AN ILLUSTRATION OF GENERALISED ARMA (GARMA) TIME SERIES MODELING OF FOREST AREA IN MALAYSIA

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1 International Conference Mathematical and Computational Biology 2011 International Journal of Modern Physics: Conference Series Vol. 9 (2012) c World Scientific Publishing Company DOI: /S AN ILLUSTRATION OF GENERALISED ARMA (GARMA) TIME SERIES MODELING OF FOREST AREA IN MALAYSIA THULASYAMMAL RAMIAH PILLAI Laboratory of Computational Statistics and Operations Research, Institute for Mathematical Research, Universiti Putra Malaysia, UPM Serdang, Selangor Darul Ehsan, Malaysia and Faculty of Engineering and Computer Technology, AIMST University, Malaysia thulasyram@hotmail.com MAHENDRAN SHITAN Laboratory of Computational Statistics and Operations Research, Institute for Mathematical Research, Universiti Putra Malaysia, UPM Serdang, Selangor Darul Ehsan, Malaysia and Department of Mathematics, Faculty of Science, UPM Serdang, Selangor Darul Ehsan, Malaysia mahendran@math.upm.edu.my Forestry is the art and science of managing forests, tree plantations, and related natural resources. The main goal of forestry is to create and implement systems that allow forests to continue a sustainable provision of environmental supplies and services. Forest area is land under natural or planted stands of trees, whether productive or not. Forest area of Malaysia has been observed over the years and it can be modeled using time series models. A new class of GARMA models have been introduced in the time series literature to reveal some hidden features in time series data. For these models to be used widely in practice, we illustrate the fitting of GARMA (1, 1; 1, δ) model to the Annual Forest Area data of Malaysia which has been observed from 1987 to The estimation of the model was done using Hannan-Rissanen Algorithm, Whittle s Estimation and Maximum Likelihood Estimation. Keywords: Forestry; GARMA model; Hannan-Rissanen algorithm; Whittle s estimation; Maximum likelihood estimation. Mathematical Subect Classification: 62M10 1. Introduction Forest ecosystems have come to be seen as the most important component of the biosphere and forestry has emerged as a vital field of science, applied art, and 390

2 Garma Time Series Modelling of Forest Area in Malaysia 391 technology. Forestry is the art and science of managing forests, tree plantations and related natural resources. The main goal of forestry is to create and implement systems that allow forests to continue a sustainable provision of environmental supplies and services. 1 Forests once surrounded the Mediterranean Sea. As the trees were steadily cut, the land was converted to an arid, inhospitable, treeless desert. 2 Due to that, silviculture, a related science, involves the growing and tending of trees and forests was introduced. 1 This had led to the numerous studies on forestry. Ref. 3 had used the time series autoregressive moving average (ARMA) approach to develop stochastic models of tree crown profiles for five conifer species of the Sierran mixed conifer habitat type. Ref. 4 had done a study to estimate stochastic models for the variation in growth of scots pine (Pinus sylvestris), Norway spruce (Picea abies) and birch(betula pendula and Betula pubescens). Non-seasonal AR(1) and seasonal AR(1, 1) models were used to produce growth scenarios in the case study in which variation in growth was integrated into forest planning. Forest area of Malaysia has been observed over the years and it can be modeled using time series models. A new class of GARMA has been introduced in the time series literature to reveal some hidden features in time series data. In this paper, we illustrate the fitting of GARMA (1, 1; 1, δ) model to the Annual Forest Area data of Malaysia which has been observed from 1987 to The estimation of the model was done using Hannan-Rissanen Algorithm, Whittle s Estimation and Maximum Likelihood Estimation. In Section 2, we give a brief account of time series models while in Section 3 the estimation of parameters of the model is discussed. In Section 4 we illustrate the modeling of Forest Area of Malaysia using GARMA(1,1;1,δ) model. Finally the conclusion are drawn in Section Time Series Modelling A time series is a set of observations X t, each one being recorded at a specific time t and denoted by {X t }. It can be represented as a realization of the process based on the general model called Classical Decomposition Model, and specified as: X t = m t +s t +Y t (1) t = 1,2,...,n, where m t is a trend component, s t is a seasonal component and Y t is a random noise component which is stationary. 5 Time series modeling help us to predict data series that are typically not deterministic but contain a random component. The deterministic components, m t and s t need to be estimated and eliminated as to make the residue or noise component Y t to be stationary time series. A non-stationary time series needs to be transformed to a stationary time series, in order to analyze its properties and to use it for prediction purposes. 5

3 392 T. R. Pillai and M. Shitan Autoregressive Moving Average (ARMA)processes are widely used in forecasting. The family of standard AR(1) processes 6,7 generated by, X t αx t 1 = Z t (2) where α < 1, {X t } is a time series, {Z t } is a sequence of uncorrelated random variables (not necessarily independent) with zero mean and variance σ 2, known as white noise and denoted by WN(0,σ 2 ). Using the backshift operator,b(i.e.b X t = X t, 0) and the identity operator I = B 0, equation (2) can be written as, (I αb)x t = Z t. (3) A new class of ARMA type models with indices has been introduced by Ref. 8 to control the degree of frequency, Generalised AR(1) (GAR(1)) process generated by, (I αb) δ X t = Z t, (4) with the introduction of the new parameter δ where δ > 0. This GAR(1) and its properties were discussed in Ref. 8. The Moving Average or MA(1) is generated by, X t = (I βb)z t, (5) where β < 1. A new generalised version of (5) with an additional parameter (or index) δ > 0 satisfying X t = (I βb) δ Z t (6) was introduced by Ref. 9. This class of models called Generalised MA(1). The standard ARMA(1, 1) can be written as, (I αb)x t = (I βb)z t (7) where α, β < 1. Ref. 8 also introduced a new, generalised version of (7) with the additional parameters δ 1 0 and δ 2 0 satisfying (I αb) δ1 X t = (I βb) δ2 Z t. (8) This new class of models known as the Generalised Autoregressive Moving Average (GARMA) Model has been introduced by Ref. 8 in order to reveal some hidden features in time series data. These type of models could be used to describe data with different frequency components for suitably chosen indices. Recently, Ref. 10 have discussed about GARM A(1, 1; 1, δ) and its properties in detail which is generated by, (I αb)x t = (I βb) δ Z t. (9) The obective of this paper is to illustrate the fitting of GARMA (1, 1; 1, δ) model to the Forest area data of Malaysia which has been observed from 1987 to We use the Hannan-Rissanen Algorithm Estimator technique as described below to find a suitable set of starting up values for the Whittle s estimation and Maximum Likelihood Estimation (MLE).

4 Garma Time Series Modelling of Forest Area in Malaysia Estimation of Parameters 3.1. Hannan-Rissanen algorithm estimator The Hannan-Rissanen Algorithm technique is one of the preliminary techniques used for ARMA (p,q) models where p > 0 and q > 0. ARMA (p,q) is generated by, X t φ 1 X t 1... φ p X t p = Z t θ 1 Z t 1... θ q Z t q. (10) Firstly,ahigh-orderAR(m) model withm > max(p,q) isfitted tothe datausing the Yule-Walker estimates. If ( φ m1,..., φ mm ) is the vector of estimated coefficients, then the estimated residuals are computed from the equations Ẑ t = X t φ m1 X t 1... φ mm X t m,t = m+1,...,n. (11) Secondly, the vector of parameters, w = (φ 1,...,φ p,θ 1,...,θ q ) is estimated by minimizing the sum of squares S(w) = Σ n t=m+1+q (X t φ 1 X t 1,...,φ p X t p +θ 1 Ẑ t θ q Ẑ t q ) 2 (12) with respect to w. This gives the Hannan-Rissanen Algorithm estimator ŵ = (Z Z) 1 Z X n, where X n = (X m+1+q,...,x n ) and Z is the (n m q) (p+q) matrix. ARMA (p, q) model is fitted using the Hannan-Rissanen estimates. See Ref. 7 for details. The fitted model is X t φ 1 X t 1... φ p X t p = Z t θ 1 Z t 1... θ q Z t q. (13) Thirdly, ŵ values can be manipulated to obtain the parameter values for GARMA(1,1;1,δ). ŵ = ( φ 1, θ 1, θ 2,..., θ q ) is computed using ARMA(1,q) model. The fitted ARMA(1,q) model is X t φ 1 X t 1 = Z t θ 1 Z t 1... θ q Z t q. (14) The GARMA(1,1;1,δ) model also can be recorded as below after the expansion of the right side expressions of equation (9), X t αx t 1 = Z t βδz t 1... (15) After comparing the equation (14) and equation (15), we can deduce that the φ 1 value is equivalent to α and the θ 1 value is equivalent to βδ. The estimation of β and δ are done by assuming that β = δ. If the value of β > 1, then we assume β = 0.6 and δ = θ 1 / β. Hannan-Rissanen Algorithm is used to provide preliminary estimates of the GARMA parameters as such these aforementioned assumptions are made. The corresponding estimate for σ 2 is given as, σ 2 = S(ŵ/(n m q)). (16)

5 394 T. R. Pillai and M. Shitan 3.2. Whittle s estimation The Whittle s estimates were obtained by minimizing the function, ln( 1 T Σ I T (w ) g(w ) )+ 1 T Σ lng(w), (17) where I T (w ) is the periodogram of the series given by, I T (w = 2π T ) = 1 T ΣT s=1 exp( i2πs T ) 2 (18) and g(w = 2π T ) = 1 βexp( i2πs T ) 2 / 1 αexp( i 2πs T ) 2δ, = [ T 1 2 ],...,[T 2 ] is the true spectrum of the process. The corresponding estimate for σ 2 is given as, See Ref. 7 for details. σ 2 = 1 T Σ I T (w ) g(w ) 3.3. Maximum likelihood estimation The maximum likelihood estimates (MLE) for the parameters of the model are obtained by numerically minimizing the function, (19) 2lnf(x) = Tln(2π)+ln Σ +x Σ 1 x, (20) where T is the number of observations, x is the observed vector and Σ denotes the covariance matrix. The entries of Σ are given as, 10 and γ 0 = σ2 ( ) δ 1 α 2[ ( αβ) F ( δ, δ; +1;β 2) =1 ( ) δ + ( αβ) F ( δ, δ; +1;β 2) ] γ h = σ2 1 α 2[βh +α h + h =1 =0 =0 =1 ( ) δ ( αβ) F ( δ,h+ δ;h+ +1;β 2) h+ ( ) δ ( αβ) F ( δ, δ; +1;β 2) ( ) δ α h ( β) F ( δ, δ; +1;β 2) ], h 1. (21) (22)

6 Garma Time Series Modelling of Forest Area in Malaysia 395 where F(a,b;c;z) = 1+ ab 1!c z+ a(a+1)b(b+1) 2!c(c+1) z 2 + a(a+1)(a+2)b(b+1)(b+2) z !c(c+1)(c+2) (23) The initial start up values for the numerical minimization are the approximate Hannan-Rissanen Algorithm estimates. 4. Modeling of GARMA(1,1;1,δ) Model to the Forest Area Data of Malaysia In this section, we illustrate the modeling of Forest Area of Malaysia using GARMA(1,1;1,δ) model. The data was obtained from the official website of Department of Statistics, Government of Malaysia. The data consisted of yearly observations of forest areas measured in thousand hectars from 1987 to A plot of the time series is shown in Figure 1 and it is clear that it is nonstationary. Fig. 1. Forest Area of Malaysia In order to achieve stationarity, the data set was twice-differenced at lag 1 and mean corrected using ITSM and a plot of this is shown in Fig. 2. Plot of the sample autocorrelation function (ACF) and the sample partial autocorrelation function (PACF) are also shown in Fig. 3. From Figs. 2 and 3, the time series appears to be stationary. Computer programs were written using SPLUS language to model the stationary forestry area of Malaysia using GARMA(1,1;1,δ) model. The estimation of the parameters using Hannan-Rissanen Algorithm, Whittle s Estimation and Maximum Likelihood Estimation was done. The results are shown in Table 1 below. The Hannan-Rissanen Algorithm estimation is obtained for the GARMA(1,1;1,δ) model and the fitted model is ( B)Y t = ( B) Z t, Z t WN(0,13805)

7 396 T. R. Pillai and M. Shitan Fig. 2. Forest Area of Malaysia which was twice differenced at lag 1 and mean corrected Fig. 3. Plot of ACF and PACF of Forest Area of Malaysia Table 1. Estimation Results for parameters of GARMA(1, 1; 1, δ) Method α β δ σ 2 Hannan-Rissanen Algorithm Whittle s Estimation Maximum Likelihood Estimation where, Y t = (1 B)(1 B)(X t 2200). On the other hand, the GARMA(1,1;1,δ) fitted models are, ( B)Y t = ( B) Z t, Z t WN(0,4093), by the Whittle s estimation method and ( B)Y t = ( B) Z t, Z t WN(0,3563) by the maximum likelihood method.

8 Garma Time Series Modelling of Forest Area in Malaysia Conclusion In this paper, the obective of our study was to illustrate the fitting of GARM A(1, 1; 1, δ) model to forest area of Malaysia. The estimation of the parameters was done using Hannan-Rissanen Algorithm, Whittle s Estimation and Maximum Likelihood Estimation. Forecasting can be done using the method given above.wehopethatmoreresearcheswouldconsiderfittinggarma(1,1;1,δ)type models besides the standard ARMA models. The authors hope that the GARMA (1, 1; 1, δ) will receive wider applicability in the future. References 1. accessed on 6 March Martin and Davd, Journal of Ecoforestry, 21, 4 6 (2008). 3. S. J. Gill and G. S. Biging, Journal of Agricultural, Biological and Environmental Statistics, 7(4), (2002). 4. K. Pasanen, Silva Fennica, 32(I), (1998). 5. P. J. Brockwell and R. A. Davis, Introduction to Time Series and Forecasting, 2nd Edition (Springer, New York, 2002). 6. B. Abraham and J. Ledolter, Statistical Methods for Forecasting (John Wiley, New York, 1983). 7. P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods (Springer-Verlag, New York, 1991). 8. M. S. Peiris, Statistical Methods, 5(2), (2003). 9. M. S. Peiris, D. Allen and A. Thavaneswaran, Journal of Applied Statistical Science, 13(3), (2004). 10. R. P. Thulasyammal, S. Mahendran and M. S. Peiris, Journal of Statistics: Advances in Theory and Applications, 2(1), (2009).