positiveand this can only happen if �1+�2 +�3 < 1 (see Proposition 3.1).

3. Mean, Autocorrelation Function, and Variance of the SINAR�1� 1�Process

Some properties of the stationary SINAR�1� 1� process defined by (2) are discussedin this subsection.

To find the mean and autocorrelation function and variance of SINAR�1� 1�process we need to establish the following results:

Var�� � X� = �2Var�X�+ ��1− ��E�X� (3)

Cov�X� � � Y� = �Cov�X� Y� (4)

Cov�� � X� � Y� = � Cov�X� Y�� (5)

These results can be easily proved by using Lemma 1 from Silva and Oliveira (2004).Now the mean of a stationary SINAR�1� 1� model is given in Proposition 3.1.

Proposition 3.1. Let �Xij� i� j ∈ �� be a stationary SINAR�1� 1� process satisfying (2).If �1 + �2 + �3 < 1, then the mean of the process is given as

�X = �

1− �1 − �2 − �3� (6)

Proof. Taking expectation on both sides of (2) gives

E�Xij� = E��1 � Xi−1�j�+ E��2 � Xi�j−1�+ E��3 � Xi−1�j−1�+ E�ij��

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2776 Ghodsi et al.

Figure 1. Two-dimensional plane into quadrants.

then by using Lemma 1 (see Silva and Oliveira, 2004) we have

�1− �1 − �2 − �3��X = ��

from which (6) follows. �

In order to derive the autocovariance function (ACF) of the model we dividethe two-dimensional plane into quadrants, as shown in Fig. 1.

The ACF of a stationary SINAR�1� 1� process is given in Proposition 3.2.

Proposition 3.2. Let �Xij� i� j ∈ �� be a stationary SINAR�1� 1� process, satisfying (2).Then the autocorrelation function ��k� l�, of the process for k ≥ 0 and l ≤ 0 (in thefourth quadrant) is given by

��k� l� = �k�−l� (7)

where 0 < � < 1 and 0 < � < 1 which satisfy the following:

� = �1+ �21 − �22 − �23�−√�1+ �21 − �22 − �23�

2 − 4��1 + �2�3�2

2��1 + �2�3�(8)

and

� = �2 + �3�

1− �1�(9)

where at most only one of �i �i = 1� 2� 3� can be zero.

Proof. Let ��k� l� = Cov�Xi−k�j−l� Xij�. For �k� l� �= �0� 0� we can write

��k� l� = Cov�Xi−k�j−l� Xij�

= Cov�Xi−k�j−l� �1 � Xi−1�j + �2 � Xi�j−1 + �3 � Xi−1�j−1 + ij�

= Cov�Xi−k�j−l� �1 � Xi−1�j�+ Cov�Xi−k�j−l� �2 � Xi�j−1�

+ Cov�Xi−k�j−l� �3 � Xi−1�j−1�+ Cov�Xi−k�j−l� ij�

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First-Order SINAR Model 2777

since Cov�Xi−k�j−l� ij� = 0 for k ≥ 1 or l ≥ 1, using (4) we obtain

��k� l� = �1��k− 1� l�+ �2��k� l− 1�+ �3��k− 1� l− 1�� (10)

To solve the difference equation in (10) for k ≥ 1 and l ≤ −1, the characteristicequation is obtained by assuming a solution of ��k� l� of the form of a constant, C,times �k�−l, i.e.,

��k� l� = C�k�−l (11)

(see Mickens, 1990, Ch. 5 and Basu and Reinsel, 1993, Proposition 3.2).In order to determine the values of � and � from (10) by substituting ��k� l� =

C�k�−l we obtain,

C�k�−l = �1C�k−1�−l + �2C�

k�−l+1 + �3C�k−1�−l+1

or

�1�−1 + �2�+ �3�

−1� = 1� (12)

Note also that for a stationary spatial process ��k� l� = Cov�Xi+k�j+l� Xij�. Soreplacing Xi+k�j+l by �1 � Xi+k−1�j+l + �2 � Xi+k�j+l−1 + �3 � Xi+k−1�j+l−1 + i+k�j+l andusing (4) when k ≤ −1 or l ≤ −1 give,

��k� l� = �1��k+ 1� l�+ �2��k� l+ 1�+ �3��k+ 1� l+ 1�� (13)

Substituting ��k� l� = C�k�−l in (13) leads to

�1�+ �2�−1 + �3��

−1 = 1 (14)

and solving (14) for � yields

� = �2 + �3�

1− �1�� (15)

provided 1− �1� �= 0.Now by multiplying � on both sides of (12) we have

�1 + �2��+ �3� = � (16)

and replacing � from (15) in (16) gives

�1 + �2�

(�2 + �3�

1− �1�

)+ �3

(�2 + �3�

1− �1�

)= ��

which can be simplified to

�1�1− �1��+ �2���2 + �3��+ �3��2 + �3�� = ��1− �1���

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2778 Ghodsi et al.

Hence, we obtain the following quadratic equation in �,

��1 + �2�3��2 − �1+ �21 − ��22 + �23���+ ��1 + �2�3� = 0� (17)

and the roots are

� = �1+ �21 − �22 − �23�±√�

2��1 + �2�3�� (18)

where � = �1+ �21 − �22 − �23�2 − 4��1 + �2�3�

2 and �1 + �2�3 > 0.Next, we will show that these roots are real and different. From the definition

of the model and Proposition 3.1 we know that 0 < �1 + �2 + �3 < 1. Thus, 1−�1 > �2 + �3 > 0 then we can write �1− �1�

2 > ��2 + �3�2 or 1+ �21 − 2�1 > �22 +

�23 + 2�2�3 and simplifying this equation leads to

1+ �21 − �22 − �23 > 2��1 + �2�3� > 0� (19)

From (19) we have �1+ �21 − �22 − �23�2 > 4��1 + �2�3�

2 and hence � = �1+ �21 −�22 − �23�

2 − 4��1 + �2�3�2 > 0 which shows that the roots of (17) are real with

different values.Let �1 and �2 be the roots of (17) and let �1 < �2. By using the Vieta’s formula

we obtain that

�1 + �2 =1+ �21 − �22 − �23

�1 + �2�3> 0�

�1�2 = 1�

which implies that 0 < �1 < 1 < �2.Since we require that ���k� l�� ≤ 1 and ��k� l� = C�k�−l for k ≥ 1 and l ≤ −1

would require ��� < 1 and ��� < 1. It follows that 0 < � < 1 and the only solutionfor � is

�1 =�1+ �21 − �22 − �23�−

√�

2��1 + �2�3��

Also, in order to show that 0 < � < 1, first note that 1− �1� �= 0 because 0 <� < 1 and 0 ≤ �1 < 1 hence 0 ≤ �1� < 1 which implies that 1− �1� > 0. So from(15) we obtain that � > 0. Next, using (15) we can easily show that � cannot be 1.

It can be also shown that the solution of (10) for k ≥ 1 and l = 0 is as

��k� 0� = C1�k (20)

and the solution for k = 0 and l ≤ −1 is as

��0� l� = C2�−l (21)

where C1 and C2 are arbitrary constants and 0 < �� � < 1. Note that ��0� l� =��0�−l� for l ≤ −1.

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First-Order SINAR Model 2779

From (11), (20), (21) and using (10) after some algebra we find that,

��k� l� = ��k� l�

��0� 0�= �k�−l� for k ≥ 0� l ≤ 0

which gives (7). �

Note that the autocorrelations for the first quadrant can be obtained recursivelyusing (10) as follows:

��k� l� = �1��k− 1� l�+ �2��k� l− 1�+ �3��k− 1� l− 1� (22)

and the autocorrelations for the second and third quadrants follow ��−k� l� =��k�−l� and ��−k�−l� = ��k� l�, respectively.

Remark 3.1. It can be shown that the correlation structures of the SINAR�1� 1�model in special cases are as follows: (i) when �2 = �3 = 0, ��k� 0� = �

�k�1 for

k = 0�±1�±2� � � � and ��k� l� = 0 for k = 0�±1�±2� � � � and l �= 0; (ii) when�1 = �3 = 0, ��0� l� = �

�l�2 for l = 0�±1�±2� � � � and ��k� l� = 0 for k �= 0 and l =

0�±1�±2� � � � ; (iii) when �1 = �2 = 0, ��m�m� = ��m�3 for m = 0�±1�±2� � � � and

��k� l� = 0 for k �= l; (iv) when �1 = �2 = �3 = 0, ��0� 0� = 1 and ��k� l� = 0 fork� l �= 0.

The variance of the model is established in Proposition 3.3.

Proposition 3.3. Let �Xij� i� j ∈ �� be a stationary SINAR�1� 1� process satisfying (2).Then the variance ��0� 0� of the process is given as

��0� 0� = �X

∑3i=1 �i�1− �i�+ 2

1− ��1 + �2�3��− ��2 + �1�3��− �23(23)

where �X is defined by (6) and � and � are defined by (8) and (9), respectively.

Proof. From (2) we obtain that

��0� 0� = Cov�Xij� Xij�

= Cov��1 � Xi−1�j + �2 � Xi�j−1 + �3 � Xi−1�j−1 + ij�

�1 � Xi−1�j + �2 � Xi�j−1 + �3 � Xi−1�j−1 + ij��

By using (3) and (5) we have,

��0� 0� = �1��1��0� 0�+ �1− �1��X + �2��1�−1�+ �3��0�−1��

+ �2��1��−1� 1�+ �2��0� 0�+ �1− �2��X + �3��−1� 0��

+ �3��1��0� 1�+ �2��1� 0�+ �3��0� 0�+ �1− �3��X�

+ 2�

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2780 Ghodsi et al.

Using (10) yields

��0� 0� = �1��1� 0�+ �2��0� 1�+ �3��1� 1�+ �X

3∑i=1

�i�1− �i�+ 2 (24)

and replacing ��1� 1� = �1��0� 1�+ �2��1� 0�+ �3��0� 0� from (10) gives

��0� 0� = �1��1� 0�+ �2��0� 1�+ �1�3��0� 1�+ �2�3��1� 0�+ �23��0� 0�

+ �X

3∑i=1

�i�1− �i�+ 2

= ��1 + �2�3���1� 0�+ ��2 + �1�3���0� 1�+ �23��0� 0�

+ �X

3∑i=1

�i�1− �i�+ 2�

Substituting ��1� 0� = ���0� 0� and ��0� 1� = ���0� 0� in the last equation leadsto (23).

In order to show that ��0� 0� in (23) is a positive quantity, we can see that ��1 +�2�3��+ ��2 + �1�3��+ �23 < ��1 + �2�3�+ ��2 + �1�3�+ �3 < �1 + �2 + �3 < 1 since0 ≤ �� �� �1� �2� �3 < 1 . Therefore, the denominator of (23) is positive and we alsoknow that the numerator of (23) is positive. Hence, ��0� 0� is positive. �

4. Yule-Walker Estimation

From (22), the autocorrelation function of the SINAR�1� 1� process satisfies thefollowing equations:

��1� 0� = �1 + �2��1�−1�+ �3��0�−1�

��0� 1� = �1��−1� 1�+ �2 + �3��−1� 0�

��1� 1� = �1��0� 1�+ �2��1� 0�+ �3�

Since ��0�−1� = ��0� 1�, ��−1� 0� = ��1� 0�, and ��−1� 1� = ��1�−1� the Yule-Walker equations are given as,

��1� 0� = �1 + �2��1�−1�+ �3��0� 1�

��0� 1� = �1��1�−1�+ �2 + �3��1� 0�

��1� 1� = �1��0� 1�+ �2��1� 0�+ �3�

The Yule-Walker estimators of the parameters �1, �2, and �3 are obtained from theseequations by replacing ��k� l� by the sample autocorrelation, ��k� l� as follows:

� = P−1�� (25)

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First-Order SINAR Model 2781

where �′ = ��1� �2� �3�, �

′ = ���1� 0�� ��0� 1�� �(1,1)� and

P =

1 ��1�−1� ��0� 1�

��1�−1� 1 ��1� 0�

��0� 1� ��1� 0� 1

note that ��k� l� = ��k� l�/��0� 0� which ��k� l� for a given data set �Xij� i =1� � � � � n1 & j = 1� � � � � n2� is given by

��k� l� = 1n1n2

∑i

∑j

�Xij −�X��Xi+k�j+l −�X�� (26)

where max�1� 1− k� < i < min�n1� n1 − k�, max�1� 1− l� < j < min�n2� n2 − l� andk� l ∈ �.

Using (6) and (23), � and 2 can be estimated as follows:

� = �X�1− �1 − �2 − �3� (27)

2 = ��0� 0�

{1− ��1 + �2�3��− ��2 + �1�3��− �23

}

−�X3∑

i=1

�i�1− �i�� (28)

where �X = ∑n1i=1

∑n2j=1 Xij/�n1n2� and � and � are defined by (8) and (9).

5. Numerical Results

In this section we present some numerical results of SINAR�1� 1� model for theselected parameter values.

5.1. Theoretical Mean, Variance and Autocorrelation Values

By using (6) and (23), the theoretical mean and variance of the model with �1 = 0�2,�2 = 0�2, �3 = 0�5, � = 1 and 2

= 1 are 10 and 14.889, respectively.In Table 1 we list out the theoretical autocorrelation values. These values are

plotted in Fig. 2. It can be seen from the table that the autocorrelation valuesdecay when k and l increase. Also the ACF values are diagonally symmetric because�1 = �2. Note that ACF generally is symmetric with respect to the origin.

The theoretical autocorrelation values are also plotted in Fig. 2 to give visualimpression.

5.2. A Simulation Study

We simulated realisations using (2) with �1 = 0�2, �2 = 0�2, �3 = 0�5 and assumingij ∼ Poisson�� = 1�. The size of generated process is 20× 20. A typical realisationof the SINAR�1� 1� process is shown in Fig. 3.

The mean and variance of the simulated process are 10.628 and 12.824,respectively. These values are close to true values mentioned in Sec. 5.1.

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2782 Ghodsi et al.

Table 1Some values of the autocorrelation function of the SINAR�1� 1� model for the

selected parameter values �1 = 0�2, �2 = 0�2, �3 = 0�5, � = 1, and 2 = 1

k

l −7 −5 −3 −1 0 1 3 5 7

7 0.000 0.000 0.001 0.004 0.008 0.016 0.058 0.150 0.1855 0.000 0.001 0.004 0.016 0.031 0.062 0.195 0.270 0.1503 0.001 0.004 0.016 0.063 0.125 0.238 0.412 0.195 0.0581 0.004 0.016 0.063 0.250 0.500 0.700 0.238 0.062 0.0160 0.008 0.031 0.125 0.500 1.000 0.500 0.125 0.031 0.008

−1 0.016 0.062 0.238 0.700 0.500 0.250 0.063 0.016 0.004−3 0.058 0.195 0.412 0.238 0.125 0.063 0.016 0.004 0.001−5 0.150 0.270 0.195 0.062 0.031 0.016 0.004 0.001 0.000−7 0.185 0.150 0.058 0.016 0.008 0.004 0.001 0.000 0.000

The sample autocorrelation values of the simulated process are also listed out inTable 2 and these values are plotted in Fig. 4. Notice the similarity between Fig. 2(theoretical set) and Fig. 4 (sample set).

Next, we investigate the performance of the Yule-Walker estimators through asimulation study.

Let s represent the number of simulations and let � be a generic term for thevector ��1� �2� �3� �e�

2e�. In this simulation study the number of replications, s, is

400. The following computations were carried out from the simulation study.

1. Mean of the point estimates, ¯� = ∑si=1 �i/s.

2. Estimated bias = ¯�− �.

3. Estimated root mean squared errors(RMSE) =√∑s

i=1��i − ��2/s.

Figure 2. Autocorrelation function of the SINAR�1� 1� model for the selected parametervalues: �1 = 0�2, �2 = 0�2, �3 = 0�5, � = 1, and 2

= 1.

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First-Order SINAR Model 2783

Figure 3. Typical realization of the SINAR�1� 1� model (true parameter values are �1 = 0�2,�2 = 0�2, �3 = 0�5).

Table 2Some values of the sample autocorrelation function, ��k� l�, of the simulated

SINAR�1� 1� model for the selected parameter values �1 = 0�2, �2 = 0�2, �3 = 0�5

k

l −7 −5 −3 −1 0 1 3 5 7

7 −0�013 0�008 −0�095 −0�032 −0�064 −0�054 0�086 0�081 0�0885 −0�036 −0�085 −0�116 −0�053 0�031 0�132 0�067 0�148 0�0673 −0�078 −0�104 −0�036 0�075 0�073 0�074 0�229 0�122 0�0101 −0�098 −0�026 0�090 0�117 0�349 0�529 0�140 0�068 0�0090 −0�020 −0�012 0�156 0�329 1�000 0�329 0�156 −0�012 −0�020

−1 0�009 0�068 0�140 0�529 0�349 0�117 0�090 −0�026 −0�098−3 0�010 0�122 0�229 0�074 0�073 0�075 −0�036 −0�104 −0�078−5 0�067 0�148 0�067 0�132 0�031 −0�053 −0�116 −0�085 −0�036−7 0�088 0�081 0�086 −0�054 −0�064 −0�032 −0�095 0�008 −0�013

Figure 4. The sample autocorrelation function, ��k� l�, of the simulated SINAR�1� 1� model(true parameter values: �1 = 0�2, �2 = 0�2, �3 = 0�5).

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2784 Ghodsi et al.

Table 3Mean, bias, and RMSE of the estimates

�1 �2 �3

Set Size Mean Bias RMSE Mean Bias RMSE Mean Bias RMSE

(i) 8× 8 0.211 0.011 0.103 0.206 0.006 0.101 0.301 −0.199 0.22214× 14 0.215 0.015 0.068 0.217 0.017 0.066 0.382 −0.118 0.13520× 20 0.215 0.015 0.047 0.216 0.016 0.046 0.414 −0.085 0.09640× 40 0.211 0.011 0.023 0.211 0.011 0.025 0.455 −0.045 0.05070× 70 0.208 0.008 0.109 0.207 0.007 0.014 0.474 −0.026 0.029

100× 100 0.205 0.005 0.009 0.205 0.005 0.010 0.482 −0.018 0.020(ii) 8× 8 0.309 −0.091 0.137 0.223 −0.077 0.128 0.103 0.003 0.077

14× 14 0.362 −0.038 0.073 0.273 −0.027 0.070 0.083 −0.017 0.05420× 20 0.382 −0.018 0.048 0.294 −0.006 0.047 0.081 −0.018 0.04540× 40 0.392 −0.008 0.023 0.301 0.001 0.024 0.086 −0.014 0.02870× 70 0.397 −0.003 0.013 0.300 0.000 0.013 0.092 −0.008 0.018

100× 100 0.398 −0.002 0.009 0.299 −0.000 0.010 0.095 −0.005 0.011(iii) 8× 8 0.124 0.024 0.082 0.119 0.019 0.083 0.104 0.004 0.071

14× 14 0.101 0.001 0.059 0.101 0.001 0.060 0.093 −0.007 0.05920× 20 0.094 −0.006 0.048 0.092 −0.008 0.045 0.091 −0.009 0.04340× 40 0.094 −0.006 0.025 0.096 −0.004 0.025 0.095 −0.005 0.02670× 70 0.098 −0.002 0.015 0.099 −0.001 0.0151 0.097 −0.003 0.015

100× 100 0.099 −0.001 0.011 0.099 −0.001 0.010 0.097 −0.002 0.011

� 2

Set Size Mean Bias RMSE Mean Bias RMSE

(i) 8× 8 2.851 1.851 2.357 2.854 1.854 2.55714× 14 1.827 0.827 1.120 2.126 1.126 1.59420× 20 1.534 0.534 0.728 1.789 0.789 1.09740× 40 1.221 0.221 0.324 1.404 0.404 0.53770× 70 1.110 0.110 0.169 1.224 0.224 0.293

100× 100 1.074 0.074 0.116 1.159 0.159 0.212(ii) 8× 8 1.796 0.796 1.060 1.543 0.543 0.996

14× 14 1.413 0.413 0.594 1.378 0.378 0.61120× 20 1.257 0.257 0.385 1.264 0.264 0.4440× 40 1.102 0.102 0.165 1.150 0.150 0.22170× 70 1.055 0.055 0.095 1.089 0.089 0.131

100× 100 1.035 0.035 0.063 1.062 0.062 0.091(iii) 8× 8 0.942 -0.058 0.205 0.929 -0.071 0.296

14× 14 1.005 0.005 0.152 0.998 -0.002 0.18720× 20 1.038 0.038 0.121 1.033 0.033 0.13940× 40 1.022 0.022 0.065 1.021 0.021 0.07570× 70 1.007 0.007 0.036 1.005 0.005 0.040

100× 100 1.007 0.007 0.027 1.004 0.004 0.028

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First-Order SINAR Model 2785

Figure 5. Spatial plotted of the yeast data.

The simulations were performed for three different sets of �′ = ��1� �2� �3�

values. The three sets of �-values used were: (i) �′ = �0�2� 0�2� 0�5�; (ii) �′ =�0�4� 0�3� 0�1�; and (iii) �′ = �0�1� 0�1� 0�1�. Note we have fixed � = 1 and 2

= 1 forall three sets.

In Table 3, the results are presented and it can be seen that the bias and RMSEof the estimated parameters are quite small. Also, we can often see that the bias andRMSE values of the estimated parameters decrease when the grid size increases.

5.3. A Real Data Example

To illustrate the fitting of the SINAR�1� 1� model, we consider the data set from aregular grid presented by Student (1906) on the yeast cell counts (Hand et al., 1994).Yeast cell counts were made on each of 400 small regions on a microscope slide.The 400 squares were arranged as a 20× 20 grid and each small square was of side1/20mm.

Table 4Some values of the sample spatial autocorrelation for the yeast data

k

l −7 −5 −3 −1 0 1 3 5 7

7 0�030 0�038 −0�029 −0�052 0�010 −0�037 −0�010 0�000 −0�0185 −0�037 0�019 0�022 −0�080 0�016 −0�049 0�025 0�038 −0�0343 −0�034 0�011 0�041 −0�007 0�023 −0�007 0�081 0�029 −0�0421 −0�027 −0�021 0�027 0�039 0�019 −0�041 0�016 −0�059 −0�0460 −0�003 0�073 −0�013 0�034 1�000 0�034 −0�013 0�073 −0�003

−1 −0�046 −0�059 0�016 −0�041 0�019 0�039 0�027 −0�021 −0�027−3 −0�042 0�029 0�081 −0�007 0�023 −0�007 0�041 0�011 −0�034−5 −0�034 0�038 0�025 −0�049 0�016 −0�080 0�022 0�019 −0�037−7 −0�018 0�000 −0�010 −0�037 0�010 −0�052 −0�029 0�038 0�030

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2786 Ghodsi et al.

Figure 6. Sample spatial autocorrelation for the yeast data.

Minimum and maximum of the data are 1 and 11, respectively. Also, the samplemean and variance of the data are 4.675 and 4.464, respectively. The data have beenplotted in Fig. 5. The sample spatial autocorrelations for the data are displayed inTable 4 and these values are plotted in Fig. 6.

The Yule-Walker estimation of the parameters using (25)–(28) and Table 4 willbe, �1 = 0�040, �2 = 0�021, �3 = max�0�−0�037� = 0, � = 4�567, and 2

= 4�358and the fitted model is as follows:

Xij = 0�04 � Xi−1�j + 0�021 � Xi�j−1 + ij�

6. Conclusion

In this article, we defined a new stationary first-order, non negative integer-valued spatial autoregressive (SINAR�1� 1�) model based on the binomial thinningmechanism. We derived the mean, variance, and autocorrelation function of thismodel. We also derived the Yule-Walker estimator for the model and the simulationstudy was performed. The theoretical values were also compared with the samplevalues and mostly there was close agreement between the theoretical and thesample values. The Yule-Walker estimates seem reasonable in the sense that thebias and RMSE were rather small. Finally, in order to illustrate the fitting of theSINAR�1� 1� model we used in this study the student’s Yeast cell counts data asan example. This study has demonstrated that there is a need for modeling integer-valued processes by using thinning binomial operation and hence contributes to thetheory and practice of spatial models. We hope that this model be able to attractwider applicability in the analysis of spatial models.

The authors are currently working on other related expects of this model onestimation issues which will be reported in a future article.

Acknowledgments

We are thankful to the referees for the comments that improved the article. Theauthors would like to thank the Department of Mathematics, University Putra

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First-Order SINAR Model 2787

Malaysia and The Institute for Mathematical Research, University Putra Malaysiafor their support. The first author would also like to thank the Hakim SabzevariUniversity, Iran for their financial support.

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