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
T[size=85]here is a system of [b]n[/b] points in a certain region. For the "average overall" characteristic of such a system, [i]point estimates[/i] are usually used: the geometric [color=#ff7700][url=https://en.wikipedia.org/wiki/Geometric_median]median[/url][/color] and the [color=#333333]geometric[/color][color=#ff00ff] center[/color] (or [color=#ff00ff]center of mass, the [url=https://en.wikipedia.org/wiki/Centroid]centroid[/url] [/color][color=#333333]of a finite set of n points[/color]). In the first case, it is a [b]point[/b] that minimizes the function of the [color=#1e84cc]sum of the distances[/color] between itself and each point in the set (for three points - [b]Fermat point[/b]), and the [color=#1e84cc]sum of the squares of this distances[/color] -in the second case. [br] I propose a more extended interpretation of the concept of [i]point estimates[/i] of the location ([color=#ff7700]geometric medians[/color] and [color=#ff00ff]geometric centers[/color]) of a discrete set of [b]n[/b] points not in the entire domain e.g. ℝ³, but in some [i] domain[/i]. That is, you need to find [b]points[/b] in this [i]domain[/i] that [color=#ff0000]extremize[/color] the [color=#1e84cc]sum of the distances[/color] function from them to [b]n[/b] points of the set.[br] In applets, a limited number of particles from ℝ³ are considered, and a circle and a sphere are considered as such a [i]domain[/i], i.e. [i]point estimates[/i] are sought on a circle / sphere. The search for [i]point estimates[/i] on a circle allows you to understand their meaning more clearly . Thus, the search for [i]point estimates[/i] is reduced to finding the [color=#ff0000]critical points[/color] of the [color=#1e84cc]distance sum function f (x, y, z)[/color] subject to a [u]constraining equation[/u] g(x,y)=x²+y -R² in the case of [i]estimation[/i] on a circle or g(x,y,z)=x²+y²+z²-R²-in the case of [i]estimation[/i] 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. [br] Algorithms are proposed for finding [b]points[/b] corresponding to [color=#ff0000]extremes[/color]: [color=#0000ff]minima[/color], [color=#ff0000]maxima[/color], and, in the case of the sphere, also [color=#93c47d]saddle[/color] [b]points[/b] of the [color=#1e84cc]sum of distances function[/color].[/size]
[b][b]Key points in images[/b]. Examples of point estimators of location (geometric medians and geometric centers) in bounded closed domains for a discrete set of points from ℝ³ [/b]
[size=150][b]1. Point estimators[/b] (geometric [i][color=#ff7700]median[/color][/i] and geometric [color=#ff00ff][i]center[/i][/color]) defined for a discrete set of points from [b]ℝ³[/b] [/size]
[size=85]The [b][u][url=https://en.wikipedia.org/wiki/Geometric_median]geometric median[/url][/u][/b] of a discrete set of sample points in a [url=https://en.wikipedia.org/wiki/Euclidean_space]Euclidean space[/url] is the [i][b]Point[/b][/i] [color=#1e84cc]minimizing the [/color][color=#ff7700]sum of distances(d[sub]i[/sub])[/color][i] [color=#1e84cc]to the sample points[/color]. [/i][color=#333333]For triangles, this point is called the [u][url=https://en.wikipedia.org/wiki/Fermat_point]Fermat point[/url][/u].[br]The [b][u][url=https://en.wikipedia.org/wiki/Geometric_median#Computation]geometric center[/url][/u][/b] of a discrete set of sample points in a [url=https://en.wikipedia.org/wiki/Euclidean_space]Euclidean space[/url] is the [i][b]Point[/b][/i] [i][color=#1e84cc]minimizing the [/color][color=#ff00ff]sum of the squares of the distances(d[sub]i[/sub][sup]2[/sup])[/color][color=#1e84cc] [i]to the sample points[/i][/color][/i], can be found by a simple formula — its coordinates are the averages of the coordinates of the points. It is also known as the [url=https://en.wikipedia.org/wiki/Centroid]centroid[/url], [url=https://en.wikipedia.org/wiki/Center_of_mass]center of mass[/url] of a finite set of points.[/color][/size]
Example 1.1
[size=85][b][color=#333333][u][url=https://www.geogebra.org/m/wc8yfreb]Example[/url] 1.1:[/u][/color][/b] [b]Point estimators[/b] [b]of location[/b] (Geometric [color=#ff7700][url=https://en.wikipedia.org/wiki/Geometric_median]median[/url] [/color]and Geometric [color=#9900ff][i][url=https://en.wikipedia.org/wiki/Centroid]center[/url][/i][/color]) of a discrete set of 6 points from [b]ℝ³.[br][/b]Here, [url=https://en.wikipedia.org/wiki/Arg_min][b]arg min[/b][/url] means the value of the argument y which [color=#1e84cc]minimizes[/color] the[color=#ff7700] sum of distances[/color][i] [color=#1e84cc]to the sample points[/color][/i]. In this case, it is the [color=#20124d]point- [/color][color=#ff7700]F[/color][sub]o[/sub]:[color=#333333]=[/color][color=#333333]y[/color]. For the [i][color=#ff00ff]sum of the squares of the distances[/color] [i][color=#1e84cc]to the sample points[/color][/i][/i], such point is [i][color=#ff00ff]Cm:[color=#333333]=[/color][color=#333333]y[/color][/color][color=#333333].[/color][/i][/size]
[size=85][size=150][b]2. Point estimators [/b][b]of location [/b](geometric [i][color=#ff7700]medians[/color][/i] and geometric [color=#ff00ff][i]centers[/i][/color]) defined in bounded closed domain: [b][u]on a circle[/u][/b] for a discrete set of points from [b]ℝ³[/b] [/size][/size]
Example 2.1a
[size=85][b][u] [url=https://www.geogebra.org/m/rvfx4euj]Example[/url] 2.1a[/u][/b]: [b]Point estimators[/b] [b]of location[/b] (geometric [color=#ff7700]medians[/color] [b][color=#ff7700]GMᵢ[/color][/b]) defined on a circle for a discrete set of 6 points from [b]ℝ³[/b] [/size]
Example 2.1b
[size=85][b][u] [url=https://www.geogebra.org/m/rvfx4euj]Example[/url] 2.1b[/u][/b]: [b]Point estimators[/b] [b]of location[/b] (geometric [color=#ff00ff]centers[/color] [b][color=#ff00ff]GCᵢ[/color][/b]) defined on a circle for a discrete set of 6 points from ℝ³[/size]
Example 2.2a
[size=85][b][u] Example 2.2a[/u][/b]: [b]Point estimators[/b] [b]of location[/b] (geometric [color=#ff7700]medians[/color] [color=#ff7700][b]GMᵢ[/b][/color]) defined on a circle for a discrete set of 6 points from [b]ℝ³[/b] [/size]
Example 2.2b
[size=85][b][u] Example 2.2b[/u][/b]: [b]Point estimators[/b] [b]of location[/b] (geometric [color=#ff00ff]centers[/color] [b][color=#ff00ff]GCᵢ[/color][/b]) defined on a circle for a discrete set of 6 points from [b]ℝ³[/b][/size]
Example 2.3
[b] [size=85][u][url=https://www.geogebra.org/m/tnut2f9y]Example[/url] 2.3[/u] [/size][size=85]Generating a uniform distribution of points on a circle.[/size][/b][size=85] Let L be a circle of radius R around the point O: L:={x∈[size=100]ℝ[/size][sup]2[/sup]: ||x||=R}. [i][i]There is a set lP={A1, A2,...,An} [/i]of n [b][u]movable free points on a circle[/u][/b].[/i][br][b]Problem[/b]: use the [b][i]method of Lagrange multipliers[/i][/b] find such their distribution corresponding to the [i][b][color=#ff0000]maximum[/color][/b][/i] [b]sum of all their mutual distances[/b]. [br][i] This means need to find out such [i]mutual arrangement[/i] of "[i][i][color=#ff0066]repulsive" set of [/color][u][color=#0000ff]particles on a circle[/color][/u][color=#ff0066],[/color][/i] [/i]when each point of this set is [b][color=#ff7700]Geometric median[/color][/b] [color=#333333][b]([/b][/color][b][color=#ff7700]GM[/color][color=#333333])[/color][/b] of the remaining n-1 points. [i]We assume that the equilibrium - [/i][i]stationary state [/i][i][color=#333333]in system of "charges" is reached if the sum of their mutual distances is [/color][color=#ff7700]maximal[/color][color=#333333]. [/color][/i][i][i]Iterative approach[/i][/i] of [i][i]particle placement[/i][/i] is applied for [i]achieving [/i][i]a stationary state.[/i][/i][/size]
[size=85][size=150][b]3.[/b] [b]Point estimators[/b] [b]of location[/b] (geometric [i][color=#ff7700]medians[/color][/i] and geometric [color=#ff00ff][i]centers[/i][/color]) defined in bounded closed domain: [u][b]on a sphere[/b][/u] for a discrete set of points from [b]ℝ³[/b] [/size][/size]
Example 3.1
[size=85][color=#333333][b] [u][url=https://www.geogebra.org/m/bqj29x9m]Example[/url] 3.1[/u]:[/b] [b]Point estimators[/b] [b]of location[/b] (geometric [/color][color=#ff7700]medians [/color][color=#ff7700][b]GMᵢ [/b][/color][color=#333333]and geometric [/color][color=#ff00ff]centers [/color][color=#ff00ff][b]GCᵢ[/b][/color][color=#333333]) defined [u][b]on a sphere[/b][/u] for a discrete set of 6 points from [b]ℝ³[/b] [br][/color][b][color=#9900ff]1[/color][/b]. 6 Points in ℝ³ and the search their [b]Point Estimators[/b]: [i]geometric [color=#ff7700]medians[/color][/i] and [i]geometric [color=#ff00ff]centers[/color][/i] on the surface of the sphere.[br][color=#9900ff][b]2[/b][/color]. Regardless of the number of points they have only two geometric [color=#ff00ff]centers[/color] on the surface of the sphere: two antipodal points.[br][color=#9900ff][b]3[/b][/color]. The existing distribution of 6 points in this example give ten geometric [color=#ff7700]medians[/color] on the surface of the sphere: 2 [color=#ff0000]maxima[/color], 4 [color=#0000ff]minima[/color] and 4 [color=#38761d]saddle[/color] critical points for the [i]sum-distance function[/i]. The vectors ∇f and ∇g are parallel at these points.[br][b][color=#9900ff]4[/color][/b]. Table of coordinates of the critical points of [color=#1e84cc]distance sum-distance function [/color]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 [i]Lagrange multipliers[/i] 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.[br][b][color=#9900ff]5[/color][/b]. (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'[sub]φ[/sub](φ, θ)=0; f'[sub]θ[/sub](φ,θ)=0 over a rectangular region φ∈[-π,π], θ∈[-0.5π,0.5π] -are solutions (critical points) of the Lagrange equations.[br][b][color=#9900ff]6[/color][/b]. 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 [color=#ff0000]maxima[/color]/[color=#0000ff]minima[/color] and [color=#38761d]saddles[/color] -its critical points.[/size]
Example 3.2
[size=100][size=85] [b] [u][size=85][url=https://www.geogebra.org/m/dvrjqmkv]Example[/url] 3.2: [/size][/u][/b]The uniformly distribution of [b]Point estimators[/b] [b]of location -[/b]geometric [color=#ff7700][i]medians[/i][/color] on a sphere, „induces“ by the discrete sample of uniformly distribution points in the 3-D space. Description is in [url=https://www.geogebra.org/m/y8dnkeuu]https://www.geogebra.org/m/y8dnkeuu[/url]. Here, the 12 vertices of the Icosahedron "induce" the vertices of the other three polyhedra:[/size][b][br]20 ●[color=#ff0000]Dodecahedron[/color]← 12 ●[color=#0000ff]Icosahedron[/color] →30 ●[/b][color=#6aa84f][b]Icosidodecahedron.[/b][/color][/size]
Example 3.3
[size=85][i][b] [size=85][u][url=https://www.geogebra.org/m/znzekxmj]Example[/url] 3.3[/u] [/size][size=85]Generating a uniform distribution of points on a sphere.[/size][/b][size=85][color=#333333] Let S be a sphere of radius R around the point O: S:={x∈[/color][size=100]ℝ[/size][sup]³[/sup][color=#333333]: ||x||=R}. [/color][i]There is a set lP={A1, A2,...,An} [/i]of [b][u]n movable free points on a sphere[/u][/b].[br][b] Problem[/b]: use the [b][i]method of Lagrange multipliers[/i][/b] find such their distribution corresponding to the [i][b][color=#ff0000]maximum[/color][/b][/i] [b]sum of all their mutual distances[/b]. [br][i] This means need to find out such [i]mutual arrangement[/i] of "[i][i][color=#ff0066]repulsive" set of [/color][u][color=#0000ff]particles on a sphere[/color][/u][color=#ff0066],[/color][/i] [/i]when each point of this set is [i][b][color=#ff7700]Geometric median[/color][/b] [color=#333333]([/color][b][color=#ff7700]GM[/color][color=#333333])[/color][/b][/i] of the remaining n-1 points. [i]We assume that the equilibrium - [/i][i]stationary state [/i][i][color=#333333]in system of "charges" is reached if the sum of their mutual distances is [/color][color=#ff7700]maximal[/color][color=#333333]. [/color][/i][i][i]Iterative approach[/i][/i] of [i][i]particle placement[/i][/i] is applied for [i]achieving [/i][i]a stationary state.[/i][/i][/size][/i][/size]