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When I read the paper “Multiscale methods for data on graphs and irregular multidimensional situations”, by Maarten Jansen, Guy P. Nason and B. W. Silverman, I find the term “irregularly spaced spatial data”. When I google it, I find that there is no clear definition in spite of its “wide” usage (in specific domains).

Can someone explain this concept to me?

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  • $\begingroup$ E.g. a satellite image or a climate model usually yields a regular raster of observations. However, weather stations are usually irregular spaced. $\endgroup$ – Anony-Mousse Sep 15 '15 at 7:54
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A lot of techniques assume that data is sampled at regularly-spaced intervals. You might count how much litter is near each mile marker on the highway, or sample points in a forest on a regularly spaced grid (100, 200, 300, ... meters north and 100, 200, 300 meters east of some landmark). This also occurs in time--my EEG machine records a data point every millisecond. We call the interval between adjacent samples the sampling period.

However, a lot of data is not or cannot be sampled with a fixed sampling period. Perhaps the terrain doesn't allow us to place weather stations exactly 50 miles apart. We often study peoples' heights and weights, but these are only opportunistically measured at doctors' appointments (which are often not exactly 1 year apart). These data are irregularly sampled.

The paper you linked describes methods for dealing with the latter kind of data, where the sampling period is not constant. One possible approach is to interpolate your data onto a grid and then use a techniques intended for gridded data. The paper argues that while this works in 1 dimension, it is less satisfactory in multiple dimensions and their lifting-based approach works better.

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Good answers by Matt (+1) and others. Just to have a picture to drive the message (visually) home. In the following figure assuming that the squares represent sampling points the grey boxes follow an obvious regularly spaced design; the red box are just random samples that are irregularly spaced.

enter image description here

Both designs have their pros and cons. Do not dismiss the irregular design as "worse". For example, certain adaptive sampling designs can be extremely helpful for density estimation but highly irregular strictly speaking. That is because you mostly care for regions of high volatility. Numerical integration schemes are a standard example. On the one hand, the trapezoid rule (and in general all the Newton–Cotes formulas) is based an equally spaced sampling technique. On the other hand Monte Carlo integration methods might a strongly irregular sample that sometimes can deviate a lot from being uniform and equally spaced (eg. importance sampling).

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    $\begingroup$ +1 I myself would add to your list the observation that methods in global optimization tend to begin from "design points" which tend to be drawn from (variations on) latin hypercube designs. The results are not a regular grid, but are considered to represent the space "well enough". $\endgroup$ – Sycorax Sep 14 '15 at 19:51
  • $\begingroup$ Yeah, valid point; this is another example but I think I would deviate a lot from the original question. One example of irregular sampling compared to something that is trivially known to be regular (trapezoid rule) should be enough. +1 for the comment though. $\endgroup$ – usεr11852 Sep 14 '15 at 20:18
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This usually means that there is no clear underlying structure of the position of the points. I.e. it is not a rectangular grid or anything that can be represented compactly which has a clear structure.

Imagine that you have weather stations around a country and you are monitoring temperature. These weather stations are most likely no on any specifically defined grid. They are irregularly spaced and thus if one wants to do any spatial inference, one needs to create some spatial graph/mesh, most often made of triangles. Then one can do inference and interpolations based on the values at the known weather stations.

This is highly dependent on which mesh/graph you select, so there are different techniques to generate them.

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It's a british way of saying that your data does not come evenly spaced. Say, you measure the temperature on the road, and obtain the observation every 1 mile apart. This would be regularly spaced data. As opposed to taking measurements at every gas station, which would not be equally spaced, of course.

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    $\begingroup$ I'm not sure what's especially "british" [sic] about this phrase. I'm an American speaker and perfectly understood what the authors meant. $\endgroup$ – Sycorax Sep 14 '15 at 15:52
  • $\begingroup$ "British" way is to put simple thoughts in complicated and fancy sentences. $\endgroup$ – Aksakal Sep 14 '15 at 19:41
  • $\begingroup$ I beg to differ. $\endgroup$ – usεr11852 Sep 14 '15 at 19:58
  • $\begingroup$ @usεr11852, qualitative analysis of dialogues in Downtown Abbey vs. Friends is my proof. $\endgroup$ – Aksakal Sep 14 '15 at 20:00
  • $\begingroup$ You realise that Downtown Abbey takes part in the 1910's-1920's, so any linguistic analysis would have to correct for chronological bias right? $\endgroup$ – usεr11852 Sep 14 '15 at 20:07

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