An Automatic Method for the Analysis of Experiments using Hierarchical Garrote

Description

'HiGarrote()' provides an automatic method for analyzing experimental data. This function applies the nonnegative garrote method to select important effects while preserving their hierarchical structures. It first estimates regression parameters using generalized ridge regression, where the ridge parameters are derived from a Gaussian process prior placed on the input-output relationship. Subsequently, the initial estimates will be used in the nonnegative garrote for effects selection.

Usage

HiGarrote(
  D,
  y,
  heredity = "weak",
  quali_id = NULL,
  quanti_id = NULL,
  quali_sum_idx = NULL,
  user_def_coding = NULL,
  user_def_coding_idx = NULL,
  model_type = 1
)

Arguments

Value

The function returns a list with:

nng_estimate

A vector for the nonnegative garrote estimates of the identified effects.

U

A model matrix of D.

pred_info

A list containing information needed for future predictions.

References

Yu, W. Y. and Joseph, V. R. (2025). Automated Analysis of Experiments using Hierarchical Garrote. Journal of Quality Technology, 1-15. doi:10.1080/00224065.2025.2513508.

Examples

# Cast fatigue experiment
data(cast_fatigue)
X <- cast_fatigue[,1:7]
y <- cast_fatigue[,8]
fit_Hi <- HiGarrote::HiGarrote(X, y)
fit_Hi$nng_estimate

# Blood glucose experiment
data(blood_glucose)
X <- blood_glucose[,1:8]
y <- blood_glucose[,9]
fit_Hi <- HiGarrote::HiGarrote(X, y, quanti_id = 2:8) 
fit_Hi$nng_estimate


# Router bit experiment --- Use default Helmert coding
data(router_bit)
X <- router_bit[, 1:9]
y <- router_bit[,10]
fit_Hi <- HiGarrote::HiGarrote(X, y, quali_id = c(4,5))
fit_Hi$nng_estimate

# Router bit experiment --- Use sum coding
fit_Hi <- HiGarrote::HiGarrote(X, y, quali_id = c(4,5), quali_sum_idx = c(4,5))
fit_Hi$nng_estimate

# Router bit experiment --- Use user-defined coding system for qualitative factors
fit_Hi <- HiGarrote::HiGarrote(X, y, quali_id = c(4,5),
 user_def_coding = list(matrix(c(-1,-1,1,1,1,-1,-1,1,-1,1,-1,1), ncol = 3)),
 user_def_coding_idx = list(c(4,5)))
fit_Hi$nng_estimate

# Resin experiment --- Use model_type = 2
data(resin)
X <- resin[,1:9]
y <- log(resin$Impurity)
fit_Hi <- HiGarrote::HiGarrote(X, y, quanti_id = c(1:9), model_type = 2)
fit_Hi$nng_estimate

# Epoxy experiment --- Use model_type = 3
data(epoxy)
X <- epoxy[,1:23]
y <- epoxy[,24]
fit_Hi <- HiGarrote::HiGarrote(X, y, model_type = 3)
fit_Hi$nng_estimate

# Experiments with replicates
# Generate simulated data
data(cast_fatigue)
X <- cast_fatigue[,1:7]
U <- data.frame(model.matrix(~.^2, X)[,-1])
error <- matrix(rnorm(24), ncol = 2) # two replicates for each run
y <- 20*U$A + 10*U$A.B + 5*U$A.C + error
fit_Hi <- HiGarrote::HiGarrote(X, y)
fit_Hi$nng_estimate

Blood Glucose Experiment

Description

Hamada and Wu (1992) analyzed an 18-run experiment designed to study blood glucose readings of a clinical testing device. The experiment contains one two-level factor and seven three-level quantitative factors, which are denoted by A through H.

Usage

data(blood_glucose)

Format

A data frame with 18 rows and 9 columns.

Source

Hamada, M. and Wu, C. F. J. (1992). Analysis of Designed Experiments with Complex Aliasing. Journal of Quality Technology, 24(3), 130-137. doi:10.1080/00224065.1992.11979383.

Cast Fatigue Experiment

Description

Hunter et al. (1982) used a 12-run Plackett-Burman design to investigate the effects of seven two-level factors on the fatigue life of weld-repaired castings. The seven factors are denoted by capital letters A through G.

Usage

data(cast_fatigue)

Format

A data frame with 12 rows and 8 columns.

Source

Hunter, G. B., Hodi, F. S., and Eagar, T. W. (1982). High Cycle Fatigue of Weld Repaired Cast Ti-6AI-4V. Metallurgical Transactions A, 13(9), 1589-1594. doi:10.1007/BF02644799.

Epoxy Experiment

Description

Lin (1993) employed a half-fraction of a 28-run PB design to investigate an experiment aimed at developingan epoxy adhesive system. The original design matrix contains 14 runs and 24 factors. Since factors 13 and 16 were assigned to the same columns in the original design matrix, only factor 13 is listed here.

Usage

data(epoxy)

Format

A data frame with 14 rows and 24 columns.

Source

Lin, D. K. J. (1993). A New Class of Supersaturated Designs. Technometrics, 35(1), 28-31. doi:10.2307/1269286.

Nonnegative Garrote Method with Hierarchical Structures

Description

'nnGarrote()' implements the nonnegative garrote method, as described in Yuan et al. (2009), for selecting important variables while preserving hierarchical structures. The method begins by obtaining the least squares estimates of the regression parameters under a linear model. These initial estimates are then used in the nonnegative garrote to perform variable selection. Note that this method is suitable only when the number of observations is much larger than the number of variables, ensuring that the least squares estimation remains reliable.

Usage

nnGarrote(X, y, heredity = "weak", model_type = 1)

Arguments

Value

The function returns a list with:

nng_estimate

A vector for the nonnegative garrote estimates of the identified variables.

U

A scaled model matrix.

pred_info

A list of information required for further prediction.

References

Yuan, M., Joseph, V. R., and Zou H. (2009). Structured Variable Selection and Estimation. The Annals of Applied Statistics, 3(4), 1738–1757. doi:10.1214/09-AOAS254.

Examples

# Generate data
x1 <- runif(100)
x2 <- runif(100)
x3 <- runif(100)
error <- rnorm(100)
X <- data.frame(x1, x2, x3)
U <- model.matrix(~. + x1:x2 + x1:x3 + x2:x3 + I(x1^2) + I(x2^2) + I(x3^2) - 1, X)
U <- data.frame(scale(U))
colnames(U) <- c("x1", "x2", "x3", "x1:x1", "x2:x2", "x3:x3", "x1:x2", "x1:x3", "x2:x3")
y <- 3 + 3*U$x1 + 3*U$`x1:x1` + 3*U$`x1:x2`+ 3*U$`x1:x3` + error

# Fit nnGarrote
fit_nng <- HiGarrote::nnGarrote(X, y)
fit_nng$nng_estimate

Make Predictions from a "HiGarrote" Object

Description

This function makes predictions from a linear model constructed using the important effects selected by HiGarrote.

Usage

## S3 method for class 'HiGarrote'
predict(object, new_D, ...)

Arguments

Value

The function returns a list with:

new_U

A model matrix of new_D.

prediction_nng

Predictions for new_D. The coefficients of the predictive equation are based on nonnegative garrote estimates.

prediction_lm

Predictions for new_D. The coefficients of the predictive equation are estimated via ordinary least squares.

Examples

# Cast fatigue experiment
data(cast_fatigue)
X <- cast_fatigue[1:10,1:7]
y <- cast_fatigue[1:10,8]
fit_Hi <- HiGarrote::HiGarrote(X, y)

# make predictions
new_D <- cast_fatigue[11:12,1:7]
pred_Hi <- predict(fit_Hi, new_D)

Make Predictions from a "nnGarrote" Object

Description

This function makes predictions from a linear model constructed using the important effects selected by nnGarrote.

Usage

## S3 method for class 'nnGarrote'
predict(object, new_X, ...)

Arguments

Value

The function returns a list with:

new_U

A model matrix of new_X.

prediction_nng

Predictions for new_X. The coefficients of the predictive equation are based on nonnegative garrote estimates.

prediction_lm

Predictions for new_X. The coefficients of the predictive equation are estimated via ordinary least squares.

Examples

# Generate data
x1 <- runif(100)
x2 <- runif(100)
x3 <- runif(100)
error <- rnorm(100)
X <- data.frame(x1, x2, x3)
U <- model.matrix(~. + x1:x2 + x1:x3 + x2:x3 + I(x1^2) + I(x2^2) + I(x3^2) - 1, X)
U <- data.frame(scale(U))
colnames(U) <- c("x1", "x2", "x3", "x1:x1", "x2:x2", "x3:x3", "x1:x2", "x1:x3", "x2:x3")
y <- 3 + 3*U$x1 + 3*U$`x1:x1` + 3*U$`x1:x2`+ 3*U$`x1:x3` + error

# training and testing set
train_idx <- sample(1:100, 80)
X_train <- X[train_idx,]
y_train <- y[train_idx]
X_test <- X[-train_idx,]
y_test <- y[-train_idx]

# fit nnGarrote
fit_nng <- HiGarrote::nnGarrote(X_train, y_train)

# predict
pred_nng <- predict(fit_nng, X_test)

Resin Experiment

Description

Jones and Lanzerath (2021) described a 21-run experiment aimed at creating a special resin for car vents designed to prevent moisture accumulation within a car’s housing. The experiment is carried out using a definitive screening design (DSD) with nine continuous factors, labeled A through J, with the exclusion of I.

Usage

data(resin)

Format

A data frame with 21 rows and 12 columns.

Source

Jones, B. and Lanzerath, M. (2021). A Novel Application of a Definitive Screening Design: A Case Study. Quality Engineering, 33(3), 563-569. doi:10.1080/08982112.2021.1892758.

Router Bit Experiment

Description

Phadke (1986) described a 32-run experiment aimed at increasing the lifespan of router bits used in a routing process to cut printed wiring boards from a panel. The experiment contains seven two-level factors and two four-level qualitative factors which are denoted by A through J with the exclusion of I.

Usage

data(router_bit)

Format

A data frame with 32 rows and 10 columns.

Source

Phadke, M. S. (1986). Design Optimization Case Studies. AT&T Technical Journal, 65(2), 51-68. doi:10.1002/j.1538-7305.1986.tb00293.x.