(Cressie, 1991, p. 58) wrote: "When the random process are Gaussian, second-order stationary and strong stationary coincide, because a Gaussian process is characterized by its mean and its covariance function".
I have to assume the insect sampling data as a Gaussian process, but the strong stationary can't be satisfied? Does someone have experience with this?
I would recommend you to go through this: Dudley, R. M. (2010). Sample Functions of the Gaussian Process. It's true that a Gaussian process can be completely defined by second order stationarity, but only if the process is assumed to have mean zero. Maintaining a zero mean over time series for insect response is highly unlikely, moreover the covariance function is dependent on the distance matrix. Autocorrelation must be satisfied according to the habitat geometry (linear e.g. rivers etc.). Therefore the simplistic way of writing the covariance function would be:
Cx(t1, t2) = Cx(t1 - t2, 0) = Cx(T); T= t1 - t2
Hi Mr. Avit Bhowmik,
Thank you for your answer. I have understood this term, your suggestion on autocorelation is also useful!
Thanks,
Viet
With population data (I assume you are talking about observational data), as mentioned, you need to consider the autocorrelation in the data. Increasing autocorrelation increases the time it takes for a process to converge to its stationary distribution (see Fowler & Ruokolainen 2013 Plos ONE).
Thank Mr Lasse Ruokolainen,
The assumption of stationary is always important, but the real data always depends on the characteristics of autocorrelation.
Thanks also for your reference!
Viet
In many cases non-stationarity pertains to (1) the mean and/or (2) the covariance function (see the equation in Bhowmik). In both of these the non-stationarity makes it more difficult to model the covariance function using only data. (1) is easier to deal with and is often encountered. Very commonly the mean is assumed to be a polynomial function of the spatial (or temporal) coordinates and one begins by fitting a trend surface, i.e. a regression surface to the data. Then use the residual data to model the covariance function. This is not the theoretical best solution but is the most likely practical solution. If you are wanting to use kriging then you can use universal kriging but with the variogram/covariance function fitted to the residuals. Note that kriging does not require a Gaussian distribution. If the covariance function is non-stationary (and just because the mean is not stationary) then the problem is much more difficult if the objective is to use kriging.
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