cobs: COnstrained B-Splines Nonparametric Regression Quantiles
In cobs99: Constrained B-splines -- outdated 1999 version

Description Usage Arguments Details Value Warning References See Also Examples

Computes constrained quantile curves using linear or quadratic splines. The median spline (L_1 loss) is a robust (constrained) smoother.

cobs(x, y, constraint = c("none", "increase", "decrease",
                          "convex", "concave", "periodic"),
     knots, nknots = if(lambda == 0) 6 else 20, method = "quantile",
     degree = 2, tau = 0.5, lambda = 0, ic = "aic",
     n.sub = n1000cut(n),
     knots.add = FALSE, pointwise,
     print.warn = TRUE, print.mesg = TRUE, trace = print.mesg,
     coef = rep(0,nvar), w = rep(1,n),
     maxiter = 20*n, lstart = 7872, toler.kn = 1e-6,
     eps = .Machine$double.eps, factor = 1)

n1000cut(n)

`x`	vector of covariate; missing values are omitted.
`y`	vector of response variable. It must have the same length as `x`.
`constraint`	character (string) specifying the kind of constraint; must be one of the values in the default list, above; may be abbreviated.
`knots`	vector of locations of the knot mesh; if missing, `nknots` number of `knots` will be created using the specified `method` and automatic knot selection will be carried out for regression B-spline (`lambda=0`); if not missing and `length(knots)==nknots`, the provided knot mesh will be used in the fit and no automatic knot selection will be performed; otherwise, automatic knots selection will be performed on the provided `knots`.
`nknots`	maximum number of knots; defaults to 6 for regression B-spline, 20 for smoothing B-spline.
`method`	character specifying the method for generating `nknots` number of `knots` when `knots` is not provided; "quantile" (equally spaced in percentile levels) or "uniform" (equally spaced knots); defaults to "quantile".
`degree`	degree of the splines; 1 for linear spline and 2 for quadratic spline; defaults to 2.
`tau`	desired quantile level; defaults to 0.5 (median).
`lambda`	penalty parameter; lambda = 0: no penalty (regression B-spline); lambda > 0: smoothing B-spline with the given lambda; lambda < 0: smoothing B-spline with lambda chosen by an information criterion, see `ic`.
`ic`	information criterion used in knot deletion and addition for regression B-spline method when lambda=0; "aic" (Akaike-type) or "sic" (Schwarz-type); default to "aic".
`n.sub`	integer, not larger than sample size `n`; the default has `n.sub == n` as long as `n` is less than 1000.
`knots.add`	logical indicating if an additional step of stepwise knot addition should be performed for regression B-splines.
`pointwise`	an optional three-column matrix with each row specifies one of the following constraints: `( 1,xi,yi)`: fitted value at xi will be >= yi; `(-1,xi,yi)`: fitted value at xi will be <= yi; `( 0,xi,yi)`: fitted value at xi will be = yi; `( 2,xi,yi)`: derivative of the fitted function at xi will be yi.
`print.warn`	logical flag for printing of interactive warning messages; true by default; probably needs to be set to `FALSE` if performing simulation.
`print.mesg`	logical flag or integer for printing of intermediate messages; true by default. Probably needs to be set to `FALSE` in simulations.
`trace`	integer >= 0 indicating how much the Fortran routine `drqssbc` should print intermediate messages; defaults to `print.mesg`.
`coef`	initial guess of the B-spline coefficients; default to a vector of zeros.
`w`	vector of weights the same length as x (y) assigned to both x and y; default to uniform weights adding up to one; using normalized weights that add up to one will speed up computation.
`maxiter`	upper bound of the number of iteration; default to 20*n.
`lstart`	starting value for lambda when performing parametric programming in lambda if `lambda < 0`; defaults to `log(big)^2`.
`toler.kn`	numeric tolerance for shifting the boundary knots outside; default 1e-6 used to be built in.
`eps`	tolerance passed to `qbsks` and `drqssbc`.
`factor`	determines how big a step to the next smaller lambda should be while performing parametric programming in lambda; the default `1` will give all unique lambda's; use of a bigger factor (> 1 and < 4) will save time for big problems.

`n`	integer, the sample size.

cobs() computes the constraint quantile smoothing B-spline with penalty when lambda is not zero.
If lambda < 0, an optimal lambda will be chosen using Schwarz type information criterion.
If lambda > 0, the supplied lambda will be used.
If lambda = 0, cobs computes the constraint quantile regression B-spline with no penalty using the provided knots or those selected by Akaike or Schwarz information criterion.

an object of class cobs, a list with components

`call`	the `cobs(..)` call used for creation.
`tau, degree`	as input
`constraint`	as input (but no more abbreviated).
`call`	the `cobs(..)` call used for creation.
`coef`	B-spline coefficients.
`knots`	the final set of knots used in the computation.
`ifl`	exit code: 1 – ok; 2 – problem is infeasible, check specification of the `pointwise` argument; 3 – maxiter is reached before finding a solution, either increase maxiter and restart the program with `coef` set to the value upon previous exit or use a smaller `lstart` value when lambda<0 or use a smaller `lambda` value when lambda>0; 4 – program aborted, numerical difficulties due to ill-conditioning.
`icyc`	length 2: number of cycles taken to achieve convergence for final lambda, and total number of cycles for all lambdas.
`k`	the effective dimensionality of the final fit.
`k0`	(usually the same)
`x.ps`	the pseudo design matrix X (as returned by `qbsks`).
`resid`	vector of residuals from the fit.
`fitted`	vector of fitted values from the fit.
`SSy`	the sum of squares around centered `y` (e.g. for computation of R^2.)
`lambda`	the penalty parameter used in the final fit.
`pp.lambda`	vector of all unique lambda's obtained from parametric programming when `lambda` < 0 on input.
`sic`	vector of Schwarz information criteria evaluated at `pp.lambda`.

This is still a beta version, and we do appreciate comments and suggestions; library(help = cobs) shows the authors.

He, X. and Ng, P. (1999) COBS: Qualitatively Constrained Smoothing via Linear Programming; Computational Statistics 14, 315–337.

Koenker, R. and Ng, P. (1996) A Remark on Bartels and Conn's Linearly Constrained L1 Algorithm, ACM Transaction on Mathematical Software 22, 493–495.

Ng, P. (1996) An Algorithm for Quantile Smoothing Splines, Computational Statistics & Data Analysis 22, 99–118.

Bartels, R. and Conn A. (1980) Linearly Constrained Discrete L_1 Problems, ACM Transaction on Mathematical Software 6, 594–608.

A postscript version of the paper that describes the details of COBS can be downloaded from http://www.cba.nau.edu/pin-ng/cobs.html

smooth.spline for unconstrained smoothing splines; bs for unconstrained (regression) B-splines.

x <- seq(-1,1,,50)
y <- (f.true <- pnorm(2*x)) + rnorm(50)/10
## specify pointwise constraints (boundary conditions)
con <- rbind(c( 1,min(x),0), # f(min(x)) >= 0
             c(-1,max(x),1), # f(max(x)) <= 1
             c(0,  0,   0.5))# f(0)      = 0.5

## obtain the median regression B-spline using automatically selected knots
Rbs <- cobs(x,y,constraint="increase",pointwise=con)
Rbs

plot(x,y)
lines(predict(Rbs), col = 2, lwd = 1.5)
lines(spline(x,f.true), col = "gray40")

Rbsub <- cobs(x,y,constraint="increase",pointwise=con, n.sub = 45)

summary(Rbsub)
lines(predict(Rbsub), col = 4, lwd = 1)


## compute the median smoothing B-spline using automatically chosen lambda
Sbs <- cobs(x,y,constraint="increase",pointwise=con,lambda=-1)
Sbs
plot(Sbs$pp.lambda[-1], Sbs$sic[-1], log = "x",
     main = "SIC ~ lambda", xlab = expression(lambda), ylab = "SIC")
axis(1, at = Sbs$lambda, label = expression(hat(lambda)),
     col.axis = 2, mgp = c(3, 0.5, 0))

Sb1 <- cobs(x,y,constraint="increase",pointwise=con,lambda=-1, degree=1)
summary(Sb1)
pxx <- predict(Sb1, xx <- seq(-1.2, 1.2, len = 201), interval = "both")
plot(x,y, main = deparse(Sb1$call),
     xlim = range(xx), ylim = range(y, pxx[,"fit"]))
lines(pxx, col = 2)
rug(Sb1$knots, col = 4, lwd = 1.6)# (too many knots)
matlines(pxx[,1], pxx[,-(1:2)], col= rep(c("green","blue"),c(2,2)), lty=2)