cobs (version 1.3-8)

cobs: COnstrained B-Splines Nonparametric Regression Quantiles


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"),
     w = rep(1,n),
     knots, nknots = if(lambda == 0) 6 else 20,
     method = "quantile", degree = 2, tau = 0.5,
     lambda = 0, ic = c("AIC", "SIC", "BIC", "aic", "sic", "bic"),
     knots.add = FALSE, repeat.delete.add = FALSE, pointwise = NULL, = TRUE, = TRUE,
     print.warn = TRUE, print.mesg = TRUE, trace = print.mesg,
     lambdaSet = exp(seq(log(lambda.lo), log(lambda.hi), length.out = lambda.length)),
     lambda.lo = f.lambda*1e-4, lambda.hi = f.lambda*1e3, lambda.length = 25,
     maxiter = 100,
     rq.tol = 1e-8, = 1e-6, tol.0res = 1e-6, nk.start = 2)


an object of class cobs, a list with components


the cobs(..) call used for creation.

tau, degree

same as input


as input (but no more abbreviated).


as input.


B-spline coefficients.


the final set of knots used in the computation.


exit code := 1 + ierr and ierr is the error from (package quantreg); consequently, ifl = 1 means “success”; all other values point to algorithmic problems or failures.


length 2: number of cycles taken to achieve convergence for final lambda, and total number of cycles for all lambdas.


the effective dimensionality of the final fit.


(usually the same)

the pseudo design matrix \(X\) (as returned by qbsks2).


vector of residuals from the fit.


vector of fitted values from the fit.


the sum of squares around centered y (e.g. for computation of \(R^2\).)


the penalty parameter used in the final fit.


vector of all lambdas used for lambda search when lambda < 0 on input.


vector of Schwarz information criteria evaluated at pp.lambda; note that it is not quite sure how good these are for determining an optimal lambda.



vector of covariate; missing values are omitted.


vector of response variable. It must have the same length as x.


character (string) specifying the kind of constraint; must be one of the values in the default list above; may be abbreviated. More flexible constraints can be specified via the pointwise argument (below).


vector of weights the same length as x (y) assigned to both x and y; default to all weights being one.


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.


maximum number of knots; defaults to 6 for regression B-splines, 20 for smoothing B-splines.


character string specifying the method for generating nknots number of knots when knots is not provided; either "quantile" (equally spaced in percentile levels) or "uniform" (equally spaced knots); defaults to "quantile".


degree of the splines; 1 for linear spline (equivalent to \(L_1\)-roughness) and 2 for quadratic spline (corresponding to \(L_{\infty}\) roughness); defaults to 2.


desired quantile level; defaults to 0.5 (median).


penalty parameter \(\lambda\)
\(\lambda = 0\): no penalty (regression B-spline);
\(\lambda > 0\): smoothing B-spline with the given \(\lambda\);
\(\lambda < 0\): smoothing B-spline with \(\lambda\) chosen by a Schwarz-type information criterion.


string indicating the information criterion used in knot deletion and addition for the regression B-spline method, i.e., when lambda == 0;
"AIC" (Akaike-type, equivalently "aic") or
"SIC" (Schwarz-type, equivalently "BIC", "sic" or "bic"). Defaults to "AIC".

Note that the default was "SIC" up to cobs version 1.1-6 (dec.2008).


logical indicating if an additional step of stepwise knot addition should be performed for regression B-splines.


logical indicating if an additional step of stepwise knot deletion should be performed for regression B-splines.


an optional three-column matrix with each row specifies one of the following constraints:

( 1,xi,yi):

fitted value at xi will be \(\ge\) yi;


fitted value at xi will be \(\le\) yi;

( 0,xi,yi):

fitted value at xi will be \(=\) yi;

( 2,xi,yi):

derivative of the fitted function at xi will be yi.

logical indicating if the x and y input vectors after removing NAs should be kept in the result.

logical indicating if the pseudo design matrix \(\tilde{X}\) should be returned (as sparse matrix). That is needed for interval prediction, predict.cobs(*, interval=..).


flag for printing of interactive warning messages; true by default; set to FALSE if performing simulation.


logical flag or integer for printing of intermediate messages; true by default. Probably needs to be set to FALSE in simulations.


integer \(\ge 0\) indicating how much diagnostics the low-level code in drqssbc2 should print; defaults to print.mesg.


numeric vector of lambda values to use for grid search; in that case, defaults to a geometric sequence (a “grid in log scale”) from lambda.lo to lambda.hi of length lambda.length.

lambda.lo, lambda.hi

lower and upper bound of the grid search for lambda (when lambda < 0). The defaults are meant to keep everything scale-equivariant and are hence using the factor \(f = \sigma_x^d\), i.e., f.lambda <- sd(x)^degree. Note however that currently the underlying algorithms in package quantreg are not scale equivariant yet.


number of grid points in the grid search for optimal lambda.


upper bound of the number of iterations; defaults to 100.


numeric convergence tolerance for the interior point algorithm called from or

numeric tolerance for shifting the boundary knots outside; defaults to \(10^{-6}\).


tolerance for testing \(|r_i| = 0\), passed to qbsks2 and drqssbc2.


number of starting knots used in automatic knot selection. Defaults to the minimum of 2 knots.


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.


Ng, P. and Maechler, M. (2007) A Fast and Efficient Implementation of Qualitatively Constrained Quantile Smoothing Splines, Statistical Modelling 7(4), 315-328.

Koenker, R. and Ng, P. (2005) Inequality Constrained Quantile Regression, Sankhya, The Indian Journal of Statistics 67, 418--440.

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

See Also

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


Run this code
x <- seq(-1,3,,150)
y <- (f.true <- pnorm(2*x)) + rnorm(150)/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)
plot(Rbs, lwd = 2.5)
lines(spline(x, f.true), col = "gray40")
lines(predict(cobs(x,y)), col = "blue")
mtext("cobs(x,y)   # completely unconstrained", 3, col= "blue")

## compute the median  SMOOTHING  B-spline using automatically chosen lambda
Sbs <- cobs(x,y, constraint="increase", pointwise= con, lambda= -1)
plot(Sbs) ## by default  includes  SIC ~ lambda

Sb1 <- cobs(x,y, constraint="increase", pointwise= con, lambda= -1,
            degree = 1)

plot(Sb1, which = 2) # only the  data + smooth
rug(Sb1$knots, col = 4, lwd = 1.6)# (too many knots)
xx <- seq(min(x) - .2, max(x)+ .2, len = 201)
pxx <- predict(Sb1, xx, interval = "both")
lines(pxx, col = 2)
mtext(" + pointwise and simultaneous 95% - confidence intervals")
matlines(pxx[,1], pxx[,-(1:2)], col= rep(c("green3","blue"), c(2,2)), lty=2)

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