Example of Point estimators of location (geometric medians) on a sphere for a discrete set of points from ℝ³

Author:
Roman Chijner
Topic:
Algebra, Calculus, Circle, Differential Calculus, Differential Equation, Optimization Problems, Geometry, Function Graph, Intersection, Linear Programming or Linear Optimization, Sphere, Surface, Vectors
Exploration of Geometric medians of a discrete set of 6 points defined in bounded closed domain: on a sphere. Compare with applet: Geometric median and Geometric center of a discrete set of sample points in a Euclidean space and applet: Geometric medians and Geometric centers of a discrete set of sample points defined in bounded closed domain: circle.
[size=85][color=#9900ff][b]1[/b][/color]. 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. [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. [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. [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 φ∈[-π,π], θ∈[-0.5π,0.5π] on the surface of the sphere. 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. [b][color=#9900ff]5[/color][/b]. (φ;θ) -plane of the angular coordinates of points on the sphere. 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. [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]
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 φ∈[-π,π], θ∈[-0.5π,0.5π] on the surface of the sphere. 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. 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.

New Resources

Discover Resources

Discover Topics