Package {MOFAT}

MOFAT: Maximum One-Factor-at-a-Time Designs

Description

Identifying important factors from a large number of potentially important factors in a highly nonlinear and computationally expensive black-box model is a difficult problem. Xiao, Joseph, and Ray (2022) doi:10.1080/00401706.2022.2141897 proposed Maximum One-Factor-at-a-Time (MOFAT) designs for this purpose. A MOFAT design is a collection of one-factor-at-a-time (OFAT) designs constructed to achieve maximal screening capability. Song and Joseph (2026) doi:10.48550/arXiv.2609.02087 extended the MOFAT framework to incorporate nominal and discrete numeric factors. The resulting designs are implemented in the mofatQQ function. Yu and Joseph (2026) doi:10.48550/arXiv.2606.02533 proposed Space-Filling One-Factor-at-a-Time (SOFT) designs, which achieve optimal space-filling properties while retaining the screening capability. Therefore, SOFT designs are useful for both factor screening and surrogate modeling. The work on MOFAT is supported by a U.S. National Science Foundation (NSF) grant CMMI-1921646 https://www.nsf.gov/awardsearch/show-award/?AWD_ID=1921646, and the work on MOFATQQ and SOFT is supported by a U.S. NSF grant DMS-2310637.

Author(s)

Maintainer: Difan Song dfsong96@gmail.com

Authors:

• Difan Song dfsong96@gmail.com

• Wei-Yang Yu wyu322@gatech.edu

• Qian Xiao qian.xiao@sjtu.edu.cn

• V. Roshan Joseph roshan@gatech.edu

Q Criterion

Description

Evaluate the Q criterion for a given design. This criterion measures the design's space-filling performance and serves as the objective function used to construct SOFT designs. See Yu and Joseph (2026) for further details.

Usage

Q(alpha, beta, design, qmc, backend = "auto")

Arguments

Details

The Q criterion for a design \mathbf{D}=[\mathbf{x}_1,\ldots,\mathbf{x}_n]^T is defined as

\min_{\mathbf{x}\in\mathbf{E}} \mathbf{r}(\mathbf{x})^T \mathbf{R}^{-1}\mathbf{r}(\mathbf{x}),

where \mathbf{E} is the evaluation set provided by qmc. Here, \mathbf{r}(\mathbf{x}) = [R(\mathbf{x}_1,\mathbf{x}), \ldots, R(\mathbf{x}_n,\mathbf{x})]^T and \mathbf{R}_{i,j} = R(\mathbf{x}_i,\mathbf{x}_j), where

R(\mathbf{x},\mathbf{x}') = \prod_{j=1}^{p} \left\{1+\frac{(x_j-x'_j)^2}{\beta}\right\}^{-\alpha}.

Larger values of Q indicate better space-filling performance.

Value

A list with the following components:

Author(s)

Wei-Yang Yu and V. Roshan Joseph

References

Yu, W.-Y. and Joseph, V. R. (2026). Space-Filling One-Factor-At-A-Time Designs. doi:10.48550/arXiv.2606.02533

See Also

qmc_generate

Examples

design <- soft(p = 2, l = 4)$design
qmc <- qmc_generate(n = 1000, p = 2)
Q(1, 1/(2*4)^2, design, qmc)

Augmented SOFT (A-SOFT) Designs

Description

Construct an augmented SOFT (A-SOFT) design by combining a SOFT component with sequentially selected unrestricted space-filling points. The resulting design has the form

D = \begin{pmatrix} D_{\mathrm{soft}} \\ D_{\mathrm{aug}} \end{pmatrix},

where D_{\mathrm{soft}} is the initial SOFT design and D_{\mathrm{aug}} contains the augmented points.

Usage

a_soft(
  p,
  l,
  structure = "standard",
  n_aug,
  m = 500L,
  N = 10000L,
  backend = "auto",
  ofat_design = NULL
)

Arguments

Details

When ofat_design = NULL, the SOFT component contains n_{\mathrm{soft}} = l(p+1) runs, and the final design contains l(p+1) + n_{\mathrm{aug}} runs. Alternatively, a user-supplied initial design can be provided through ofat_design, in which case the supplied design is held fixed and augmented directly.

The augmented points are selected sequentially from a Sobol candidate pool by maximizing the Q criterion. The augmented points themselves are not constrained to follow the OFAT structure. CPU and optional GPU computation are supported. See Yu and Joseph (2026) for further details.

The initial SOFT design and the sequential augmentation use different kernel scales for optimizing the Q criterion. For the initial SOFT optimization, the kernel parameters are fixed internally at \alpha=1 and \beta=(2l)^{-2}. During sequential augmentation, they are fixed at \alpha=1 and \beta=(2l+n_{\mathrm{aug}})^{-2}.

Regardless of whether ofat_design is supplied, candidate bounds for the augmentation are fixed at [0.25/(2l+n_{\mathrm{aug}}), 1-0.25/(2l+n_{\mathrm{aug}})].

Value

A list with the following components:

Author(s)

Wei-Yang Yu and V. Roshan Joseph

References

Yu, W.-Y. and Joseph, V. R. (2026). Space-Filling One-Factor-At-A-Time Designs. doi:10.48550/arXiv.2606.02533

See Also

soft, Q, qmc_generate

Examples

result <- a_soft(p = 2, l = 4, n_aug = 8)
result$design

existing <- mofat(p = 2, l = 4, method = "projection")
extended <- a_soft(n_aug = 8, ofat_design = existing)
extended$design

Screening Measures

Description

This function can be used for computing screening measures.

Usage

measure(design, y)

Arguments

Details

The measure function computes the screening measures such as the total Sobol' indices (Sobol' 1993) and \mu^* measure of Campolongo et al. (2007). The design matrix should have the Sobol' design structure. Please see Xiao et al. (2022) and Yu and Joseph (2026) for further details.

Value

Author(s)

Wei-Yang Yu, Difan Song, Qian Xiao, and V. Roshan Joseph

References

Sobol’, I. M. (1993). On Sensitivity Estimation for Nonlinear Mathematical Models. Mathematical Modeling and Computational Experiments, 1, 407–414.

Campolongo, F., Cariboni, J., and Saltelli, A. (2007). An Effective Screening Design for Sensitivity Analysis of Large Models. Environmental Modelling and Software, 22, 1509–1518.

Xiao, Q., Joseph, V. R., and Ray, D. M. (2022). Maximum One-Factor-At-A-Time Designs for Screening in Computer Experiments. Technometrics, 65, 261–270.

Yu, W.-Y. and Joseph, V. R. (2026). Space-Filling One-Factor-At-A-Time Designs. doi:10.48550/arXiv.2606.02533

See Also

mofat, mofatQQ, soft

Examples

#Friedman function
fun <- function (X)
{
 Y <- 10*sin(pi*X[1]*X[2]) + 20*(X[3] - 0.5)^2 + 10*X[4] + 5*X[5]
 return(Y)
}
design = mofat(p=10, l=3)
y = apply(design, 1, fun)

#Screening measures
measure(design, y)

MOFAT

Description

This function can be used for generating MOFAT designs.

Usage

mofat(p, l, method = "best", nstart = 1, init = "MaxPro")

Arguments

Details

The mofat function generates the MOFAT design for a given number of factors (p\ge 2) and number of base runs (l \ge 3). The total number of runs in the MOFAT design will be l(p+1). A MOFAT design can be viewed as an optimized version of Morris screening design (Morris 1991) by exploiting its connections with the Monte Carlo-based design of Sobol' (1993). Please see Xiao et al. (2022) for details.

Three choices for the method are given: "uniform", "projection", and "best". Option "uniform" gives l equally-spaced levels for the entire design, which are also balanced. "projection" option adjusts the levels of the two base matrices A and B such that there are 2l or 2l-1 levels in the design depending on l is even or odd. Option "best" (default) chooses the best among the first two options using maximin distance criterion.

Value

Author(s)

Qian Xiao and V. Roshan Joseph

References

Morris, M. D. (1991), “Factorial sampling plans for preliminary computational experiments,” Technometrics, 33, 161–174.

Sobol’, I. M. (1993), “On sensitivity estimation for nonlinear mathematical models,” Mathematical Modeling and Computational Experiments, 1, 407–414.

Xiao, Q., Joseph, V. R., and Ray, D. M. (2022). “Maximum One-Factor-At-A-Time Designs for Screening in Computer Experiments”. Technometrics, 65, 261–270.

Examples

#MOFAT with three base runs
mofat(p=10, l=3, method="uniform")
mofat(p=10, l=3, method="projection")

#MOFAT with five base runs
mofat(p=10,l=5)
dim(mofat(p=125,l=5))

MOFAT Designs for Quantitative and Qualitative Factors

Description

This function generates MOFAT-based designs for continuous, discrete numeric, and nominal factors.

Usage

mofatQQ(
  l,
  p_continuous = 0,
  b_continuous = NULL,
  p_disnum = 0,
  l_disnum = NULL,
  p_nom = 0,
  l_nom = NULL,
  nstart = 1,
  ...
)

Arguments

Details

The mofatQQ function first generates a base MOFAT design with mofat, then transforms the columns according to factor type. The total number of factors must be at least two. Continuous factors are kept on the unit scale or rescaled to user-supplied bounds. Discrete numeric factors are mapped either to integer levels or to user-supplied numeric levels. Nominal factors are encoded by integer level labels, and the nominal columns are placed to the right of the numeric columns in the returned design. Each discrete numeric and nominal factor must have at least two levels.

When nstart > 1, the function generates multiple candidate mixed designs and selects the one with the smallest projected space-filling criterion. In that criterion, continuous and discrete numeric distances are measured on a rescaled numeric scale, while nominal differences are treated as binary mismatches.

Value

A design matrix with columns ordered as continuous, discrete numeric, and nominal factors. Nominal factors are returned as integer level labels.

Author(s)

Difan Song and V. Roshan Joseph

References

Xiao, Q., Joseph, V. R., and Ray, D. M. (2022). Maximum One-Factor-At-A-Time Designs for Screening in Computer Experiments. Technometrics, 65, 261–270.

Song, D., and Joseph, V. R. (2026). Efficient Screening Designs for Expensive Black-Box Models with Qualitative and Quantitative Factors. doi:10.48550/arXiv.2609.02087

See Also

mofat, measure

Examples

# Mixed continuous and nominal factors
D <- mofatQQ(
  l = 5,
  p_continuous = 2,
  b_continuous = list(c(-1, 1), c(0, 10)),
  p_nom = 1,
  l_nom = 3,
  nstart = 2,
  init = "rLHD"
)
dim(D)

Inspect MOFAT Computational Backends

Description

Report whether the CPU, Metal, and CUDA backends were compiled and whether they are usable in the current R process. CUDA probing includes helper ABI validation and a small runtime self-test. Results are cached for the R session unless refresh = TRUE is requested.

Usage

mofat_backend_info(refresh = FALSE)

Arguments

Value

A named list with cpu, metal, and cuda components. Every component contains compiled, available, device, and reason. CUDA additionally reports helper, toolkit, runtime, driver, and compute-capability details when available.

Examples

info <- mofat_backend_info()
vapply(info, function(x) isTRUE(x$available), logical(1))
info$cuda$reason

Evaluation Set for the Q Criterion

Description

Generate Sobol points and augment them with corner points to evaluate the Q criterion. Duplicate points are removed from the resulting evaluation set. See Yu and Joseph (2026) for further details.

Usage

qmc_generate(n, p, threshold = 10)

Arguments

Value

A matrix with p columns containing the unique Sobol and corner evaluation points. The number of rows may differ from n because corner points are added and duplicate rows are removed.

Author(s)

Wei-Yang Yu and V. Roshan Joseph

References

Yu, W.-Y. and Joseph, V. R. (2026). Space-Filling One-Factor-At-A-Time Designs. doi:10.48550/arXiv.2606.02533

See Also

Q

Examples

qmc <- qmc_generate(n = 1000, p = 2)

SOFT Designs

Description

Generate a SOFT design under either the standard or strict OFAT structure. For p factors and l base runs, the resulting design contains l(p+1) runs. The design is constructed by maximizing the Q criterion under either the standard or strict OFAT structure. Computation can use an available GPU backend, with CPU execution used when GPU acceleration is unavailable. See Yu and Joseph (2026) for further details.

Usage

soft(p, l, structure = "standard", m = 500L, backend = "auto")

Arguments

Details

When optimizing Q, we set \alpha = 1 and \beta = (2*l)^{-2} internally.

Value

A list with the following components:

Author(s)

Wei-Yang Yu and V. Roshan Joseph

References

Yu, W.-Y. and Joseph, V. R. (2026). Space-Filling One-Factor-At-A-Time Designs. doi:10.48550/arXiv.2606.02533

See Also

Q, qmc_generate

Examples

result <- soft(p = 2, l = 4)
result$design