Package deepgp

Description

Performs Bayesian posterior inference for deep Gaussian processes following Sauer, Gramacy, and Higdon (2023). See Sauer (2023) for comprehensive methodological details and https://bitbucket.org/gramacylab/deepgp-ex/ for a variety of coding examples. Models are trained through MCMC including elliptical slice sampling of latent Gaussian layers and Metropolis-Hastings sampling of kernel hyperparameters. Vecchia-approximation for faster computation is implemented following Sauer, Cooper, and Gramacy (2023). Optional monotonic warpings are implemented following Barnett et al. (2024). Downstream tasks include sequential design through active learning Cohn/integrated mean squared error (ALC/IMSE; Sauer, Gramacy, and Higdon, 2023), optimization through expected improvement (EI; Gramacy, Sauer, and Wycoff, 2022), and contour location through entropy (Booth, Renganathan, and Gramacy, 2024). Models extend up to three layers deep; a one layer model is equivalent to typical Gaussian process regression. Incorporates OpenMP and SNOW parallelization and utilizes C/C++ under the hood.

Important Functions

• fit_one_layer: conducts MCMC sampling of hyperparameters for a one layer GP

• fit_two_layer: conducts MCMC sampling of hyperparameters and hidden layer for a two layer deep GP

• fit_three_layer: conducts MCMC sampling of hyperparameters and hidden layers for a three layer deep GP

• continue: collects additional MCMC samples

• trim: cuts off burn-in and optionally thins samples

• predict: calculates posterior mean and variance over a set of input locations (optionally calculates EI or entropy)

• plot: produces trace plots, hidden layer plots, and posterior predictive plots

• ALC: calculates active learning Cohn over set of input locations using reference grid

• IMSE: calculates integrated mean-squared error over set of input locations

Author(s)

Annie S. Booth annie_booth@ncsu.edu

References

Sauer, A. (2023). Deep Gaussian process surrogates for computer experiments. *Ph.D. Dissertation, Department of Statistics, Virginia Polytechnic Institute and State University.* http://hdl.handle.net/10919/114845

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

Gramacy, R. B., Sauer, A. & Wycoff, N. (2022). Triangulation candidates for Bayesian optimization. *Advances in Neural Information Processing Systems (NeurIPS), 35,* 35933-35945. arXiv:2112.07457

Booth, A., Renganathan, S. A. & Gramacy, R. B. (2024). Contour location for reliability in airfoil simulation experiments using deep Gaussian processes. *In Review.* arXiv:2308.04420

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

# See vignette, ?fit_one_layer, ?fit_two_layer, ?fit_three_layer, 
# ?ALC, or ?IMSE for examples
# Many more examples including real-world computer experiments are available at: 
# https://bitbucket.org/gramacylab/deepgp-ex/

Active Learning Cohn for Sequential Design

Description

Acts on a gp, dgp2, or dgp3 object. Current version requires squared exponential covariance (cov = "exp2"). Calculates ALC over the input locations x_new using specified reference grid. If no reference grid is specified, x_new is used as the reference. Optionally utilizes SNOW parallelization. User should select the point with the highest ALC to add to the design.

Usage

ALC(object, x_new, ref, cores)

## S3 method for class 'gp'
ALC(object, x_new = NULL, ref = NULL, cores = 1)

## S3 method for class 'dgp2'
ALC(object, x_new = NULL, ref = NULL, cores = 1)

## S3 method for class 'dgp3'
ALC(object, x_new = NULL, ref = NULL, cores = 1)

Arguments

Details

Not yet implemented for Vecchia-approximated fits or Matern kernels.

All iterations in the object are used in the calculation, so samples should be burned-in. Thinning the samples using trim will speed up computation. This function may be used in two ways:

• Option 1: called on an object with only MCMC iterations, in which case x_new must be specified

• Option 2: called on an object that has been predicted over, in which case the x_new from predict is used

In Option 2, it is recommended to set store_latent = TRUE for dgp2 and dgp3 objects so latent mappings do not have to be re-calculated. Through predict, the user may specify a mean mapping (mean_map = TRUE) or a full sample from the MVN distribution over w_new (mean_map = FALSE). When the object has not yet been predicted over (Option 1), the mean mapping is used.

SNOW parallelization reduces computation time but requires more memory storage. C code derived from the "laGP" package (Robert B Gramacy and Furong Sun).

Value

list with elements:

• value: vector of ALC values, indices correspond to x_new

• time: computation time in seconds

References

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

Seo, S, M Wallat, T Graepel, and K Obermayer. 2000. Gaussian Process Regression: Active Data Selection and Test Point Rejection. In Mustererkennung 2000, 2734. New York, NY: SpringerVerlag.

Gramacy, RB and F Sun. (2016). laGP: Large-Scale Spatial Modeling via Local Approximate Gaussian Processes in R. Journal of Statistical Software 72 (1), 1-46. doi:10.18637/jss.v072.i01

Examples

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

# --------------------------------------------------------
# Example 1: toy step function, runs in less than 5 seconds
# --------------------------------------------------------

f <- function(x) {
    if (x <= 0.4) return(-1)
    if (x >= 0.6) return(1)
    if (x > 0.4 & x < 0.6) return(10*(x-0.5))
}

x <- seq(0.05, 0.95, length = 7)
y <- sapply(x, f)
x_new <- seq(0, 1, length = 100)

# Fit model and calculate ALC
fit <- fit_two_layer(x, y, nmcmc = 100, cov = "exp2")
fit <- trim(fit, 50)
fit <- predict(fit, x_new, cores = 1, store_latent = TRUE)
alc <- ALC(fit)


# --------------------------------------------------------
# Example 2: damped sine wave
# --------------------------------------------------------

f <- function(x) {
    exp(-10*x) * (cos(10*pi*x - 1) + sin(10*pi*x - 1)) * 5 - 0.2
}

# Training data
x <- seq(0, 1, length = 30)
y <- f(x) + rnorm(30, 0, 0.05)

# Testing data
xx <- seq(0, 1, length = 100)
yy <- f(xx)

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

# Conduct MCMC (can replace fit_two_layer with fit_one_layer/fit_three_layer)
fit <- fit_two_layer(x, y, D = 1, nmcmc = 2000, cov = "exp2")
plot(fit)
fit <- trim(fit, 1000, 2)

# Option 1 - calculate ALC from MCMC iterations
alc <- ALC(fit, xx)

# Option 2 - calculate ALC after predictions
fit <- predict(fit, xx, cores = 1, store_latent = TRUE)
alc <- ALC(fit)

# Visualize fit
plot(fit)
par(new = TRUE) # overlay ALC
plot(xx, alc$value, type = 'l', lty = 2, axes = FALSE, xlab = '', ylab = '')

# Select next design point
x_new <- xx[which.max(alc$value)]

Integrated Mean-Squared (prediction) Error for Sequential Design

Description

Acts on a gp, dgp2, or dgp3 object. Current version requires squared exponential covariance (cov = "exp2"). Calculates IMSE over the input locations x_new. Optionally utilizes SNOW parallelization. User should select the point with the lowest IMSE to add to the design.

Usage

IMSE(object, x_new, cores)

## S3 method for class 'gp'
IMSE(object, x_new = NULL, cores = 1)

## S3 method for class 'dgp2'
IMSE(object, x_new = NULL, cores = 1)

## S3 method for class 'dgp3'
IMSE(object, x_new = NULL, cores = 1)

Arguments

Details

Not yet implemented for Vecchia-approximated fits or Matern kernels.

All iterations in the object are used in the calculation, so samples should be burned-in. Thinning the samples using trim will speed up computation. This function may be used in two ways:

• Option 1: called on an object with only MCMC iterations, in which case x_new must be specified

• Option 2: called on an object that has been predicted over, in which case the x_new from predict is used

In Option 2, it is recommended to set store_latent = TRUE for dgp2 and dgp3 objects so latent mappings do not have to be re-calculated. Through predict, the user may specify a mean mapping (mean_map = TRUE) or a full sample from the MVN distribution over w_new (mean_map = FALSE). When the object has not yet been predicted over (Option 1), the mean mapping is used.

SNOW parallelization reduces computation time but requires more memory storage.

Value

list with elements:

• value: vector of IMSE values, indices correspond to x_new

• time: computation time in seconds

References

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

Binois, M, J Huang, RB Gramacy, and M Ludkovski. 2019. "Replication or Exploration? Sequential Design for Stochastic Simulation Experiments." Technometrics 61, 7-23. Taylor & Francis. doi:10.1080/00401706.2018.1469433

Examples

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

# --------------------------------------------------------
# Example 1: toy step function, runs in less than 5 seconds
# --------------------------------------------------------

f <- function(x) {
    if (x <= 0.4) return(-1)
    if (x >= 0.6) return(1)
    if (x > 0.4 & x < 0.6) return(10*(x-0.5))
}

x <- seq(0.05, 0.95, length = 7)
y <- sapply(x, f)
x_new <- seq(0, 1, length = 100)

# Fit model and calculate IMSE
fit <- fit_one_layer(x, y, nmcmc = 100, cov = "exp2")
fit <- trim(fit, 50)
fit <- predict(fit, x_new, cores = 1, store_latent = TRUE)
imse <- IMSE(fit)


# --------------------------------------------------------
# Example 2: Higdon function
# --------------------------------------------------------

f <- function(x) {
    i <- which(x <= 0.48)
    x[i] <- 2 * sin(pi * x[i] * 4) + 0.4 * cos(pi * x[i] * 16)
    x[-i] <- 2 * x[-i] - 1
    return(x)
}

# Training data
x <- seq(0, 1, length = 30)
y <- f(x) + rnorm(30, 0, 0.05)

# Testing data
xx <- seq(0, 1, length = 100)
yy <- f(xx)

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

# Conduct MCMC (can replace fit_three_layer with fit_one_layer/fit_two_layer)
fit <- fit_three_layer(x, y, D = 1, nmcmc = 2000, cov = "exp2")
plot(fit)
fit <- trim(fit, 1000, 2)

# Option 1 - calculate IMSE from only MCMC iterations
imse <- IMSE(fit, xx)

# Option 2 - calculate IMSE after predictions
fit <- predict(fit, xx, cores = 1, store_latent = TRUE)
imse <- IMSE(fit)

# Visualize fit
plot(fit)
par(new = TRUE) # overlay IMSE
plot(xx, imse$value, col = 2, type = 'l', lty = 2, axes = FALSE, 
     xlab = '', ylab = '')

# Select next design point
x_new <- xx[which.min(imse$value)]

Continues MCMC sampling

Description

Acts on a gp, gpvec, dgp2, dgp2vec, dgp3, or dgp3vec object. Continues MCMC sampling of hyperparameters and hidden layers using settings from the original object. Appends new samples to existing samples. When vecchia = TRUE, this function provides the option to update Vecchia ordering/conditioning sets based on latent layer warpings through the specification of re_approx = TRUE.

Usage

continue(object, new_mcmc, verb, re_approx, ...)

## S3 method for class 'gp'
continue(object, new_mcmc = 1000, verb = TRUE, ...)

## S3 method for class 'dgp2'
continue(object, new_mcmc = 1000, verb = TRUE, ...)

## S3 method for class 'dgp3'
continue(object, new_mcmc = 1000, verb = TRUE, ...)

## S3 method for class 'gpvec'
continue(object, new_mcmc = 1000, verb = TRUE, re_approx = FALSE, ...)

## S3 method for class 'dgp2vec'
continue(object, new_mcmc = 1000, verb = TRUE, re_approx = FALSE, ...)

## S3 method for class 'dgp3vec'
continue(object, new_mcmc = 1000, verb = TRUE, re_approx = FALSE, ...)

Arguments

Details

See fit_one_layer, fit_two_layer, or fit_three_layer for details on MCMC. The resulting object will have nmcmc equal to the previous nmcmc plus new_mcmc. It is recommended to start an MCMC fit then investigate trace plots to assess burn-in. The primary use of this function is to gather more MCMC iterations in order to obtain burned-in samples.

Specifying re_approx = TRUE updates random orderings and nearest-neighbor conditioning sets (only for vecchia = TRUE fits). In one-layer, there is no latent warping but the Vecchia approximation is still re-randomized and nearest-neighbors are adjusted accordingly. In two- and three-layers, the latest samples of hidden layers are used to update nearest-neighbors. If you update the Vecchia approximation, you should later remove previous samples (updating the approximation effectively starts a new chain). When re_approx = FALSE the previous orderings and conditioning sets are used (maintaining the continuity of the previous chain).

Value

object of the same class with the new iterations appended

Examples

# See ?fit_two_layer for an example

Calculates CRPS

Description

Calculates continuous ranked probability score (lower CRPS indicate better fits, better uncertainty quantification).

Usage

crps(y, mu, s2)

Arguments

References

Gneiting, T, and AE Raftery. 2007. Strictly Proper Scoring Rules, Prediction, and Estimation. Journal of the American Statistical Association 102 (477), 359-378.

Examples

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

MCMC sampling for one layer GP

Description

Conducts MCMC sampling of hyperparameters for a one layer GP. Length scale parameter theta governs the strength of the correlation and nugget parameter g governs noise. In Matern covariance, v governs smoothness.

Usage

fit_one_layer(
  x,
  y,
  nmcmc = 10000,
  sep = FALSE,
  verb = TRUE,
  g_0 = 0.001,
  theta_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

Details

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 and theta 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 and theta 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.

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

Value

a list of the S3 class gp or gpvec 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: vector of MCMC samples for theta

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

• ll: vector of MVN log likelihood for each 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

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 fixed)
fit <- fit_one_layer(x, y, nmcmc = 2000, true_g = 1e-6)
plot(fit)
fit <- trim(fit, 1000, 2)
fit <- predict(fit, xx, cores = 1)
plot(fit)

# Example 2: full model (nugget estimated, EI calculated)
fit <- fit_one_layer(x, y, nmcmc = 2000)
plot(fit) 
fit <- trim(fit, 1000, 2)
fit <- predict(fit, xx, cores = 1, EI = TRUE)
plot(fit)
par(new = TRUE) # overlay EI
plot(xx[order(xx)], fit$EI[order(xx)], type = 'l', lty = 2, 
axes = FALSE, xlab = '', ylab = '')

# Example 3: Vecchia approximated model (nugget estimated)
fit <- fit_one_layer(x, y, nmcmc = 2000, vecchia = TRUE, m = 10)
plot(fit)
fit <- trim(fit, 1000, 2)
fit <- predict(fit, xx, cores = 1)
plot(fit)

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
)

Arguments

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.

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

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/

# 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)

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

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)

Plots object from deepgp package

Description

Acts on a gp, gpvec, dgp2, dgp2vec, dgp3, or dgp3vec object. Generates trace plots for outer log likelihood, length scale, and nugget hyperparameters. Generates plots of hidden layers for one-dimensional inputs or monotonic warpings. Generates plots of the posterior mean and estimated 90% prediction intervals for one-dimensional inputs; generates heat maps of the posterior mean and point-wise variance for two-dimensional inputs.

Usage

## S3 method for class 'gp'
plot(x, trace = NULL, predict = NULL, ...)

## S3 method for class 'gpvec'
plot(x, trace = NULL, predict = NULL, ...)

## S3 method for class 'dgp2'
plot(x, trace = NULL, hidden = NULL, predict = NULL, ...)

## S3 method for class 'dgp2vec'
plot(x, trace = NULL, hidden = NULL, predict = NULL, ...)

## S3 method for class 'dgp3'
plot(x, trace = NULL, hidden = NULL, predict = NULL, ...)

## S3 method for class 'dgp3vec'
plot(x, trace = NULL, hidden = NULL, predict = NULL, ...)

Arguments

Details

Trace plots are useful in assessing burn-in. If there are too many hyperparameters to plot them all, then it is most useful to visualize the log likelihood (e.g., plot(fit$ll, type = "l")).

Hidden layer plots are colored on a gradient - red lines represent earlier iterations and yellow lines represent later iterations - to help assess burn-in of the hidden layers. Only every 100th sample is plotted.

Examples

# See ?fit_one_layer, ?fit_two_layer, or ?fit_three_layer
# for examples

Predict posterior mean and variance/covariance

Description

Acts on a gp, dgp2, or dgp3 object. Calculates posterior mean and variance/covariance over specified input locations. Optionally calculates expected improvement (EI) or entropy over candidate inputs. Optionally utilizes SNOW parallelization.

Usage

## S3 method for class 'gp'
predict(
  object,
  x_new,
  lite = TRUE,
  return_all = FALSE,
  EI = FALSE,
  entropy_limit = NULL,
  cores = 1,
  ...
)

## S3 method for class 'dgp2'
predict(
  object,
  x_new,
  lite = TRUE,
  store_latent = FALSE,
  mean_map = TRUE,
  return_all = FALSE,
  EI = FALSE,
  entropy_limit = NULL,
  cores = 1,
  ...
)

## S3 method for class 'dgp3'
predict(
  object,
  x_new,
  lite = TRUE,
  store_latent = FALSE,
  mean_map = TRUE,
  return_all = FALSE,
  EI = FALSE,
  entropy_limit = NULL,
  cores = 1,
  ...
)

## S3 method for class 'gpvec'
predict(
  object,
  x_new,
  m = object$m,
  ordering_new = NULL,
  lite = TRUE,
  return_all = FALSE,
  EI = FALSE,
  entropy_limit = NULL,
  cores = 1,
  ...
)

## S3 method for class 'dgp2vec'
predict(
  object,
  x_new,
  m = object$m,
  ordering_new = NULL,
  lite = TRUE,
  store_latent = FALSE,
  mean_map = TRUE,
  return_all = FALSE,
  EI = FALSE,
  entropy_limit = NULL,
  cores = 1,
  ...
)

## S3 method for class 'dgp3vec'
predict(
  object,
  x_new,
  m = object$m,
  ordering_new = NULL,
  lite = TRUE,
  store_latent = FALSE,
  mean_map = TRUE,
  return_all = FALSE,
  EI = FALSE,
  entropy_limit = NULL,
  cores = 1,
  ...
)

Arguments

Details

All iterations in the object are used for prediction, so samples should be burned-in. Thinning the samples using trim will speed up computation. Posterior moments are calculated using conditional expectation and variance. As a default, only point-wise variance is calculated. Full covariance may be calculated using lite = FALSE.

Expected improvement is calculated with the goal of minimizing the response. See Chapter 7 of Gramacy (2020) for details. Entropy is calculated based on two classes separated by the specified limit. See Sauer (2023, Chapter 3) for details.

SNOW parallelization reduces computation time but requires more memory storage.

Value

object of the same class with the following additional elements:

• x_new: copy of predictive input locations

• mean: predicted posterior mean, indices correspond to x_new locations

• s2: predicted point-wise variances, indices correspond to x_new locations (only returned when lite = TRUE)

• mean_all: predicted posterior mean for each sample (column indices), only returned when return_all = TRUE

• s2_all predicted point-wise variances for each sample (column indices), only returned when return-all = TRUE

• Sigma: predicted posterior covariance, indices correspond to x_new locations (only returned when lite = FALSE)

• EI: vector of expected improvement values, indices correspond to x_new locations (only returned when EI = TRUE)

• entropy: vector of entropy values, indices correspond to x_new locations (only returned when entropy_limit is numeric)

• w_new: list of hidden layer mappings (only returned when store_latent = TRUE), list index corresponds to iteration and row index corresponds to x_new location (two or three layer models only)

• z_new: list of hidden layer mappings (only returned when store_latent = TRUE), list index corresponds to iteration and row index corresponds to x_new location (three layer models only)

Computation time is added to the computation time of the existing object.

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

# See ?fit_one_layer, ?fit_two_layer, or ?fit_three_layer
# for examples

Calculates RMSE

Description

Calculates root mean square error (lower RMSE indicate better fits).

Usage

rmse(y, mu)

Arguments

Examples

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

Calculates score

Description

Calculates score, proportional to the multivariate normal log likelihood. Higher scores indicate better fits. Only applicable to noisy data. Requires full covariance matrix (e.g. predict with lite = FALSE).

Usage

score(y, mu, sigma)

Arguments

References

Gneiting, T, and AE Raftery. 2007. Strictly Proper Scoring Rules, Prediction, and Estimation. Journal of the American Statistical Association 102 (477), 359-378.

Examples

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

Calculates squared pairwise distances

Description

Calculates squared pairwise euclidean distances using C.

Usage

sq_dist(X1, X2 = NULL)

Arguments

Details

C code derived from the "laGP" package (Robert B Gramacy and Furong Sun).

Value

symmetric matrix of squared euclidean distances

References

Gramacy, RB and F Sun. (2016). laGP: Large-Scale Spatial Modeling via Local Approximate Gaussian Processes in R. Journal of Statistical Software 72 (1), 1-46. doi:10.18637/jss.v072.i01

Examples

x <- seq(0, 1, length = 10)
d2 <- sq_dist(x)

Trim/Thin MCMC iterations

Description

Acts on a gp, gpvec, dgp2, dgp2vec, dgp3vec, or dgp3 object. Removes the specified number of MCMC iterations (starting at the first iteration). After these samples are removed, the remaining samples are optionally thinned.

Usage

trim(object, burn, thin)

## S3 method for class 'gp'
trim(object, burn, thin = 1)

## S3 method for class 'gpvec'
trim(object, burn, thin = 1)

## S3 method for class 'dgp2'
trim(object, burn, thin = 1)

## S3 method for class 'dgp2vec'
trim(object, burn, thin = 1)

## S3 method for class 'dgp3'
trim(object, burn, thin = 1)

## S3 method for class 'dgp3vec'
trim(object, burn, thin = 1)

Arguments

Details

The resulting object will have nmcmc equal to the previous nmcmc minus burn divided by thin. It is recommended to start an MCMC fit then investigate trace plots to assess burn-in. Once burn-in has been achieved, use this function to remove the starting iterations. Thinning reduces the size of the resulting object while accounting for the high correlation between consecutive iterations.

Value

object of the same class with the selected iterations removed

Examples

# See ?fit_one_layer, ?fit_two_layer, or ?fit_three_layer
# for examples