# How to define marked point processes?

Background: In mining engineering, sampling is used to get the rate of concentration of minerals in rocks. Sampling procedure is carried out in the field on the area being explored at the predefined coordinates (x,y) according to designed maps. So there are lots of samples from surface materials i.e., rocks/soils/waters etc.

Questions:
Can we call the sample locations as a point process?
Note that they are not random!

Can we assign each characteristic extracted from each sample (e.g., concentrations, moisture, size, texture etc) as marks?
Is this approach going to be a marked point process?

A point process is a collection of random variables that are positions in some space (like locations on a plane). A marked point process is a point process in which some additional features are measured at each point.

For your situation, the locations of the points are by design rather than random, and so while you could call it a marked point process, the point process part isn't particularly interesting, and probably thinking along the marked point process line isn't going to be particularly useful.

I take it that you are trying to characterize the distribution of the measured features across the region. In this case, I would look at other kinds of spatial analysis.

• The most basic concept of modern geostatistics, kriging (GLS-type prediction), comes from the name Krige, who was a mining engineer. Books like <a href="amazon.com/Statistics-Spatial-Data-Wiley-Probability/dp/…>, although heavy (and not only technically, but in hand, too) would help. – StasK Sep 30 '11 at 1:54
• @StasK - Great addition. The link to the book got messed up. Heavy indeed (928 pgs). – Karl Sep 30 '11 at 2:23
• @StasK - Sorry if my example again made misleading. In this question my focus was on point process and particularly to understand what makes a work with points to be considered as a point process. Marks are important key concepts in these cases. Kriging is only an estimation and does not relate to point process. Since usually and basically Poisson point processes are used which satisfy complete randomness, my question was how about if don't consider randomness for locations? – Developer Sep 30 '11 at 4:27
• @Developer, it is not a point process, then. It's akin to saying, I generated numbers from 1 to 100 uniformly, and now I want to use them as dependent variable in regression. In some rare and weird occasions it will make sense, but not generally. What you have is a spatial design, not a point process. – StasK Sep 30 '11 at 13:42
• @StasK - You can still call it a point process, though it is degenerate. – Karl Sep 30 '11 at 14:11

The aim of such cases is to model the measured characteristic in spatial locations. These type of data are called geostatistical data. So you have to apply the geostatistics theory as the approperiate methods for modeling geostatistical data. Then, the answer of your question is no, this approach could not to be a marked point process.

• As I said to StasK, it is still a point process, though it is degenerate. – Karl Sep 30 '11 at 14:13