Thanks Henning Lebrenz , it's a good job, thumbs up!

another quick question Henning Lebrenz !

as I see you're experienced in Kriging, is it correct to call Kriging as a statistical learning method? if yes, could you please explain ?

Thanks

This method is the most important and widespread interpolation method based on models and statistical relationships.The raster layer produced by this method provides a very accurate surface (this method produces the best and most accurate output for mountainous areas).Kriging's method, in contrast to the IDW method, which is a local interpolation method, is a global method. In this method, all observations of the area are used.

Hossein

Your claims are all wrong. The only way you could determine "accuracy" is if you can compare "true values" with "interpolated values". At the locations where you know the true values there is no need to interpolate. The usual forms of the kriging estimator and the kriging equation do not take elevation into account so any claim that it is best for mountainous areas has no theoretical justification.

You need to consult a good book on geostatistics, whatever you are using as a reference is wrong.

Sajad

You seem to be assuming that there is an assumption about a probability distribution in the derivation of the kriging equations. That is completely wrong, there is no assumption about a distribution. However there are a number of papers about spatio temporal kriging which requires the use of a spatio temporal variogram or covariance function.

Thanks Donald Myers , it was a helpful answer. I usually see a simple normal distribution and i thought that it is a certain part of Kriging method.

thanks a lot

Donald Myers ,

What are your thoughts about using 'Regression Kriging' with elevation (or DEM & its derivatives viz. slope, aspect etc.) as a co-variate for interpolation in mountainous regions?

Vaibhav

If you look at the literature on interpolation of precipitation you will find examples of using "regression kriging" incorporating elevation. You will also find examples of treating elevation as a co-variable in cokriging. The question of which is better seems to depend on the particular data set (and perhaps on other unobserved variables). If you look at the book "Geostatistics" by J.-P. Chiles and P. Delfiner, page 322 .... you will see a discussion of incorporating derivatives into cokriging.

Also if you look at the literature on Radial Basis Functions you will see examples of the use of derivatives

Donald Myers

Thanks for sharing your views.

Thanks a lot

I think you have to revisit some basic presumptions of kriging. First of all, when you say "kriging" you actually refer to a family of methods, not one method. For example, ordinary kriging assumes that within the domain of interest you have CONSTANT (but unknown) mean and CONSTANT variance. So, it is clear that by nature, ordinary kriging cannot be used to model temporally varying distributions. "temporally varying distribution" is just another way to say that you would like to krig the data where variance is non stationary, which opposes basic presumptions of ordinary kriging. Non-statinarity is a hell of a problem. For example, variance can be evolving but its evolutionary operator might be statinary. In this case, probably something can be done. But, in cases where evolutionary operator is also non-statinary, and its evolutionary operator too, you fall into absolute non-stationarity and phylosophical regression, and the scenario is hopeless.

There are two most common approaches to handle non-stationarities in kriging, in general: a) nonstatinarity of the mean is handled by regression or universal kriging, b) non-statinarity of the variance is handled by moving window approach (Hass, 1990). The latter relies on the assumption that there is always domain small enough for which the stationarity assumption holds, so you subsample the data around estimation location.

I see that many answers you received mix questions of the stationarity, with the question of the dimensionality and nature of the kriging kernel. You can krig in space and time, and, in principle, in arbitrary number of dimensions, BUT assumption of the statinary variance has to hold if you apply ordinary kriging machinery. We recently published a paper on hyperdimensional kernels, which provides just one way to handle anisotropy, or space-time problems:

Towards Hyper-Dimensional Variography Using the Product-Sum Covariance Model (https://www.mdpi.com/2073-4433/10/3/148/htm)

You can use it to krig having different variances in all spatial directions + time.

However, it is NOT a way to handle non-stationary variance, which also applies to non-stationarity in the temporal domain.

If you would like to krig in spatial domain only, but having different variance along vertical axis (along altitude, somebody mentioned) you can take a look into another paper where we treated this problem explicitly, and actually provided comparison between couple of ways to do it:

Article Elliptic Cylinder Airborne Sampling and Geostatistical Mass ...

Thanks alot

What I and Jovan are trying to tell you is that information about the multivariate probability distribution is not incorporated in the derivation of the kriging equations (whether they be for Simple kriging, Ordinary kriging or Universal kriging nor even for Cokriging). Nor is that information used in estimating/modeling the variogram. If you look at the paper of Matheron's (Advances in Applied Probability,, 1973) you will that even when using a generalized covariance (a variogram is a zero order generalized covariance) there are no assumptions about the probability distribution other than the form of the stationarity but that pertains to the existence of the variogram, i.e that it is only a function of the separation vector. However there is a moderate amount in the literature about spatial temporal kriging but pertains to the variogram or covariance function