Finding discriminant functions of minimum risk binary classification systems is a novel geometric locus problem -- that requires solving a system of fundamental locus equations of binary classification -- subject to deep-seated statistical laws. We show that a discriminant function of a minimum risk binary classification system is the solution of a locus equation that represents the geometric locus of the decision boundary of the system, wherein the discriminant function is connected to the decision boundary by an intrinsic eigen-coordinate system in such a manner that the discriminant function is represented by a geometric locus of a novel principal eigenaxis -- formed by a dual locus of likelihood components and principal eigenaxis components. We demonstrate that a minimum risk binary classification system acts to jointly minimize its eigenenergy and risk by locating a point of equilibrium wherein critical minimum eigenenergies exhibited by the system are symmetrically concentrated in such a manner that the geometric locus of the novel principal eigenaxis of the system exhibits symmetrical dimensions and densities, such that counteracting and opposing forces and influences of the system are symmetrically balanced with each other -- about the geometric center of the locus of the novel principal eigenaxis -- whereon the statistical fulcrum of the system is located. Thereby, a minimum risk binary classification system satisfies a state of statistical equilibrium wherein the total allowed eigenenergy and the expected risk exhibited by the system are jointly minimized within the decision space of the system, so that the system exhibits the minimum probability of classification error.
翻译:最低风险二进制分类系统存在差异性功能,这是一个新颖的几何轨道问题 -- -- 需要解决二进制分类基本核心方程式系统 -- -- 需要遵循深层次的统计法。我们显示,最低风险二进制分类系统的一个差异性功能,是代表系统决定界限几何极极地的立体方程式的解决方案,其中差异性功能与决定界限的内在偏差协调系统相连,其内在的偏差系统以内在的偏差坐标系统与决定界限相联,其形式是:盘点功能由新颖的主要等离子体的几何极分级中心 -- -- 由概率组成部分和主要偏差组成部分的双重中心形成 -- -- 其最小风险二进制分类系统的最小风险和风险。 我们显示,最小风险二进制分类系统通过定位一个平衡点来共同尽量减少其易分能和风险。 该系统展示的临界最小精度,其新精度主要等分数和密度的地标点表明,系统内部的精确度系统的准确度和对等值系统内部的准确度系统,其准确度系统内部的准确度系统对准的准确度系统,其准确度系统对等值的准确度系统对准,其准确度系统对准。