# SECR and circular home ranges

Do the models used for SECR (spatially explicit capture recapture) assume that animals' home ranges are approximately circular?

I've seen this asserted a couple of times, in particular in Tobler and Powell (2013, p.110), and I've myself drawn circular home ranges when discussing the interpretation of the capture parameters, but I don't think it is a necessary assumption.

I've outlined the main assumptions of SECR here. I think the shape of the animal's home range is unimportant provided these two assumptions are met:

• Each animal has an Activity Centre and probability of capture depends on the distance between the trap and the AC.
• Detectors are randomly placed with respect to the location of activity centres.

### Circular home range

Let's start off by considering an animal which actually does have a circular home range, but with fuzzy edges. It spends most of its time near its AC (so probability of capture is highest in that area), and decreasing amounts of time as the distance from the AC increases (so capture probability declines), but it moves in all directions around its AC. This is depicted in the plot on the left below, where the density of the grey dots indicates the probability that the animal will be in that part of the plot.

If we set a trap in the animals home range, the probability of capture depends on the distance between the trap and the AC, d, as shown in the plot on the right above. The histogram shows the density of the dots, and the blue curve is a half-normal function fitted to the histogram.

And if the home range really is circular like this, the capture probability is the same at all points on a circle centred on the AC, such as the blue circle in the plot on the left.

### An elongated home range

Now let's look at an elongated home range like the one shown by the cloud of points below left. Now the capture probability is not the same at all points on the circle. Can we still say something useful about capture probability if all we know is the distance between the AC and the trap?

The key is that traps are placed randomly with respect to AC locations. That means that for a given radius, d, all points on the circle are equally probable as trap locations. And we can then average (integrate) capture probability around the whole circle. The average density of the dots for different distances is shown by the histogram below right.

If we only had one animal and one trap, the results we'd get vary greatly depending on trap location on the circle. But with a large number of traps and many animals, the averaging will work quite well.

The half-normal detection function (the blue curve in the plot above right) doesn't fit the histogram well: it's too high for small distances and too low for large distances. The exponential function (the solid red line) is better, but the opposite problem: too low for small distances and a bit too high for large distances. We need something in between.

The most important bit of the exponential function - its kernel - is: $$e^{-x/\sigma}$$, where $$\sigma$$ is the scale parameter, with bigger values corresponding to curves which are wider. Now, we can write1 the half-normal kernel like this: $$e^{{-(x/\sigma)}^2}$$. Looking at those two equations gives the idea of using a hybrid function, $$e^{{-(x/\sigma)}^z}$$, where z is the shape parameter and lies between 1 and 2. When z = 1 we have the exponential shape, and when z = 2 it's a half-normal shape. The dashed line in the plot above is a hybrid curve with z = 1.000006, only just over 1, but enough to improve the fit.

### Irregular shapes

The plot below shows an irregularly-shaped home range, a combination of a circular area with a pan-handle. We can identify an AC and the probability of capture averaged over the whole circle decreases as the radius of the circle increases, as shown in the histogram below.

An exponential detection function fits the histogram very well, but a hybrid function with shape parameter = 1.19 is even better.

### Hard-edged home ranges

So far we've looked at "fuzzy" home ranges, where animals might move large distances from their activity centre, but do so only rarely. A home range with a well-defined boundary might be a better model for species which are vigorously territorial. And we might also suppose that they use the area within their boundary fairly uniformly. This is shown below.

Small circles around the AC lie entirely inside the home range, and for those the detection probability is the same (the slight differences in the heights of the first 8 bars in the histogram is because the dots in the plot are randomly drawn). The half-normal detection function doesn't capture this initial flatness; we need the function labelled "w exponential" in Murray Efford's secr package. With this, detection probability is uniform up to a threshold determined by the shape parameter, then declines exponentially.

### Hard-edged circular home ranges

As a final example, let's look back at circular home ranges, this time with a hard edge instead of a fuzzy edge.

The detection function now is uniform for distances up to a threshold, the radius of the home range, and zero beyond that. In the secr package documentation, Murray Efford says: "[The uniform function] is defined only for simulation as it poses problems for likelihood maximisation by gradient methods." Far from being ideal, this type of home range would actually cause problems for analysis with SECR software! Fortunately, perfectly circular, hard-edged home ranges are unlikely to occur in nature.

### Conclusion

SECR analysis methods can deal with home ranges which are not circular, provided traps are placed randomly with respect to the animals' activity centres. For example, if animals occupy elongated home ranges aligned along a road and you place your traps along the road, estimates of capture parameters will be biased.

With an elongated home range, capture probability for a given distance can vary widely. This will affect the precision of our  estimates of capture parameters, and means that we need larger samples then when home ranges are more compact.

Our discussion suggests that differently shaped home ranges need different detection functions. But in practice we are working with many animals with different shapes and sizes of home range, so it's likely that overall a half-normal detection function will be fine. You can try different detection functions and compare the model fit with AIC.

So: SECR does not assume that home ranges are circular, but it does assume that traps are placed randomly relative to the AC in terms of direction as well as distance.

### Code

A ZIP file with the R code to produce the graphs shown is here. [Added 15 Aug 2013.]

### References

Tobler, M W; G V N Powell. 2013. Estimating jaguar densities with camera traps: Problems with current designs and recommendations for future studies. Biological Conservation 159:109-118

### Footnote

1. The usual form for the normal distribution is $$e^{-x^2/2\sigma^2}$$. The form in the main text is equivalent, but the value of $$\sigma$$ is greater by a factor of $$\sqrt{2}$$.