tautfit

About tautfit

tautfit implements the approach to nonparametric regression of Davies, Kovac & Meise (2009). Instead of choosing a bandwidth or a penalty, it first describes every function that is consistent with the data, then picks the simplest of them.

The confidence region Aₙ

A function g is consistent with the data when no interval of the data shows a systematic residual: for every interval I in a multiresolution family (λ = 2 gives dyadic blocks of 1, 2, 4, … consecutive x values), |Σ_I(y − g)| / √|I| ≤ c, with c = σ·√(τ·log n) (eqs. 6–7). σ is estimated from the data, and τ = 3 gives about 95% coverage for large n, or τ can be calibrated by Monte Carlo for the actual n. The residual view under the chart shows every interval’s |w(I)|/c; cells above 1 are violations. Intervals are intervals of x, so tied observations are never separated.

Fits: the simplest functions in Aₙ

• Taut string: approximately the fewest local extremes of any function in Aₙ (§3, with local squeezing).
• min TV(g), TV(g′), TV(g″): least total variation of the function, its slope or its curvature, solved as linear programs (§4.1). They give a step function, a linear spline and a smooth curve.
• min ‖g′‖∞, ‖g″‖∞: the smallest bound on the slope or curvature (eq. 46). min ‖g″‖∞ gives K_min, the least curvature any consistent function needs.
• min TV(g″) + taut-string shape: smooth, with the taut string’s monotone pieces (§4.2).

The LP fits can also be constrained to be constant, linear, decreasing, increasing, convex or concave: the simplest function in Aₙ with that shape. If there is none, the readouts give the smallest τ that would admit one and how much of the time Aₙ at that τ covers the truth, which is a test of the shape.

Confidence bands

Aₙ alone does not bound f at a point, but with a shape assumption it does: monotone, piecewise monotone, convex/concave, piecewise convex, or |f″| ≤ K (§5). The fast bounds take O(n log n); exact LP bounds (2n linear programs, n ≤ 300) can be drawn for comparison. A band that is empty somewhere means the assumption is inconsistent with the data. Their coverage rests on Aₙ holding for all intervals, so the readout says which τ supports it.

Departures from the paper

A few formulas in the arXiv version needed corrections (for example, the KKT conditions of the taut string, the convex lower bound and the smoothness bounds), and the coverage claim of eq. 11 holds only marginally for all intervals. The project plan lists each issue with the evidence: PLAN.md, section 2.

Using the page

• Paste or drop CSV/TSV data, or choose a simulated signal. Hover the chart and the residual view for exact values.
• Hover, focus or tap an ⓘ for an explanation of a control or readout; a click keeps it open.
• Gallery shows several simulated signals at once with the same settings, each chosen above its chart; a pane's ↗ opens it in the single view with its readouts. Panes leave out the exact LP bounds, and the Aₙ boxes when they follow the residual view.
• The ↺ ↶ ↷ buttons above the settings reset them to their defaults, undo and redo (⌘Z and ⇧⌘Z, or Ctrl+Z and Ctrl+Shift+Z); they leave the data as they are.
• The address bar always holds the current settings, so reloading keeps them and Copy link shares them. Pasted data is included when it compresses to a short enough link.
• Export saves x, y, every fit shown and the band as CSV, one row per observation, and the main chart as SVG, PNG or JPEG, as drawn and in the current color scheme (the residual view is not included).

P. L. Davies, A. Kovac, M. Meise (2009). Nonparametric regression, confidence regions and regularization. Ann. Statist. 37(5B):2597–2625. arXiv:0711.0690. LP solver: HiGHS (MIT license), compiled to WebAssembly.