design {dgpsi}

Sequential design of a (D)GP emulator or a bundle of (D)GP emulators

Description

This function implements the sequential design of a (D)GP emulator or a bundle of (D)GP emulators.

Usage

design(
  object,
  N,
  x_cand,
  y_cand,
  n_cand,
  limits,
  int,
  f,
  reps,
  freq,
  x_test,
  y_test,
  reset,
  target,
  method,
  eval,
  verb,
  autosave,
  new_wave,
  cores,
  ...
)

## S3 method for class 'gp'
design(
  object,
  N,
  x_cand = NULL,
  y_cand = NULL,
  n_cand = 200,
  limits = NULL,
  int = FALSE,
  f = NULL,
  reps = 1,
  freq = c(1, 1),
  x_test = NULL,
  y_test = NULL,
  reset = FALSE,
  target = NULL,
  method = vigf,
  eval = NULL,
  verb = TRUE,
  autosave = list(),
  new_wave = TRUE,
  cores = 1,
  ...
)

## S3 method for class 'dgp'
design(
  object,
  N,
  x_cand = NULL,
  y_cand = NULL,
  n_cand = 200,
  limits = NULL,
  int = FALSE,
  f = NULL,
  reps = 1,
  freq = c(1, 1),
  x_test = NULL,
  y_test = NULL,
  reset = FALSE,
  target = NULL,
  method = vigf,
  eval = NULL,
  verb = TRUE,
  autosave = list(),
  new_wave = TRUE,
  cores = 1,
  train_N = 100,
  refit_cores = 1,
  pruning = TRUE,
  control = list(),
  ...
)

## S3 method for class 'bundle'
design(
  object,
  N,
  x_cand = NULL,
  y_cand = NULL,
  n_cand = 200,
  limits = NULL,
  int = FALSE,
  f = NULL,
  reps = 1,
  freq = c(1, 1),
  x_test = NULL,
  y_test = NULL,
  reset = FALSE,
  target = NULL,
  method = vigf,
  eval = NULL,
  verb = TRUE,
  autosave = list(),
  new_wave = TRUE,
  cores = 1,
  train_N = 100,
  refit_cores = 1,
  ...
)

Arguments

object

can be one of the following:

• the S3 class gp.

• the S3 class dgp.

• the S3 class bundle.

N

the number of steps for the sequential design.

x_cand

a matrix (with each row being a design point and column being an input dimension) that gives a candidate set in which the next design point is determined. If x_cand = NULL, the candidate set will be generated using n_cand, limits, and int. Defaults to NULL.

y_cand

a matrix (with each row being a simulator evaluation and column being an output dimension) that gives the realizations from the simulator at input positions in x_cand. Defaults to NULL.

n_cand

an integer that gives

• the size of the candidate set in which the next design point is determined, if x_cand = NULL;

• the size of a sub-set to be sampled from the candidate set x_cand at each step of the sequential design to determine the next design point, if x_cand is not NULL.

Defaults to 200.

limits

a two-column matrix that gives the ranges of each input dimension, or a vector of length two if there is only one input dimension. If a vector is provided, it will be converted to a two-column row matrix. The rows of the matrix correspond to input dimensions, and its first and second columns correspond to the minimum and maximum values of the input dimensions. Set limits = NULL if x_cand is supplied. This argument is only used when x_cand is not supplied, i.e., x_cand = NULL. Defaults to NULL.

int

a bool or a vector of bools that indicates if an input dimension is an integer type. If a bool is given, it will be applied to all input dimensions. If a vector is provided, it should have a length equal to the input dimensions and will be applied to individual input dimensions. Defaults to FALSE.

f

an R function that represents the simulator. f needs to be specified with the following basic rules:

• the first argument of the function should be a matrix with rows being different design points and columns being input dimensions.

• the output of the function can either

  • a matrix with rows being different outputs (corresponding to the input design points) and columns being output dimensions. If there is only one output dimension, the matrix still needs to be returned with a single column.

  • a list with the first element being the output matrix described above and, optionally, additional named elements which will update values of any arguments with the same names passed via .... The list output can be useful if some additional arguments of f and aggregate need to be updated after each step of the sequential design.

See Note section below for further information. This argument is used when y_cand = NULL. Defaults to NULL.

reps

an integer that gives the number of repetitions of the located design points to be created and used for evaluations of f. Set the argument to an integer greater than 1 if f is a stochastic function that can generate different responses given a same input and the supplied emulator object can deal with stochastic responses, e.g., a (D)GP emulator with nugget_est = TRUE or a DGP emulator with a likelihood layer. The argument is only used when f is supplied. Defaults to 1.

freq

a vector of two integers with the first element giving the frequency (in number of steps) to re-fit the emulator, and the second element giving the frequency to implement the emulator validation (for RMSE). Defaults to c(1, 1).

x_test

a matrix (with each row being an input testing data point and each column being an input dimension) that gives the testing input data to evaluate the emulator after each step of the sequential design. Set to NULL for the LOO-based emulator validation. Defaults to NULL. This argument is only used if eval = NULL.

y_test

the testing output data that correspond to x_test for the emulator validation after each step of the sequential design:

• if object is an instance of the gp class, y_test is a matrix with only one column and each row being an testing output data point.

• if object is an instance of the dgp class, y_test is a matrix with its rows being testing output data points and columns being output dimensions.

Set to NULL for the LOO-based emulator validation. Defaults to NULL. This argument is only used if eval = NULL.

reset

a bool or a vector of bools indicating whether to reset hyperparameters of the emulator to their initial values when it was initially constructed after the input-output update and before the re-fit. If a bool is given, it will be applied to every step of the sequential design. If a vector is provided, its length should be equal to N and will be applied to individual steps of the sequential design. Defaults to FALSE.

target

a numeric or a vector that gives the target RMSEs at which the sequential design is terminated. Defaults to NULL, in which case the sequential design stops after N steps. See Note section below for further information about target.

method

an R function that give indices of designs points in a candidate set. The function must satisfy the following basic rules:

• the first argument is an emulator object that can be either an instance of

  • the gp class (produced by gp());

  • the dgp class (produced by dgp());

  • the bundle class (produced by pack()).

• the second argument is a matrix with rows representing a set of different design points.

• the output of the function

  • is a vector of indices if the first argument is an instance of the gp class;

  • is a matrix of indices if the first argument is an instance of the dgp class. If there are different design points to be added with respect to different outputs of the DGP emulator, the column number of the matrix should equal to the number of the outputs. If design points are common to all outputs of the DGP emulator, the matrix should be single-columned. If more than one design points are determined for a given output or for all outputs, the indices of these design points are placed in the matrix with extra rows.

  • is a matrix of indices if the first argument is an instance of the bundle class. Each row of the matrix gives the indices of the design points to be added to individual emulators in the bundle.

See alm(), mice(), pei(), and vigf() for examples on customizing method. Defaults to vigf().

eval

an R function that calculates the customized evaluating metric of the emulator. The function must satisfy the following basic rules:

• the first argument is an emulator object that can be either an instance of

  • the gp class (produced by gp());

  • the dgp class (produced by dgp());

  • the bundle class (produced by pack()).

• the output of the function can be

  • a single metric value, if the first argument is an instance of the gp class;

  • a single metric value or a vector of metric values with the length equal to the number of output dimensions, if the first argument is an instance of the dgp class;

  • a single metric value metric or a vector of metric values with the length equal to the number of emulators in the bundle, if the first argument is an instance of the bundle class.

If no customized function is provided, the built-in evaluation metric, RMSE, will be calculated. Defaults to NULL. See Note section below for further information.

verb

a bool indicating if the trace information will be printed during the sequential design. Defaults to TRUE.

autosave

a list that contains configuration settings for the automatic saving of the emulator:

• switch: a bool indicating whether to enable the automatic saving of the emulator during the sequential design. When set to TRUE, the emulator in the final iteration is always saved. Defaults to FALSE.

• directory: a string specifying the directory path where the emulators will be stored. Emulators will be stored in a sub-directory of directory named 'emulator-id'. Defaults to './check_points'.

• fname: a string representing the base name for the saved emulator files. Defaults to 'check_point'.

• freq: an integer indicating the frequency of automatic savings, measured in the number of iterations. Defaults to 5.

• overwrite: a bool value controlling the file saving behavior. When set to TRUE, each new automatic saving overwrites the previous one, keeping only the latest version. If FALSE, each automatic saving creates a new file, preserving all previous versions. Defaults to FALSE.

new_wave

a bool indicating if the current execution of design() will create a new wave of sequential designs or add the sequential designs to the last existing wave. This argument is only used if there are waves existing in the emulator. By creating new waves, one can better visualize the performance of the sequential designs in different executions of design() in draw() and can specify a different evaluation frequency in freq. However, it can be beneficiary to turn this option off to restrict a large number of waves to be visualized in draw() that could run out of colors. Defaults to TRUE.

cores

an integer that gives the number of cores to be used for emulator validations. If set to NULL, the number of cores is set to ⁠(max physical cores available - 1)⁠. Defaults to 1. This argument is only used if eval = NULL.

...

any arguments (with names different from those of arguments used in design()) that are used by f, method, and eval can be passed here. design() will pass relevant arguments to f, method, and eval based on the names of additional arguments provided.

train_N

an integer or a vector of integers that gives the number of training iterations to be used to re-fit the DGP emulator at each step of the sequential design:

• If train_N is an integer, then at each step the DGP emulator will re-fitted (based on the frequency of re-fit specified in freq) with train_N iterations.

• If train_N is a vector, then its size must be N even the re-fit frequency specified in freq is not one.

Defaults to 100.

refit_cores

the number of cores/workers to be used to re-fit GP components (in the same layer of a DGP emulator) at each M-step during the re-fitting. If set to NULL, the number of cores is set to ⁠(max physical cores available - 1)⁠. Only use multiple cores when there is a large number of GP components in different layers and optimization of GP components is computationally expensive. Defaults to 1.

pruning

a bool indicating if dynamic pruning of DGP structures will be implemented during the sequential design after the total number of design points exceeds min_size in control. The argument is only applicable to DGP emulators (i.e., object is an instance of dgp class) produced by dgp() with struc = NULL. Defaults to TRUE.

control

a list that can supply any of the following components to control the dynamic pruning of the DGP emulator:

• min_size, the minimum number of design points required to trigger the dynamic pruning. Defaults to 10 times of the input dimensions.

• threshold, the R2 value above which a GP node is considered redundant. Defaults to 0.97.

• nexceed, the minimum number of consecutive iterations that the R2 value of a GP node must exceed threshold to trigger the removal of that node from the DGP structure. Defaults to 3.

The argument is only used when pruning = TRUE.

Details

See further examples and tutorials at https://mingdeyu.github.io/dgpsi-R/.

Value

An updated object is returned with a slot called design that contains:

• S slots, named ⁠wave1, wave2,..., waveS⁠, that contain information of S waves of sequential designs that have been applied to the emulator. Each slot contains the following elements:

  • N, an integer that gives the numbers of steps implemented in the corresponding wave;

  • rmse, a matrix that gives the RMSEs of emulators constructed during the corresponding wave, if eval = NULL;

  • metric, a matrix that gives the customized evaluating metric values of emulators constructed during the corresponding wave, if a customized function is supplied to eval;

  • freq, an integer that gives the frequency that the emulator validations are implemented during the corresponding wave.

  • enrichment, a vector of size N that gives the number of new design points added after each step of the sequential design (if object is an instance of the gp or dgp class), or a matrix that gives the number of new design points added to emulators in a bundle after each step of the sequential design (if object is an instance of the bundle class).

  If target is not NULL, the following additional elements are also included:

  • target, the target RMSE(s) to stop the sequential design.

  • reached, a bool (if object is an instance of the gp or dgp class) or a vector of bools (if object is an instance of the bundle class) that indicate if the target RMSEs are reached at the end of the sequential design.

• a slot called type that gives the type of validations:

  • either LOO ('loo') or OOS ('oos') if eval = NULL. See validate() for more information about LOO and OOS.

  • 'customized' if a customized R function is provided to eval.

• two slots called x_test and y_test that contain the data points for the OOS validation if the type slot is 'oos'.

• If y_cand = NULL and there are NAs returned from the supplied f during the sequential design, a slot called exclusion is included that records the located design positions that produced NAs via f. The sequential design will use this information to avoid re-visiting the same locations (if x_cand is supplied) or their neighborhoods (if x_cand is NULL) in later runs of design().

See Note section below for further information.

Note

• The validation of an emulator is forced after the final step of a sequential design even N is not multiples of the second element in freq.

• Any loo or oos slot that already exists in object will be cleaned, and a new slot called loo or oos will be created in the returned object depending on whether x_test and y_test are provided. The new slot gives the validation information of the emulator constructed in the final step of the sequential design. See validate() for more information about the slots loo and oos.

• If object has previously been used by design() for sequential designs, the information of the current wave of the sequential design will replace those of old waves and be contained in the returned object, unless

  • the validation type (LOO or OOS depending on whether x_test and y_test are supplied or not) of the current wave of the sequential design is the same as the validation types (shown in the type of the design slot of object) in previous waves, and if the validation type is OOS, x_test and y_test in the current wave must also be identical to those in the previous waves;

  • both the current and previous waves of the sequential design supply customized evaluation functions to eval. Users need to ensure the customized evaluation functions are consistent among different waves. Otherwise, the trace plot of RMSEs produced by draw() will show values of different evaluation metrics in different waves.

  In above two cases, the information of the current wave of the sequential design will be added to the design slot of the returned object under the name waveS.

• If object is an instance of the gp class and eval = NULL, the matrix in the rmse slot is single-columned. If object is an instance of the dgp or bundle class and eval = NULL, the matrix in the rmse slot can have multiple columns that correspond to different output dimensions or different emulators in the bundle.

• If object is an instance of the gp class and eval = NULL, target needs to be a single value giving the RMSE threshold. If object is an instance of the dgp or bundle class and eval = NULL, target can be a vector of values that gives the RMSE thresholds for different output dimensions or different emulators. If a single value is provided, it will be used as the RMSE threshold for all output dimensions (if object is an instance of the dgp) or all emulators (if object is an instance of the bundle). If a customized function is supplied to eval, the user needs to ensure that the length of target is equal to that of the output from eval.

• When defining f, it is important to ensure that:

  • the column order of the first argument of f is consistent with the training input used for the emulator;

  • the column order of the output matrix of f is consistent with the order of emulator output dimensions (if object is an instance of the dgp class), or the order of emulators placed in object (if object is an instance of the bundle class).

• The output matrix produced by f may include NAs. This is especially beneficial as it allows the sequential design process to continue without interruption, even if errors or NA outputs are encountered from f at certain input locations identified by the sequential designs. Users should ensure to handle any errors within f by appropriately returning NAs.

• When defining eval, the output metric needs to be positive if draw() is used with log = T. And one needs to ensure that a lower metric value indicates a better emulation performance if target is set.

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

Examples

## Not run: 

# load packages and the Python env
library(lhs)
library(dgpsi)

# construct a 2D non-stationary function that takes a matrix as the input
f <- function(x) {
  sin(1/((0.7*x[,1,drop=F]+0.3)*(0.7*x[,2,drop=F]+0.3)))
}

# generate the initial design
X <- maximinLHS(5,2)
Y <- f(X)

# generate the validation data
validate_x <- maximinLHS(30,2)
validate_y <- f(validate_x)

# training a 2-layered DGP emulator with the initial design
m <- dgp(X, Y)

# specify the ranges of the input dimensions
lim_1 <- c(0, 1)
lim_2 <- c(0, 1)
lim <- rbind(lim_1, lim_2)

# 1st wave of the sequential design with 10 steps
m <- design(m, N=10, limits = lim, f = f, x_test = validate_x, y_test = validate_y)

# 2nd wave of the sequential design with 10 steps
m <- design(m, N=10, limits = lim, f = f, x_test = validate_x, y_test = validate_y)

# 3rd wave of the sequential design with 10 steps
m <- design(m, N=10, limits = lim, f = f, x_test = validate_x, y_test = validate_y)

# draw the design created by the sequential design
draw(m,'design')

# inspect the trace of RMSEs during the sequential design
draw(m,'rmse')

# reduce the number of imputations for faster OOS
m_faster <- set_imp(m, 5)

# plot the OOS validation with the faster DGP emulator
plot(m_faster, x_test = validate_x, y_test = validate_y)

## End(Not run)