Package 'MOFAT' reference manual

Title:	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.
Authors:	Difan Song [aut, cre], Wei-Yang Yu [aut], Qian Xiao [aut], V. Roshan Joseph [aut]
Maintainer:	Difan Song <[email protected]>
License:	GPL (>= 2)
Version:	2.0
Built:	2026-10-26 23:30:44 UTC
Source:	https://github.com/cran/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 [email protected]

Authors:

Difan Song [email protected]
Wei-Yang Yu [email protected]
Qian Xiao [email protected]
V. Roshan Joseph [email protected]

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

Arguments

p

number of factors; an integer of at least 2. Not required when ofat_design is supplied; inferred as ncol(ofat_design), overriding any supplied value.

l

even number of base runs; an integer of at least 2. Not required when ofat_design is supplied; inferred as nrow(ofat_design) / (p + 1), overriding any supplied value. Also determines the augmentation kernel scale.

structure

indicating the OFAT structure. Choose from "standard" or "strict". The default is "standard". Ignored when ofat_design is supplied.

n_aug

number of points to augment.

m

number of Sobol evaluation points for initial SOFT optimization. The default is 500. Ignored when ofat_design is supplied.

N

size of the Sobol candidate pool for the sequential augmentation. The default is 10000.

backend

one of "auto", "cpu", "cuda", or "metal". Invalid, unavailable, or failed GPU backends silently use CPU.

ofat_design

NULL, or a nonempty numeric matrix with p columns, l * (p + 1) runs for an even integer l, and values in [0, 1]. When supplied, p and l are inferred from its dimensions, and this matrix is augmented directly instead of generating an initial SOFT design. Its range does not affect the Sobol candidate bounds.

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:

design

The design matrix containing the generated or supplied initial design followed by the sequentially augmented points.

backend_used

The computational backend actually used.

device

The CPU or GPU device used for computation.

fallback_reason

The reason for falling back to the CPU backend, or NULL if no fallback occurred.

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

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

Arguments

design

design matrix, which should have the Sobol' design structure

y

response vector

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

mustar

$\mu^*$ measure

t

Total Sobol' index

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

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

Arguments

p

number of factors

l

number of base runs

method

choose among "uniform", "projection", and "best"

nstart

number of random starts used when l > 3

init

initialization method for the base matrix when l > 3; choose among "rLHD", "MaxPro", and "MmLHD"

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

design

MOFAT design

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 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))

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)
mofat_backend_info(refresh = FALSE)

Arguments

refresh

logical scalar. If TRUE, discard the cached CUDA probe and repeat helper loading, device discovery, and the self-test.

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

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

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

Arguments

l

number of base runs

p_continuous

number of continuous factors

b_continuous

optional list of length p_continuous; each element should be a length-two numeric vector giving the lower and upper bounds used to rescale a continuous factor

p_disnum

number of discrete numeric factors

l_disnum

specification of the discrete numeric levels. Supply either a numeric vector giving the number of levels for each factor, or a list whose $j$ th element contains the numeric levels for factor $j$

p_nom

number of nominal factors

l_nom

specification of the nominal levels. Supply either a numeric vector giving the number of levels for each nominal factor, or a list whose element lengths define the number of nominal levels. The returned design always encodes nominal factors as integer level labels

nstart

number of candidate mixed-factor designs to generate

...

additional arguments passed to mofat, such as method or init

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

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)
# 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)

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

Arguments

alpha

positive MIM kernel exponent.

beta

positive MIM kernel scale.

design

the design matrix.

qmc

the evaluation set.

backend

one of "auto", "cpu", "cuda", or "metal". Invalid, unavailable, or failed GPU backends silently use CPU.

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:

Q

the $Q$ criterion of the design.

normalized_Q

the normalized criterion defined as $Q^{1/p}$ .

backend_used

The computational backend actually used.

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

Examples

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

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)
qmc_generate(n, p, threshold = 10)

Arguments

n

number of Sobol points.

p

number of factors.

threshold

dimension cutoff for corner augmentation. When p <= threshold, all $2^p$ binary corners are added. Otherwise, representative corners derived from the Sobol points are added.

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

Examples

qmc <- qmc_generate(n = 1000, p = 2)
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")
soft(p, l, structure = "standard", m = 500L, backend = "auto")

Arguments

p

number of factors; an integer of at least 2.

l

even number of base runs; an integer of at least 2.

structure

indicating the OFAT structure. Choose from "standard" or "strict". The default is "standard".

m

number of Sobol evaluation points generated before vertex augmentation. The default is 500.

backend

one of "auto", "cpu", "cuda", or "metal". Invalid, unavailable, or failed GPU backends silently use CPU.

Details

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

Value

A list with the following components:

design

The SOFT design matrix.

Q

The $Q$ criterion of the design.

normalized_Q

The normalized criterion defined as $Q^{1/p}$ .

qmc_size

The number of unique evaluation points after adding corner points and removing duplicates. This value may differ from the input m.

backend_used

The computational backend actually used.

device

The CPU or GPU device used for computation.

fallback_reason

The reason for falling back to the CPU backend, or NULL if no fallback occurred.

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

Examples

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

Package 'MOFAT'

Help Index

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

Description

Author(s)

Augmented SOFT (A-SOFT) Designs

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Screening Measures

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

MOFAT

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Inspect MOFAT Computational Backends

Description

Usage

Arguments

Value

Examples

MOFAT Designs for Quantitative and Qualitative Factors

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Q Criterion

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Evaluation Set for the Q Criterion

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

SOFT Designs

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples