A symbolic routine to define that a surface is decreasing in two predictors in a formula argument to wps.
1 
x1 
A numeric predictor which has the same length as the response vector. 
x2 
A numeric predictor which has the same length as the response vector. 
numknots 
A vector of the number of knots used to constrain x_1 and x_2. It will not be used if the user specifies the knots argument and each predictor is within the range of its knots. The default is numknots = c(0, 0). 
knots 
A list of two vectors of knots used to constrain x_1 and x_2. Userdefined knots will be used if each predictor is within the range of its knots. Otherwise, numknots and space will be used to create knots. The default is knots = list(k1 = 0, k2 = 0). 
space 
A vector of the character specifying the method to create knots for x_1 and x_2. It will not be used if the user specifies the knots argument. If "E" is used, then equally spaced knots will be created; if "Q" is used, then a vector of equal quantiles will be created with duplicate elements removed. The number of knots is numknots when numknots is a positive integer > 4. Otherwise it is of the order n^{1/6}. The default is space = c("E", "E"). 
"dd" returns the vectors "x1" and "x2", and imposes on each vector six attributes: name, shape, numknots, knots, space and decreasing.
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 "dd"; the shape attribute is "wps_dd"(doublydecreasing); the decreasing values for both vectors are TRUE. According to the value of the vector itself and its shape, numknots, knots, space and decreasing attributes, the cone edges will be made by warpedplane spline basis functions in Meyer (2016). The cone edges are a set of basis employed in the hinge algorithm.
Note that "dd" does not make the corresponding cone edges itself. It sets things up to a subroutine called makedelta_wps in wps.
See references cited in this section for more details.
The vectors x_1 and x_2. Each of them has six attributes, i.e., name: names of x_1 and x_2; shape: "wps_dd"(doublydecreasing); numknots: the numknots argument in "dd"; knots: the knots argument in "dd"; space: the space argument in "dd"; decreasing: two logical values indicating the monotonicity of the isotonicallyconstrained surface with respect to x_1 and x_2, which are both TRUE.
Mary C. Meyer and Xiyue Liao
Meyer, M. C. (2016) Estimation and inference for isotonic regression in two dimensions, using warpedplane splines.
ii
, di
, wps
1 2 3 4 5 6 7 8 9 10 11 12 13  # generate data
n < 100
set.seed(123)
x1 < runif(n)
x2 < runif(n)
y < 4 * (x1 + x2) + rnorm(n, sd = .2)
# regress y on x1 and x2 under the shaperestriction: "doublydecreasing"
# using the penalized estimator
ans < wps(y ~ dd(x1, x2), pnt = TRUE)
# make a 3D plot of the constrained surface
plotpersp(ans)

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