No more than $n$ moose, but how many?

Question

Introduction

I am thinking about how to estimate the number of individual moose from wildlife camera photos. I have the latitude and longitude position of each observation, along with a datetime of the observation. But unlike mark and recapture methods, we're not marking the moose. It can be quite difficult to tell from the photos alone whether two observations are the same moose.

These are the attributes I know (so far) that we have.

datetime	Class	Location	Image ID
<datetime>	Bull/Cow/Juvenile	Lat/Long	#

Background Information

If we have $n$ moose observations, then there are at-most (I almost left it as "at-moost") $n$ individuals. But of course moose may wander back in front of a camera. Letting $k$ be the number of individuals, we have $k \in [1,n]$ in our sample.

Often we can tell the difference between bull, cow, and juvenile. Occasionally the camera angle is very poor (like just seeing a big moose nose) to tell between bulls and cows, so there is some missing data. But to an extent the bull/cow/juvenile puts some fuzzy constraints. If we have two bull observations, it is difficult to discern them. But if we have a bull observation and cow observation we can put a very high prior on those two observations not being the same animal. So if we have seen at least one one bull and cow, that tightens the interval to $k \in [2,n]$. Since juveniles eventually grow up (or perish), also observing at least one juvenile in close time proximity to the bull and cow would give us $k \in [3,n]$. We have such examples so this is a very slight improvement on the constraint.

Moose don't travel in herds for the most part, but on occasion multiple moose can be observed together in a single image. This can used to improve the lower bound on the number of moose in the sample. Offhand I don't know what this number is. Let's call it $h$, and refine that $k \in [\max (3, h), n]$. This also puts a hard constraint that if we have seen two or more moose together then the probability/likelihood of being assigned to be the same moose should be zero.

Moose age, and eventually die. So there are survivorship curve aspects that might suggest that two observations very far apart in time become less likely to be the same moose just due to survival. But the probability of being the same individual wouldn't have to be strictly decreasing with time between observations. A juvenile observed earlier in the sample may later be observed as either a bull or cow.

In some cases the images idiosyncratic differences about the moose may be identified. Some of these would be heritable and not phenotypically plastic. A rare example are "ghost moose", which are white rather than brown. Other traits may be acquired, such as substantial scars on the face/ears. Those sorts of observations have implications for observations in the future as such features may fade only slowly, but then again in future images they may not always be visible due to angle/occlusion from the camera's field of view. Furthermore, moose observed with mortal wounds or in very poor health (e.g. from tick infestation) are less likely to survive long enough to be seen again.

But that brings us to the last of the prior information I have worked out so far: proximity in space and time. If two moose observations are extremely close in time and space, all else being equal, it is quite likely (but not guaranteed) to be the same moose. Moose might be able to travel quite far in a day, but using some existing radio collar data it should be possible to put priors on how far they can travel per unit time. I think the simplest approach here is to include the average velocities, although the thought of simulating the movement of moose between observations using diffusion or quantum-inspired uncertainty or random walks also crossed my mind.

Question

Mainly I am having trouble defining the adaptive likelihood for this problem.

$$Pr \left[\text{k} | \text{data} \right] \propto Pr \left[ \text{data} | k \right] Pr[k] $$

The basic notion in mind is that for a given $k$ there is a probability distribution over the possible partitions of the $n$ observations. But I don't know how to sample effectively over partitions since the number of partitions for a modest $n$ will be very large (according to the Stirling number of the second kind).

What mathematical model(s) would be suitable for this problem? Or do I have an unresolvable specification/identification problem?

---

• Sampling period 10-15 years.

Answer 1

Implicit in your question is the monitored area. When you say "there are X individuals" you are implicitly taking into account that these moose inhabit a certain contiguous geographical area, its perimeter loosely defined by the locations of the "outer-most" cameras.

For the sake of simplicity, let's assume moose roaming behavior is identical so all individuals have a single "roaming strategy".

To further simplify, I'll make an additional assumption of independence, which means moose don't care about the positions and movements of other individuals. With some care, this assumption can be lifted, if necessary.

Since we're looking at the accumulation of events (event=moose in frame) along time, a Poisson probability comes to mind. To apply Poisson as-is, we need to add a further assumption that the probability of an event occurs, does not depend on recent past occurences. Without knowledge of moose walking patterns, I cannot say whether this makes sense, but if it doesn't, then a probability distribution other than Poisson can help account for that.

Having said all that, what it boils down to is this: If we are given the Poission parameter, which is the number of times a moose typically appears on camera per time period given that it is in the monitored area, then $Pr[data|k] = Pr$[actual count of events recorded | there are k moose in monitored area] ~ Poisson($\lambda k$).

The catch here, of course, is that $\lambda$ is unknown, and there's no straightforward way to estimate it. If your record includes the times of entering the frame and leaving the frame for each individual animal, it may be easier to come up with an estimate for lambda then if you simply have "crossing frame" events.

Another issue implicit in this approach, is time scales. There are two (competing) time scales governing the dynamics and the ability to estimate moose population effectively.

The first is the moose movement time scales. Let's pretend for the sake of explanation, that a moose only moves a couple hundred meters a day. That means that it may not be seen by any camera for months (i.e. very small Poisson's $\lambda$). This would require us to accumulate events over long periods before we can estimate $k$. On the other hand, if moose walk many kilometers each days, the chances of one individual wandering into a frame in any given week are much larger, and we can track trends over shorter periods.

The other time scale is that of population size trends. Moose are born, mature, age and die; Moose from nearby areas can walk into the monitored area and temporarily increase the population in the monitored area (and decrease it when they leave). If these changes occur quickly, then the question "how many moose are in the monitored area" may become meaningless for a large time frame, as the population numbers may have changed considerably over that period.