Deep Gaussian process emulator construction

This function builds and trains a DGP emulator.

Usage

dgp(
  X,
  Y,
  struc = NULL,
  depth = 2,
  node = ncol(X),
  name = "sexp",
  lengthscale = 1,
  bounds = NULL,
  prior = "ga",
  share = TRUE,
  nugget_est = FALSE,
  nugget = NULL,
  scale_est = TRUE,
  scale = 1,
  connect = TRUE,
  likelihood = NULL,
  training = TRUE,
  verb = TRUE,
  check_rep = TRUE,
  vecchia = FALSE,
  M = 25,
  ord = NULL,
  N = ifelse(vecchia, 200, 500),
  cores = 1,
  blocked_gibbs = TRUE,
  ess_burn = 10,
  burnin = NULL,
  B = 10,
  internal_input_idx = NULL,
  linked_idx = NULL,
  id = NULL
)

Arguments

X

a matrix where each row is an input training data point and each column is an input dimension.

Y

a matrix containing observed training output data. The matrix has its rows being output data points and columns being output dimensions. When likelihood (see below) is not NULL, Y must be a matrix with only one column.

struc

a list that specifies a user-defined DGP structure. It should contain L (the number of DGP layers) sub-lists, each of which represents a layer and contains a number of GP nodes (defined by kernel()) in the corresponding layer. The final layer of the DGP structure (i.e., the final sub-list in struc) can be a likelihood layer that contains a likelihood function (e.g., Poisson()). When struc = NULL, the DGP structure is automatically generated and can be checked by applying summary() to the output from dgp() with training = FALSE. If this argument is used (i.e., user provides a customized DGP structure), arguments depth, node, name, lengthscale, bounds, prior, share, nugget_est, nugget, scale_est, scale, connect, likelihood, and internal_input_idx will NOT be used. Defaults to NULL.

The argument will be removed in the next release. To customize DGP specifications, please adjust the other arguments in the dgp() function.

depth

number of layers (including the likelihood layer) for a DGP structure. depth must be at least 2. Defaults to 2. This argument is only used when struc = NULL.

node

number of GP nodes in each layer (except for the final layer or the layer feeding the likelihood node) of the DGP. Defaults to ncol(X). This argument is only used when struc = NULL.

name

a character or a vector of characters that indicates the kernel functions (either "sexp" for squared exponential kernel or "matern2.5" for Matérn-2.5 kernel) used in the DGP emulator:

1. if a single character is supplied, the corresponding kernel function will be used for all GP nodes in the DGP hierarchy.

2. if a vector of characters is supplied, each character of the vector specifies the kernel function that will be applied to all GP nodes in the corresponding layer.

Defaults to "sexp". This argument is only used when struc = NULL.

lengthscale

initial lengthscales for GP nodes in the DGP emulator. It can be a single numeric value or a vector:

1. if it is a single numeric value, the value will be applied as the initial lengthscales for all GP nodes in the DGP hierarchy.

2. if it is a vector, each element of the vector specifies the initial lengthscales that will be applied to all GP nodes in the corresponding layer. The vector should have a length of depth if likelihood = NULL or a length of depth - 1 if likelihood is not NULL.

Defaults to a numeric value of 1.0. This argument is only used when struc = NULL.

bounds

the lower and upper bounds of lengthscales in GP nodes. It can be a vector or a matrix:

1. if it is a vector, the lower bound (the first element of the vector) and upper bound (the second element of the vector) will be applied to lengthscales for all GP nodes in the DGP hierarchy.

2. if it is a matrix, each row of the matrix specifies the lower and upper bounds of lengthscales for all GP nodes in the corresponding layer. The matrix should have its row number equal to depth if likelihood = NULL or to depth - 1 if likelihood is not NULL.

Defaults to NULL where no bounds are specified for the lengthscales. This argument is only used when struc = NULL.

prior

prior to be used for Maximum a Posterior for lengthscales and nuggets of all GP nodes in the DGP hierarchy:

• gamma prior ("ga"),

• inverse gamma prior ("inv_ga"), or

• jointly robust prior ("ref").

Defaults to "ga". This argument is only used when struc = NULL.

share

a bool indicating if all input dimensions of a GP node share a common lengthscale. Defaults to TRUE. This argument is only used when struc = NULL.

nugget_est

a bool or a bool vector that indicates if the nuggets of GP nodes (if any) in the final layer are to be estimated. If a single bool is provided, it will be applied to all GP nodes (if any) in the final layer. If a bool vector (which must have a length of ncol(Y)) is provided, each bool element in the vector will be applied to the corresponding GP node (if any) in the final layer. The value of a bool has following effects:

• FALSE: the nugget of the corresponding GP in the final layer is fixed to the corresponding value defined in nugget (see below).

• TRUE: the nugget of the corresponding GP in the final layer will be estimated with the initial value given by the correspondence in nugget (see below).

Defaults to FALSE. This argument is only used when struc = NULL.

nugget

the initial nugget value(s) of GP nodes (if any) in each layer:

1. if it is a single numeric value, the value will be applied as the initial nugget for all GP nodes in the DGP hierarchy.

2. if it is a vector, each element of the vector specifies the initial nugget that will be applied to all GP nodes in the corresponding layer. The vector should have a length of depth if likelihood = NULL or a length of depth - 1 if likelihood is not NULL.

Set nugget to a small value and the bools in nugget_est to FASLE for deterministic emulations where the emulator interpolates the training data points. Set nugget to a reasonable larger value and the bools in nugget_est to TRUE for stochastic emulations where the computer model outputs are assumed to follow a homogeneous Gaussian distribution. Defaults to 1e-6 if nugget_est = FALSE and 0.01 if nugget_est = TRUE. If likelihood is not NULL and nugget_est = FALSE, the nuggets of GPs that feed into the likelihood layer default to 1e-4. This argument is only used when struc = NULL.

scale_est

a bool or a bool vector that indicates if variance of GP nodes (if any) in the final layer are to be estimated. If a single bool is provided, it will be applied to all GP nodes (if any) in the final layer. If a bool vector (which must have a length of ncol(Y)) is provided, each bool element in the vector will be applied to the corresponding GP node (if any) in the final layer. The value of a bool has following effects:

• FALSE: the variance of the corresponding GP in the final layer is fixed to the corresponding value defined in scale (see below).

• TRUE: the variance of the corresponding GP in the final layer will be estimated with the initial value given by the correspondence in scale (see below).

Defaults to TRUE. This argument is only used when struc = NULL.

scale

the initial variance value(s) of GP nodes (if any) in the final layer. If it is a single numeric value, it will be applied to all GP nodes (if any) in the final layer. If it is a vector (which must have a length of ncol(Y)), each numeric in the vector will be applied to the corresponding GP node (if any) in the final layer. Defaults to 1. This argument is only used when struc = NULL.

connect

a bool indicating whether to implement global input connection to the DGP structure. Setting it to FALSE may produce a better emulator in some cases at the cost of slower training. Defaults to TRUE. This argument is only used when struc = NULL.

likelihood

the likelihood type of a DGP emulator:

1. NULL: no likelihood layer is included in the emulator.

2. "Hetero": a heteroskedastic Gaussian likelihood layer is added for stochastic emulation where the computer model outputs are assumed to follow a heteroskedastic Gaussian distribution (i.e., the computer model outputs have varying noises).

3. "Poisson": a Poisson likelihood layer is added for stochastic emulation where the computer model outputs are assumed to a Poisson distribution.

4. "NegBin": a negative Binomial likelihood layer is added for stochastic emulation where the computer model outputs are assumed to follow a negative Binomial distribution.

5. "Categorical": a categorical likelihood layer is added for stochastic emulation (i.e., classification), where the computer model outputs are assumed to follow a categorical distribution.

When likelihood is not NULL, the value of nugget_est is overridden by FALSE. Defaults to NULL. This argument is only used when struc = NULL.

training

a bool indicating if the initialized DGP emulator will be trained. When set to FALSE, dgp() returns an untrained DGP emulator, to which one can apply summary() to inspect its specifications (especially when a customized struc is provided) or apply predict() to check its emulation performance before the training. Defaults to TRUE.

verb

a bool indicating if the trace information on DGP emulator construction and training will be printed during the function execution. Defaults to TRUE.

check_rep

a bool indicating whether to check the repetitions in the dataset, i.e., if one input position has multiple outputs. Defaults to TRUE.

vecchia

a bool indicating whether to use Vecchia approximation for large-scale DGP emulator construction and prediction. Defaults to FALSE.

M

the size of the conditioning set for the Vecchia approximation in the DGP emulator training. Defaults to 25.

ord

an R function that returns the ordering of the input to each GP node contained in the DGP emulator for the Vecchia approximation. The function must satisfy the following basic rules:

• the first argument represents the input to a GP node scaled by its lengthscales.

• the output of the function is a vector of indices that gives the ordering of the input to the GP node.

If ord = NULL, the default random ordering is used. Defaults to NULL.

N

number of iterations for the training. Defaults to 500 if vecchia = FALSE and 200 if vecchia = TRUE. This argument is only used when training = TRUE.

cores

the number of processes to be used to optimize GP components (in the same layer) at each M-step of the training. If set to NULL, the number of processes is set to (max physical cores available - 1) if vecchia = FALSE and max physical cores available %/% 2 if vecchia = TRUE. Only use multiple processes when there is a large number of GP components in different layers and optimization of GP components is computationally expensive. Defaults to 1.

blocked_gibbs

a bool indicating if the latent variables are imputed layer-wise using ESS-within-Blocked-Gibbs. ESS-within-Blocked-Gibbs would be faster and more efficient than ESS-within-Gibbs that imputes latent variables node-wise because it reduces the number of components to be sampled during the Gibbs, especially when there is a large number of GP nodes in layers due to higher input dimensions. Default to TRUE.

ess_burn

number of burnin steps for the ESS-within-Gibbs at each I-step of the training. Defaults to 10. This argument is only used when training = TRUE.

burnin

the number of training iterations to be discarded for point estimates of model parameters. Must be smaller than the training iterations N. If this is not specified, only the last 25% of iterations are used. Defaults to NULL. This argument is only used when training = TRUE.

B

the number of imputations to produce the later predictions. Increase the value to account for more imputation uncertainties with slower predictions. Decrease the value for lower imputation uncertainties but faster predictions. Defaults to 10.

internal_input_idx

The argument will be removed in the next release. To set up connections of emulators for linked emulations, please use the updated lgp() function instead.

Column indices of X that are generated by the linked emulators in the preceding layers. Set internal_input_idx = NULL if the DGP emulator is in the first layer of a system or all columns in X are generated by the linked emulators in the preceding layers. Defaults to NULL. This argument is only used when struc = NULL.

linked_idx

The argument will be removed in the next release. To set up connections of emulators for linked emulations, please use the updated lgp() function instead.

Either a vector or a list of vectors:

• If linked_idx is a vector, it gives indices of columns in the pooled output matrix (formed by column-combined outputs of all emulators in the feeding layer) that feed into the DGP emulator. The length of the vector shall equal to the length of internal_input_idx when internal_input_idx is not NULL. If the DGP emulator is in the first layer of a linked emulator system, the vector gives the column indices of the global input (formed by column-combining all input matrices of emulators in the first layer) that the DGP emulator will use. If the DGP emulator is to be used in both the first and subsequent layers, one should initially set linked_idx to the appropriate values for the situation where the emulator is not in the first layer. Then, use the function set_linked_idx() to reset the linking information when the emulator is in the first layer.

• When the DGP emulator is not in the first layer of a linked emulator system, linked_idx can be a list that gives the information on connections between the DGP emulator and emulators in all preceding layers. The length of the list should equal to the number of layers before the DGP emulator. Each element of the list is a vector that gives indices of columns in the pooled output matrix (formed by column-combined outputs of all emulators) in the corresponding layer that feed into the DGP emulator. If the DGP emulator has no connections to any emulator in a certain layer, set NULL in the corresponding position of the list. The order of input dimensions in X[,internal_input_idx] should be consistent with linked_idx. For example, a DGP emulator in the 4th-layer that is fed by the output dimension 2 and 4 of emulators in layer 2 and all output dimension 1 to 3 of emulators in layer 3 should have linked_idx = list( NULL, c(2,4), c(1,2,3) ). In addition, the first and second columns of X[,internal_input_idx] should correspond to the output dimensions 2 and 4 from layer 2, and the third to fifth columns of X[,internal_input_idx] should correspond to the output dimensions 1 to 3 from layer 3.

Set linked_idx = NULL if the DGP emulator will not be used for linked emulations. However, if this is no longer the case, one can use set_linked_idx() to add linking information to the DGP emulator. Defaults to NULL.

id

an ID to be assigned to the DGP emulator. If an ID is not provided (i.e., id = NULL), a UUID (Universally Unique Identifier) will be automatically generated and assigned to the emulator. Default to NULL.

Value

An S3 class named dgp that contains five slots:

• id: A number or character string assigned through the id argument.

• data: a list that contains two elements: X and Y which are the training input and output data respectively.

• specs: a list that contains

  1. L (i.e., the number of layers in the DGP hierarchy) sub-lists named layer1, layer2,..., layerL. Each sub-list contains D (i.e., the number of GP/likelihood nodes in the corresponding layer) sub-lists named node1, node2,..., nodeD. If a sub-list corresponds to a likelihood node, it contains one element called type that gives the name (Hetero, Poisson, NegBin, or Categorical) of the likelihood node. If a sub-list corresponds to a GP node, it contains four elements:

     • kernel: the type of the kernel function used for the GP node.

     • lengthscales: a vector of lengthscales in the kernel function.

     • scale: the variance value in the kernel function.

     • nugget: the nugget value in the kernel function.

  2. internal_dims: the column indices of X that correspond to the linked emulators in the preceding layers of a linked system. The slot will be removed in the next release.

  3. external_dims: the column indices of X that correspond to global inputs to the linked system of emulators. It is shown as FALSE if internal_input_idx = NULL. The slot will be removed in the next release.

  4. linked_idx: the value passed to argument linked_idx. It is shown as FALSE if the argument linked_idx is NULL. The slot will be removed in the next release.

  5. seed: the random seed generated to produce the imputations. This information is stored for the reproducibility when the DGP emulator (that was saved by write() with the light option light = TRUE) is loaded back to R by read().

  6. B: the number of imputations used to generate the emulator.

  7. vecchia: whether the Vecchia approximation is used for the GP emulator training.

  8. M: the size of the conditioning set for the Vecchia approximation in the DGP emulator training.

  internal_dims and external_dims are generated only when struc = NULL. M is generated only when vecchia = TRUE.

• constructor_obj: a 'python' object that stores the information of the constructed DGP emulator.

• container_obj: a 'python' object that stores the information for the linked emulation.

• emulator_obj: a 'python' object that stores the information for the predictions from the DGP emulator.

The returned dgp object can be used by

• predict() for DGP predictions.

• continue() for additional DGP training iterations.

• validate() for LOO and OOS validations.

• plot() for validation plots.

• lgp() for linked (D)GP emulator constructions.

• window() for model parameter trimming.

• summary() to summarize the trained DGP emulator.

• write() to save the DGP emulator to a .pkl file.

• set_imp() to change the number of imputations.

• design() for sequential designs.

• update() to update the DGP emulator with new inputs and outputs.

• alm(), mice(), pei(), and vigf() to locate next design points.

Details

See further examples and tutorials at https://mingdeyu.github.io/dgpsi-R/ and learn how to customize a DGP structure.

Note

Any R vector detected in X and Y will be treated as a column vector and automatically converted into a single-column R matrix. Thus, if X is a single data point with multiple dimensions, it must be given as a matrix.

Examples

if (FALSE) { # \dontrun{

# load the package and the Python env
library(dgpsi)

# construct a step function
f <- function(x) {
  if (x < 0.5) return(-1)
  if (x >= 0.5) return(1)
  }

# generate training data
X <- seq(0, 1, length = 10)
Y <- sapply(X, f)

# set a random seed
set_seed(999)

# training a DGP emulator
m <- dgp(X, Y)

# continue for further training iterations
m <- continue(m)

# summarizing
summary(m)

# trace plot
trace_plot(m)

# trim the traces of model parameters
m <- window(m, 800)

# LOO cross validation
m <- validate(m)
plot(m)

# prediction
test_x <- seq(0, 1, length = 200)
m <- predict(m, x = test_x)

# OOS validation
validate_x <- sample(test_x, 10)
validate_y <- sapply(validate_x, f)
plot(m, validate_x, validate_y)

# write and read the constructed emulator
write(m, 'step_dgp')
m <- read('step_dgp')
} # }