Intro:
I have dropped math since high school, therefore I am trying to describe my question in layman's terms. Please forgive me for not typing the convention notations. I even do not know what is the correct term to put in the Google search bar for doing research. Therefore, any suggestion is greatly appreciated.
Scenario:
Imagine I am doing a field study to record the number of animals in a field. It is impossible to count everything, and I am going to sample a 1 km2 area. It is assumed that the number of my observation is depending on the following 3 variables (for simplicity):
- Underlying number of that animal.
- Whether the animal is active (i.e. how often they "appear" or "come" to the sample area).
- The length of time I perform the observation.
The second point is relevant because I am not able to distinguish individual animals, therefore, if the same animal comes to the area twice, I will just count twice and assume they are not the same. However, I do assume that it is less likely to count the same animal if it is less active (see distributions 5 & 6 below).
The third point is relevant because I am going to film the area for 10 minutes (as an example). Assuming I will look at the video many times and am able to count everything, the observed count will be higher if I lengthen the observation period (so that both less active and abundant animals might also go into the field for counting).
Assuming I counted 100 different fields (exp1), I am now having a dataset as a count matrix that is animal by field. Each row and column will form a count distribution. For example, I will get a distribution of the number of rabbits per field across 100 fields (distribution 1); I will also get a distribution of the number of animals among the animal list in each field (distribution 2).
For some reasons (e.g. by image recognition), I am able to virtually "repeat" the experiment (exp2) and this time I am able to distinguish individual animals (so that I am not going to double count them). I am thus able to get a similar data matrix as described above. I am also able to get 2 more distributions, (distribution 3): the exact number of rabbits across the fields; (distribution 4): the exact number of different types of animals in each fill.
Questions:
I am interested to know the actual underlying number of animals in each field (imagine I am monitoring these fields indefinitely). Although it can be assumed that more counts will be observed in a longer observation period, they should converge to a "steady state" that the observation is "saturated".
My naive statistics knowledge tells me that each of these experiments can be used to interpolate the actual number. However, I am thinking about if I can get a better estimation by combining these two pieces of info to get a better estimation. This is because:
- I can now also get an idea of how "active" each animal is, with the
assumption that the activity is correlated with double counting. And this activity will also form two distributions, (
distribution 5): the activity of each animal across different fields; (distribution 6): the activity of different types of animals within a field. - It also gives me a sense of how much I did not count due to the 10-minute observation period. For example, in some fields, I did not observe any rabbit is due to it is not active enough to jump into the field.
Therefore, I am looking for a method for transforming the data from exp (I think that is better as it has accounted for the double counting issue) according to distributions 5 and 6.
Outro:
It is also possible that it is not possible to get the imputed data, in this case, it would be great if you can let me know what is missing. If one of the missing pieces is I need longer observation time. Then, it will be helpful to just assume a 10-minute observation is enough to get the steady state/ saturation.
For simplicity, I allow the imputed data to have fractional counts (I assume that they are averaged from multiple stochastic processes). I am open to seeing if any of the common distributions (e.g. normal, poison, negative binomial) works before going to complex stuff. Optionally, it would be great if you can show me some steps of how to calculate (either by hand or by a computer program). For example, pointing me to some well-written (non-math) literature would be helpful.
Please let me know if the question is not clear enough, I am more than happy to edit with more information.