I don't really know where to ask this one... In fact, I am not sure I can define it properly.

Here goes...

Let's say I take measurements.

In order to "normalize" these measurements, I divide their values by a reference measurement.

So, if my reference measurement has value 100, and one measurement has value 150, I rescale it as follows: 150 / 100 = 1.5. That particular measurement's value is 1.5 time the reference value.

So far, trivial.

Now, time for a picture:

enter image description here

The triangles are measurement values. A and B are reference values.

The domain the measurements are taken in indicates that the closer a measurement value is to A, the more A should be the reference value to rescale it. Similarly, the closer a measurement value is to B, the more B should be the reference value to rescale it.

So, values 1 and 2 should be restated in terms of A, while values 4 and 5 should be restated in terms of B.

But, in general, all values should probably be restated in function of A and B. This is particular true for value 3 in the picture, as it is almost equidistant from A and B.

In other words, there must be a function f(x){A, B}, that will allow to order all measurements.

What do you call that type of problem? Any reference or pointer I could use to restate what I mean?


Update 1:

Here is a metaphor that just came to me... Let's say I must define a scale that measure the total cost of moving an object at a certain speed.

A and B become two reference "cost per unit of speed"s. For speeds close to B, the total cost can be accurately estimated in terms of B. For speeds close to A, the total cost can be accurately estimated in terms of A. For speeds in between A and B, the total cost must be a function of A and B.

The farther away a speed is from A/B, the less A/B is a factor in the cost of moving the object at that speed.

  • $\begingroup$ I think that you can just interpolate your reference measurement, and use the interpolated values to normalize the other measurements. $\endgroup$
    – nicoguaro
    Nov 15 '16 at 15:06

Have you thought of barycentric coordinates? There is a unique way to write $x=\alpha A + \beta B$ with $\alpha+\beta=1$.

Barycentric coordinates are usually employer in larger dimensions, but seem to be the concept you are looking for. You can sort your observations according to either $\alpha$ or $\beta$, this will give you their "closeness" to either. If $A$ and $B$ are well chosen, values should cluster around $[0,1]$, or even around $\{0,1\}$. See e.g. https://en.wikipedia.org/wiki/Barycentric_coordinate_system


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