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I know what a Gaussian field is. However, I am not quite sure what is meant by stationary. I have seen this stationary thing at lots of places like stationary autoregressive processes etc but don't actually know what is meant by stationary.

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For time series stationarity means that the joint distribution of variables in the sequence depends only on their separation in time and not on the actual time. This implies that the mean and variance are constant and the covariance between the variables at two time points depends only on the difference in time between the points. With spatial data it would mean that the distribution of a set of points on a grid only depend on how they are separated. So if you shift a set of points k units in the x direction and m units in the y direction their joint distribution will not change.

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    $\begingroup$ +1--but you don't need to restrict points to a grid. In many spatial applications you don't have this luxury, so it's essential that the theory and concepts also apply to "irregular point patterns." $\endgroup$
    – whuber
    Jul 3, 2012 at 17:18
  • $\begingroup$ @whuber Sure. The idea is that for any configuration of points a shift of all the points by a fixed vector would not chnage their joint distribution. $\endgroup$ Jul 3, 2012 at 17:22
  • $\begingroup$ This answer is actually a good short version. It might help to look at the simple definitions of stationary processes. Random fields are a generalization of stochastic processes, and the idea of stationary is analagous between the two. You can find these definitions in most first year graduate probability books. $\endgroup$
    – Fraijo
    Jul 3, 2012 at 19:35
  • $\begingroup$ I think you can consider random fileds as stochastic processes with a spatial index rather than a 1 dimension sequence of integers or time points. $\endgroup$ Jul 3, 2012 at 19:47
  • $\begingroup$ So strictly speaking a stocahstic process is a random field with a single real-valued "time" parameter, but that really gets away from the point of the question. My only point was that if you want to ignore the measure theory/ differential geometry/ functional analysis definitions of stationary random fields then you can just view them as stationary stochastic processes. It's eaiser to understand the latter. $\endgroup$
    – Fraijo
    Jul 3, 2012 at 19:58

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