高自由度S形曲线函数研究

Investigation of the Function with High Degree of Freedom for the S-shape Curve

摘要: S形曲线函数对空间几何及概率统计等的发展具有促进意义,本研究在分析Logistic函数基本特性的基础上调制出了高自由度的S形曲线函数y=A_2+\fracA_1-A_21+\left(\fracx-x_0\Delta x_1\right)^p_1+\left(\fracx-x_0\Delta x_2\right)^p_2。其特征参数为极值A₁和A₂、极值A₁所在位置x₀,形状调制参数为Δx₁、p₁、Δx₂和p₂,函数曲线形状为S形的参数条件为A₁≠A₂、Δx₁＞0、Δx₂＞0、p₁＞1和p₂＞1。并推导了调制Logistic函数的一阶导数函数(即切线斜率函数)、法线斜率函数和二阶导数函数。

Abstract: The S-shape curve function with high degree of freedom is proposed based on the analysis of the basic characteristics of the Logistic function, and the S-shape curve function is y=A_2+\fracA_1-A_21+\left(\fracx-x_0\Delta x_1\right)^p_1+\left(\fracx-x_0\Delta x_2\right)^p_2. The characteristic parameters of the function are extreme values of A₁ and A₂, and the location of extreme value A₁ is x₀. The shape modulatory parameters of function are Δx₁, p₁, Δx₂ and p₂. The conditions assuring the S shape of the function’s curve are A₁≠A₂, Δx₁>0, Δx₂>0, p₁>1 and p₂>1. The functions of the first derivative (the slope of the tangent), the slope of the normal and the second derivative of the modulatory Logistic function are derived.