ShepFuns: Test function ShepFun1 from SHEPPACK

In IRSN/smint: Smooth Multivariate Interpolation for Gridded and Scattered Data

Test function ShepFun1 from SHEPPACK

Description

Test functions for/from SHEPPACK

Usage

ShepFun1(x)

ShepFun2(x)

ShepFun3(x)

ShepFun4(x)

ShepFun5(x)

Arguments

x	A numeric vector with arbitrary length.

Details

These functions are described in the article cited in the references section.

f_1(\mathbf{x}) = 1 + \frac{2}{d} \, \left| d/2 - \sum_i x_i \right|

f_2(\mathbf{x}) = 1 - \frac{2}{d}\, \sum_i \left|x_i - 0.5 \right|

f_3(\mathbf{x}) = 1 - 2 \, \max_{i} \left|x_i - 0.5 \right|

f_4(\mathbf{x}) = \prod_i \left[ 1 - 2 \left| x_i - 0.5 \right| \right]

f_5(\mathbf{x}) = 1 - c_5 \, \left[ \sum_i \left|x_i - 0.5 \right| + \prod_i \left| x_i - 0.5 \right| \right]

where c_5 = d/2 + (0.5)^d, and all sums or products are for i=1 to d. All these functions are defined on [0,\,1]^d and take values in [0,1]. The four functions f_i for i > 1 have an unique maximum located at \mathbf{x}^\star with all coordinates x_j^\star = 0.5 and f_i(\mathbf{x}^\star) =1.

Value

Function's value.

Note

These functions are also exported as elements of the ShepFuns list.

References

W.I. Thacker, J. Zhang, L.T. Watson, J.B. Birch, M.A. Iyer and M.W. Berry (2010). Algorithm 905: SHEPPACK: Modified Shepard Algorithm for Interpolation of Scattered Multivariate Data ACM Trans. on Math. Software (TOMS) Vol. 37, n. 3. link

Examples

## interpolate 'Shepfun3' for d = 4
d <- 4
GDd <- Grid(nlevels = rep(8, d))
fGrid <- apply_Grid(GDd, ShepFun3)
Xoutd <- matrix(runif(200 * d), ncol = d)
GI <- interp_Grid(X = GDd, Y = fGrid, Xout = Xoutd)
F <- apply(Xoutd, 1, ShepFun3)
max(abs(F - GI))

## 3D plot
if (requireNamespace("lattice")) {
  X <- as.data.frame(Grid(nlevels = c("x1" = 30, "x2" = 30)))
  df <- data.frame(x1 = numeric(0), x2 = numeric(0),
                   f = numeric(0), i = numeric(0))
  for (i in 1:5) {
     f <- apply(X, 1, ShepFuns[[i]])
     df <- rbind(df, data.frame(x1 = X$x1, x2 = X$x2, f = f, i = i))
  }
  pl <- lattice::wireframe(f ~ x1 * x2 | i, data = df,
                  outer = TRUE, shade = FALSE, zlab = "",
                  screen = list(z = 20, x = -30),
                  strip = lattice::strip.custom(strip.names = c(TRUE),
                                       strip.levels = c(TRUE)),
                  main = "", horizontal = TRUE, col = "SpringGreen4")
  pl
}

IRSN/smint documentation built on Dec. 9, 2023, 9:53 p.m.