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I have data that are strictly increasing and would like to fit a smoothing spline that is monotonically increasing as well with the smooth.spline() function if possible, due to the ease of use of this function.
For example, my data can be effectively reproduced with the example:
testx <- 1:100
testy <- abs(rnorm(length(testx)))^3
testy <- cumsum(testy)
plot(testx,testy)
sspl <- smooth.spline(testx,testy)
lines(sspl,col="blue")
which is not necessarily increasing everywhere. Any suggestions?
919
To tell if a function is monotonically increasing, simply find its derivative and see if it is always positive on its domain. If the derivative of a function is always positive (or greater than or equal to zero), then the function is monotonically increasing.
Smoothing splines are a powerful approach for estimating functional relationships between a predictor X and a response Y. Smoothing splines can be fit using either the smooth. spline function (in the stats package) or the ss function (in the npreg package).
Cubic smoothing splines embody a curve fitting technique which blends the ideas of cubic splines and curvature minimization to create an effective data modeling tool for noisy data.
Here, \lambda is the smoothing parameter guiding the trade-off fitting the data and roughness of function. To estimate the. we perform the generalized cross-validation or restricted marginal likelihood. No smoothing, the spline converges to interpolating spline.
You could use shape-constrained splines for this, e.g. using the scam package:
require(scam)
fit = scam(testy~s(testx, k=100, bs="mpi", m=5),
family=gaussian(link="identity"))
plot(testx,testy)
lines(testx,predict(fit),col="red")
Or if you would like to use L1 loss as opposed to L2 loss, which is less sensitive to outliers, you could also use the cobs package for this...
Advantage of this method compared to the solution above is that it also works if the original data perhaps are not 100% monotone due to the presence of noise...
74
This doesn't use smooth.spline() but the splinefun(..., method="hyman") will fit a monotonically increasing spline and is also easy to use. So for example:
testx <- 1:100
testy <- abs(rnorm(length(testx)))^3
testy <- cumsum(testy)
plot(testx,testy)
sspl <- smooth.spline(testx,testy)
lines(sspl,col="blue")
tmp <- splinefun(x=testx, y=cumsum(testy), method="hyman")
lines(testx[-1], diff(tmp(testx)), col="red")
Yields the following figure (red are the values from the monotonically increasing spline)
From the help file of splinefun: "Method "hyman" computes a monotone cubic spline using Hyman filtering of an method = "fmm" fit for strictly monotonic inputs. (Added in R 2.15.2.)"
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