Some Aspects of the Spatial Unilateral Autoregressive Moving Average Model for Regular Grid Data
Spatial statistics has received much attention in the last three decades and has covered various disciplines. It involves methods which take into account the locational information for exploring and modelling the data. Many models have been considered for spatial processes and these include the S...
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my-upm-ir.652013-05-27T06:45:25Z Some Aspects of the Spatial Unilateral Autoregressive Moving Average Model for Regular Grid Data 2005-02 Awang, Norhashidah Spatial statistics has received much attention in the last three decades and has covered various disciplines. It involves methods which take into account the locational information for exploring and modelling the data. Many models have been considered for spatial processes and these include the Simultaneous Autoregressive model, the Conditional Autoregressive model and the Moving Average model. However, most researchers focused only on first-order models. In this thesis, a second-order spatial unilateral Autoregressive Moving Average (ARMA) model, denoted as ARMA(2,1;2,1) model, is introduced and some properties of this model are studied. This model is a special case of the spatial unilateral models which is believed to be useful in describing and modelling spatial correlations in the data. It is also important in the field of digital filtering and systems theory and for data whenever there is a natural ordering to the sites. Some explicit stationarity conditions for this model are established and some numerical computer simulations are conducted to verify the results. The general iv explicit correlation structure for this model over the fourth quadrant is obtained which is then specialised to AR(2,1), MA(2,1) and the second-order separable models. The results from simulation studies show that the theoretical correlations are in good agreement with the empirical correlations. A procedure using the maximum likelihood (ML) method is provided to estimate the parameters of the AR(2,1) model. This procedure is then extended to the case of spatial AR model of any order. For the AR(2,1) model, in terms of the absolute bias and the RMSE value, the results from simulation studies show that this estimator outperforms the other estimators, namely the Yule-Walker estimator, the ‘unbiased’ Yule-Walker estimator and the conditional Least Squares estimator. The ML procedure is then demonstrated by fitting the AR(1,1) and AR(2,1) models to two sets of data. Since the AR(2,1) model has the second-order terms which are only in one direction, two types of data orientation are taken into consideration. The results show that there is a preferred orientation of these data sets and the AR(2,1) model gives better fit. Finally, some directions for further research are given. In this research, inter alia, the field of spatial modelling has been advanced by establishing the explicit stationarity conditions for the ARMA(2,1;2,1) model, by deriving the explicit correlation structure over the fourth lag quadrant for ARMA(2,1;2,1) model and its special cases and by providing a modified practical procedure to estimate the parameters of the spatial unilateral AR model. Computational grids (Computer systems) - Case studies 2005-02 Thesis http://psasir.upm.edu.my/id/eprint/65/ http://psasir.upm.edu.my/id/eprint/65/1/1000548935_t_FS_2005_8.pdf application/pdf en public phd doctoral Universiti Putra Malaysia Computational grids (Computer systems) - Case studies Faculty of Science English |
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Universiti Putra Malaysia |
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PSAS Institutional Repository |
| language |
English English |
| topic |
Computational grids (Computer systems) - Case studies |
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Computational grids (Computer systems) - Case studies Awang, Norhashidah Some Aspects of the Spatial Unilateral Autoregressive Moving Average Model for Regular Grid Data |
| description |
Spatial statistics has received much attention in the last three decades and has
covered various disciplines. It involves methods which take into account the
locational information for exploring and modelling the data. Many models have
been considered for spatial processes and these include the Simultaneous
Autoregressive model, the Conditional Autoregressive model and the Moving
Average model. However, most researchers focused only on first-order models. In
this thesis, a second-order spatial unilateral Autoregressive Moving Average
(ARMA) model, denoted as ARMA(2,1;2,1) model, is introduced and some
properties of this model are studied. This model is a special case of the spatial
unilateral models which is believed to be useful in describing and modelling spatial
correlations in the data. It is also important in the field of digital filtering and
systems theory and for data whenever there is a natural ordering to the sites.
Some explicit stationarity conditions for this model are established and some
numerical computer simulations are conducted to verify the results. The general
iv
explicit correlation structure for this model over the fourth quadrant is obtained
which is then specialised to AR(2,1), MA(2,1) and the second-order separable
models. The results from simulation studies show that the theoretical correlations are
in good agreement with the empirical correlations. A procedure using the maximum
likelihood (ML) method is provided to estimate the parameters of the AR(2,1)
model. This procedure is then extended to the case of spatial AR model of any order.
For the AR(2,1) model, in terms of the absolute bias and the RMSE value, the
results from simulation studies show that this estimator outperforms the other
estimators, namely the Yule-Walker estimator, the ‘unbiased’ Yule-Walker
estimator and the conditional Least Squares estimator. The ML procedure is then
demonstrated by fitting the AR(1,1) and AR(2,1) models to two sets of data. Since
the AR(2,1) model has the second-order terms which are only in one direction, two
types of data orientation are taken into consideration. The results show that there is a
preferred orientation of these data sets and the AR(2,1) model gives better fit.
Finally, some directions for further research are given.
In this research, inter alia, the field of spatial modelling has been advanced by
establishing the explicit stationarity conditions for the ARMA(2,1;2,1) model, by
deriving the explicit correlation structure over the fourth lag quadrant for
ARMA(2,1;2,1) model and its special cases and by providing a modified practical
procedure to estimate the parameters of the spatial unilateral AR model. |
| format |
Thesis |
| qualification_name |
Doctor of Philosophy (PhD.) |
| qualification_level |
Doctorate |
| author |
Awang, Norhashidah |
| author_facet |
Awang, Norhashidah |
| author_sort |
Awang, Norhashidah |
| title |
Some Aspects of the Spatial Unilateral Autoregressive Moving Average Model for Regular Grid Data |
| title_short |
Some Aspects of the Spatial Unilateral Autoregressive Moving Average Model for Regular Grid Data |
| title_full |
Some Aspects of the Spatial Unilateral Autoregressive Moving Average Model for Regular Grid Data |
| title_fullStr |
Some Aspects of the Spatial Unilateral Autoregressive Moving Average Model for Regular Grid Data |
| title_full_unstemmed |
Some Aspects of the Spatial Unilateral Autoregressive Moving Average Model for Regular Grid Data |
| title_sort |
some aspects of the spatial unilateral autoregressive moving average model for regular grid data |
| granting_institution |
Universiti Putra Malaysia |
| granting_department |
Faculty of Science |
| publishDate |
2005 |
| url |
http://psasir.upm.edu.my/id/eprint/65/1/1000548935_t_FS_2005_8.pdf |
| _version_ |
1747810159224160256 |