Solved – Strictly increasing interpolation / spline

interpolationr

I have points in the x-y-plane that are strictly increasing most of the time. The problem is that there are cases with one or two outliers (Knots where an out-of-the-box spline would be decreasing). Without deleting any data points, is there a way to interpolate / create a spline that is strictly increasing everywhere? Also, I would like the interpolation to be $C^1$. (Which package could do this in R?)

Best Answer

R package cobs allows you to fit shape-constrained splines, including monotonically increasing ones; syntax would be something like:

require(cobs)
fit = cobs(x,y,
      constraint= "increase", 
       lambda=0, 
       degree=1, # for L1 roughness
       knots=seq(min(x),max(x),length.out=10), # desired nr of knots 
       tau=0.5) # to predict median
preds = predict(fit,interval="none",z=xvals)[,2]

And R packages ConSpline, scar, scam and cgam offer also alternative options to fit shape-constrained splines, including monotonically increasing ones....

Related Solutions

Solved – How to interpolate (spline, LOESS, etc.) with a mix of critical and non-critical points

You may want to use weighted loess, as in:

a = data.frame(x=1:10, y=c(2,5,3,4,5,9,7,8,9,10))
fit1 = loess(y~x, a)  # unweighted loess fit
fit2 = loess(y~x, a, weights = c(0.1,1,0.1,0.1,0.1,0.1,1,0.1,0.1,0.1))
plot(1:10, predict(fit1), col="blue", type="l", ylim=c(1,10)) # unweighted fit
lines(1:10, predict(fit3), col="darkgreen") # weighted fit
points(y~x, a, col="brown", type="b") # all points 
points(y~x, a[c(2,7),], col="red") # points of high importance

plot

One issue with this approach is that weighted regression assumes that weights are known exactly, while this is very rarely the case. Some ideas on estimating weights for the case of weighted linear regression can be found here. Quite often however even a vague idea on differences in weights may work better than nothing. For example, here even though there was no guarantee that the loess fitted line will go exactly through the "important" points, this was very nearly the case with a rather arbitrary choice of weights.