interpTwo: Monotone interpolation between a function and a reference...

In smoothemplik: Smoothed Empirical Likelihood

Monotone interpolation between a function and a reference parabola

Description

Create *piece-wise monotone* splines that smoothly join an arbitrary function 'f' to the quadratic reference curve (x-\mathrm{mean})^{2}/\mathrm{var} at a user–chosen abscissa at. The join occurs over a finite interval of length gap, guaranteeing a C1-continuous transition (function and first derivative are continuous) without violating monotonicity.

Usage

interpTwo(x, f, mean, var, at, gap)

Arguments

x	A numeric vector of evaluation points.
f	Function: the original curve to be spliced into the parabola. It must be vectorised (i.e.\ accept a numeric vector and return a numeric vector of the same length).
mean	Numeric scalar defining the shift of the reference parabola.
var	Numeric scalar defining the vertical scaling of the reference parabola.
at	Numeric scalar: beginning of the transition zone, i.e.\ the boundary where 'f' stops being evaluated and merging into the parabola begins.
gap	Positive numeric scalar. Width of the transition window; the spline is constructed on '[at, at+gap]' (or '[at-gap, at]' when 'at < mean') when the reference parabola is higher. If the reference parabola is lower, it is the distance from the point 'z' at which 'f(z) = parabola(z)' to allow some growth and ensure monotonicity.

Details

This function calls 'interpToHigher()' when the reference parabola is *above* 'f(at)'; the spline climbs from 'f' up to the parabola, and 'interpToLower()' when the parabola is *below* 'f(at)', and the transition interval has to be extended to ensure that the spline does not descend.

Internally, the helpers build a **monotone Hermite cubic spline** via Fritsch–Carlson tangents. Anchor points on each side of the transition window are chosen so that the spline’s one edge matches 'f' while the other edge matches the reference parabola, ensuring strict monotonicity between the two curves.

Value

A numeric vector of length length(x) containing the smoothly interpolated values.

See Also

[splinefun()]

Examples

xx <- -4:5  # Global data for EL evaluation
w <- 10:1
w <- w / sum(w)

f <- Vectorize(function(m) -2*EL0(xx, mu = m, ct = w, chull.fail = "none")$logelr)
museq <- seq(-6, 6, 0.1)
LRseq <- f(museq)
plot(museq, LRseq, bty = "n")
rug(xx, lwd = 4)

wm <- weighted.mean(xx, w)
wv <- weighted.mean((xx-wm)^2, w) / sum(w)
lines(museq, (museq - wm)^2 / wv, col = 2, lty = 2)

xr <- seq(4, 6, 0.1)
xl <- seq(-6, -3, 0.1)
lines(xl, interpTwo(xl, f, mean = wm, var = wv, at = -3.5, gap = 0.5), lwd = 2, col = 4)
lines(xr, interpTwo(xr, f, mean = wm, var = wv, at = 4.5, gap = 0.5), lwd = 2, col = 3)
abline(v = c(-3.5, -4, 4.5, 5), lty = 3)

smoothemplik documentation built on Aug. 22, 2025, 1:11 a.m.