# An Extended Interpretation of the Concept of point estimators of location (geometric medians and geometric centers) in bounded closed domains for a discrete set of points from ℝ³

- Author:
- Roman Chijner

There is a system of

**n**points in a certain region. For the "average overall" characteristic of such a system,*point estimates*are usually used: the geometric median and the geometric center (or center of mass, the centroid of a finite set of n points). In the first case, it is a**point**that minimizes the function of the sum of the distances between itself and each point in the set (for three points -**Fermat point**), and the sum of the squares of this distances -in the second case. I propose a more extended interpretation of the concept of*point estimates*of the location (geometric medians and geometric centers) of a discrete set of**n**points not in the entire domain e.g. ℝ³, but in some*domain*. That is, you need to find**points**in this*domain*that extremize the sum of the distances function from them to**n**points of the set. In applets, a limited number of particles from ℝ³ are considered, and a circle and a sphere are considered as such a*domain*, i.e.*point estimates*are sought on a circle / sphere. The search for*point estimates*on a circle allows you to understand their meaning more clearly . Thus, the search for*point estimates*is reduced to finding the critical points of the distance sum function f (x, y, z) subject to a__constraining equation__g(x,y)=x²+y -R² in the case of*estimation*on a circle or g(x,y,z)=x²+y²+z²-R²-in the case of*estimation*on the sphere. The problem is solved using Lagrange multipliers. There is a system of equations: ∇f(x,y,z)= λ∇g(x,y,z). A local optimum occurs when ∇f(x,y,z) and ∇g(x,y,z) are parallel, and so ∇f is some multiple of ∇g. Algorithms are proposed for finding**points**corresponding to extremes: minima, maxima, and, in the case of the sphere, also saddle**points**of the sum of distances function.

**Key points in images**. Examples of point estimators of location (geometric medians and geometric centers) in bounded closed domains for a discrete set of points from ℝ³**1. Point estimators**(geometric

*median*and geometric

*center*) defined for a discrete set of points from

**ℝ³**

The

**of a discrete set of sample points in a Euclidean space is the**__geometric median__*minimizing the sum of distances(d***Point**_{i})*to the sample points.*For triangles, this point is called the__Fermat point__. The**of a discrete set of sample points in a Euclidean space is the**__geometric center__**Point***minimizing the sum of the squares of the distances(d*, can be found by a simple formula — its coordinates are the averages of the coordinates of the points. It is also known as the centroid, center of mass of a finite set of points._{i}^{2})*to the sample points*## Example 1.1

__Example 1.1:__**Point estimators**

**of location**(Geometric median and Geometric

*center*) of a discrete set of 6 points from

**ℝ³.**Here,

**arg min**means the value of the argument y which minimizes the sum of distances

*to the sample points*. In this case, it is the point- F

_{o}:=y. For the

*sum of the squares of the distances*, such point is

*to the sample points**Cm:=y.*

**2. Point estimators**

**of location**(geometric

*medians*and geometric

*centers*) defined in bounded closed domain:

**for a discrete set of points from**

__on a circle__**ℝ³**

## Example 2.1a

**:**

__Example 2.1a__**Point estimators**

**of location**(geometric medians

**GMᵢ**) defined on a circle for a discrete set of 6 points from

**ℝ³**

## Example 2.1b

**:**

__Example 2.1b__**Point estimators**

**of location**(geometric centers

**GCᵢ**) defined on a circle for a discrete set of 6 points from ℝ³

## Example 2.2a

**:**

__Example 2.2a__**Point estimators**

**of location**(geometric medians

**GMᵢ**) defined on a circle for a discrete set of 6 points from

**ℝ³**

## Example 2.2b

**:**

__Example 2.2b__**Point estimators**

**of location**(geometric centers

**GCᵢ**) defined on a circle for a discrete set of 6 points from

**ℝ³**

## Example 2.3

**Let L be a circle of radius R around the point O: L:={x∈ℝ**

__Example 2.3__Generating a uniform distribution of points on a circle.^{2}: ||x||=R}.

*There is a set lP={A1, A2,...,An}*of n**.**__movable free points on a circle__**Problem**: use the

**find such their distribution corresponding to the**

*method of Lagrange multipliers*

**maximum****sum of all their mutual distances**.

*This means need to find out such*

*mutual arrangement*of "*when each point of this set is**repulsive" set of*__particles on a circle__,**Geometric median****(****GM)**of the remaining n-1 points.*We assume that the equilibrium -**stationary state**in system of "charges" is reached if the sum of their mutual distances is maximal.**of**Iterative approach**is applied for**particle placement**achieving**a stationary state.***3.**

**Point estimators**

**of location**(geometric

*medians*and geometric

*centers*) defined in bounded closed domain:

__for a discrete set of points from__

**on a sphere****ℝ³**

## Example 3.1

__Example 3.1__:**Point estimators**

**of location**(geometric medians

**GMᵢ**and geometric centers

**GCᵢ**) defined

__for a discrete set of 6 points from__

**on a sphere****ℝ³**

**1**. 6 Points in ℝ³ and the search their

**Point Estimators**:

*geometric medians*and

*geometric centers*on the surface of the sphere.

**2**. Regardless of the number of points they have only two geometric centers on the surface of the sphere: two antipodal points.

**3**. The existing distribution of 6 points in this example give ten geometric medians on the surface of the sphere: 2 maxima, 4 minima and 4 saddle critical points for the

*sum-distance function*. The vectors ∇f and ∇g are parallel at these points.

**4**. Table of coordinates of the critical points of distance sum-distance function f(φ,θ) over a rectangular region on the surface of the sphere: φ∈[-π,π](x-Axis), θ∈[-0.5π,0.5π](y-Axis). They are found using

*Lagrange multipliers*as finding the Extreme values of the function f(x,y,z) subject to a g(x,y,z)=0 (constraining equation: g(x,y,z)=x²+y²+z²-R²). The table also contains partial derivatives and Angles between the vectors ∇f and ∇g at these critical points.

**5**. (φ;θ) -plane of the angular coordinates of points on the sphere: φ∈[-π,π], θ∈[-0.5π,0.5π]. The colored Isolines are qualitatively indicate the type of critical points. The intersection of implicit functions of the equations of zero partial derivatives: f'

_{φ}(φ, θ)=0; f'

_{θ}(φ,θ)=0 over a rectangular region φ∈[-π,π], θ∈[-0.5π,0.5π] -are solutions (critical points) of the Lagrange equations.

**6**. Graphic of the distance sum function f(φ, θ) over a rectangular region φ∈[-π,π], θ∈[-0.5π,0.5π] with the positions of the corresponding maxima/minima and saddles -its critical points.

## Example 3.2

**The uniformly distribution of**

__Example 3.2:__**Point estimators**

**of location -**geometric

*medians*on a sphere, „induces“ by the discrete sample of uniformly distribution points in the 3-D space. Description is in https://www.geogebra.org/m/y8dnkeuu. Here, the 12 vertices of the Icosahedron "induce" the vertices of the other three polyhedra:

**20 ●Dodecahedron← 12 ●Icosahedron →30 ●**

**Icosidodecahedron.**

## Example 3.3

**Let S be a sphere of radius R around the point O: S:={x∈ℝ**__Example 3.3__Generating a uniform distribution of points on a sphere.^{³}: ||x||=R}.*There is a set lP={A1, A2,...,An}*of**.**__n movable free points on a sphere__**Problem**: use the**find such their distribution corresponding to the***method of Lagrange multipliers***maximum****sum of all their mutual distances**.*This means need to find out such**mutual arrangement*of "*when each point of this set is**repulsive" set of*__particles on a sphere__,*of the remaining n-1 points.***Geometric median**(**GM)***We assume that the equilibrium -**stationary state**in system of "charges" is reached if the sum of their mutual distances is maximal.**of**Iterative approach**is applied for**particle placement**achieving**a stationary state.*