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I am having a hard time understanding what is the difference between kriging and gaussian processes. I mean wiki says they are the same but their formulas for prediction are so different.

I am a bit confused why they are called similar. Clarifications?

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There are some subtle differences between ordinary and simple kriging, maybe that confuses you. GP regression in the way it is usually presented is analogous to simple kriging. In the Gaussian process Wikipedia entry it says that the article refers explicitly to a "zero-meaned distribution"; that is the same assumption found in simple kriging.

Also generally speaking kriging is usually performed in a 2 or 3 dimensional spaces, (eg. pollutant concentration along some given area) while most GPR toy examples are one dimensional (eg. $CO_2$ concentration in the atmosphere against time).

Ultimately kriging/GPR is an interpolation technique and most (not all) of the difference among the variants of it lays on the assumption about the mean trend $\mu(X)$ (or E[$X_t$] if you like this notation better).

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    $\begingroup$ This isn't really true. Often you see in the GP literature that without loss of generality they use zero mean assumptions but then add the structure of the mean into the kernel (for example with the addition of a linear kernel etc). GPs are certainly not used in only one dimension as can be seen in pretty much any paper on the subject. The 1D scenario is used only for intuition purposes in introductory texts. In fact in most 1D cases you can encode the GP into a Kalman filter which is computationally more efficient. $\endgroup$ – j__ Nov 9 '15 at 8:15
  • $\begingroup$ @j__ For the first part of your comment: I agree partially but unfortunately it is mostly a terminology issue that people tend to abuse it at times. I present the canonical distinction I have seen in books. For the second part: Allow me to disagree. I have seen multiple applications of GPR 1D cases (eg. in FX rates modelling, in Phylogenetics, and in ODE solutions - these just doing a quick Google search). i appreciate your comment that generally a statistical framework (cont.) $\endgroup$ – usεr11852 Nov 9 '15 at 9:31
  • $\begingroup$ comes into its own when applied in multivariate settings but that does not discredit 1D applications. $\endgroup$ – usεr11852 Nov 9 '15 at 9:33
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    $\begingroup$ well I do see where you are coming from. I guess I would say it is more common for GPs to act in general N dimensional spaces as opposed to being restricted to 2/3 which is the case with Kriging. A special case is the 1D setting. That may be a good middle ground we can agree on ;) $\endgroup$ – j__ Nov 9 '15 at 9:40
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    $\begingroup$ Yes, 1-D cases tend to be unique. (Awful pun) $\endgroup$ – usεr11852 Nov 9 '15 at 9:58
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GPs are known as kriging in geostatistics. To learn about the history of Gaussian Processes watch this video http://youtu.be/4r463NLq0jU?t=26s

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