File:One parameter is always enough.pdf

English: We construct an elementary equation fθ(x) with a single real valued parameter θ ∈ [0, 1] that, as θ varies, is capable of fitting any scatter plot on any number of points to within a fixed precision. Specifically, given ϵ > 0, we may construct fθ so that for any collection of ordered pairs {(xj,yj)}nj=0 with n,xi∈ℕ and yi ∈ (0, 1), there exists a θ ∈ [0, 1] giving |fθ(xj) − yj| < ϵ for all j simultaneously. To achieve this, we apply results about the logistic map, an iterated map in dynamical systems theory that can be solved exactly. The existence of an equation fθ 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.