Several classes of models have been proposed to describe the spatial processes. A special class of non-separable spatial unilateral model for modeling spatial data on two dimensional rectangular regular grid is the spatial autoregressive model, denoted by AR( PI ,1). In this paper, we establish a procedure to estimate the parameters of this model using the maximum likelihood method. We show that this procedure is practical and easy to be implemented We illustrate the procedure by fitting the AR(l, J) model to two set of data on

the regular rectangular grid

Awang, Norhashidah

Shitan, Mahendran

English

Repository@USM

The 3rd International Conference on Mathematics and Statistics (ICoMS-3)Institut Pertanian Bogar, Indonesia, 5-6 August 2008MAXIMUM; LIKELIHOOD ESTIMATION FOR THENON-SEPARABLE SPATIAL UNILATERALAUTOREGRESSIVE MODELINorhashidah Awang, 2Mahendran Shitan1 School of Mathematical Sciences, Universiti Sains Malaysia,11800 USM, Penang, Malaysia2 Department of Mathematics, Faculty of Science, Universiti Putra Malaysia,43400 UPM Serdang, Selangor Darnl Ehsan, MalaysiaAbstract. Several classes ofmodels have been proposed to describe the spatial processes. Aspecial class of non-separable spatial unilateral model for modelling spatial data on two-dimensional rectangular regular grid is the spatial autoregressive model, denoted byAR( PI ,1). In this paper, we establish a procedure to estimate the parameters of this modelusing the maximum likelihood method. We show that thiS procedure is practical and easy tobe implemented We illustrate the procedure byfitting the AR(l, J) model to two set ofdata onthe regular rectangular grid. The results show that the model is adequate in capturing thespatial correlation in the data.Keywords: Parameter estimation; spatial autoregressive model; spatial unilateral model1. IntroductionA large amount of research in modelling spatial processes has been conducted and they have coveredvarious applications. Spatial series may be considered as a generalisation of time series, howevermodelling it is more difficult since the dependence in spatial series spreads in all directions and itencounters larger proportion of edge effects as compared to time series.In this paper, consideration will be given to modelling spatial process in two-dimensional regular gridwhere a random variable is defined at each intersection point. Various classes of models have beenproposed to describe the process. Among the earliest models of this particular process are theSimultaneous Autoregressive (SAR) model (Whittle, 1954), the Conditional Autoregressive (CAR) model(Besag, 1974) and the Moving Average (MA) model (Haining, 1978).Many methods and procedures have been developed and proposed to overcome the estimationproblems in spatial modelling. Martin (1979, 1990 and 1996) studied a class of models called separablemodels, while Tjostheim (1978 and 1983) and Basu and Reinsel (1992, 1993 and 1994) considered theunilateral models. A special characteristic of separable models is that it has a product correlationstructure, which in tum simplifies the estimation. However, these models are only applied for data whichexhibit separable correlation structure. The unilateral models can be analysed using extensions of timeseries theory in some special cases and are claimed useful in systems theory and digital filtering (seeTjostheim, 1978).A special class of non-separable unilateral model is the spatial autoregressive model, denoted byAR( Pl ,1). Shitan and Brockwell (1996) have established the procedure to estimate the parameters of thismodel where the approach is to transform the two-dimensional spatial series to a multiple time series,treating one of the coordinates as a time index and the other as a multivariate index and then performedthe multivariate least squares estimation procedures. In this paper, we look at the problem of estimatingthe parameters of this model from a different perspective. Our approach is to use maximum likelihoodmethod with some modifications at the border to simplify the parameter estimation.(2.1)In the next section, the construction of the weight matrices is presented. The derivation of theprocedure of estimation is discussed in Section 3. In Section 4, the numerical examples to illustrate theprocedure are given and finally the conclusions are presented in Section 5.2. Construction of the weight matricesWe consider a spatial process in two-dimensional regular grid of size mxn. The non-separable spatialunilateral autoregressive, AR( Pl ,1) model is defined as,lit = aIOYi-l,j + a01 Yi,j-l + all Ji-l,j-l + ... +ap,oJi-PI,j + apllli-PI,j-1 + £ij'i =1, 2, ... , m and j =1, 2, ... , n,where {Yij} is a sequence of two-dimensional random variable with zero mean and the errors £ij areassumed to be nonnally distributed with mean 0 and common variance d at site labelled (i,j), and aij'sare the parameters to be estimated.By assuming that the unobserved values to be zeroes, and letting the observation vector,Y = (Yll , il2' , YIn' Y21 , Yn , , Y2n' ... , Yml , Ym2 , ..., Ymn)' =(YI , Y2 , .. ·, Ym )', whereYi = Oil, >'i2, , >in)' , i = 1. 2, , m and the error vector,E = (611, £12, , £In' £21,622, , 62n' ... , 6ml' Cm2, ... , Gmn )' = (EI' E2' ..., Em)' , where£j =(cil' £i2, £in)', i =1, 2 , m , we can rewrite equation (2.1) in the matrix form as,~Q",1·····~..····I.._.~......I-..~----I-~~-:---L ...~...··-!.·······_~-------··! ..:·:·:··~······?_·t····?_·..l·····~_.cJ)1 i cJ)o i 0 I 0 1 ... 1 0 ! 0 i ... i 0 ! 0 ! 0·..·_..·-r···..··....·T·········--r········T-..·····r..····--..··T·····..··..······..·T-·..···..l........···--T--·..·-·t·----------cJ)2 ! cJ)1 ! cJ)o ! 0 !... i 0 ! 0 i' .. i 0 ! 0 ! 0=~:::~Ii.~:.=l··-;·~~~~[:.::I~=r-=:=..··[~~·~~~:~:.:I~:.J: ..~~~:..l~:~~T:~=o : 0 i 0 i 0 i··· ! (1) I (1) I!' .. i cJ)2 ! (1)1 i cJ)o! ! ! ! ! PI! Pt-! ! ! !YI E1Y2 £2Y3 + 1:3:Ym £m(2.2)where (J) j 's are nxn matrices defined as.0 0 0 0 0 ajO 0 0 0 0aOI 0 0 0 0 ajl ajO 0 0 00 aOI 0 0 0 0 ajl ajO 0 0())o= and ())j=0 0 aOI 0 0 0 0 ajl 0 00 0 0 aOI 0 0 0 0 ajl ajofor j = 1, 2, ... , PI'Equation (2.2) can be written more compactly as,Y = (J)Y + £, (2.3)where <J) is NxN matrix, N = mn. It is clear that <J) is a lower triangular matrix with zeroes on the maindiagonal. Then, if we decompose en into 2PI + 1 matrices such that it isolates different parameters, weobtainY = (alOWIO + aOlWOl + allWlI + ... + aplOWplO + apllWpll)Y + £, (2.4)where, en = alOWIO + a Ol W01 + allWll + ... + aplOWplO + ap,lWpll and W;bj = 1,2, ... ,PI ; k = 0, 1 are the NxN lower triangular weight matrices with elements ones and zeroes.3. Maximum likelihood procedure for estimating the parametersEquation (2.4) can then be written as,Y= (I -(alOWlO + aOlW01 + allWll + ,.. + ap,OWp10 + ap,IWp,I))-1 E (3.1)or(3.2)where 1 is an NxN identity matrix.Therefore, the covariance matrix of Y , V is given as,V = a 2 ( 1 -«I> r l [( } -«I> r l J. (3.3)The square root of the determinant of V is given as,IvI 1I2 = (a 2)NI2\(I_«I»-II. (3.4)Since (I - «1» is the lower triangular matrix with diagonal elements I, I(1 - «I> r 11 = 1. This leads toIVI l/2 = (a2 )NI2. (3.5)Therefore, the likelihood function 1 is given as,1= I eXP{-~YV-IY}(2n")NI2\VI1/2 2~ (27fF N 12(<r2 )-N /2 exp {- ~2 Y'[(I _",)-1(1_<1»-1r Y}= (2Jl")~N12(a2 )-N12 exp{- _I_ y ,(} -(1))(1 -«I»V} ,2u2Thus we obtain the log likelihood, LasN N 2 1" (36)L = - -In(27l") - -In(a ) - -Y (I -cJ) )(1 -«I»V. .2 2 2a2The partial derivative ofL with respect to a jk ,j = 1,2, ... , Pl; k = 0, 1 is given byaL _~[ -Y'Wjk Y + ajkY'W.;k WjkY + I I arsY'W;,I' WjkY] (3.7)Cajk U VrFjVs*kforj = 1,2, ... , 5.Equating (3.7) to zero leads to[ ajkY'WjkWjkY + I. I arsy'w;.I,WjkY ] = Y'Wjk Y ' (3.8)Vr*.1 Vs'#kTherefore, denoting Zjk = W jk Y , the maximum likelihood for a jk ' S can be obtained by solving theequationZ;o ZlO ZOI ZlO Z~IO ZlO Z~ll ZlO alO Y'ZlOZ;o ZOl Z(n ZOI Z~10 ZOl Z~lIZOl aOl Y'ZOlZio Zp,o ZOI ZPI O Z~jO Zp10 Z~ll Zp10 ap,o Y'Zp,oZiOZP1I ZOI ZPI 1 Z~IOZpll Z~lIZPll a p,1 Y'Zp,1(4.3)(4.1)(4.2)Zio zlQ ZOI zlQ Z~IO ZlQ Z~II ZlQ -1alQ Y'ZlQam Zio Zm ZOI Z01 Zp1OZ Ol ZpI1 Z01 Y'ZOIor (3.9)aplo Z;o ZP1 0 ZOI Zp10 Z~IO ZPIO Z~11 Zplo Y'ZpI Oa pl 1 ZioZplI ZOlZpl1 Z~IOZpII Z~IIZPll Y'Zp114. Numerical examplesWe fit the AR(l, 1) model to two well-known data sets observed on regular grids to illustrate theprocedure mentioned above. The AR(l,l) model is defined asYij = alOlf-l,j + aOIJi,j-l + al1 Ji-l,j-l +Eij'and hence, equation (2.2) becomesYI «Ilo ! 0 ! 0 ! 0 !... I 0 I 0 I ... I 0 ! 0 lOY} I:}~i.: :~!~::r!:2:~:I:_:~~~::1:::~~:r::~::.:1~:~-"-"[:::~:.:r::::~l:?::::I:~~~::::I::~"~~: y 2 1:2Y3 0 ! «Ill ! (1)0 ! 0 !'" ! 0 ! 0 !'" i 0 ! 0 ! 0 Y3 + 1:3Ym ~~~~:rH~~~~J~FfH~;~~~ Ym EmFrom equation (3.9), the estimate of the parameters can be obtained by solving the equation(alQ J (ZiO ZlQ ZOI ZlQ Z; 1 ZIOJ-I (Y' ZIOJaOl = Zio ZOl ZOl ZOI Zi 1ZOI Y' ZOIall zio Zll ZOI Zll Zi 1Zll Y' Zl1where ZlQ =WlO Y, ZOI =WOI Y and ZII =WI I Y . The matrices WlQ ' WOI and WI I are the NxNweight matrices given as[f.t"-&+-g-"t'::'I""g'i."~ .."i'3] [~-f"~+~...!.:~.:"t-'&-"I-"g".t"g.] [~t-"g"-l":--f":-H"':""I-"g ..tg]WIO = llt"T"!'''(rt~-:t""'ll"o''i"o W = 'O"'!"O""!"fi"\";";":"t"O"i""Ir!"O" and WI = 'o"1"'l)T"fri";"::TijT(fm .~r~T~:l-"~"T~T;":l~' 01 "~T~T~:r~"T~T;-'-lri 0 "~r~l~'l-~-~"r~Tril~Here, I is an nxn identity matrix and D is an nxn matrix defined by[~ .r..~. ::: .z l z].000 .. ·010The computations are performed by a computer program written in S-Plus.For the first numerical example, we consider the data set obtained in Cressie (1993) on wheat (yield ofgrain) where the uniformity trial was conducted by Mercer and Hall (1911). The data are observed on25x20 grid. Since the data is stationary and fulfils the normality assumption (from the density and theKolmogorov-Smimov test), no transformation is needed. Table I displays the sample spatial correlationsof the mean-corrected data for s = 0 to 4 and t = -4 to 4, where s is the lag in west-east direction (betweencolumns) and t is the lag in south-north direction (between rows). The table suggests that there exists aconsiderable amount of spatial correlation in the data, and the correlation is much stronger along thesouth-north direction (between rows) than along the west-east direction (between columns), for example,POI = 0.494 and P02 =0.357 (correlations along the south-north direction at lags I and 2, respectively)whereas Ao = 0.280 and P20 =0.140 (correlations along the west-east direction at lags I and 2,respectively).Table I: Sample spatial correlations, Pst for the wheat(yield of grain) mean-corrected data of size 25 x 20obtained in Cressie (1993).s0 1 2 3 4-4 0.282 0.066 -0075 0.095 -0.026-3 0.303 0.063 -0.011 0.094 -0.006-2 0.357 0.114 0.001 0.117 0.022-1 0.494 0.167 0.021 0.134 0.056t 0 1.000 0.280 0.140 0.166 0.065I 0.494 0.212 0.112 0.160 0.0462 0.357 0.152 0.081 0.192 0.0843 0.303 0.096 0.057 0.176 0.0454 0.282 0.106 0.062 0.157 0.068The maximum likelihood estimate of the parameters is obtained using the equation (4.3) and is shownin Table 2, together with the estimates of (-2 In L) and if. The goodness of the fitted model is examinedby the spatial correlations of the residuals. The correlations of the residuals from the fitted model aresmall and these suggest that the model is adequate in capturing the spatial correlations in the data.In the second numerical example, we analyse the data set on the yield of barley from an agricultureuniformity trial experiment at Plant Breeding Institute, Cambridge, United Kingdom and the data isobtained in Kempton and Howes (1981). The data set are of 7x28. The plot of the data shows that it is notstationary and therefore a transformation is needed to make it stationary. In this analysis, we apply thespatial variate differencing where the first row differencing is performed and the result shows that theresulting series has no apparent trend, suggesting that the transformation is sufficient to make the datastationary. Furthermore, the density plot and the result of the Kolmogorov-Smirnov test suggest that thedata satisfy the normality assumption. Table 3 displays the sample spatial correlation of the mean-corrected original data, whereas Table 4 displays the sample spatial correlation of the mean-correctedvalues of the differenced data. Having satisfied the stationary and normality assumption, we fit theAR(l,l) model to this differenced data and the results are displayed in Table 5. Again, the correlations ofthe residuals from this model are small and these suggest that the model is adequate in capturing thespatial correlations in the data.Table 3: Sample spatial correlations, Pst for theyield ofbarley mean-corrected data of size 7x28from Kempton and Howes (1981).s0 1 2 3 4-4 0.470 0.220 0.053 0.035 -0.035-3 0.570 0.231 0.037 0.015 -0.065-2 0.677 0.241 0.041 0.026 -0.063-1 0.796 0.253 0.035 0.DI5 -0.075( 0 1.000 0.264 0.013 -0.Q25 -0.0971 0.796 0.190 -0.045 -0.060 -0.1012 0.677 0.137 -0061 -0.064 -0.0753 0.570 0.088 -0.063 -0.065 -0.0554 0.470 0.044 -0.079 -0.073 -0.051Table 4: Sample spatial correlations, Pst for themean-corrected of first row-differenced yields ofsize 7x27s0 I 2 3 4-4 -0.064 0.010 0.079 -0.052 0.021-3 -0.030 0.035 -0014 -0.024 -0068-2 0.005 -0.020 -0.018 0.030 0.014-)-0.206 -0.042 0.020 0.048 0.004I 0 1.000 0.182 0.077 0.000 0.0051 -0.206 -0.052 -0108 -0.070 -0.0942 0.005 0.029 -0.012 -0.022 ~0.0193 -0.030 -0.060 0.061 0.002 0.0434 -0.064 0.006 -0.027 -0.058 -0.0345. ConclusionIn this paper, we have proposed a procedure to estimate the parameters of the spatial non-separableunilateral autoregressive models, i.e. the AR( PI ,I) model using the maximum likelihood method. Theprocedure is practical and easy to be implemented. We have illustrated the procedure by fitting the modelfor PI = I to two set of data on the regular rectangular grid. The results show that the model is adequatein capturing the spatial correlation in the data; hence, we conclude that this class of model and theestimation method proposed can provide as an alternative for modelling spatial data on regularrectangular grid.6. ReferencesBasu, S. and Reinsel, G. C. (1992), A note on properties of spatial Yule-Walker estimators, Journal ofStatistical Computing and Simulation, 41, pp. 243-255.Basu, S. and Reinsel, G. C. (1993), Properties of the spatial unilateral first order ARMA model, Advancesin Applied Probability, 25, pp. 631-648.Basli, S. and Reinsel, G. C. (1994), Regression models with spatially correlated errors, Journal of theAmerican Statistical Association: Theory and Method, 89(425), pp. 88-99.Besag, 1. E. (1974), Spatial interaction and the statistical analysis of lattice systems, Journal ojthe RoyalStatistical Society B, 36, pp. 192-236.Cressie, N. A. C. (1993), Statistics for Spatial Data, Revised Edition, Wiley, New York.Haining, R. P. (I 978b), The moving average model for spatial interaction, Transactions Institute ofBritish Geographer, 3, pp. 202-225.Kempton, R. A. and Howes, C. W. (1981), The use of neighbouring plot values in the analysis of varietytrials, Applied Statistics, 30, pp. 59-70.Martin, R. 1. (1979), A subclass oflattice processes applied to a problem of planar sampling, Biometrika,66, pp. 209-217.Martin, R. J. (1990), The use of time series models and methods in the analysis of agricultural field trials,Communication in Statistics - Theory and Method, 19(1), pp. 55-81.Martin, R. 1. (1996), Some results on unilateral ARMA lattice processes, Journal ofStatistical Planningand Inference, 50, pp. 395-411.Mercer, W. B. and Hall, A. D. (1911), The experimental error of field trials, Journal of AgriculturalScience, 4, pp. 107-132.Shitan, M. and Brockwell, P. J. (1996), An alternative estimation procedure of the spatial AR(pj,I)model, Research Report No.2, Dept of Statistics and Operations Research, RMIT, Australia.Tjostheim, D. (1978), Statistical spatial series modelling, Advances in Applied Probability, 10, pp. 130-154.Tjostheim, D. (1983), Statistical spatial series modelling II. Some further results on unilateral latticeprocesses, Advances in Applied Probability, 15, pp. 562-584.Whittle, P. (1954), On stationary processes in the plane, Biometrika. 41, pp.434-449.

Maximum; Likelihood Estimation For The Non-Separable Spatial Unilateral Autoregressive Model.

http://eprints.usm.my/11688/1/MAXIMUM_LIKELIHOOD_ESTIMATION_FOR_THE_NON-SEPARABLE_SPATIAL_UNILATERAL_AUTOREGRESSIVE_MODEL.pdf

Maximum; Likelihood Estimation For The Non-Separable Spatial Unilateral Autoregressive Model.

Abstract

Similar works

Full text

Available Versions

Repository@USM