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| LINEAR PREDICTION THEORY OF A HOMOGENEOUSRANDOM FIELD WITH DISCRETE PARAMETERS |
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Citation： |
XU YEJI.LINEAR PREDICTION THEORY OF A HOMOGENEOUSRANDOM FIELD WITH DISCRETE PARAMETERS[J].Chinese Annals of Mathematics B,1981,2(1):33~46 |
| Page view： 857
Net amount： 1227 |
Authors： |
XU YEJI; |
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| Abstract： |
A homogeneons random field with discrete parameters is defined to he a family
of random variables \[\{ x(m,n)\} ,m,n, \pm 1, \pm 2,...,\]，such that Ex(m,n)=0,and
that B(m,n)=E[x(m+m',n+n')x(m',n')] exists and is independent of m' and n'.
At this time, we have
\[B(m,n) = \int_{ - \pi }^\pi {\int_{ - \pi }^\pi {{e^{i(m\lambda + n\mu )}}} } dF(\lambda ,\mu )\]
Generally, linear prediction problems of a homogeneous random field with discrete
parameters are as follows:
Let T and T' are two sets of (m,n), {x(m,n)} have been observed if \[(m,n) \in T\].But {x(m',n')} are unknown quantities if \[(m',n') \in T'\] . We want to predict x(m', n'),
\[(m',n') \in T'\], by the linear combination of the \[\{ x(m,n),(m,n) \in T\} \] and its limit in
terms of square mean, such that its error of square mean is minimum.
In the present paper，linear predictions of three types have Ьeen discussed
\[\begin{array}{l}
{1^{\rm{o}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {T^'} = \{ (m',0), - \infty < m' < \infty \} ;T = \{ (m,n), - \infty < m < \infty ,n \ne 0\} \{2^{\rm{o}}}{T^'} = \{ (m',0),m' > 0\} ;T = \{ (m,n), - \infty < m < \infty ,n \ne 0,(m,0),m < 0\} \{3^{\rm{o}}}{T^'} = \{ (0,0)\} ;T = \{ (m,n), - \infty < m < \infty ,n \le 0\} - \{ (0,0)\}
\end{array}\]
The errors of linear prediction have been obtained respectively as follows: \[\begin{array}{l}
{1^{\rm{o}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {d^2} = \mathop {\lim }\limits_{l \to \infty } \mathop {\inf }\limits_{a_{m,n}^{(l)}} E|x(m,0) - \sum\limits_{m = - l}^l {\sum\limits_{\scriptstylen = - l\hfill\atop
\scriptstylen \ne 0\hfill}^l {a_{m,n}^{(l)}} } x(m,n){|^2}\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = 4{\pi ^2}\int_{ - \pi }^\pi {[\int_{ - \pi }^\pi {\frac{{d\mu }}{{\frac{{dF(\lambda ,\mu )}}{{dF(\lambda ,\pi )d\mu }}}}} } {]^{ - 1}}dF(\lambda ,\pi ).\{2^{\rm{o}}}{\sigma ^2}(1) = \mathop {\lim }\limits_{l \to \infty } \inf {\kern 1pt} {\kern 1pt} {\kern 1pt} E|x(0,0) - \sum\limits_{m = - l}^l {\sum\limits_{\scriptstylen = - l\hfill\atop
\scriptstylen \ne 0\hfill}^l {b_{m,n}^{(l)}} } x(m,n) - \sum\limits_{m = - l}^{ - 1} {b_{m,0}^{(l)}} x(m,0){|^2}\ = 2\pi \exp {\{ \frac{1}{{2\pi }}\int_{ - \pi }^\pi {\log 4{\pi ^2}(\int_{ - \pi }^\pi {[\frac{{dF(\lambda ,\mu )}}{{d\lambda d\mu }}} } ]^{ - 1}}d\mu {)^{ - 1}}d\lambda \} \{3^{\rm{o}}}{{\tilde \sigma }^2} = \mathop {\lim }\limits_{l \to \infty } \mathop {\inf }\limits_{c_{m,n}^{(l)}} E|x(0,0) - \sum\limits_{m = - l}^l {\sum\limits_{n = - l}^l {c_{m,n}^{(l)}} } x(m,n) - \sum\limits_{m = - l}^l {c_{m,0}^{(l)}} x(m,n){|^2}
\end{array}\] |
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