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In ArcMap 9.3 I've used Kernel Density to map various incidents, but the resulting shapefile doesn't display any units of measurement. Is there a good, not-to-technical source that would explain in lay terminology the interpretation of the output values in terms of the input cell size and search radius?

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2 Answers 2

Here is the web answer.
Esri webhelp 9.3 How Kernel Density Works
Density Calculations (differences)
Kernel Density

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Links are great, but we encourage you--the respondent--to provide a summary so that your reply can stand alone. –  whuber Sep 6 '11 at 21:19
    
I understand the encouragement but I thought the question was for "good source". Thanks for the great answer - you are a great source. –  Brad Nesom Sep 7 '11 at 3:53
    
You're right Brad, the question was indeed phrased that way (and therefore I have upvoted your answer). But I would still encourage you to flesh out your recommendations, nonetheless :-). –  whuber Sep 7 '11 at 12:52

This is almost a duplicate of How to interpret GRASS v.kernel results?, but it differs slightly in asking for an interpretation in terms of the search radius. Let's talk about that.

A kernel density is a convolution. In nontechnical terms this means that the value of each cell in the input grid is spread around its vicinity. The "kernel" is a function that describes the shape of the spreading. Think of the value as recording the height of sand poured into a box based on the cell. If you were to remove the box, the sand would slump. The kernel says what shape it would acquire; the amount of sand determines how high that shape is. Independently repeat this process for each cell in the grid, allowing the piles of sand to accumulate vertically (without introducing any additional slumping from the overlap).

From this description we can deduce the answers to the two questions posed here:

  1. Depending on the software, the output values give either the total quantity of sand in each cell or--more usually--they give the amount per unit area. (This is what "density" means.) Using output per unit area is better because it does not appreciably change when you change the output cellsize. For instance, if you halve the output cellsize, each cell occupies only one-quarter its former footprint, so typically it's covered by only about one-quarter of the sand. When you express the output as sand per unit area, though, that doesn't change: you get one-quarter of the sand in one-quarter of the original area, whence the ratio is the same.

  2. The "search radius" (an idiosyncratic term adopted by certain GIS vendors; in the literature related quantities are used, known as a kernel "half-width" or a "full width at half maximum"), describes the amount of spreading. Regardless of how this is expressed, if you want to spread the original cell values twice as far, you will end up covering four times as much area. When you're spreading the value of a single cell, the resulting pile will be just one-quarter as high at each point. However, in most cases the spread-out density bears a more complex relationship to the less-spread density, because the piles of "sand"--although individually smaller--receive contributions from cells that are further away. On the whole, the effects balance out. What you see is that greater spreading creates output grids that vary in a smoother manner, whereas less spreading creates output grids that are locally more variable.

These figures illustrate the effects of changing the radius (for a Gaussian kernel) on a sparse input grid having values of 0 or 1.

An image and some of its Gaussian kernel densities

Images Darkness depicts grid values (black = 1, white = 0). All images are 16 by 16.

The same figure shown as 3D plots of grid values

3D plots Height depicts grid values. All plots are on a common scale for comparison. This plotting method shows the original piles of "sand" as cones rather than as boxes.

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My eyes tell me the total volume of "sand" appears to decrease as we move from left to right in the bottom panel. That is partly an artifact of how the software draws the surfaces, but it is also partly real: a great deal of the sand is being spread outside the study area in the last two images and has disappeared from view. This "edge effect" is present in many kernel density maps. –  whuber Jul 2 '13 at 17:57

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