deepgp (version 1.1.3)

fit_three_layer: MCMC sampling for three layer deep GP

Description

Conducts MCMC sampling of hyperparameters, hidden layer z, and hidden layer w for a three layer deep GP. Separate length scale parameters theta_z, theta_w, and theta_y govern the correlation strength of the inner layer, middle layer, and outer layer respectively. Nugget parameter g governs noise on the outer layer. In Matern covariance, v governs smoothness.

Usage

fit_three_layer(
  x,
  y,
  nmcmc = 10000,
  D = ifelse(is.matrix(x), ncol(x), 1),
  verb = TRUE,
  w_0 = NULL,
  z_0 = NULL,
  g_0 = 0.001,
  theta_y_0 = 0.1,
  theta_w_0 = 0.1,
  theta_z_0 = 0.1,
  true_g = NULL,
  settings = NULL,
  cov = c("matern", "exp2"),
  v = 2.5,
  vecchia = FALSE,
  m = min(25, length(y) - 1),
  ordering = NULL
)

Value

a list of the S3 class dgp3 or dgp3vec with elements:

• x: copy of input matrix

• y: copy of response vector

• nmcmc: number of MCMC iterations

• settings: copy of proposal/prior settings

• v: copy of Matern smoothness parameter (v = 999 indicates cov = "exp2")

• g: vector of MCMC samples for g

• theta_y: vector of MCMC samples for theta_y (length scale of outer layer)

• theta_w: matrix of MCMC samples for theta_w (length scale of middle layer)

• theta_z: matrix of MCMC samples for theta_z (length scale of inner layer)

• tau2: vector of MLE estimates for tau2 (scale parameter of outer layer)

• w: list of MCMC samples for middle hidden layer w

• z: list of MCMC samples for inner hidden layer z

• ll: vector of MVN log likelihood of the outer layer for reach Gibbs iteration

• time: computation time in seconds

Arguments

x

vector or matrix of input locations

y

vector of response values

nmcmc

number of MCMC iterations

D

integer designating dimension of hidden layers, defaults to dimension of x

verb

logical indicating whether to print iteration progress

w_0

initial value for hidden layer w (must be matrix of dimension nrow(x) by D or dimension nrow(x) - 1 by D). Defaults to the identity mapping.

z_0

initial value for hidden layer z (must be matrix of dimension nrow(x) by D or dimension nrow(x) - 1 by D). Defaults to the identity mapping.

g_0

initial value for g

theta_y_0

initial value for theta_y (length scale of outer layer)

theta_w_0

initial value for theta_w (length scale of middle layer), may be single value or vector of length D

theta_z_0

initial value for theta_z (length scale of inner layer), may be single value or vector of length D

true_g

if true nugget is known it may be specified here (set to a small value to make fit deterministic). Note - values that are too small may cause numerical issues in matrix inversions.

settings

hyperparameters for proposals and priors (see details)

cov

covariance kernel, either Matern or squared exponential ("exp2")

v

Matern smoothness parameter (only used if cov = "matern")

vecchia

logical indicating whether to use Vecchia approximation

m

size of Vecchia conditioning sets (only used if vecchia = TRUE)

ordering

optional ordering for Vecchia approximation, must correspond to rows of x, defaults to random, is applied to x, w, and z

Details

pmx = TRUE option not yet implemented for three-layer DGP.

Maps inputs x through hidden layer z then hidden layer w to outputs y. Conducts sampling of the hidden layers using Elliptical Slice sampling. Utilizes Metropolis Hastings sampling of the length scale and nugget parameters with proposals and priors controlled by settings. When true_g is set to a specific value, the nugget is not estimated. When vecchia = TRUE, all calculations leverage the Vecchia approximation with specified conditioning set size m. Vecchia approximation is only implemented for cov = "matern".

NOTE on OpenMP: The Vecchia implementation relies on OpenMP parallelization for efficient computation. This function will produce a warning message if the package was installed without OpenMP (this is the default for CRAN packages installed on Apple machines). To set up OpenMP parallelization, download the package source code and install using the gcc/g++ compiler.

Proposals for g, theta_y, theta_w, and theta_z follow a uniform sliding window scheme, e.g.

g_star <- runif(1, l * g_t / u, u * g_t / l),

with defaults l = 1 and u = 2 provided in settings. To adjust these, set settings = list(l = new_l, u = new_u). Priors on g, theta_y, theta_w, and theta_z follow Gamma distributions with shape parameters (alpha) and rate parameters (beta) controlled within the settings list object. Defaults have been updated with package version 1.1.3. Default priors differ for noisy/deterministic settings and depend on whether monowarp = TRUE. All default values are visible in the internal deepgp:::check_settings function. These priors are designed for x scaled to [0, 1] and y scaled to have mean 0 and variance 1. These may be adjusted using the settings input.

In the current version, the three-layer does not have any equivalent setting for monowarp = TRUE or pmx = TRUE as in fit_two_layer.

When w_0 = NULL and/or z_0 = NULL, the hidden layers are initialized at x (i.e. the identity mapping). The default prior mean of the inner hidden layer z is zero, but may be adjusted to x using settings = list(z_prior_mean = x). The prior mean of the middle hidden layer w is set at zero is is not user adjustable. If w_0 and/or z_0 is of dimension nrow(x) - 1 by D, the final row is predicted using kriging. This is helpful in sequential design when adding a new input location and starting the MCMC at the place where the previous MCMC left off.

The output object of class dgp3 or dgp3vec is designed for use with continue, trim, and predict.

References

Sauer, A. (2023). Deep Gaussian process surrogates for computer experiments. *Ph.D. Dissertation, Department of Statistics, Virginia Polytechnic Institute and State University.*

Sauer, A., Gramacy, R.B., & Higdon, D. (2023). Active learning for deep Gaussian process surrogates. *Technometrics, 65,* 4-18. arXiv:2012.08015

Sauer, A., Cooper, A., & Gramacy, R. B. (2023). Vecchia-approximated deep Gaussian processes for computer experiments. *Journal of Computational and Graphical Statistics, 32*(3), 824-837. arXiv:2204.02904

Examples

# Additional examples including real-world computer experiments are available at: 
# https://bitbucket.org/gramacylab/deepgp-ex/
# \donttest{
# G function in 2 dimensions (https://www.sfu.ca/~ssurjano/gfunc.html)
f <- function(xx, a = (c(1:length(xx)) - 1) / 2) { 
  new1 <- abs(4 * xx - 2) + a
  new2 <- 1 + a
  prod <- prod(new1 / new2)
  return((prod - 1) / 0.86)
}

# Training data
d <- 2
n <- 30
x <- matrix(runif(n * d), ncol = d)
y <- apply(x, 1, f)

# Testing data
n_test <- 500
xx <- matrix(runif(n_test * d), ncol = d)
yy <- apply(xx, 1, f)

i <- interp::interp(xx[, 1], xx[, 2], yy)
image(i, col = heat.colors(128))
contour(i, add = TRUE)
contour(i, level = -0.5, col = 4, add = TRUE) # potential failure limit
points(x)

# Example 1: full model (nugget estimated, entropy calculated)
fit <- fit_three_layer(x, y, nmcmc = 2000)
plot(fit)
fit <- trim(fit, 1000, 2)
fit <- predict(fit, xx, entropy_limit = -0.5, cores = 1)
plot(fit)
i <- interp::interp(xx[, 1], xx[, 2], fit$entropy)
image(i, col = heat.colors(128), main = "Entropy")

# Example 2: Vecchia approximated model (nugget fixed)
# (Vecchia approximation is faster for larger data sizes)
fit <- fit_three_layer(x, y, nmcmc = 2000, vecchia = TRUE, 
                       m = 10, true_g = 1e-6)
plot(fit)
fit <- trim(fit, 1000, 2)
fit <- predict(fit, xx, cores = 1)
plot(fit)
# }