First-order planar autoregressive model
This paper establishes the conditions for the existence of a stationary solution to the first-order autoregressive equation on a plane as well as properties of the stationary solution. The first-order autoregressive model on a plane is defined by the equation \[ {X_{i,j}}=a{X_{i-1,j}}+b{X_{i,j-1}...
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| Language: | English |
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2024-08-01
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| Series: | Modern Stochastics: Theory and Applications |
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| Online Access: | https://www.vmsta.org/doi/10.15559/24-VMSTA263 |
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| author | Sergiy Shklyar |
| author_facet | Sergiy Shklyar |
| author_sort | Sergiy Shklyar |
| collection | DOAJ |
| description | This paper establishes the conditions for the existence of a stationary solution to the first-order autoregressive equation on a plane as well as properties of the stationary solution. The first-order autoregressive model on a plane is defined by the equation
\[ {X_{i,j}}=a{X_{i-1,j}}+b{X_{i,j-1}}+c{X_{i-1,j-1}}+{\epsilon _{i,j}}.\]
A stationary solution X to the equation exists if and only if $(1-a-b-c)(1-a+b+c)(1+a-b+c)(1+a+b-c)\gt 0$. The stationary solution X satisfies the causality condition with respect to the white noise ϵ if and only if $1-a-b-c\gt 0$, $1-a+b+c\gt 0$, $1+a-b+c\gt 0$ and $1+a+b-c\gt 0$. A sufficient condition for X to be purely nondeterministic is provided.
An explicit expression for the autocovariance function of X on the axes is provided. With Yule–Walker equations, this facilitates the computation of the autocovariance function everywhere, at all integer points of the plane. In addition, all situations are described where different parameters determine the same autocovariance function of X. |
| format | Article |
| id | doaj-art-60af18926b974aaaa59fd7ff4df1874e |
| institution | Kabale University |
| issn | 2351-6046 2351-6054 |
| language | English |
| publishDate | 2024-08-01 |
| publisher | VTeX |
| record_format | Article |
| series | Modern Stochastics: Theory and Applications |
| spelling | doaj-art-60af18926b974aaaa59fd7ff4df1874e2025-01-10T11:16:09ZengVTeXModern Stochastics: Theory and Applications2351-60462351-60542024-08-011218312110.15559/24-VMSTA263First-order planar autoregressive modelSergiy Shklyar0Taras Shevchenko National University of Kyiv, Kyiv, Ukraine; Scientific Centre for Aerospace Research of the Earth, Kyiv, UkraineThis paper establishes the conditions for the existence of a stationary solution to the first-order autoregressive equation on a plane as well as properties of the stationary solution. The first-order autoregressive model on a plane is defined by the equation \[ {X_{i,j}}=a{X_{i-1,j}}+b{X_{i,j-1}}+c{X_{i-1,j-1}}+{\epsilon _{i,j}}.\] A stationary solution X to the equation exists if and only if $(1-a-b-c)(1-a+b+c)(1+a-b+c)(1+a+b-c)\gt 0$. The stationary solution X satisfies the causality condition with respect to the white noise ϵ if and only if $1-a-b-c\gt 0$, $1-a+b+c\gt 0$, $1+a-b+c\gt 0$ and $1+a+b-c\gt 0$. A sufficient condition for X to be purely nondeterministic is provided. An explicit expression for the autocovariance function of X on the axes is provided. With Yule–Walker equations, this facilitates the computation of the autocovariance function everywhere, at all integer points of the plane. In addition, all situations are described where different parameters determine the same autocovariance function of X.https://www.vmsta.org/doi/10.15559/24-VMSTA263autoregressive modelscausalitydiscrete random fieldspurely nondeterministic random fieldsstationary random fields60G60 |
| spellingShingle | Sergiy Shklyar First-order planar autoregressive model Modern Stochastics: Theory and Applications autoregressive models causality discrete random fields purely nondeterministic random fields stationary random fields 60G60 |
| title | First-order planar autoregressive model |
| title_full | First-order planar autoregressive model |
| title_fullStr | First-order planar autoregressive model |
| title_full_unstemmed | First-order planar autoregressive model |
| title_short | First-order planar autoregressive model |
| title_sort | first order planar autoregressive model |
| topic | autoregressive models causality discrete random fields purely nondeterministic random fields stationary random fields 60G60 |
| url | https://www.vmsta.org/doi/10.15559/24-VMSTA263 |
| work_keys_str_mv | AT sergiyshklyar firstorderplanarautoregressivemodel |