cgam (version 1.21)

s.decr: Specify a Smooth and Decreasing Shape-Restriction in a CGAM Formula

Description

A symbolic routine to define that the systematic component \(\eta\) is smooth and decreasing in a predictor in a formula argument to cgam. This is the smooth version.

Usage

s.decr(x, numknots = 0, knots = 0, var.knots = 0, space = "Q", db.exp = FALSE)

Value

The vector x with five attributes, i.e, name: the name of x; shape: 10("smooth and decreasing"); numknots: the numknots argument in "s.decr"; knots: the knots argument in "s.decr"; space: the space argument in "s.decr".

Arguments

x

A numeric predictor which has the same length as the response vector.

numknots

The number of knots used to constrain \(x\). It will not be used if the user specifies the knots argument. The default is numknots = \(0\).

knots

The knots used to constrain \(x\). User-defined knots will be used when given. Otherwise, numknots and space will be used to create knots. The default is knots = \(0\).

var.knots

The knots used in variance function estimation. User-defined knots will be used when given. The default is var.knots = \(0\).

space

A character specifying the method to create knots. It will not be used if the user specifies the knots argument. If space == "E", then equally spaced knots will be created; if space == "Q", then a vector of equal \(x\) quantiles will be created based on \(x\) with duplicate elements removed. The number of knots is numknots when numknots \(> 0\). Otherwise it is of the order \(n^{1/7}\). The default is space = "Q".

db.exp

The parameter will be used in variance function estimation. If db.exp = TRUE, then the errors are assumed to follow a normal distribution; otherwise, the errors are assumed to follow a double-exponential distribution. The default is db.exp = FALSE.

Author

Mary C. Meyer and Xiyue Liao

Details

"s.decr" returns the vector "x" and imposes on it five attributes: name, shape, numknots, knots, space, var.knots and db.exp.

The name attribute is used in the subroutine plotpersp; the numknots, knots and space attributes are the same as the numknots, knots and space arguments in "s.decr"; the shape attribute is 10("smooth and decreasing"). According to the value of the vector itself and its shape, numknots, knots and space attributes, the cone edges will be made by I-spline basis functions in Meyer (2008). The cone edges are a set of basis employed in the hinge algorithm.

Note that "s.decr" does not make the corresponding cone edges itself. It sets things up to a subroutine called makedelta in cgam.

var.knots and db.exp will be used for monotonic variance function estimation.

See references cited in this section for more details.

References

Meyer, M. C. (2013b) A simple new algorithm for quadratic programming with applications in statistics. Communications in Statistics 42(5), 1126--1139.

Meyer, M. C. (2008) Inference using shape-restricted regression splines. Annals of Applied Statistics 2(3), 1013--1033.

See Also

decr

Examples

  data(cubic)

  # extract x
  x <- - cubic$x

  # extract y
  y <- cubic$y

  # regress y on x under the shape-restriction: "smooth and decreasing"
  ans <- cgam(y ~ s.decr(x))
  knots <- ans$knots[[1]]

  # make a plot
  par(mar = c(4, 4, 1, 1))
  plot(x, y, cex = .7, xlab = "x", ylab = "y")
  lines(x, ans$muhat, col = 2)
  legend("topleft", bty = "n", "smooth and decreasing fit", col = 2, lty = 1)
  legend(-.3, 8, bty = "o", "knots", pch = "X")
  points(knots, 1:length(knots)*0+min(y), pch = "X")