One parameter is always enough

• Published in 2018
• Added on 2018-06-06

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.

Comment

John von Neumann said that with four free parameters he could make an elephant, and with five he could make it wiggle its trunk. Steven T. Piantadosi did better than that and made an equation with a single parameter that can fit any scatter plot to a given precision.

Links

• https://colala.berkeley.edu/papers/piantadosi2018one.pdf

Other information

BibTeX entry

@article{OneParameterIsAlwaysEnough,
	key = {OneParameterIsAlwaysEnough},
	type = {article},
	title = {One parameter is always enough},
	author = {Steven T. Piantadosi},
	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.},
	comment = {John von Neumann said that with four free parameters he could make an elephant, and with five he could make it wiggle its trunk. Steven T. Piantadosi did better than that and made an equation with a single parameter that can fit any scatter plot to a given precision.},
	date_added = {2018-06-06},
	date_published = {2018-07-13},
	urls = {https://colala.berkeley.edu/papers/piantadosi2018one.pdf},
	collections = {animals,attention-grabbing-titles,fun-maths-facts,modelling,probability-and-statistics},
	url = {https://colala.berkeley.edu/papers/piantadosi2018one.pdf},
	urldate = {2018-06-06},
	year = 2018
}