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I am always on the lookout for useful methods to sample or partition study areas (usually in the form of raster datasets) into smaller units. Recently, I read an ESRI blog post about a new tool for creating sampling hexagons. Although the hexagons are an eye catcher, my first thought is that they are more complicated and contain more vertices than, for example, a fishnet grid which could accomplish the same goals. What are the benefits of working with hexagonal grids over rectangular grids for study area sampling or partitioning raster datasets?

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The idea with hexagons is to reduce sampling bias from edge effects of the grid shape, which is related to high perimeter:area ratios. A circle is the lowest ratio, but cannot form a continuous grid, and hexagons are the closest shape to a circle that can still form a grid.
Also, if you are working over a larger area, a square grid will suffer more from distortion due to curvature than shapes like hexagons.

There are a number of tools and extensions for creating and using hex grids for ecological/landscape analysis, Patch analyst (Rempel et al., 2003) being a good example, that also provides a large volume of landscape metric measurement capacity. The former Hawth's Tools, now redesigned as the Geospatial Modeling Environment has a wide array of tools that were developed to fill in gaps in arcgis functionality, including repeating grids. A number of third-party extensions have been made for this sort of thing, usually by the researchers who need them, so they frequently don't have the resources to rebuild their products after every new GIS version is released, so it often seems like there is nothing available

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To put it succinctly, hex grids minimize edge artifacts, double the level of detail of neighbor effects, and they look really cool :) - also note that QGIS has a great plugin (MMQGIS) that works swimmingly for hex grid creation in the current version of the platform. –  Bill Morris Jan 8 at 22:02
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One of the benefits, that I've seen when doing wildlife or habitat modelling especially, is that hexagons allow patterns in the data (ex, edge of a field or any other patch) to be seen more easily than what squares would of offered.

Think of a soccer ball too, though not always hexagons, those geometric shapes fit to a curved surface quite nicely.

In your image, try creating smaller hexagons and they would get close to the actual shape of the polygon. Then try computing a rectangular/square grid over the same region with a similar width or height and you can see the difference.

looping over hexagon bin sizes

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When you say "you can see the difference", I suspect you may be able to quantify that difference quite easily too by using Select Layer By Location of the polygon over the hexagon and fishnet polygons to keep only whole hexagons/rectangles and then Summary Statistics on area to see how close each is to the known area of the polygon. –  PolyGeo Jan 8 at 21:52
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The hexagon is the most complex regular polygon that can fill a plane (without gaps or overlap).

I can see two advantages:

  • It is closer to a circle than the square in terms of shape index, so you suffer less from orientation bias.

  • The "length of contact" is the same on each side (with a square, the neighbours include the four squares at the corners). EDIT: As mentioned by @Jason, the distance between centroids is also the same in all six directions.

There are also two drawbacks :

  • there are six adjacent neighbours instead of eight with the square (if you account for the corners)

  • most importantly, you cannot subdivide hexagons to upscale or downscale your sampling with hexagon (with square, it is easy o aggregate or split to new squares).

In your case, there is another drawback because you want to partition a raster. Indeed, raster cells are square-based as is the raster extent. So, if you try to partition a raster using hexagon, it will not be possible to avoid partially included pixels, and any clipped raster based on hexagon will result in a proportion of NoData pixels.

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"It is closer to a circle than the square" -- as a result, and more importantly, the center point of each neighboring shape is equidistant, whereas with squares above/below/right/left neighbors' center points are N units away and diagonal neighbors are sqrt(2)*N units away. –  Jason Scheirer Jan 8 at 22:32
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A key disadvantage of grid squares is that the sample rate is substantially lower along the diagonal vectors to those of the four sides (Jasons point above).

If you have some regular linear pattern to your data the orientation of the grid affects the effective sample rate of each context.

For example if you have a series of ridges and valleys, orienting the grid along these might only sample the valley or the tops and thus the type of vegetation or fauna to be found. Some other angle relative to the valleys would give a shifting sample rate between high and low over the region. A good example of such a problematic vector in an aquatic might be tidal range, sea depth, undersea ridges an so forth.

Obviously, the effect can be mitigated or exacerbated by choice of sampling resolution, but ideally the sample rate to variance ratio should be stable over space. Hexagons, being closer to a circle, are less likely to accidentally cause such variable sample rate bias.

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