Probability and average area of intersection of rectangles and ellipse

[chowell262]

Hello all,
I am neither a physicist nor a mathematician, I am an archaeologist trying to develop a mathematical model for archaeological site detection. The problem is set up like this (and hopefully this will make some sense):

The detector moves in parallel transects that can be of any interval but is consistent within a given test.
The detector has a swath width of a fixed distance.
The target is an archaeological site typically elliptical in shape which has an average density of artifacts which are visible to the detector per certain unit of area.
The orientation of transects and the site are independent of each other.
Assume an even (non clustered) distribution of artifacts within the site.

To find the chance of detection I have been using the formula for a Poisson distribution:
p(x) = exp(-1*lambda) lambda^x / x! for x=0,1,2,...
lamda is the expected density of artifacts per unit of space

Which I believe I can then multiply by the average area of intersection between the transects (rectangles) and the site (ellipse). I have found formulas for determining the probability of intersection of ellipses with transects but they do not calculate the area of intersection:
P=(1-b/l)(2l/πs(1-sqr root((1-s/l)² + s/l(Acs(s/l)))) + (b/l)(2l/πs)
where P is probability of intersection
b=width of ellipse
l is length of the ellipse
s is the transect spacing or interval

This formula should account for multiple intersections of transects if the ellipse is much larger than the transect spacing.
I assume that the formula might be adapted to provide spatial data but I do not know quite how to make that work to hook it into the Poisson distribution formula.
I suspect that I might be forgetting something fundamental so if so let me know.
Thanks very much in advance for your help!

 

[DrDu]

That sounds interesting.
Let's see whether I understand you correctly.
You are scanning an area and you are asking for the probability to detect at least one artifact given that the elliptic site falls completely inside the region scanned, don't you?
In principle, the detection probability will depend on the orientation of the ellipse relative to the scanning direction, which is unknown, however. So you would have to average over all orientations of the ellipsis relative to the scanning direction. The result will be independent of the form of the site. What counts is only its area. The area you scan effectively is the area of the ellipsis $$A_0$$ times the swath width (if I interprete this term correctly), d, divided by the transsect spacing s, or $$A=A_0 d/s$$.
The average density of artifacts inside the circle per unit of area is lambda. Before you can use this in the Poisson formula, you have to multiply it with the area A.
Then the probability to detect x artifacts is $$P(x)=\exp(-\lambda A) (\lambda A)^x/x!$$.

 

[chowell262]

Dr. Du,
Thank you very much for your reply, my apologies for not responding earlier but sometimes things take awhile to percolate through my mind and to make sure I am framing things in the proper way. The assumptions that you make about what I was trying to say are correct. There are some simplifications involved in that the transect is assumed to be completely within the ellipse. Also the simplification from an ellipse to a circle is fine as it will actually work better with my data.
I came up with a more complicated formula for interception though:
If D ≤ 2t, then A = w*D
If D ≤ 4t, then A = s*D +(4sqrt((D/2)^2 + t^2))s
If D > 4t but ≤ 6t then A = s*D + (4sqrt((D/2)^2+t^2 ))s + (4sqrt((D/2)^2+t^2 ))s
Where D= site diameter, t = transect width, s = detector swath width, A = area intersected
This assumes that a transect goes through the middle of the circle which is much less likely for circles with D< t, however both of our formulas seemed to make this assumption. I think then that the amount of area encountered by the transects is properly described as best case or maximum area of overlap.

When I run the numbers, the results are similar though not exactly the same, so since both are more accurately estimates I would guess the difference is not significant.

I do have one question about combining probabilities since I have two probabilities that I am working with. One of these is the probability of their being an artifact in the site based on lambda or site density calculated with a Poisson Distribution. The other is the probability of intersection of the transects with the site. In the case of the above type of detector that number is greater than 1 and thus no longer a probability. However I have another intersection method which samples at a point on a grid with regular spacing between points. The chance for intersection is lower than 1 in many cases and is thus still a probability. For your solution you multiplied the area of intersection by lambda in the Poisson formula to generate the overall probability of detection.

My question then is how to combine probabilities for the two different cases. Are they computed where you include the result of the intersection formula into the Poisson distribution as you provided or would you multiply the result of the intersection formula by the result of the Poisson Distribution? Does a result greater or less than 1 matter when combining it with the probability of the Poisson Distribution? I want to make sure that I am correctly talking about the results of this formula.

Thanks again for your interest and help!

 

[DrDu]

Huh, I have forgotten already many details. However, although I accidentially wrote circle somewhere in the text, my considerations are not bound to any assumed shape of the site. You don't know where the site is located. Hence you have to average over its location and orientation. I don't think that the result depends on the shape of the object any more after averaging.

 

[blue_raver22]

I find your approach to developing a mathematical model for archaeological site detection to be commendable. It is important to use mathematical tools and formulas to aid in scientific research and analysis.

Regarding your question about the probability and average area of intersection of rectangles and an ellipse, I would suggest taking a closer look at the formula you have found for determining the probability of intersection (P). This formula seems to take into account the width and length of the ellipse (b and l) as well as the transect spacing (s) in order to calculate the probability of intersection. However, it is not clear how this formula accounts for the average area of intersection between the rectangles and the ellipse.

One approach you could take is to use a Monte Carlo simulation to estimate the average area of intersection. This involves creating a computer program that randomly generates transects and ellipses (based on the parameters you have specified) and calculates the area of intersection for each combination. By running this simulation multiple times, you can obtain an average area of intersection that can then be used in your overall model.

Additionally, it may be worth considering other factors that could affect the probability of detection, such as the depth at which artifacts are buried or the sensitivity of the detector. These variables could also be incorporated into your model to provide a more accurate estimation of the chance of detection.

Overall, I would suggest further exploring the formula you have found and considering a Monte Carlo simulation to estimate the average area of intersection. Additionally, incorporating other variables into your model may improve its accuracy and effectiveness. Best of luck with your research!