CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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One observation does not determine a natural cubic smoothing fit

Statement refuted

The formula $\sum_{i} (y_{i} - g (x_{i}))^{2} + λ \int_{a}^{b} ∣ D^{2} g ∣^{2}$ has a unique minimizing function for every nonempty data set whenever $λ > 0$ , including data with only one distinct input.

Facts & Assumptions

Given: The same squared-residual plus roughness criterion as the natural cubic smoothing spline, but with one distinct observed input.

[F1]

On data sets with at least two distinct inputs, the natural cubic smoothing criterion is $\sum_{i} (y_{i} - g (x_{i}))^{2} + λ \int_{a}^{b} ∣ D^{2} g ∣^{2}$ (Natural cubic smoothing spline); the counterexample below evaluates this same formula on one input, outside that definition's domain.

[A1]

The extended criterion has one input $x_{1} = 1 / 2$ , response $y_{1} = 1$ , and $λ > 0$ .

Counterexample

Take the domain $[0, 1]$ , one observation $(x_{1}, y_{1}) = (1 / 2, 1)$ , and $λ > 0$ . Extend the displayed criterion formula to this data set. For every $c \in R$ , set $g_{c} (x) = 1 + c (x - 1 / 2)$ . Then $g_{c} (1 / 2) = 1$ and $D^{2} g_{c} = 0$ .

Given: The one-point criterion $J (g) = (1 - g (1 / 2))^{2} + λ \int_{0}^{1} ∣ g^{''} (x) ∣^{2} d x$ .

1.1A1F1algebra

For every $c$ , both terms in [F1] vanish at $g_{c}$ , so $J (g_{c}) = 0$ . The criterion is a sum of nonnegative terms, hence zero is its minimum.

2.1A1step 1.1algebra∎

If $c \neq d$ , then $g_{c} (1) = 1 + c / 2 \neq 1 + d / 2 = g_{d} (1)$ , so $g_{c}$ and $g_{d}$ are distinct minimizers. The one-observation extension is not unique, proving the lower bound of two distinct inputs is necessary.

Depends on

Natural cubic smoothing spline

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ryan Tibshirani, Nonparametric Regression: Splines and RKHS Methods, §2.3 (standard reference, not scraped)