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Let's say I have a known vector of two numbers: c(A,B)

Is there a scoring term, or a combination of scoring terms, that can measure the unique closeness of a random vector c(a,b) to the known vector? In other words, is there a scoring term that can be used to determine that a is very close to A AND that b is very close to B?

Heres an example of a scoring term, $S$, I thought of:

$$S = \lvert a - A \rvert + \lvert b - B \rvert$$

It's not ideal because if our vector is c(1,5), then the vectors c(2,4) and c(0,4) would give the same scoring metric value of $2$. I would like the scoring metric to give me more information. Such as, for this example, $a$ is approaching $A$ from above for c(2,4)and $a$ is approaching $A$ from below for c(0,4).

I would also be interested to see how a scoring term, or combination of scoring terms, could be generalized to a vector of length $n$.

Nova
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    In your example, if $(A,B) = (5,1)$, then wouldn't you have $c(5,1)=0$ and $c(1,5)=4+4=8$? – Théophile Aug 06 '20 at 01:01
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    There is no best answer until you define your objective. You are probably interested in a metric, which measures the distance between points. Yours is called the Manhattan metric. Théophile gives the Euclidean metric. Neither is better until you define the objective. – Ross Millikan Aug 06 '20 at 01:09
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    Your call for a unique value emphasizes the need to say what you want. Why shouldn't $(0,4)$ and $(2,4)$ be the same distance from $(1,5)$. Any rotationally symmetric metric will do that. Which do you want to be closer? Why? – Ross Millikan Aug 06 '20 at 01:55

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A natural choice would be the Euclidean distance to $(A,B)$: $$c(a,b) = \sqrt{(a-A)^2+(b-B)^2}.$$ This generalizes to $n$ dimensions.

Théophile
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