constrained.smoothspline: constrained.smoothspline
In CausalKinetiX: Learning Stable Structures in Kinetic Systems

Description Usage Arguments Details Value Author(s) References Examples

Fit a smoothing spline with constraints on derivatives

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constrained.smoothspline(y, times, pen.degree, constraint = "fixed",
  derivative.values = NA, initial.value = NA, times.new,
  num.folds = "leave-one-out", lambda = "optim")

`y`	a vector of response variables.
`times`	a vector of time points at which y was measured, same length as y.
`pen.degree`	desired degree of the derivative in smoothing penalty.
`constraint`	one of 'none', 'fixed' or 'bounded' depending on whether the derivatives should not be constrained, fixed to a constant or bounded in an interval, respectively.
`derivative.values`	either a vector with the same length as y if contraint=='fixed' or a matrix with 2 columns conatining the lower and upper bounds on the derivatives if contraint=='bounded'.
`initial.value`	optional paramter that specifies an initial value of the spline incase it should be fixed.
`times.new`	optional vector of new time points at which the spline should be evaluated.
`num.folds`	either a numeric value of the number of folds or the string "leave-one-out" for a leave one out type cross-validation in determining the penalty parameter.
`lambda`	either a numeric value if the penalty parameter is fixed explicitely or one of the values 'optim' or 'grid.search' depending on the desired optimization procedure.

For further details see the references.

a list consisting of the following elements

`smooth.vals`	predicted values at points times
`residual`	residuals
`smooth.vals.new`	predicted values at time points times.new
`smooth.deriv`	predicted derivative values at points times
`pen.par`	penality parameter used for smoothing

Niklas Pfister, Stefan Bauer and Jonas Peters

Pfister, N., S. Bauer, J. Peters (2018). Identifying Causal Structure in Large-Scale Kinetic Systems ArXiv e-prints (arXiv:1810.11776).

## Example
x <- seq(0,4,length.out=200)
x.long <- seq(0,4,length.out=200)
y <- x^2+rnorm(200,0,2)
dy <- 2*x
dybdd <- cbind(dy-0.5,dy+0.5)
plot(x,y)

ptm <- proc.time()
fit <- constrained.smoothspline(y=y,
                                times=x,
                                pen.degree=2,
                                constraint="none",
                                derivative.values=NA,
                                times.new=x.long,
                                num.folds=5,
                                lambda="optim")
print(proc.time()-ptm)
fit2 <- smooth.spline(x,y)
lines(x.long,fit[[3]],col="blue")
lines(fit2, col="green")
lines(x.long, x.long^2, col="black")