fit_two_layer: MCMC sampling for two layer deep GP

In deepgp: Bayesian Deep Gaussian Processes using MCMC

MCMC sampling for two layer deep GP

Description

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

Usage

fit_two_layer(
  x,
  y,
  nmcmc = 10000,
  D = ifelse(is.matrix(x), ncol(x), 1),
  monowarp = FALSE,
  pmx = FALSE,
  verb = TRUE,
  w_0 = NULL,
  g_0 = 0.001,
  theta_y_0 = 0.1,
  theta_w_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
)

Arguments

x	vector or matrix of input locations
y	vector of response values
nmcmc	number of MCMC iterations
D	integer designating dimension of hidden layer, defaults to dimension of x
monowarp	indicates whether warpings should be forced to be monotonic. Input may be a matrix of grid points (or a vector which will be applied to every dimension) for interpolation of the cumulative sum, an integer specifying the number of grid points to use over the range [0, 1], or simply the boolean TRUE which triggers 50 grid points over the range [0, 1].
pmx	"prior mean x", logical indicating whether w should have prior mean of x (TRUE, requires D = ncol(x)) or prior mean zero (FALSE). pmx = TRUE is recommended for higher dimensions. May be numeric, in which case the specified argument is used as the scale (tau2) in the latent w layer (default is 1). Small values encourage identity mappings.
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.
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 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 and w

Details

Maps inputs x through hidden layer w to outputs y. Conducts sampling of the hidden layer 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".

When monowarp = TRUE, each input dimension is warped separately and monotonically. This requires D = ncol(x). Monotonic warpings trigger separable lengthscales on the outer layer (theta_y). As a default, monotonic warpings use the reference grid: seq(0, 1, length = 50). The grid size may be controlled by passing a numeric integer to monowarp (i.e., monowarp = 100 uses the grid seq(0, 1, length = 100)). Alternatively, any user-specified grid may be passed as the argument to monowarp.

When pmx = TRUE, the prior on the latent layer is set at x (rather than the default of zero). This requires D = ncol(x). If pmx is set to a numeric value, then that value is used as the scale parameter on the latent layer. Specifying a small value here encourages an identity mapping.

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, and theta_w 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, and theta_w 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.

When w_0 = NULL, the hidden layer is initialized at x (i.e. the identity mapping). If w_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 dgp2 or dgp2vec is designed for use with continue, trim, and predict.

Value

a list of the S3 class dgp2 or dgp2vec 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 inner layer)

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

• w: list of MCMC samples for hidden layer w

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

• time: computation time in seconds

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

Barnett, S., Beesley, L. J., Booth, A. S., Gramacy, R. B., & Osthus D. (2024). Monotonic warpings for additive and deep Gaussian processes. *In Review.* arXiv:2408.01540

Examples

# Additional examples including real-world computer experiments are available at: 
# https://bitbucket.org/gramacylab/deepgp-ex/

# Booth function (inspired by the Higdon function)
f <- function(x) {
  i <- which(x <= 0.58)
  x[i] <- sin(pi * x[i] * 6) + cos(pi * x[i] * 12)
  x[-i] <- 5 * x[-i] - 4.9
  return(x)
}

# Training data
x <- seq(0, 1, length = 25)
y <- f(x)

# Testing data
xx <- seq(0, 1, length = 200)
yy <- f(xx)

plot(xx, yy, type = "l")
points(x, y, col = 2)

# Example 1: full model (nugget estimated, using continue)
fit <- fit_two_layer(x, y, nmcmc = 1000)
plot(fit)
fit <- continue(fit, 1000) 
plot(fit, hidden = TRUE) # trace plots and ESS samples 
fit <- trim(fit, 1000, 2)
fit <- predict(fit, xx, cores = 1)
plot(fit)

# Example 2: Vecchia approximated model (nugget estimated)
# (Vecchia approximation is faster for larger data sizes)
fit <- fit_two_layer(x, y, nmcmc = 2000, vecchia = TRUE, m = 10)
plot(fit, hidden = TRUE) # trace plots and ESS samples
fit <- trim(fit, 1000, 2)
fit <- predict(fit, xx, cores = 1)
plot(fit)

# Example 3: Vecchia approximated model, re-approximated after burn-in 
fit <- fit_two_layer(x, y, nmcmc = 1000, vecchia = TRUE, m = 10)
fit <- continue(fit, 1000, re_approx = TRUE)
plot(fit, hidden = TRUE) # trace plots and ESS samples
fit <- trim(fit, 1000, 2)
fit <- predict(fit, xx, cores = 1)
plot(fit)

# Example 4: full model with monotonic warpings (nugget estimated)
fit <- fit_two_layer(x, y, nmcmc = 2000, monowarp = TRUE)
plot(fit, hidden = TRUE) # trace plots and ESS samples
fit <- trim(fit, 1000, 2)
fit <- predict(fit, xx, cores = 1)
plot(fit)

deepgp documentation built on Sept. 11, 2024, 8:30 p.m.