lineqGPR-package: Gaussian Processes with Linear Inequality Constraints

In lineqGPR: Gaussian Process Regression Models with Linear Inequality Constraints

Description

A package for Gaussian process interpolation, regression and simulation under linear inequality constraints based on (López-Lopera et al., 2018). Constrained models and constrained additive models are given as objects with "lineqGP" and "lineqAGP" S3 class, respectively. Implementations according to (Maatouk and Bay, 2017) are also provided as objects with "lineqDGP" S3 class.

Details

This package was not yet installed at build time.

Warning

lineqGPR may strongly evolve in the future in order to incorporate other packages for Gaussian process regression modelling (see, e.g., kergp, DiceKriging, DiceDesign). It could be also scaled to higher dimensions and for a large number of observations.

Note

This package was developed within the frame of the Chair in Applied Mathematics OQUAIDO, gathering partners in technological research (BRGM, CEA, IFPEN, IRSN, Safran, Storengy) and academia (CNRS, Ecole Centrale de Lyon, Mines Saint-Etienne, University of Grenoble, University of Nice, University of Toulouse) around advanced methods for Computer Experiments.

	Important functions or methods
create	Creation function of GP models under linear inequality constraints.
augment	Augmentation of GP models according to local and covariance parameters.
lineqGPOptim	Covariance parameter estimation via maximum likelihood.
predict	Prediction of the objective function at new points using a Kriging model under
	linear inequality constraints.
simulate	Simulation of kriging models under linear inequality constraints.
plot	Plot for a constrained Kriging model.
ggplot	GGPlot for a constrained Kriging model.

Author(s)

Andrés Felipe López-Lopera (IMT, Toulouse) with contributions from Olivier Roustant (INSA, Toulouse) and Yves Deville (Alpestat).

Maintainer: Andrés Felipe López-Lopera, <[e-mail]>

References

López-Lopera, A. F., Bachoc, F., Durrande, N., and Roustant, O. (2018), "Finite-dimensional Gaussian approximation with linear inequality constraints". SIAM/ASA Journal on Uncertainty Quantification, 6(3): 1224–1255. [link]

Bachoc, F., Lagnoux, A., and Lopez-Lopera, A. F. (2019), "Maximum likelihood estimation for Gaussian processes under inequality constraints". Electronic Journal of Statistics, 13 (2): 2921-2969. [link]

Maatouk, H. and Bay, X. (2017), "Gaussian process emulators for computer experiments with inequality constraints". Mathematical Geosciences, 49(5): 557-582. [link]

Roustant, O., Ginsbourger, D., and Deville, Y. (2012), "DiceKriging, DiceOptim: Two R Packages for the Analysis of Computer Experiments by Kriging-Based Metamodeling and Optimization". Journal of Statistical Software, 51(1): 1-55. [link]

Examples

## ------------------------------------------------------------------
## Gaussian process regression modelling under boundedness constraint
## ------------------------------------------------------------------
library(lineqGPR)

#### generating the synthetic data ####
sigfun <- function(x) return(1/(1+exp(-7*(x-0.5))))
x <- seq(0, 1, 0.001)
y <- sigfun(x)
DoE <- splitDoE(x, y, DoE.idx = c(201, 501, 801))

#### GP with inactive boundedness constraints ####
# creating the "lineqGP" model
model <- create(class = "lineqGP", x = DoE$xdesign, y = DoE$ydesign,
                constrType = c("boundedness"))
model$localParam$m <- 100
model$bounds <- c(-10,10)
model <- augment(model)

# sampling from the model
sim.model <- simulate(model, nsim = 1e3, seed = 1, xtest = DoE$xtest)
plot(sim.model, xlab = "x", ylab = "y(x)", ylim = range(y),
     main = "Unconstrained GP model")
lines(x, y, lty = 2)
legend("topleft", c("ytrain","ytest","mean","confidence"),
       lty = c(NaN,2,1,NaN), pch = c(20,NaN,NaN,15),
       col = c("black","black","darkgreen","gray80"))

#### GP with active boundedness constraints ####
# creating the "lineqGP" model
model <- create(class = "lineqGP", x = DoE$xdesign, y = DoE$ydesign,
                constrType = c("boundedness"))
model$localParam$m <- 100
model$bounds <- c(0,1)
model <- augment(model)

# sampling from the model
sim.model <- simulate(model, nsim = 1e3, seed = 1, xtest = DoE$xtest)
plot(sim.model, bounds = model$bounds,
     xlab = "x", ylab = "y(x)", ylim = range(y),
     main = "Constrained GP model under boundedness conditions")
lines(x, y, lty = 2)
legend("topleft", c("ytrain","ytest","mean","confidence"),
       lty = c(NaN,2,1,NaN), pch = c(20,NaN,NaN,15),
       col = c("black","black","darkgreen","gray80"))


## ------------------------------------------------------------------
## Gaussian process regression modelling under multiple constraints
## ------------------------------------------------------------------
library(lineqGPR)

#### generating the synthetic data ####
sigfun <- function(x) return(1/(1+exp(-7*(x-0.5))))
x <- seq(0, 1, 0.001)
y <- sigfun(x)
DoE <- splitDoE(x, y, DoE.idx = c(201, 501, 801))

#### GP with boundedness and monotonicity constraints ####
# creating the "lineqGP" model
model <- create(class = "lineqGP", x = DoE$xdesign, y = DoE$ydesign,
                constrType = c("boundedness","monotonicity"))
model$localParam$m <- 50
model$bounds[1, ] <- c(0,1)
model <- augment(model)

# sampling from the model
sim.model <- simulate(model, nsim = 1e2, seed = 1, xtest = DoE$xtest)
plot(sim.model, bounds = model$bounds,
     xlab = "x", ylab = "y(x)", ylim = range(y),
     main = "Constrained GP model under boundedness & monotonicity conditions")
lines(x, y, lty = 2)
legend("topleft", c("ytrain","ytest","mean","confidence"),
       lty = c(NaN,2,1,NaN), pch = c(20,NaN,NaN,15),
       col = c("black","black","darkgreen","gray80"))


## ------------------------------------------------------------------
## Gaussian process regression modelling under linear constraints
## ------------------------------------------------------------------
library(lineqGPR)
library(Matrix)

#### generating the synthetic data ####
targetFun <- function(x){
  y <- rep(1, length(x))
  y[x <= 0.4] <- 2.5*x[x <= 0.4]
  return(y)
}
x <- seq(0, 1, by = 0.001)
y <- targetFun(x)
DoE <- splitDoE(x, y, DoE.idx = c(101, 301, 501, 701))

#### GP with predefined linear inequality constraints ####
# creating the "lineqGP" model
model <- create(class = "lineqGP", x = DoE$xdesign, y = DoE$ydesign,
                constrType = c("linear"))
m <- model$localParam$m <- 100

# building the predefined linear constraints
bounds1 <- c(0,Inf)
LambdaB1 <- diag(2*m/5)
LambdaM <- diag(2*m/5)
LambdaB2 <- diag(3*m/5)
lsys <- lineqGPSys(m = 2*m/5, constrType = "monotonicity",
                   l = bounds1[1], u = bounds1[2], lineqSysType = "oneside")
LambdaM[-seq(1),] <- lsys$M
model$Lambda <- as.matrix(bdiag(rbind(LambdaM,LambdaB1),LambdaB2))
model$lb <- c(-Inf, rep(0, 2*m/5-1), rep(0, 2*m/5), rep(0.85, 3*m/5))
model$ub <- c(rep(0.1, 2*m/5), rep(1.1, 2*m/5), rep(1.1, 3*m/5))
model <- augment(model)

# sampling from the model
sim.model <- simulate(model, nsim = 1e3, seed = 1, xtest = DoE$xtest)
plot(sim.model, bounds = c(0,1.1),
     xlab = "x", ylab = "y(x)", ylim = c(0,1.1),
     main = "Constrained GP model under linear conditions")
lines(x, y, lty = 2)
abline(v = 0.4, lty = 2)
lines(c(0.4, 1), rep(0.85, 2), lty = 2)
legend("bottomright", c("ytrain","ytest","mean","confidence"),
       lty = c(NaN,2,1,NaN), pch = c(20,NaN,NaN,15),
       col = c("black","black","darkgreen","gray80"))


## ------------------------------------------------------------------
## Note:
## 1. More examples are given as demos (run: demo(package="lineqGPR")).
## 2. See also the examples from inner functions of the package
## (run: help("simulate.lineqGP")).
## ------------------------------------------------------------------

lineqGPR documentation built on Jan. 11, 2020, 9:23 a.m.