Two-parameter superposable S-curves

TL;DR

This paper addresses modeling sigmoidal and bell-shaped data with an algebraic two-parameter S-curve family derived from a singular perturbation of $y=mx$. The main approach introduces $S_{\text{a-m}}$ curves defined by $ay^3+y=mx$ and extends them via linear superposition to fit nonlinear patterns, with analysis of limiting cases $a\to0$ and $a\to\infty$. The authors apply these models to the Iris dataset, deriving parameter-based measures such as $a$, $m$, and $m/(1+a)$, and discuss the method's ability to represent distributions from uniform to degenerate and to capture unimodal and multimodal shapes, along with limitations like sensitivity to initialization and increasing parameter count. The work provides a novel, bounded, interpretable framework for distribution fitting and pattern recognition in real data.

Abstract

Straight line equation $y=mx$ with slope $m$, when singularly perturbed as $ay^3+y=mx$ with a positive parameter $a$, results in S-shaped curves or S-curves on a real plane. As $a\rightarrow 0$, we get back $y=mx$ which is a cumulative distribution function of a continuous uniform distribution that describes the occurrence of every event in an interval to be equally probable. As $a\rightarrow\infty$, the derivative of $y$ has finite support only at $y=0$ resembling a degenerate distribution. Based on these arguments, in this work, we propose that these S-curves can represent maximum entropy uniform distribution to a zero entropy single value. We also argue that these S-curves are superposable as they are only parametrically nonlinear but fundamentally linear. So far, the superposed forms have been used to capture the patterns of natural systems such as nonlinear dynamics of biological growth and kinetics of enzyme reactions. Here, we attempt to use the S-curve and its superposed form as statistical models. We fit the models on a classical dataset containing flower measurements of iris plants and analyze their usefulness in pattern recognition. Based on these models, we claim that any non-uniform pattern can be represented as a singular perturbation to uniform distribution. However, our parametric estimation procedure have some limitations such as sensitivity to initial conditions depending on the data at hand.

Figures (10)

• Figure 1: S- and bell- shaped curves of modified exponential function and modified (singularly perturbed) straight lines.
• Figure 2: Fitting superposed $S_{\text{a-m}}$ curves on S- and bell- shaped curves obtained by modifying the exponential function. We get better fits as more $S_{\text{a-m}}$ curves are added to the superposition.
• Figure 3: Variation of parameters $a$ and $m$ based on the fits on $S_{\text{exp}}$ for two different intervals $x\in[-3,3]$ (shown as '$\times$') and $x\in[-5,5]$ (shown as '$\bullet$') with $n$ as the $x-$ axis.
• Figure 4: Variation of parameters $a$ and $m$ based on the fits on Gauss error function whose derivative is $\tilde{y}$ for two different initial conditions $p_i=1,\ m_i=1$ (shown as '$\times$') and $p_i=0,\ m_i=\text{slope at }x_{ci}\text{ between data points}$ (shown as '$\bullet$') with $n$ as the $x-$ axis.
• Figure 5: Percentage nonlinearity measures against the number of $S_{\text{a-m}}$ curves in superposition for (a) sigmoid function under different $x$ intervals (b) Gauss error function for different initial conditions (c) sigmoid and Gauss error function under the same initial conditions and $x$ interval.