# A Note on the Properties of Generalised Separable Spatial Autoregressive   Process

Authors: Mahendran Shitan; Shelton Peiris
Journal: Journal of Probability and Statistics
 (2009)

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## Abstract

Spatial modelling has its applications in many fields like geology, agriculture, meteorology, geography, and so forth. In time series a class of models known as Generalised Autoregressive (GAR) has been introduced by Peiris (2003) that includes an index parameterδ. It has been shown that the inclusion of this additional parameter aids in modelling and forecasting many real data sets. This paper studies the properties of a new class of spatial autoregressive process of order 1 with an index. We will call this a Generalised Separable Spatial Autoregressive (GENSSAR) Model. The   spectral density function (SDF), the autocovariance function (ACVF), and the autocorrelation function (ACF) are derived. The theoretical ACF and SDF plots are presented as three-dimensional figures.


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## Body

## 1. Introduction

Spatial modelling has its applications in many fields like geology, agriculture, meteorology, geography, and so forth. Spatial data can be classified as geostatistical data, lattice data, or point patterns. These differences are due to whether the spatial data has been observed on a continuous domain or at discrete locations. In point pattern analysis the domain is random and interest focuses on the location of events.In this paper we concentrate on lattice data observed on a regular grid. Many models have been suggested in modelling spatial dependence like the Simultaneous Autoregression (SAR) [1], Conditional Autoregression (CAR) [2, 3], Moving Average (MA) [4], and Unilateral models [5].For a two-dimensionalstationary process we have the following definitions. Let {Yij,i,j=0,±1,±2,…} be a sequence of spatial observations on a two-dimensional regular lattice. The mean function is E[Yij]=μ (a constant). The autocovariance function is γh,k = Cov[Yi+h,j+k,Yi,j] and the autocorrelation function is given as ρh,k=γh,k/γ0,0.Now, there exists a class of models that are known asseparable models which have the property of a reflection symmetric correlation structure (i.e., ρh,k=ρ-h,-k=ρh,-k=ρ-h,k). The linear by linear process X is defined as a stationary process where the autocovariance generating function of X is defined as proportional to the product of two one-dimensional processes, Y and Z (see [6]) and the relationship may be represented as X=Y*Z. As a consequence, its correlation structure can be expressed as a product of correlations (i.e., ρx,h,k=ρy,hρz,k). Basawa et al. [7] have considered separable models on a k-dimensional lattice and have shown that the correlation structure is ρ(h)=∏ikρi(hi), where h is the lag vector (h1,h2,…,hk)′.On the other hand, in the area of time series a class of models known as generalised autoregressive (GAR) models has been introduced by Peiris [8] by including an additional index parameter, δ. This is a natural extension of the standard AR model. It has been shown in Peiris [8] and Peiris et al. [9]  that the additional index parameter plays an important role in modelling and forecasting real data sets. Shitan and Peiris [10] have also studied the estimation problem of the GAR(1) model with a simulation study.In this paper we will consider a special type of spatial model called aGeneralised Separable Spatial Autoregressive (GENSSAR) Model. Some of its properties are discussed in Section 2. Finally in Section 3, some conclusions  are drawn.The GENSSAR Model
Let{Yij} be a sequence of spatial observations on a two-dimensional regular lattice that satisfies
(1.1)(1-ϕ10B1-ϕ01B2+ϕ10ϕ01B1B2)δYij=Zij,
where B1 is the usual backward shift operator acting in the ith direction, B2 is the backward shift operator acting in the jth direction, and {Zi,j} is a two-dimensional white noise process with mean zero and variance.
The term(1-ϕ10B1-ϕ01B2+ϕ10ϕ01B1B2) can be factored out as (1-ϕ10B1)(1-ϕ01B2) and hence   (1.1) can be written as
(1.2)(1-ϕ10B1)δ(1-ϕ01B2)δYij=Zij.
The inclusion of the extra index parameterδ generalises the standard separable spatial model. Hence, we call the model defined in   (1.1) as the Generalised Separable Spatial Autoregressive model or GENSSAR(1,1) model.
The following section reports some of its properties in detail.


## 2. Some Properties of GENSSAR(1,1)

The solution of this process in (1.1) is given in Proposition 1.Proposition 1.
For a process defined in   (1.1), the solution is
(2.1)Yij=∑k=0∞∑l=0∞Γ(k+δ)Γ(l+δ)Γ(k+1)Γ(δ)Γ(l+1)Γ(δ)ϕ10kϕ01lZi-k,j-l,
where Γ(·) is the gamma function.Proof.
From   (1.2), we have Yij=(1-ϕ10B1)-δ(1-ϕ01B2)-δZij. Using binomial expansion, it follows that
(2.2)Yij=[∑k=0∞(-1)k(-δk)(-ϕ10B1)k][∑l=0∞(-1)l(-δl)(-ϕ01B2)l]Zij.
Note that
(2.3)(-δk)=(-δ)(-δ-1)⋯(-δ-k+1)k!=(-1)k(δ)(δ+1)⋯(δ+k-1)k!=(-1)kΓ(k+δ)Γ(k+1)Γ(δ).
Similarly
(2.4)(-dl)=(-1)lΓ(l+d)Γ(l+1)Γ(d).
Substituting   (2.3) and (2.4) into (2.2) and upon simplification completes the proof.Proposition 2.
For a process defined in   (1.1), the spectral density is given as
(2.5)f(λ1,λ2)=σ24π2(1-2ϕ10cosλ1+ϕ102)δ(1-2ϕ01cosλ2+ϕ012)δ.Proof.
The proof is established by simplifying the following expression:(2.6)f(λ1,λ2)=σ24π2×1|(1-ϕ10e-iλ1)(1-ϕ01e-iλ2)|2δ.The following proposition provides an expression for the autocovariance function of the GENSSAR(1,1) process.Proposition 3.
For a process defined in   (1.1) the autocovariance γ(k1,k2) of the process γ(k1,k2) is given  as
(2.7)γ(k1,k2)=σ2ϕ10k1Γ(k1+δ)F(δ,k1+δ;k1+1;ϕ102)ϕ01k2Γ(k2+δ)F(δ,k2+δ;k2+1;ϕ012)Γ2(δ)Γ(k1+1)Γ(k2+1),
where F(·,·;·;·) is the hypergeometric function.Proof.
We establish the previous proposition by integrating the spectral density as given in Proposition2(2.8)γ(k1,k2)=∬-ππek1λ1+k2λ2f(λ1,λ2)dλ1dλ2=σ24π2∬-ππek1λ1+k2λ2dλ1dλ2(1-2ϕ10cosλ1+ϕ102)δ(1-2ϕ01cosλ2+ϕ012)δ=σ24π2∫-ππek1λ1dλ1(1-2ϕ10cosλ1+ϕ102)δ∫-ππek2λ2dλ2(1-2ϕ01cosλ2+ϕ012)δ=σ2π2∫0πcosk1λ1dλ1(1-2ϕ10cosλ1+ϕ102)δ∫0πcosk2λ2dλ2(1-2ϕ01cosλ2+ϕ012)δ.
Now by making use of the identity (see [8]),
(2.9)∫0πcoskxdx(1-2αcosx+α2)δ=παkΓ(k+δ)F(δ,k+δ;k+1;α2)Γ(δ)Γ(k+1),
we obtain
(2.10)γ(k1,k2)=σ2ϕ10k1Γ(k1+δ)F(δ,k1+δ;k1+1;ϕ102)ϕ01k2Γ(k2+δ)F(δ,k2+δ;k2+1;ϕ012)Γ2(δ)Γ(k1+1)Γ(k2+1),
which completes the proof.Corollary 4.
For a process defined in   (1.1) the variance of the process γ(0,0) is given as
(2.11)γ(0,0)=σ2F(δ,δ;1;ϕ102)F(δ,δ;1;ϕ012).Proof.
This result is directly from Proposition3 by letting k1=k2=0.The autocorrelation function (ACF) of the model in (1.1) is given as(2.12)ρ(k1,k2)=γ(k1,k2)γ(0,0)=ϕ10k1Γ(k1+δ)F(δ,k1+δ;k1+1;ϕ102)ϕ01k2Γ(k2+δ)F(δ,k2+δ;k2+1;ϕ012)Γ2(δ)Γ(k1+1)Γ(k2+1)F(δ,δ;1;ϕ102)F(δ,δ;1;ϕ012).Remark 5.
Note whenδ=1, we have the standard separable spatial model. Substituting δ=1 in Proposition 3, we obtain
(2.13)γ(k1,k2)=σ2ϕ10k1Γ(k1+1)F(1,k1+1;k1+1;ϕ102)ϕ01k2Γ(k2+1)F(1,k2+1;k2+1;ϕ012)Γ2(1)Γ(k1+1)Γ(k2+1)=σ2ϕ10k1F(1,k1+1;k1+1;ϕ102)ϕ01k2F(1,k2+1;k2+1;ϕ012).Using the following identity (see Abramowitz and Stegun [11, Page 556, Identity No. 15.1.8]):
(2.14)F(a,b,b,z)=1(1-z)a,
(2.13) reduces to
(2.15)γ(k1,k2)=σ2ϕ10k1ϕ01k2(1-ϕ102)(1-ϕ012).
Hence, Proposition 3 reduces to the autocovariance function of the standard separable spatial model when δ=1.In Table1, we have tabulated (to three decimal places) the ACF, ρ(k1,k2) computed by using   (2.12) with ϕ10=0.9, ϕ01=0.9, δ=1.0. This is the standard separable model. Clearly, we can see that the numerical values computed by using   (2.12) agree  with the ACF of the standard separable model which is ρ(k1,k2)=ϕ10k1ϕ01k2. Hence, this verifies  (2.12).Table 1
ACF,ρ(k1,k2) computed by using   (2.12), (ϕ10=0.9,  ϕ01=0.9,  δ=1.0, i.e., standard separable model).
k1:k201234567891001.0000.9000.8100.7290.6560.5900.5310.4780.4300.3870.34910.9000.8100.7290.6560.5900.5310.4780.4300.3870.3490.31420.8100.7290.6560.5900.5310.4780.4300.3870.3490.3140.28230.7290.6560.5900.5310.4780.4300.3870.3490.3140.2820.25440.6560.5900.5310.4780.4300.3870.3490.3140.2820.2540.22950.5900.5310.4780.4300.3870.3490.3140.2820.2540.2290.20660.5310.4780.4300.3870.3490.3140.2820.2540.2290.2060.18570.4780.4300.3870.3490.3140.2820.2540.2290.2060.1850.16780.4300.3870.3490.3140.2820.2540.2290.2060.1850.1670.15090.3870.3490.3140.2820.2540.2290.2060.1850.1670.1500.135100.3490.3140.2820.2540.2290.2060.1850.1670.1500.1350.122In Table2, we have tabulated (to three decimal places) the ACF, ρ(k1,k2) of the GENSSAR model (δ=1.8) computed by using (2.12) with ϕ10=0.9, ϕ01=0.9, δ=1.8. While Table 3  shows the ACF values of the GENSSAR model (δ=0.2) computed by (2.12) with ϕ10=0.9, ϕ01=0.9, δ=0.2.Table 2
ACF,ρ(k1,k2) computed for GENSSAR model by using   (2.12), (ϕ10=0.9,  ϕ01=0.9,  δ=1.8).
k1:k201234567891001.0000.9920.9720.9450.9160.8740.8350.7940.7520.7100.66810.9920.9830.9640.9370.9040.8670.8280.7870.7450.7040.66220.9720.9640.9450.9180.8860.8500.8120.7710.7310.6900.65030.9450.9370.9180.8920.8610.8260.7890.7500.7100.6710.63140.9120.9040.8860.8610.8310.7970.7610.7230.6850.6470.60950.8740.8670.8500.8260.7970.7650.7300.6940.6570.6210.58460.8350.8280.8120.7890.7610.7300.6970.6620.6270.5920.55870.7940.7870.7710.7500.7230.6940.6620.6300.5960.5630.53080.7520.7450.7310.7100.6850.6570.6270.5960.5650.5330.50290.7100.7040.6900.6710.6470.6210.5920.5630.5330.5040.474100.6680.6630.6500.6310.6090.5840.5580.5300.5020.4740.447Table 3
ACF,ρ(k1,k2) computed for GENSSAR model by using   (2.12), (ϕ10=0.9,  ϕ01=0.9,  δ=0.2).
k1:k201234567891001.0000.2010.1130.0770.0560.0430.0340.0270.0220.0180.01510.2010.0400.0280.0150.0110.0090.0070.0050.0040.0040.00320.1130.0230.0130.0090.0060.0050.0040.0030.0020.0020.00230.0770.0150.0080.0060.0040.0030.0030.0020.0020.0010.00140.0560.0110.0060.0040.0030.0020.0020.0020.0010.0010.00150.0430.0090.0050.0030.0020.0020.0010.0010.0010.0010.00160.0340.0070.0040.0020.0020.0010.0010.0010.0010.0010.00170.0270.0050.0030.0020.0020.0010.0010.0010.0010.0000.00080.0220.0040.0020.0020.0010.0010.0010.0010.0000.0000.00090.0180.0040.0020.0010.0010.0010.0010.0000.0000.0000.000100.0150.0030.0020.0010.0010.0010.0010.0000.0000.0000.000Figures1, 2, and 3 show the ACF for the three models considered in this paper.Figure 1
ACF plot for standard separable model (ϕ10=0.9, ϕ01=0.9, δ=1.0).Figure 2
ACF plot for GENSSAR (ϕ10=0.9, ϕ01=0.9, δ=1.8).Figure 3
ACF plot for GENSSAR (ϕ10=0.9, ϕ01=0.9, δ=0.2).From the tables and figures we can clearly see that the behaviour of ACF depends on the index parameterδ. When δ>1.0, the ACF decays slower than that of the standard separable model. On the other hand, when δ<1.0 the ACF decays faster than the standard model. Hence, the GENSSAR model can be used to model many types of autocorrelation structure.We also considered a further illustrative example whenϕ10 and ϕ01 were not equal to each other. That is, we chose the parameter values to be ϕ10=0.4, ϕ01=0.7, δ=1.8. In Table 4, we have tabulated (to three decimal places) the ACF, ρ(k1,k2) of the GENSSAR model (δ=1.8) computed by using   (2.12) with ϕ10=0.4, ϕ01=0.7, δ=1.8, and Figure 4 shows a plot of the ACF. Clearly we can see that the decay in the autocorrelation is more rapid along the k1 axis as compared to the k2 axis. Hence, we can model data whose autocorrelations decay at different rates in different directions.Table 4
ACF,ρ(k1,k2) computed for GENSSAR model by using   (2.12), (ϕ10=0.4,  ϕ01=0.7,  δ=1.8).
k1:k201234567891001.0000.9200.7870.6460.5170.4050.3130.2400.1820.1370.10210.6450.5930.5070.4170.3330.2610.2020.1540.1170.0880.06620.3470.3190.2730.2240.1790.1410.1090.0830.0630.0470.03530.1720.1580.1350.1110.0890.0700.0540.0410.0310.0240.01840.0820.0750.0640.0530.0420.0330.0260.0200.0150.0110.00850.0380.0350.0300.0240.0190.0150.0120.0090.0070.0050.00460.0170.0160.0130.0110.0090.0070.0050.0040.0030.0020.00270.0070.0070.0060.0050.0040.0030.0020.0020.0010.0010.00180.0030.0030.0030.0020.0020.0010.0010.0010.0010.0000.00090.0010.0010.0010.0010.0010.0010.0000.0000.0000.0000.000100.0010.0010.0000.0000.0000.0000.0000.0000.0000.0000.000Figure 4
ACF plot for GENSSAR (ϕ10=0.4, ϕ01=0.7, δ=1.8).For the models considered in this paper, the two-dimensional spectral densities for various parameter values are shown in Figures5, 6, 7, 8, 9, 10, and 11.Figure 5
Spectral density plot of standard separable model (ϕ10=0.9, ϕ01=0.9, δ=1.0).Figure 6
Spectral density plot of GENSSAR model (ϕ10=0.9, ϕ01=0.9, δ=1.8).Figure 7
Spectral density plot of GENSSAR model (ϕ10=0.9, ϕ01=0.9, δ=0.2).Figure 8
Spectral density plot of standard separable model (ϕ10=-0.9, ϕ01=0.9, δ=1.0).Figure 9
Spectral density plot of GENSSAR model (ϕ10=-0.9, ϕ01=0.9, δ=1.8).Figure 10
Spectral density plot of GENSSAR model (ϕ10=-0.9, ϕ01=0.9, δ=0.2).Figure 11
Spectral density plot of GENSSAR model (ϕ10=0.4, ϕ01=0.7, δ=1.8).


## 3. Conclusion

The objective of this research is to introduce a new class of models calledGENSSAR models by including an additional index parameter δ and to establish some of its properties. We have established the autocovariance function. The GENSSAR(1,1) model is a more general model than the standard separable spatial AR(1,1) process. Due to the generality of this model, it is a useful model.The authors are working on the other aspects of this model with applications and will be reported in a future paper.



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