# One parameter is always enough

• Published in 2018
We construct an elementary equation with a single real valued parameter that is capable of fitting any “scatter plot” on any number of points to within a fixed precision. Specifically, given given a fixed $\epsilon \gt 0$, we may construct $f_\theta$ so that for any collection of ordered pairs $\{(x_j,y_j)\}_{j=0}^n$ with $n,x_j \in \mathbb{N}$ and $y_j \in (0,1)$, there exists a $\theta \in [0,1]$ giving $|f_\theta(x_j)-y_j| \lt \epsilon$ for all $j$ simultaneously. To achieve this, we apply prior results about the logistic map, an iterated map in dynamical systems theory that can be solved exactly. The existence of an equation $f_\theta$ with this property highlights that “parameter counting” fails as a measure of model complexity when the class of models under consideration is only slightly broad.

### BibTeX entry

@article{OneParameterIsAlwaysEnough,
title = {One parameter is always enough},
abstract = {We construct an elementary equation with a single real valued parameter that is capable of fitting any “scatter plot” on any number of points to within a fixed precision.  Specifically, given given a fixed $\epsilon \gt 0$, we may construct $f{\_}\theta$ so that for any collection of ordered pairs $\{\{}(x{\_}j,y{\_}j)\{\}}{\_}{\{}j=0{\}}^n$ with $n,x{\_}j \in \mathbb{\{}N{\}}$ and $y{\_}j \in (0,1)$, there exists a $\theta \in [0,1]$ giving $|f{\_}\theta(x{\_}j)-y{\_}j| \lt \epsilon$ for all $j$ simultaneously. To achieve this, we apply prior results about the logistic map, an iterated map in dynamical systems theory that can be solved exactly.  The existence of an equation $f{\_}\theta$ with this property highlights that “parameter counting” fails as a measure of
model complexity when the class of models under consideration is only slightly broad.},
}