cobs (version 0.9-3)

cobs: COnstrained B-Splines Nonparametric Regression Quantiles

Description

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

Usage

cobs(x, y, constraint = c("none", "increase", "decrease",
                          "convex", "concave", "periodic"),
     knots, nknots, 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)

Arguments

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

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 "quantil

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: [object Object],[object Object],[object Object],[object Object]

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 $\ge 0$ 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 $(> 1 & < 4)$ will save time for big problems.

n

integer, the sample size.

Value

• 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: [object Object],[object Object],[object Object],[object Object]
• 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.

Warning

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

Details

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.

References

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

See Also

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

Examples

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)