Measuring Membership, or Likelihood, of given Point

[Trecius]

Hello:

I've three circles as seen in the image.

Source: http://www.picpaste.com/Quantify-PwdfQxLB.png

Within the innermost circle is a point. I'd like to determine the likelihood, or probability, that this point belongs to the innermost circle, the middle-circle, and outermost circle.

Now, this point -- as can be seen in the image -- is very near the boundary of the innermost circle. Also, it is near the boundary of the middle-circle, so one might assume that the probability of being in each of the three circles is APPROXIMATELY (.33, .33, .33).

If we were to translate the point and shift it into the center of the innermost circle, I'd assume the distribution to be (1.0, 0.0, 0.0), for the point would lie in the middle of the innermost circle.

For another example, if I were to translate the point to the left a little, I'd probably see the distribution to be (0.45, 0.35, 0.2), respectively.

Is there a way to quantify, or determine, the likelihood that this point belongs to a specific circle in this example?

Thank you.


Trecius

 

[Stephen Tashi]

Hello:

I've three circles as seen in the image.


Source: http://www.picpaste.com/Quantify-PwdfQxLB.png

Within the innermost circle is a point.

For a specific point and 3 specific circles, the point either is or is-not in a given circle. There is no model for this situation involving probability. If you think of the red mark not as a single point, but as some sort of aimpoint for where darts are thrown, you could construct a model involving probability. If you think of the red mark as a figure with a finite area and the actual point it is supposed to designate as some randomly selected point inside that area, you could construct a model involving probability.

 

[Trecius]

Yes, you are correct. It is much like an "aim point." Again, the closer to the center of the inner-most circle, you're nearing a probability of 1 for that circle while the others will be near 0. However, as you levitate outward, the "ownership," or probability will shift. As you approach the boundary of the inner-most circle, you may have a 50 / 50 split between the inner circle and its encompassing circle, so the ratio between all three may be .4, .4, .2. This is just an example.

Also, how can I build this model? It can be an abstract model, but I'd like to know a way to quantify the "ownership."

Thank you again for such a quick reply; I did not expect it.


Trecius

 

[Stephen Tashi]

If you imagine there is a small circle around the point, you could use the overlap between the area of the small circle and the larger circle to quantify how much the small circle is within the larger circle. Off hand, I don't know the formula for finding the area of two overlapping circles, but I'm sure we can look up the formula. That model would be equivalent to using a uniform distribution for where a dart fell within the small circle containing the point.

If you use a non-uniform distribution of some kind, you will have to integrate the distribution over the larger circle. This would probably require a numerical method of some kind.

Many problems involving "degrees of membership" are better modeled using "fuzzy sets" than with geometric models.

 

[D H]

Is there a way to quantify, or determine, the likelihood that this point belongs to a specific circle in this example?

The answer depends a whole lot on what those circles are supposed to represent.

As depicted, you are demanding an either/or answer where the correct answer is "yes" to all three. Just because a point is inside one circle does not mean it isn't inside another. Your point obviously is inside each of the three circles.

If those circles are a kind of Venn diagram you'll run into the same problem. Labeling the smallest circle as A, the middle-sized one as B, and the largest one as C, then once again this is not an either-or kind of question. Any A is a B, and any B is a C. Your point is an A, a B, and a C.


On the other hand, membership is indeed an either/or proposition if those circles represent three mutually exclusive events but all you have at hand is a lousy measurement that can't quite distinguish between those events. For example, suppose circle A represents the three sigma boundary of what you would expect to see from failure scenario A, circle B represents the three sigma boundary of what you would expect to see from failure scenario B, and circle C represents the three sigma boundary of what you would expect to see if everything was acting nominally.

In this case, you can use the Mahalanobis distance (google that term) as a metric to help to distinguish between these events. You are still SOL with the circles as depicted. There's no distinguishing off-nominal from nominal behavior. You have a chance if the circles are intersecting but have distinct centers.

If the measurements are lousy but the centers are distinct, it might take multiple measurements before you can say which condition is present (failure scenario A, failure scenario B, or everything's cool). This is where Bayesian methods can be of a huge advantage.