About Kernel Density Analysis

Home Range · background and theory · by Jeff Jenness

This page covers the ideas shared by the three Kernel Density tools — Kernel Density Enhanced, KD to Proportion Surface, and KD Probability Contours — so each tool's page can concentrate on its own controls. The teaching sequence is adapted from the wildlife GIS lab Jeff has taught for years; the About LoCoH page compares this method with its local-hull alternative.

Videos and labs Home-range estimation — kernel and LoCoH methods — is one of the wildlife modules of Jeff's GIS training course, with video lectures and hands-on lab exercises built around these very tools.

Kernel density, plainly

Kernel density surfaces are definitely more abstract than MCPs, but they are also fairly intuitive once you understand them. In truth, one of the most confusing things about the entire concept is the name: the word “kernel” actually refers to a mathematical equation that defines a 3-dimensional bell-shaped curve. Like many things in GIS, this concept could probably have been given a more intuitive name.

Kernel density analysis takes vector input data (points), performs a raster analysis, and produces a raster output representing the density surface. A proper understanding should start with a simpler type of analysis — point density — before moving on to the kernel version.

Point density: similar, but simpler

The goal of any density analysis is to describe the landscape in terms of how densely points are distributed across it. For example, we may be interested in how many water sources there are, per square mile, in the desert of southwestern Arizona — places with a high density of water sources might be more likely to support various vertebrate and invertebrate populations than places with low water source density.

Suppose we know of one spring on the landscape, at the blue dot below, and we generate a point density surface around it:

Aerial imagery of desert terrain with a single blue dot marking a spring — One water source.

The simple point density surface looks like the circle below, which happens to enclose exactly 1 square mile. Any place inside that circle has a density value of 1 water source per square mile; any place outside has a density of 0:

The same terrain with a large blue circle of exactly one square mile centered on the spring — The 1-square-mile density disk: 1 point per square mile inside, 0 outside.

It helps to view this in 3-D, where the density region rises like a disk lying on the landscape — a perspective that becomes especially useful when disks overlap, and even more so when comparing simple point density to kernel density:

A three-dimensional perspective of the density disk rising like a flat cylinder from the terrain — The same disk in 3-D perspective.

An important point: the density value inside the disk is based on the size of the disk — the “search radius,” one of the tool's parameters. A neighborhood twice as large (2 square miles) holds the same one point, so the density inside drops to 0.5 points per square mile:

A wider, flatter three-dimensional disk covering two square miles with half the height — Twice the disk, half the density.

A neighborhood half as large (½ square mile) gives 2 points per square mile — a smaller, taller disk:

A narrower, taller three-dimensional disk covering half a square mile with double the height — Half the disk, twice the density.

Note: a value of 2 water sources per square mile may seem odd given that there is only one water source on the landscape, but that is the nature of density analysis. It does not tell you how many points there are; it is a ratio of points per area, scaled to your area unit.

So what is the water source density at some random spot on the landscape? It depends on two factors: how many water sources are within the search radius of that spot, and the search radius you used to generate the surface. In the illustration below, point A is inside the disks at all three neighborhood sizes; B is outside the ½-square-mile disk but inside the others; C is inside only the 2-square-mile disk; D is outside all three:

The spring with three concentric density disks of half, one and two square miles, and four labeled test points A through D at increasing distances — Four test locations against the three disk sizes.

Density (Water Points per Square Mile) at 4 Sample Locations on the Landscape
Test Location	Circle Size Used to Generate Density Surface
Test Location	½ square mile	1 square mile	2 square miles
Point A	2	1	0.5
Point B	0	1	0.5
Point C	0	0	0.5
Point D	0	0	0

Please make sure you understand the numbers in this table, and how the density value at any spot on the landscape changes when a different neighborhood size generates the surface.

A clarification of the method. At this point you may be thinking that point density is calculated by drawing circles around the water locations — the output certainly looks like that. But this is a raster operation, computed cell-by-cell: the software steps through every cell of a blank output raster and counts how many water sources lie within the search radius of that cell. The search radius is applied to every raster cell, not to the water sources! One source in range gives the cell a value of 1 (scaled to the area unit); two give it 2; and so on. (Happily, the two ways of picturing this — disks stamped around the sources, or counts gathered around each cell — give exactly the same surface, because a cell is within the radius of a source precisely when the source is within the radius of the cell. Picture whichever one helps.)

This causes an interesting thing to happen when points sit close enough to produce overlapping disks. Suppose we had four water sources:

Aerial terrain with four blue points, three clustered near each other and one alone to the right — Four water sources, three of them close together.

With a 1-square-mile circle, three of the disks overlap — and much like a Venn diagram, some ground is covered by one disk, some by two, and the dark region in the middle by all three. The density anywhere equals the number of overlapping disks times the per-disk density:

The four points with their one-square-mile disks, the three left disks overlapping in progressively darker blue where two and three disks stack — Overlapping disks stack, Venn-diagram style.

A three-dimensional view of the overlapping disks stacking into stepped towers where they coincide — The stacked disks, viewed at an angle.

An important point: because of these stacked disks, point density values tend to be constant over large areas and then change abruptly at sharp edges. You are either inside the radius of a point or you are not — there is no adjustment for how far from a point you are.

Kernel density

Kernel density is very similar to simple point density. The difference is the shape of the density region around each point. Simple point density produces a disk, where the entire neighborhood shares one density value:

The point-density shape: a flat disk over the point.

Kernel density instead produces a bell-shaped curve around each point:

The kernel shape: a smooth bell over the point, reaching zero at the search radius.

The word “kernel” is the name given to the mathematical function that produces this bell shape. Its effect is to weight the density at any location by how close that location is to an actual point: the closer you are to a water source, the higher the density. As with point density you specify the neighborhood by a search radius — in kernel density, the distance at which the bell drops to zero. You will see this size called the search radius, the kernel size, or the bandwidth.

Important: density values are interpreted exactly as in simple point density — points per your area unit, a ratio rather than a count. Also important: the volume under the curve stays the same regardless of the search radius, so a larger radius produces a flatter overall curve — just as the larger disk had the lower density:

A tall narrow three-dimensional bell surface over the spring at a half-square-mile search radius — Search radius equivalent to a ½-square-mile circle: tall and tight.

A broader, lower bell surface at a one-square-mile search radius — 1 square mile: broader and flatter…

A still wider and flatter bell surface at a two-square-mile search radius — …and 2 square miles: flatter still. All three bells contain the same volume.

A nice feature of kernel density surfaces is that the value changes with proximity to a point — areas closer to a water source weigh more than areas farther away, which is more reasonable for most environmental phenomena. Animals (like people) usually care more about things when they are closer.

A bell-shaped kernel surface with sample locations at different distances from the peak showing smoothly decreasing density values — Density falls smoothly with distance — no cliff edges.

Kernel density with multiple points

With multiple points, the bell curves stack just as the disks did — but smoothly. Using the same four water sources, here are the surfaces at neighborhood sizes of 0.5, 1, and 2 square miles:

Four sharp separate bell peaks over the four points at the half-square-mile radius — 0.5 square miles: four distinct peaks.

The bells beginning to merge over the three clustered points at the one-square-mile radius — 1 square mile: the cluster's bells begin to merge.

A single broad mound over the cluster, higher than the lone point's bell, at the two-square-mile radius — 2 square miles: the cluster merges into one mound.

Notice that with the 2-square-mile circle, the density close to the cluster of three points is higher than is possible anywhere near the single point on the right:

Stacked bells: the cluster outweighs any single point.

Notice also how kernel density values change gradually over the landscape rather than breaking sharply as simple point density did. The neighborhood size again plays the critical role: larger circles smooth the data into generalized density regions, while smaller circles reveal detail over shorter distances:

A contoured kernel density map at the smallest bandwidth showing many small detailed density pockets — Small bandwidth: fine detail.

The same map at a medium bandwidth with smoother, more connected density regions — Medium bandwidth: smoother regions.

The same map at the largest bandwidth with broad generalized density zones — Large bandwidth: broad generalization.

How do you know what kernel size to use?

This, unfortunately, can be a very difficult question to answer, and can lead to some fairly sophisticated statistical analysis (Silverman 1986 is the classic reference). Some people follow a rough rule of thumb based on contour lines like those above: try progressively larger kernel sizes until the outermost contours join into a single polygon containing all the locations. That method may or may not be appropriate for your particular project. If you specify nothing, the tool sets a default for you — the same default Esri's tool uses — described next.

Sidebar: the default bandwidth (Silverman's rule of thumb)

The default has the appearance of being reasonable and defensible — but remember it is only a best guess. The software knows nothing about your dataset or what the points represent; it knows only their spatial distribution. Gather a second set of points on the same animal and you will almost certainly get a different default. Treat it as a starting value, and don't be shy about specifying your own.

That said, the default is sometimes useful and reasonable when you have nothing better. The method — adapted from Silverman's rule of thumb (Silverman 1986, pp. 44–48, which presents the one-dimensional original) — is based on the distances from each point to the centroid of all points. From that population of distances come two statistics: the standard distance,

SD = \sqrt{\frac{\sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}}{n} + \frac{\sum_{i = 1}^{n} {(y_{i} - \bar{y})}^{2}}{n}}

and the median distance D_m — simply the median of the distances to the centroid. The default bandwidth is then:

Default bandwidth = 0.9 \times \min (SD, \sqrt{\frac{1}{\ln 2}} \times D_{m}) \times n^{- 1 / 5}

Conceptually: take the smaller of the standard distance and the scaled median distance (√(1/ln 2) ≈ 1.20), then shrink it by a factor of 0.9 and by n to the −1/5 power — so the bandwidth shrinks as the number of points grows. When a population field weights the points, the statistics use their weighted versions. Kernel Density Enhanced implements this rule exactly — matching the Esri default — including the weighted variants, and computes it correctly for geodesic analyses of projected data.

From density surface to home-range polygon

Once you have a density surface, the next step is to decide which part of it represents the animal's home range. The guiding assumption: areas with higher location densities are more important to the animal, on the logic that the animal chooses to spend the most time in the areas most important to it. Suppose we have a GPS-collared animal in the Secret Canyon area north of Sedona:

A topographic map of the Secret Canyon area north of Sedona with a scatter of GPS locations — GPS locations in Secret Canyon.

From these points we generate a kernel density surface (500 m kernel, densities in points per square kilometer, ranging 0 to 189):

The kernel density surface over the Secret Canyon locations, shaded from pale low densities to dark high-density cores — The kernel density surface.

Now, what part of this map is the home range? We could take everything with density > 0 — conservative, and guaranteed to capture every location you observed — though as always, the unsampled parts of the true range come with no guarantees. It also risks delineating more area than necessary. And the stakes are real: if you are writing a land management plan and define a home range larger than the animal needs, you may block management actions unnecessarily; define one too small and the animal itself may be threatened. If you are studying the animal, a wrong threshold means including or excluding land erroneously and learning the wrong habitat associations. This problem — estimating the true area used by an animal from insufficient data — is common in wildlife analysis.

To choose a threshold, it helps to draw contour lines on the density surface — here at every 5-points-per-km² increment:

The density surface overlaid with contour lines at five point per square kilometer intervals — Density contours, every 5 pts/km².

Based on a visual examination we might decide everything at or above 5 pts/km² represents the home range, and isolate that polygon:

The extracted home-range polygon covering the higher-density parts of Secret Canyon, with three separate islands — The estimated home-range polygon. In practice we might edit it — connect the three islands, or drop the small northeastern one entirely.

A kernel density bonus: finding areas of highest use. Unlike the MCP, a kernel home range specifically excludes areas the animal avoids even when they sit inside the overall extent of the points — and better yet, the same method marks out the core areas where the animal spends the majority of its time. These high-intensity areas are probably exceptionally important to the animal for some reason; for a threatened or endangered species, management plans might protect them specifically. The procedure is identical, just with a higher threshold — say 50 pts/km²:

The contoured density surface with the high-density contours emphasized — The same contours, eyes on the high-density cores.

The extracted core-area polygons covering only the densest parts of the range — Core areas: density > 50 pts/km².

A step farther: from densities to probabilities

The problem: the major downside of kernel density analysis is that density units — points per area — are difficult to work with. The distribution of density values will rarely be the same between two animals, or even between two runs on the same animal with different parameters. There was nothing special about our thresholds of 5 and 50 pts/km² above; a different sample size or a different bandwidth would have demanded different thresholds to identify equivalent regions.

The solution: rescale the density values into proportion-of-volume-under-the-curve values — probabilities, in effect, of the animal being inside an area, or equivalently the proportion of its observed time spent there. Now we can mark out the area where the animal would be 95% of the time, or 90%, or whatever level we want — home ranges based on consistent activity levels rather than guesses at arbitrary density thresholds.

This also makes comparing animals possible. Dr. Carol Chambers has been studying home ranges of New Mexico meadow jumping mice. Kernel densities map where each mouse spends its time beautifully — but each mouse has a different number of observations spread over a differently-sized area, so their density distributions differ wildly, and no single density level identifies equivalent home ranges for both:

A kernel density surface for the first jumping mouse with its own range of density values — Mouse one's density surface…

A kernel density surface for the second jumping mouse with a very different range of density values — …and mouse two's — same method, incomparable density scales.

Rescaled to probability levels, the 95% highest-use area of each mouse extracts directly — habitat use compared at a constant activity level:

The 95 percent probability home range extracted for the first mouse — Mouse one at the 95% level…

The 95 percent probability home range extracted for the second mouse — …and mouse two at the same level — finally comparable.

When this walkthrough was first written for the wildlife GIS lab, this step came with a warning: “this section describes functions that are only available in custom code or 3rd-party tools.” These tools are that custom code, ready-made: KD to Proportion Surface rescales any kernel density raster into the 0-to-1 proportion surface, and KD Probability Contours draws the isopleths — the 95% home range, the 50% core — directly, from either form of raster. Kernel Density Enhanced can even produce the proportion surface in one step, straight from the points.

Beyond the standard kernel: what these tools add

The story above describes the standard kernel density analysis that any GIS can produce. The Kernel Density Enhanced tool reproduces Esri's tool exactly when asked — same quartic kernel, same Silverman default bandwidth — and then goes further:

Seven kernel shapes. The bell can fall off gently or sharply: quartic (the Esri shape), uniform, triangular, Epanechnikov, triweight, tricube, and cosine — all with finite support, reaching exactly zero at the search radius (infinite-tailed kernels like the Gaussian are deliberately not offered because these types of kernels don't actually have a clearly defined search radius/bandwidth).

The seven kernel profiles, peak-normalized — the choice is how sharply weight falls off with distance. Full per-kernel illustrations are on the Kernel Density Enhanced page.

True geodesic distances. An explicit planar/geodesic choice, with geographic (latitude–longitude) inputs always analyzed geodesically — densities remain honest at any latitude.

Three kinds of cell value. Classic densities (points per area unit), expected counts (cell values sum to the number of points), or the proportion-under-curve surface directly — the “step farther” above, in a single run.

No cut-off edges. By default the output extent is the extent of the points plus the bandwidth distance, so every kernel runs all the way out to zero. Esri's tool only generates the surface within the extent of the points, cutting the raster off at the edges unless you remember to set an analysis extent environment yourself.

The full environment set. Output Coordinate System, Cell Size, Processing Extent, Snap Raster, and Mask are all honored, with the surface computed natively in the target coordinate system rather than resampled into it.

No extension licenses. Esri's Kernel Density tool requires the Spatial Analyst extension; this suite runs at every ArcGIS Pro license level.

References

Silverman, B. W. 1986. Density estimation for statistics and data analysis. Monographs on Statistics and Applied Probability, Vol. 26. Chapman and Hall, London.
Worton, B. J. 1989. Kernel methods for estimating the utilization distribution in home-range studies. Ecology 70:164–168. doi.org/10.2307/1938423

Kernel Density Enhanced, KD to Proportion Surface, and KD Probability Contours — the three tools this page underpins.
About LoCoH home-range analysis — the local-hull alternative, with a discussion of when each method is the better fit.
Concave Hull — one clean footprint polygon, when the question is the outline rather than the utilization distribution.

About Kernel Density Analysis

Kernel density, plainly

Point density: similar, but simpler

Kernel density

Kernel density with multiple points

How do you know what kernel size to use?

From density surface to home-range polygon

A step farther: from densities to probabilities

Beyond the standard kernel: what these tools add

References

Related tools and pages