A word for choosing a distance metric

Question

Looking for help with a technical term analogous to clustering. In particular, given a bunch of data, I want to describe the problem of coming up with distances (not measuring but rather estimation) like "these two points are 0.2 apart" and "these two points are 0.95 apart" in the same spirit as clustering, in which you look at a whole data set and determine which points belong to the same cluster.

The standard dictionary definitions of metrification, metricization, or metrication, and metrifying, all refer to using the metric system (as opposed to, e.g., inches or feet). That is not what I am referring to.

Answer 1

The (obviously mathematical) term for constructing a metric for a topological space (or showing that one exists) is metrization; thus, metrization theorems were popular in the early twentieth century and are the height of ennui in the twenty-first (not an actual quote, merely a fairly commonly-held opinion).

Answer 2

Alas, others have greedily snatched up the metri- words for their own selfish purposes. Metrication, metrification, metrizing all refer to defining a distance metric for an abstract mathematical space so as not to change the topological properties of the space. This is not always possible, but what can you do?

Metrication also refers to the process by which a country adopts the International System of Units (aka, the metric system) and abandons its traditional units of measure. (Currently, the United States, Liberia, and Myanmar are the only countries refusing to be dragged forward into the 19th century on this issue.)

What you are doing is picking a particular distance measure , and that is called defining a metric. This is a common phrasing in mathematics, pure and applied. For instance, from Notes on Geometry by E Rees

We could now proceed to define a metric on H by....

The procedure is unimportant because it always entails the same thing. You'll have to show that your definition takes any two points you're considering and gives you a non-negative (real) number that is zero when and only when the two points are the same. That number is called the distance, which additionally must be symmetric (i.e., the order of the points doesn't matter, so the distance from Los Angeles to New York is the same as the distance from New York to Los Angeles) and must obey the triangle equality (i.e., intermediate stops cannot decrease the distance, so the distance from Los Angeles to New York is at most the distance from Los Angeles to Denver plus the distance from Denver to New York.)

Answer 3

According to https://en.wikipedia.org/wiki/Metric_(mathematics):

In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set.

In ordinary two dimensional space with x-y coordinates, the metric is the absolute value of the square root of x² + y².

You are assuming a metric in your example when you say, "Two points are 0.2 apart." The "0.2 (units)" is the distance between the two points based on the metric. You could also call "0.2 (units)" the (value of the) metric for those two particular points. So, metric may be the single word you're seeking.

If, as per @Drew's comment, you are looking for a word "that refers to the process of defining an accurate metric," you should consider:

Metrology: the science of measurement https://en.wikipedia.org/wiki/Metrology

and measurement itself:

Measurement: the assignment of a number to a characteristic of an object or event, which can be compared with other objects or events https://en.wikipedia.org/wiki/Measurement

Note that the scopes of both metrology and measurement extend far beyond the physical sciences.