Is Gower's Distance a metric?

Question

A novice here

My previous question was closed due to inadequate details

So here I've added more details

A metric should basically satisfy 3 properties

Distance is equal to zero if and only if $x$ is equal to $y$ ($d(x,y)=0 ⇔ x=y$))
Distance from $x$ to $y$ is the same as distance from $y$ to $x$ ($d(x,y)=d(y,x)$)
Distance should satisfy the triangular inequality ($d(x,y)\leq d(x,z) +d(z,y)$)

I already know that Gower's distance satisfy the first 2 properties to be a metric, but I want to know whether it satisfies the triangular inequality property.

The reason I want to know this, is because all metric spaces are Hausdorff spaces, and I want use the Gowers distance in order to find the Hausdorff distance for 2 sets of points.

In my case, a point contains data of mixed types (logical, categorical & numeral), and therefore I have to use the Gowers distance.

Any help would be appreciated.

Thank You!

Edit:

According to a suggestion on the comments, here is the formal definition of a metric

A metric on a set X is a function (called the distance function or simply distance)

$d : X × X → R$

(where R is the set of real numbers). For all $ x, y, z $ in $X$, this function is required to satisfy the following conditions:

$d(x, y) ≥ 0$ (non-negativity)
$d(x, y) = 0$ if and only if $x = y $
$d(x, y) = d(y, x) $ (symmetry)
$d(x, z) ≤ d(x, y) + d(y, z) $

Note that the first condition is implied by the others.

For people who don't know it, Gower's original paper can be found here. — Jean Marie, Nov 01 '20 at 08:52
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. — José Carlos Santos, Nov 01 '20 at 08:54
@ArcticChar the paper has been linked to, in the first comment — Shaveen De Silva, Nov 01 '20 at 09:36
The paper has 14 pages. You have more chances to get an answer if you extract the definition of Gower's distance and copy it in your question. — Miguel, Nov 01 '20 at 10:49
By the way, have you read the paper? It seems to me that it mentions the triangle inequality in Equation (4). — Miguel, Nov 01 '20 at 10:55

score 1 · Accepted Answer · edited Jun 28 '23 at 16:20

Referring to the answer in https://stats.stackexchange.com/questions/15287/hierarchical-clustering-with-mixed-type-data-what-distance-similarity-to-use

"Gower distance". Without ordinal variables present (i.e. w/o using the Podani's option) $\sqrt{1-GS}$ behaves as Euclidean distance, it fully supports euclidean space. But $1-GS$ is only a metric (supports triangular inequality), not Euclidean. With ordinal variables present (using the Podani's option) $\sqrt{1-GS}$ is only a metric, not Euclidean; and $1-GS$ isn't a metric at all. See also.

Therefore it can be used a metric

Papers

1 Gower J. C. A general coefficient of similarity and some of its properties // Biometrics, 1971, 27, 857-872

2 Podani, J. Extending Gower’s general coefficient of similarity to ordinal characters // Taxon, 1999, 48, 331-340

Is Gower's Distance a metric?

1 Answers1