mono.con {mgcv}  R Documentation 
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)
x 
The array of knot locations. 
up 
If 
lower 
This specifies the lower bound on the spline unless it is

upper 
This specifies the upper bound on the spline unless it is

Consider the natural cubic spline passing through the points (x_i,p_i), i=1..n. Then it is possible to find a relatively small set of linear constraints on p sufficient to ensure monotonicity (and bounds if required): Ap >= b. Details are given in Wood (1994).
a list containing constraint matrix A
and constraint vector b
.
Simon N. Wood simon.wood@rproject.org
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.
http://www.maths.bris.ac.uk/~sw15190/
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