Generate X and Y values from the 10-dim “first” Friedman data set used to validate the Multivariate Adaptive Regression Splines (MARS) model, and a variation involving boolean indicators. This test function has three non-linear and interacting variables, along with two linear, and five which are irrelevant. The version with indicators has parts of the response turned on based on the setting of the indicators
friedman.1.data(n = 100)
fried.bool(n = 100)Output is a data.frame with columns
describing the 10-d randomly sampled inputs
boolean version of the indicators provided only
for fried.bool, as described above
sample responses (with N(0,1) noise)
true responses (without noise)
Number of samples desired
Robert B. Gramacy, rbg@vt.edu, and Matt Taddy, mataddy@amazon.com
In the original formulation, as implemented by friedman.1.data
the function has 10-dim inputs X are drawn from Unif(0,1), and responses
are \(N(m(X),1)\) where
\(m(\mathbf{x}) = E[f(\mathbf{x})]\) and
$$m(\mathbf{x}) = 10\sin(\pi x_1 x_2) + 20(x_3-0.5)^2 + 10x_4 + 5x_5$$
The variation fried.bool uses indicators
\(I\in \{1,2,3,4\}\). The function also has 10-dim
inputs X with columns distributed as Unif(0,1) and responses
are \(N(m(\mathbf{x},I), 1)\) where
\(m(\mathbf{x},I) = E(f(\mathbf{x},I)\) and
$$m(\mathbf{x},I) = f_1(\mathbf{x})_{[I=1]} + f_2(\mathbf{x})_{[I=2]} + f_3(\mathbf{x})_{[I=3]} + m([x_{10},\cdots,x_1])_{[I=4]}$$
where
$$f_1(\mathbf{x}) = 10\sin(\pi x_1 x_2), \; f_2(\mathbf{x}) = 20(x_3-0.5)^2, \; \mbox{and } f_3(\mathbf{x}) = 10x_4 + 5x_5.$$
The indicator I is coded in binary in the output data frame as:
c(0,0,0) for I=1,
c(0,0,1) for I=2,
c(0,1,0) for I=3, and
c(1,0,0) for I=4.
Gramacy, R. B. (2007). tgp: An R Package for Bayesian Nonstationary, Semiparametric Nonlinear Regression and Design by Treed Gaussian Process Models. Journal of Statistical Software, 19(9). https://www.jstatsoft.org/v19/i09 tools:::Rd_expr_doi("10.18637/jss.v019.i09")
Robert B. Gramacy, Matthew Taddy (2010). Categorical Inputs, Sensitivity Analysis, Optimization and Importance Tempering with tgp Version 2, an R Package for Treed Gaussian Process Models. Journal of Statistical Software, 33(6), 1--48. https://www.jstatsoft.org/v33/i06/. tools:::Rd_expr_doi("10.18637/jss.v033.i06")
Friedman, J. H. (1991). Multivariate adaptive regression splines. “Annals of Statistics”, 19, No. 1, 1--67.
Gramacy, R. B., Lee, H. K. H. (2008). Bayesian treed Gaussian process models with an application to computer modeling. Journal of the American Statistical Association, 103(483), pp. 1119-1130. Also available as ArXiv article 0710.4536 https://arxiv.org/abs/0710.4536
Chipman, H., George, E., & McCulloch, R. (2002). Bayesian treed models. Machine Learning, 48, 303--324.
bgpllm, btlm,
blm, bgp, btgpllm, bgp