Finds linear constraints sufficient for monotonicity (and
optionally upper and/or lower boundedness) of a cubic regression
spline. The basis representation assumed is that given by the
gam, "cr" basis: that is the spline has a set of knots,
which have fixed x values, but the y values of which constitute the
parameters of the spline.
mono.con(x,up=TRUE,lower=NA,upper=NA)a list containing constraint matrix A and constraint vector b.
The array of knot locations.
If TRUE then the constraints imply increase, if
FALSE then decrease.
This specifies the lower bound on the spline unless it is
NA in which case no lower bound is imposed.
This specifies the upper bound on the spline unless it is
NA in which case no upper bound is imposed.
Simon N. Wood simon.wood@r-project.org
Consider the natural cubic spline passing through the points \( \{x_i,p_i:i=1 \ldots n \} \). Then it is possible to find a relatively small set of linear constraints on \(\mathbf{p}\) sufficient to ensure monotonicity (and bounds if required): \(\mathbf{Ap}\ge\mathbf{b}\). Details are given in Wood (1994).
Gill, P.E., Murray, W. and Wright, M.H. (1981) Practical Optimization. Academic Press, London.
Wood, S.N. (1994) Monotonic smoothing splines fitted by cross validation. SIAM Journal on Scientific Computing 15(5), 1126--1133.
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