Package 'MTSYS' reference manual

Title:	Methods in Mahalanobis-Taguchi (MT) System
Description:	Mahalanobis-Taguchi (MT) system is a collection of multivariate analysis methods developed for the field of quality engineering. MT system consists of two families depending on their purpose. One is a family of Mahalanobis-Taguchi (MT) methods (in the broad sense) for diagnosis (see Woodall, W. H., Koudelik, R., Tsui, K. L., Kim, S. B., Stoumbos, Z. G., and Carvounis, C. P. (2003) <doi:10.1198/004017002188618626>) and the other is a family of Taguchi (T) methods for forecasting (see Kawada, H., and Nagata, Y. (2015) <doi:10.17929/tqs.1.12>). The MT package contains three basic methods for the family of MT methods and one basic method for the family of T methods. The MT method (in the narrow sense), the Mahalanobis-Taguchi Adjoint (MTA) methods, and the Recognition-Taguchi (RT) method are for the MT method and the two-sided Taguchi (T1) method is for the family of T methods. In addition, the Ta and Tb methods, which are the improved versions of the T1 method, are included.
Authors:	Akifumi Okayama [aut, cre], Masato Ohkubo [ctb], Yasushi Nagata [ctb]
Maintainer:	Akifumi Okayama <[email protected]>
License:	MIT + file LICENSE
Version:	1.2.1
Built:	2024-11-08 02:40:55 UTC
Source:	https://github.com/okayaa/mtsys

Function to calculate a cofactor matrix

Description

calc_cofactor calculates a cofactor matrix.

Usage

calc_cofactor(data)
calc_cofactor(data)

Arguments

data

Matrix with n rows (samples) and p columns (variables). All data should be continuous values and should not have missing values.

Value

calc_cofactor returns a cofactor matrix of size p x p.

Examples

# 40 data for versicolor in the iris dataset                            
iris_versicolor <- iris[61:100, -5] 
                            
calc_cofactor(cov(iris_versicolor))
 
# 40 data for versicolor in the iris dataset                            
iris_versicolor <- iris[61:100, -5] 
                            
calc_cofactor(cov(iris_versicolor))

Function to estimate M value (M hat) for a family of T methods.

Description

calc_M_hat estimates M values (M hat) for the T method.

Usage

calc_M_hat(X, beta_hat, eta_hat)
calc_M_hat(X, beta_hat, eta_hat)

Arguments

`X`	Matrix with n rows (samples) and q columns (variables). The independent variable data after the data transformation. All data should be continuous values and should not have missing values.
`beta_hat`	Vector with length q. Estimated proportionality constants between each independent variable and the dependent variable.
`eta_hat`	Vector with length q. Estimated squared signal-to-noise ratios (S/N) coresponding to `beta_hat`.

Value

Vector with length n. Estimated M values (M hat).

Examples

# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

# The following samples are data other than the unit space data and the test
# data.
stackloss_signal <- stackloss[-c(2, 9, 10, 11, 12, 19, 20, 21), ]

# The following settings are same as the T1 method.
model <- general_T(unit_space_data = stackloss_center,
                   signal_space_data = stackloss_signal,
                   generates_transform_functions =
                                       generates_transformation_functions_T1,
                   includes_transformed_data = TRUE)

modified_eta_hat <- model$eta_hat
modified_eta_hat[3] <- 0

(modified_M_hat <- calc_M_hat(model$X, model$beta_hat, modified_eta_hat))

# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

# The following samples are data other than the unit space data and the test
# data.
stackloss_signal <- stackloss[-c(2, 9, 10, 11, 12, 19, 20, 21), ]

# The following settings are same as the T1 method.
model <- general_T(unit_space_data = stackloss_center,
                   signal_space_data = stackloss_signal,
                   generates_transform_functions =
                                       generates_transformation_functions_T1,
                   includes_transformed_data = TRUE)

modified_eta_hat <- model$eta_hat
modified_eta_hat[3] <- 0

(modified_M_hat <- calc_M_hat(model$X, model$beta_hat, modified_eta_hat))

Function to calculate overall prediction eta for the T method

Description

calc_M_hat calculates the overall prediction eta for the T method.

Usage

calc_overall_predicton_eta(M, M_hat, subtracts_V_e = TRUE)
calc_overall_predicton_eta(M, M_hat, subtracts_V_e = TRUE)

Arguments

`M`	Vector with length n. The (true) value of the dependent variable after the data trasformation.
`M_hat`	Vector with length n. The estimated values of the dependent variable after the data trasformation.
`subtracts_V_e`	If `TRUE`, then the error variance is subtracted in the numerator when calculating `eta_hat`.

Value

Numeric. Overall prediction eta which is used to measure the estimation accuracy.

Examples


# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

# The following samples are data other than the unit space data and the test
# data.
stackloss_signal <- stackloss[-c(2, 9, 10, 11, 12, 19, 20, 21), ]

# The following settings are same as the T1 method.
model <- general_T(unit_space_data = stackloss_center,
                   signal_space_data = stackloss_signal,
                   generates_transform_functions =
                                       generates_transformation_functions_T1,
                   subtracts_V_e = TRUE,
                   includes_transformed_data = TRUE)

modified_eta_hat <- model$eta_hat
modified_eta_hat[3] <- 0

modified_M_hat <- calc_M_hat(model$X, model$beta_hat, modified_eta_hat)

(modified_overall_predicton_eta <-
                            calc_overall_predicton_eta(model$M,
                                                       modified_M_hat,
                                                       subtracts_V_e = TRUE))

# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

# The following samples are data other than the unit space data and the test
# data.
stackloss_signal <- stackloss[-c(2, 9, 10, 11, 12, 19, 20, 21), ]

# The following settings are same as the T1 method.
model <- general_T(unit_space_data = stackloss_center,
                   signal_space_data = stackloss_signal,
                   generates_transform_functions =
                                       generates_transformation_functions_T1,
                   subtracts_V_e = TRUE,
                   includes_transformed_data = TRUE)

modified_eta_hat <- model$eta_hat
modified_eta_hat[3] <- 0

modified_M_hat <- calc_M_hat(model$X, model$beta_hat, modified_eta_hat)

(modified_overall_predicton_eta <-
                            calc_overall_predicton_eta(model$M,
                                                       modified_M_hat,
                                                       subtracts_V_e = TRUE))

Function to predict a diagnosis for a family of Mahalanobis-Taguchi (MT) methods

Description

diagnosis is a generic function. For details, see diagnosis.MT, diagnosis.MTA, diagnosis.RT or general_diagnosis.MT.

Usage

diagnosis(unit_space, newdata, threshold, includes_transformed_newdata)
diagnosis(unit_space, newdata, threshold, includes_transformed_newdata)

Arguments

`unit_space`	Object generated as a unit space.
`newdata`	Matrix with n rows (samples) and p columns (variables). The data are used to calculate the desired distances from the unit space. All data should be continuous values and should not have missing values.
`threshold`	Numeric specifying the threshold value to classify each sample into positive (`TRUE`) or negative (`FALSE`).
`includes_transformed_newdata`	If `TRUE`, then the transformed data for `newdata` are included in a return object.

Value

A list containing the following components is returned.

`distance`	Vector with length n. Distances from the unit space to each sample.
`le_threshold`	Vector with length n. Logical values indicating the distance of each sample is less than or equal to the threhold value (`TRUE`) or not (`FALSE`).
`threshold`	Numeric value to classify the sample into positive or negative.
`unit_space`	Object passed by `unit_space`.
`n`	The number of samples for `newdata`.
`q`	The number of variables after the data transformation.
`x`	If `includes_transformed_newdata` is `TRUE`, then the transformed data for `newdata` are included.

Diagnosis method for the Mahalanobis-Taguchi (MT) method

Description

diagnosis.MT (via diagnosis) calculates the mahalanobis distance based on the unit space generated by MT or generates_unit_space(..., method = "MT") and classifies each sample into positive (TRUE) or negative (FALSE) by comparing the values with the set threshold value.

Usage

## S3 method for class 'MT'
diagnosis(unit_space, newdata, threshold = 4,
  includes_transformed_newdata = FALSE)
## S3 method for class 'MT'
diagnosis(unit_space, newdata, threshold = 4,
  includes_transformed_newdata = FALSE)

Arguments

`unit_space`	Object of class "MT" generated by `MT` or `generates_unit_space`(..., method = "MT").
`newdata`	Matrix with n rows (samples) and p columns (variables). The data are used to calculate the desired distances from the unit space. All data should be continuous values and should not have missing values.
`threshold`	Numeric specifying the threshold value to classify each sample into positive (`TRUE`) or negative (`FALSE`).
`includes_transformed_newdata`	If `TRUE`, then the transformed data for `newdata` are included in a return object.

Value

diagnosis.MT (via diagnosis) returns a list containing the following components:

`distance`	Vector with length n. Distances from the unit space to each sample.
`le_threshold`	Vector with length n. Logical values indicating the distance of each sample is less than or equal to the threhold value (`TRUE`) or not (`FALSE`).
`threshold`	Numeric value to classify the sample into positive or negative.
`unit_space`	Object of class "MT" passed by `unit_space`.
`n`	The number of samples for `newdata`.
`q`	The number of variables after the data transformation. q equals p.
`x`	If `includes_transformed_newdata` is `TRUE`, then the transformed data for `newdata` are included.

References

Taguchi, G. (1995). Pattern Recognition and Quality Engineering (1). Journal of Quality Engineering Society, 3(2), 2-5. (In Japanese)

Taguchi, G., Wu, Y., & Chodhury, S. (2000). Mahalanobis-Taguchi System. McGraw-Hill Professional.

Taguchi, G., & Jugulum, R. (2002). The Mahalanobis-Taguchi strategy: A pattern technology system. John Wiley & Sons.

Woodall, W. H., Koudelik, R., Tsui, K. L., Kim, S. B., Stoumbos, Z. G., & Carvounis, C. P. (2003). A review and analysis of the Mahalanobis-Taguchi system. Technometrics, 45(1), 1-15.

Examples

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

unit_space_MT <- MT(unit_space_data = iris_versicolor,
                    includes_transformed_data = TRUE)

# 10 data for each kind (setosa, versicolor, virginica) in the iris dataset
iris_test <- iris[c(1:10, 51:60, 101:111), -5]

diagnosis_MT <- diagnosis(unit_space = unit_space_MT,
                          newdata = iris_test,
                          threshold = 4,
                          includes_transformed_newdata = TRUE)

(diagnosis_MT$distance)
(diagnosis_MT$le_threshold)

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

unit_space_MT <- MT(unit_space_data = iris_versicolor,
                    includes_transformed_data = TRUE)

# 10 data for each kind (setosa, versicolor, virginica) in the iris dataset
iris_test <- iris[c(1:10, 51:60, 101:111), -5]

diagnosis_MT <- diagnosis(unit_space = unit_space_MT,
                          newdata = iris_test,
                          threshold = 4,
                          includes_transformed_newdata = TRUE)

(diagnosis_MT$distance)
(diagnosis_MT$le_threshold)

Diagnosis method for the Mahalanobis-Taguchi Adjoint (MTA) method

Description

diagnosis.MTA (via diagnosis) calculates the distance based on the unit space generated by MTA or generates_unit_space(..., method = "MTA") and classifies each sample into positive (TRUE) or negative (FALSE) by comparing the values with the set threshold value.

Usage

## S3 method for class 'MTA'
diagnosis(unit_space, newdata, threshold,
  includes_transformed_newdata = FALSE)
## S3 method for class 'MTA'
diagnosis(unit_space, newdata, threshold,
  includes_transformed_newdata = FALSE)

Arguments

`unit_space`	Object of class "MTA" generated by `MTA` or `generates_unit_space`(..., method = "MTA").
`newdata`	Matrix with n rows (samples) and p columns (variables). The data are used to calculate the desired distances from the unit space. All data should be continuous values and should not have missing values.
`threshold`	Numeric specifying the threshold value to classify each sample into positive (`TRUE`) or negative (`FALSE`).
`includes_transformed_newdata`	If `TRUE`, then the transformed data for `newdata` are included in a return object.

Value

diagnosis.MTA (via diagnosis) returns a list containing the following components:

`distance`	Vector with length n. Distances from the unit space to each sample.
`le_threshold`	Vector with length n. Logical values indicating the distance of each sample is less than or equal to the threhold value (`TRUE`) or not (`FALSE`).
`threshold`	Numeric value to classify the sample into positive or negative.
`unit_space`	Object of class "MTA" passed by `unit_space`.
`n`	The number of samples for `newdata`.
`q`	The number of variables after the data transformation. q equals p.
`x`	If `includes_transformed_newdata` is `TRUE`, then the transformed data for `newdata` are included.

References

Taguchi, G. & Kanetaka, T. (2002). Engineering Technical Development in MT System - Lecture on Applied Quality. Japanese Standards Association. (In Japanese)

Taguchi, G., & Jugulum, R. (2002). The Mahalanobis-Taguchi strategy: A pattern technology system. John Wiley & Sons.

Examples

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

unit_space_MTA <- MTA(unit_space_data = iris_versicolor,
                      includes_transformed_data = TRUE)

# 10 data for each kind (setosa, versicolor, virginica) in the iris dataset
iris_test <- iris[c(1:10, 51:60, 101:111), -5]

diagnosis_MTA <- diagnosis(unit_space = unit_space_MTA,
                           newdata = iris_test,
                           threshold = 0.5,
                           includes_transformed_newdata = TRUE)

(diagnosis_MTA$distance)
(diagnosis_MTA$le_threshold)

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

unit_space_MTA <- MTA(unit_space_data = iris_versicolor,
                      includes_transformed_data = TRUE)

# 10 data for each kind (setosa, versicolor, virginica) in the iris dataset
iris_test <- iris[c(1:10, 51:60, 101:111), -5]

diagnosis_MTA <- diagnosis(unit_space = unit_space_MTA,
                           newdata = iris_test,
                           threshold = 0.5,
                           includes_transformed_newdata = TRUE)

(diagnosis_MTA$distance)
(diagnosis_MTA$le_threshold)

Diagnosis method for the Recognition-Taguchi (RT) method

Description

diagnosis.RT (via diagnosis) calculates the distance based on the unit space generated by RT or generates_unit_space(..., method = "RT") and classifies each sample into positive (TRUE) or negative (FALSE) by comparing the values with the set threshold value.

Usage

## S3 method for class 'RT'
diagnosis(unit_space, newdata, threshold,
  includes_transformed_newdata = FALSE)
## S3 method for class 'RT'
diagnosis(unit_space, newdata, threshold,
  includes_transformed_newdata = FALSE)

Arguments

`unit_space`	Object of class "RT" generated by `RT` or `generates_unit_space`(..., method = "RT").
`newdata`	Matrix with n rows (samples) and p columns (variables). The data are used to calculate the desired distances from the unit space. All data should be continuous values and should not have missing values.
`threshold`	Numeric specifying the threshold value to classify each sample into positive (`TRUE`) or negative (`FALSE`).
`includes_transformed_newdata`	If `TRUE`, then the transformed data for `newdata` are included in a return object.

Value

diagnosis.RT (via diagnosis) returns a list containing the following components:

`distance`	Vector with length n. Distances from the unit space to each sample.
`le_threshold`	Vector with length n. Logical values indicating the distance of each sample is less than or equal to the threhold value (`TRUE`) or not (`FALSE`).
`threshold`	Numeric value to classify the sample into positive or negative.
`unit_space`	Object of class "RT" passed by `unit_space`.
`n`	The number of samples for `newdata`.
`q`	The number of variables after the data transformation. q is always 2.
`x`	If `includes_transformed_newdata` is `TRUE`, then the transformed data for `newdata` are included.

References

Taguchi, G. (2006). Objective Function and Generic Function (11). Journal of Quality Engineering Society, 14(2), 5-9. (In Japanese)

Huda, F., Kajiwara, I., Hosoya, N., & Kawamura, S. (2013). Bolt loosening analysis and diagnosis by non-contact laser excitation vibration tests. Mechanical systems and signal processing, 40(2), 589-604.

Examples

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

unit_space_RT <- RT(unit_space_data = iris_versicolor,
                    includes_transformed_data = TRUE)

# 10 data for each kind (setosa, versicolor, virginica) in the iris dataset
iris_test <- iris[c(1:10, 51:60, 101:111), -5]

diagnosis_RT <- diagnosis(unit_space = unit_space_RT,
                          newdata = iris_test,
                          threshold = 0.2,
                          includes_transformed_newdata = TRUE)

(diagnosis_RT$distance)
(diagnosis_RT$le_threshold)

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

unit_space_RT <- RT(unit_space_data = iris_versicolor,
                    includes_transformed_data = TRUE)

# 10 data for each kind (setosa, versicolor, virginica) in the iris dataset
iris_test <- iris[c(1:10, 51:60, 101:111), -5]

diagnosis_RT <- diagnosis(unit_space = unit_space_RT,
                          newdata = iris_test,
                          threshold = 0.2,
                          includes_transformed_newdata = TRUE)

(diagnosis_RT$distance)
(diagnosis_RT$le_threshold)

Function to predict a forecasting for a family of Taguchi (T) methods

Description

forecasting is a generic function. For details, see forecasting.T1, forecasting.Ta, forecasting.Tb or general_forecasting.T.

Usage

forecasting(model, newdata, includes_transformed_newdata)
forecasting(model, newdata, includes_transformed_newdata)

Arguments

`model`	Object generated as a model.
`newdata`	Matrix with n rows (samples) and p columns (variables). The Data to be estimated. All data should be continuous values and should not have missing values.
`includes_transformed_newdata`	If `TRUE`, then the transformed data for `newdata` are included in a return object.

Value

A list containing the following components is returned.

`M_hat`	Vector with length n. The estimated values of the dependent variable after the data trasformation.
`y_hat`	Vector with length n. The estimated values after the inverse transformation from `M_hat`.
`model`	Object passed by `model`.
`n`	The number of samples for `newdata`.
`q`	The number of variables after the data transformation.
`X`	If `includes_transformed_newdata` is `TRUE`, then the transformed data for `newdata` are included.

Forecasting method for the T1 method

Description

forecasting.T1 (via forecasting) estimates the dependent values based on the T1 model.

Usage

## S3 method for class 'T1'
forecasting(model, newdata, includes_transformed_newdata = FALSE)
## S3 method for class 'T1'
forecasting(model, newdata, includes_transformed_newdata = FALSE)

Arguments

`model`	Object of class "T1" generated by `T1` or `generates_model`(..., method = "T1").
`newdata`	Matrix with n rows (samples) and p columns (variables). The Data to be estimated. All data should be continuous values and should not have missing values.
`includes_transformed_newdata`	If `TRUE`, then the transformed data for `newdata` are included in a return object.

Value

A list containing the following components is returned.

`M_hat`	Vector with length n. The estimated values of the dependent variable after the data transformation.
`y_hat`	Vector with length n. The estimated values after the inverse transformation from `M_hat`.
`model`	Object of class "T1" passed by `model`.
`n`	The number of samples for `newdata`.
`q`	The number of variables after the data transformation. q equals p.
`X`	If `includes_transformed_newdata` is `TRUE`, then the transformed data for `newdata` are included.

References

Taguchi, G. (2006). Objective Function and Generic Function (12). Journal of Quality Engineering Society, 14(3), 5-9. (In Japanese)

Inou, A., Nagata, Y., Horita, K., & Mori, A. (2012). Prediciton Accuracies of Improved Taguchi's T Methods Compared to those of Multiple Regresssion Analysis. Journal of the Japanese Society for Quality Control, 42(2), 103-115. (In Japanese)

Kawada, H., & Nagata, Y. (2015). An application of a generalized inverse regression estimator to Taguchi's T-Method. Total Quality Science, 1(1), 12-21.

Examples

# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

# The following samples are data other than the unit space data and the test
# data.
stackloss_signal <- stackloss[-c(2, 9, 10, 11, 12, 19, 20, 21), ]

model_T1 <- T1(unit_space_data = stackloss_center,
               signal_space_data = stackloss_signal,
               subtracts_V_e = TRUE,
               includes_transformed_data = TRUE)

# The following test samples are chosen casually.
stackloss_test <- stackloss[c(2, 12, 19), -4]

forecasting_T1 <- forecasting(model = model_T1,
                              newdata = stackloss_test,
                              includes_transformed_newdata = TRUE)

(forecasting_T1$y_hat) # Estimated values
(stackloss[c(2, 12, 19), 4]) # True values

# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

# The following samples are data other than the unit space data and the test
# data.
stackloss_signal <- stackloss[-c(2, 9, 10, 11, 12, 19, 20, 21), ]

model_T1 <- T1(unit_space_data = stackloss_center,
               signal_space_data = stackloss_signal,
               subtracts_V_e = TRUE,
               includes_transformed_data = TRUE)

# The following test samples are chosen casually.
stackloss_test <- stackloss[c(2, 12, 19), -4]

forecasting_T1 <- forecasting(model = model_T1,
                              newdata = stackloss_test,
                              includes_transformed_newdata = TRUE)

(forecasting_T1$y_hat) # Estimated values
(stackloss[c(2, 12, 19), 4]) # True values

Forecasting method for the Ta method

Description

forecasting.Ta (via forecasting) estimates the dependent values based on the Ta model.

Usage

## S3 method for class 'Ta'
forecasting(model, newdata, includes_transformed_newdata = FALSE)
## S3 method for class 'Ta'
forecasting(model, newdata, includes_transformed_newdata = FALSE)

Arguments

`model`	Object of class "Ta" generated by `Ta` or `generates_model`(..., method = "Ta").
`newdata`	Matrix with n rows (samples) and p columns (variables). The Data to be estimated. All data should be continuous values and should not have missing values.
`includes_transformed_newdata`	If `TRUE`, then the transformed data for `newdata` are included in a return object.

Value

A list containing the following components is returned.

`M_hat`	Vector with length n. The estimated values of the dependent variable after the data transformation.
`y_hat`	Vector with length n. The estimated values after the inverse transformation from `M_hat`.
`model`	Object of class "Ta" passed by `model`.
`n`	The number of samples for `newdata`.
`q`	The number of variables after the data transformation. q equals p.
`X`	If `includes_transformed_newdata` is `TRUE`, then the transformed data for `newdata` are included.

References

Kawada, H., & Nagata, Y. (2015). An application of a generalized inverse regression estimator to Taguchi's T-Method. Total Quality Science, 1(1), 12-21.

Examples

model_Ta <- Ta(sample_data = stackloss[-c(2, 12, 19), ],
               subtracts_V_e = TRUE,
               includes_transformed_data = TRUE)

forecasting_Ta <- forecasting(model = model_Ta,
                              newdata = stackloss[c(2, 12, 19), -4],
                              includes_transformed_newdata = TRUE)

(forecasting_Ta$y_hat) # Estimated values
(stackloss[c(2, 12, 19), 4]) # True values

model_Ta <- Ta(sample_data = stackloss[-c(2, 12, 19), ],
               subtracts_V_e = TRUE,
               includes_transformed_data = TRUE)

forecasting_Ta <- forecasting(model = model_Ta,
                              newdata = stackloss[c(2, 12, 19), -4],
                              includes_transformed_newdata = TRUE)

(forecasting_Ta$y_hat) # Estimated values
(stackloss[c(2, 12, 19), 4]) # True values

Forecasting method for the Tb method

Description

forecasting.Tb (via forecasting) estimates the dependent values based on the Tb model.

Usage

## S3 method for class 'Tb'
forecasting(model, newdata, includes_transformed_newdata = FALSE)
## S3 method for class 'Tb'
forecasting(model, newdata, includes_transformed_newdata = FALSE)

Arguments

`model`	Object of class "Tb" generated by `Tb` or `generates_model`(..., method = "Tb").
`newdata`	Matrix with n rows (samples) and p columns (variables). The Data to be estimated. All data should be continuous values and should not have missing values.
`includes_transformed_newdata`	If `TRUE`, then the transformed data for `newdata` are included in a return object.

Value

A list containing the following components is returned.

`M_hat`	Vector with length n. The estimated values of the dependent variable after the data transformation.
`y_hat`	Vector with length n. The estimated values after the inverse transformation from `M_hat`.
`model`	Object of class "Tb" passed by `model`.
`n`	The number of samples for `newdata`.
`q`	The number of variables after the data transformation. q equals p.
`X`	If `includes_transformed_newdata` is `TRUE`, then the transformed data for `newdata` are included.

References

Kawada, H., & Nagata, Y. (2015). An application of a generalized inverse regression estimator to Taguchi's T-Method. Total Quality Science, 1(1), 12-21.

Examples

model_Tb <- Tb(sample_data = stackloss[-c(2, 12, 19), ],
               subtracts_V_e = TRUE,
               includes_transformed_data = TRUE)

forecasting_Tb <- forecasting(model = model_Tb,
                              newdata = stackloss[c(2, 12, 19), -4],
                              includes_transformed_newdata = TRUE)

(forecasting_Tb$y_hat) # Estimated values
(stackloss[c(2, 12, 19), 4]) # True values

model_Tb <- Tb(sample_data = stackloss[-c(2, 12, 19), ],
               subtracts_V_e = TRUE,
               includes_transformed_data = TRUE)

forecasting_Tb <- forecasting(model = model_Tb,
                              newdata = stackloss[c(2, 12, 19), -4],
                              includes_transformed_newdata = TRUE)

(forecasting_Tb$y_hat) # Estimated values
(stackloss[c(2, 12, 19), 4]) # True values

General function to implement a diagnosis method for a family of Mahalanobis-Taguchi (MT) methods

Description

general_diagnosis.MT is the general function that implements a diagnosis method for a family of Mahalanobis-Taguchi (MT) methods. Each diagnosis method of a family of MT methods can be implemented by setting the parameters of this function appropriately.

Usage

general_diagnosis.MT(unit_space, newdata, threshold,
  includes_transformed_newdata = FALSE)
general_diagnosis.MT(unit_space, newdata, threshold,
  includes_transformed_newdata = FALSE)

Arguments

`unit_space`	Object generated as a unit space.
`newdata`	Matrix with n rows (samples) and p columns (variables). The data are used to calculate the desired distances from the unit space. All data should be continuous values and should not have missing values.
`threshold`	Numeric specifying the threshold value to classify each sample into positive (`TRUE`) or negative (`FALSE`).
`includes_transformed_newdata`	If `TRUE`, then the transformed data for `newdata` are included in a return object.

Value

A list containing the following components is returned.

`distance`	Vector with length n. Distances from the unit space to each sample.
`le_threshold`	Vector with length n. Logical values indicating the distance of each sample is less than or equal to the threhold value (`TRUE`) or not (`FALSE`).
`threshold`	Numeric value to classify the sample into positive or negative.
`unit_space`	Object passed by `unit_space`.
`n`	The number of samples for `newdata`.
`q`	The number of independent variables after the data transformation. According to the data transoformation function, q may be equal to p.
`x`	If `includes_transformed_newdata` is `TRUE`, then the transformed data for `newdata` are included.

Examples

                          
# 40 data for versicolor in the iris dataset                            
iris_versicolor <- iris[61:100, -5] 

# The following settings are same as the MT method.                          
unit_space <- general_MT(unit_space_data = iris_versicolor, 
                         generates_transform_function = 
                                            generates_normalization_function,
                         calc_A = function(x) solve(cor(x)),  
                         includes_transformed_data = TRUE)

# 10 data for each kind (setosa, versicolor, virginica) in the iris dataset                         
iris_test <- iris[c(1:10, 51:60, 101:111), -5]
                         
diagnosis <- general_diagnosis.MT(unit_space = unit_space, 
                                  newdata = iris_test, 
                                  threshold = 4,
                                  includes_transformed_newdata = TRUE)
                              
(diagnosis$distance)
(diagnosis$le_threshold)                          

# 40 data for versicolor in the iris dataset                            
iris_versicolor <- iris[61:100, -5] 

# The following settings are same as the MT method.                          
unit_space <- general_MT(unit_space_data = iris_versicolor, 
                         generates_transform_function = 
                                            generates_normalization_function,
                         calc_A = function(x) solve(cor(x)),  
                         includes_transformed_data = TRUE)

# 10 data for each kind (setosa, versicolor, virginica) in the iris dataset                         
iris_test <- iris[c(1:10, 51:60, 101:111), -5]
                         
diagnosis <- general_diagnosis.MT(unit_space = unit_space, 
                                  newdata = iris_test, 
                                  threshold = 4,
                                  includes_transformed_newdata = TRUE)
                              
(diagnosis$distance)
(diagnosis$le_threshold)

General function to implement a forecasting method for a family of Taguchi (T) methods

Description

general_forecasting.T is the general function that implements a forecasting method for a family of Taguchi (T) methods. Each forecasting method of a family of T methods can be implemented by setting the parameters of this function appropriately.

Usage

general_forecasting.T(model, newdata, includes_transformed_newdata = FALSE)
general_forecasting.T(model, newdata, includes_transformed_newdata = FALSE)

Arguments

`model`	Object generated as a model.
`newdata`	Matrix with n rows (samples) and p columns (variables). The data are used to calculate the desired distances from the unit space. All data should be continuous values and should not have missing values.
`includes_transformed_newdata`	If `TRUE`, then the transformed data for `newdata` are included in a return object.

Value

A list containing the following components is returned.

`M_hat`	Vector with length n. The estimated values of the dependent variable after the data trasformation.
`y_hat`	Vector with length n. The estimated values after the inverse transformation from `M_hat`.
`model`	Object passed by `model`.
`n`	The number of samples for `newdata`.
`q`	The number of variables after the data transformation.
`X`	If `includes_transformed_newdata` is `TRUE`, then the transformed data for `newdata` are included.

Examples

# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

# The following samples are data other than the unit space data and the test
# data.
stackloss_signal <- stackloss[-c(2, 9, 10, 11, 12, 19, 20, 21), ]

# The following settings are same as the T1 method.
model <- general_T(unit_space_data = stackloss_center,
                   signal_space_data = stackloss_signal,
                   generates_transform_functions =
                                       generates_transformation_functions_T1,
                   subtracts_V_e = TRUE,
                   includes_transformed_data = TRUE)

# The following test samples are chosen casually.
stackloss_test <- stackloss[c(2, 12, 19), -4]

forecasting <- general_forecasting.T(model = model,
                                     newdata = stackloss_test,
                                     includes_transformed_newdata = TRUE)

(forecasting$y_hat) # Estimated values
(stackloss[c(2, 12, 19), 4]) # True values

# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

# The following samples are data other than the unit space data and the test
# data.
stackloss_signal <- stackloss[-c(2, 9, 10, 11, 12, 19, 20, 21), ]

# The following settings are same as the T1 method.
model <- general_T(unit_space_data = stackloss_center,
                   signal_space_data = stackloss_signal,
                   generates_transform_functions =
                                       generates_transformation_functions_T1,
                   subtracts_V_e = TRUE,
                   includes_transformed_data = TRUE)

# The following test samples are chosen casually.
stackloss_test <- stackloss[c(2, 12, 19), -4]

forecasting <- general_forecasting.T(model = model,
                                     newdata = stackloss_test,
                                     includes_transformed_newdata = TRUE)

(forecasting$y_hat) # Estimated values
(stackloss[c(2, 12, 19), 4]) # True values

General function to generate a unit space for a family of Mahalanobis-Taguchi (MT) methods

Description

general_MT is a (higher-order) general function that generates a unit space for a family of Mahalanobis-Taguchi (MT) methods. Each MT method can be implemented by setting the parameters of this function appropriately.

Usage

general_MT(unit_space_data, calc_A, generates_transform_function,
  includes_transformed_data = FALSE)
general_MT(unit_space_data, calc_A, generates_transform_function,
  includes_transformed_data = FALSE)

Arguments

`unit_space_data`	Matrix with n rows (samples) and p columns (variables). Data to generate the unit space. All data should be continuous values and should not have missing values.
`calc_A`	Function that returns A in a quadratic form x'Ax. `calc_A` takes the transformed data as an (only) argument.
`generates_transform_function`	Function that takes `unit_space_data` as an (only) argument and returns a data transformation function. The data transformation function takes data as an (only) argument and returns the transformed data.
`includes_transformed_data`	If `TRUE`, then the transformed data are included in a return object.

Value

A list containing the following components is returned.

`A`	q x q matrix calculated by `calc_A`.
`calc_A`	Function passed by `calc_A`.
`transforms_data`	Data transformation function generated from `generates_transform_function` based on `unit_space_data`.
`distance`	Vector with length n. Distances from the unit space to each sample.
`n`	The number of samples.
`q`	The number of independent variables after the data transformation. According to the data transoformation function, q may be equal to p.
`x`	If `includes_transformed_data` is `TRUE`, then the transformed data are included.

Examples

# 40 data for versicolor in the iris dataset                            
iris_versicolor <- iris[61:100, -5] 

# The following settings are same as the MT method.                          
unit_space <- general_MT(unit_space_data = iris_versicolor, 
                         generates_transform_function = 
                                            generates_normalization_function,
                         calc_A = function(x) solve(cor(x)),  
                         includes_transformed_data = TRUE)
                         
(unit_space$distance)

# 40 data for versicolor in the iris dataset                            
iris_versicolor <- iris[61:100, -5] 

# The following settings are same as the MT method.                          
unit_space <- general_MT(unit_space_data = iris_versicolor, 
                         generates_transform_function = 
                                            generates_normalization_function,
                         calc_A = function(x) solve(cor(x)),  
                         includes_transformed_data = TRUE)
                         
(unit_space$distance)

General function to generate a prediction expression for a family of Taguchi (T) methods

Description

general_T is a (higher-order) general function that generates a prediction expression for a family of Taguchi (T) methods. Each T method can be implemented by setting the parameters of this function appropriately.

Usage

general_T(unit_space_data, signal_space_data, generates_transform_functions,
  subtracts_V_e = TRUE, includes_transformed_data = FALSE)
general_T(unit_space_data, signal_space_data, generates_transform_functions,
  subtracts_V_e = TRUE, includes_transformed_data = FALSE)

Arguments

`unit_space_data`	Matrix with n rows (samples) and (p + 1) columns (variables). The 1 ~ p th columns are independent variables and the (p + 1) th column is a dependent variable. Underlying data to obtain a representative point for the normalization of the `signal_space_data`. All data should be continuous values and should not have missing values.
`signal_space_data`	Matrix with m rows (samples) and (p + 1) columns (variables). The 1 ~ p th columns are independent variables and the (p + 1) th column is a dependent variable. Underlying data to generate a prediction expression. All data should be continuous values and should not have missing values.
`generates_transform_functions`	A function that takes the `unit_space_data` as an (only) argument and returns a list containing three functions. A data transformation function for independent variables is the first component, a data transformation function for a dependent variable is the second component, and an inverse function of the data transformation function for a dependent variable is the third component. The data transformation function for independent variables takes independent variable data (a matrix of p columns) as an (only) argument and returns the transformed independent variable data. The data transformation function for a dependent variable takes dependent variable data (a vector) as an (only) argument and returns the transformed dependent variable data. The inverse function of the data transformation for a dependent variable takes the transformed dependent variable data (a vector) as an (only) argument and returns the untransformed dependent variable data.
`subtracts_V_e`	If `TRUE`, then the error variance is subtracted in the numerator when calculating `eta_hat`.
`includes_transformed_data`	If `TRUE`, then the transformed data are included in a return object.

Value

A list containing the following components is returned.

`beta_hat`	Vector with length q. Estimated proportionality constants between each independent variable and the dependent variable.
`subtracts_V_e`	Logical. If `TRUE`, then `eta_hat` was calculated without subtracting the error variance in the numerator.
`eta_hat`	Vector with length q. Estimated squared signal-to-noise ratios (S/N) coresponding to `beta_hat`.
`M_hat`	Vector with length n. The estimated values of the dependent variable after the data transformation for `signal_space_data`.
`overall_prediction_eta`	Numeric. The overall squared signal-to-noise ratio (S/N).
`transforms_independent_data`	Data transformation function generated from `generates_transform_functions` based on `unit_space_data`. The function for independent variables takes independent variable data (a matrix of p columns) as an (only) argument and returns the transformed independent variable data.
`transforms_dependent_data`	Data transformation function generated in `generates_transform_functions` based on the `unit_space_data`. The function for a dependent variable takes dependent variable data (a vector) as an (only) argument and returns the transformed dependent variable data.
`inverses_transformed_dependent_data`	Inverse function generated in the `generates_transform_functions` based on `unit_space_data`. The function of the takes the transformed dependent variable data (a vector) as an (only) argument and returns the dependent variable data inversed from the transformed dependent variable data.
`m`	The number of samples for `signal_space_data`.
`q`	The number of independent variables after the data transformation. According to the data transoformation function, q may be equal to p.
`X`	If `includes_transformed_data` is `TRUE`, then the independent variable data after the data transformation for the `signal_space_data` are included.
`M`	If `includes_transformed_data` is `TRUE`, then the (true) value of the dependent variable after the data transformation for the `signal_space_data` are included.

Examples

# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

# The following samples are data other than the unit space data and the test
# data.
stackloss_signal <- stackloss[-c(2, 9, 10, 11, 12, 19, 20, 21), ]

# The following settings are same as the T1 method.
model <- general_T(unit_space_data = stackloss_center,
                   signal_space_data = stackloss_signal,
                   generates_transform_functions =
                                       generates_transformation_functions_T1,
                   subtracts_V_e = TRUE,
                   includes_transformed_data = TRUE)

(model$M_hat)

# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

# The following samples are data other than the unit space data and the test
# data.
stackloss_signal <- stackloss[-c(2, 9, 10, 11, 12, 19, 20, 21), ]

# The following settings are same as the T1 method.
model <- general_T(unit_space_data = stackloss_center,
                   signal_space_data = stackloss_signal,
                   generates_transform_functions =
                                       generates_transformation_functions_T1,
                   subtracts_V_e = TRUE,
                   includes_transformed_data = TRUE)

(model$M_hat)

Function to generate a data transformation function for the Recognition-Taguchi (RT) method

Description

generates_dimensionality_reduction_function returns the data transformation function for the Recognition-Taguchi (RT) method based on the unit_space_data. The function reduces the dimensionality of data into 2 synthetic variables.

Usage

generates_dimensionality_reduction_function(unit_space_data)
generates_dimensionality_reduction_function(unit_space_data)

Arguments

unit_space_data

Matrix with n rows (samples) and p columns (variables). Data to generate the unit space. All data should be continuous values and should not have missing values.

Value

Function is returned which takes an n x p matrix as an (only) argument and returns a dimensionality-reduced n x 2 data frame with named columns; Y_1 and Y_2.

References

Taguchi, G. (2006). Objective Function and Generic Function (11). Journal of Quality Engineering Society, 14(2), 5-9. (In Japanese)

Examples

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

reduces_dimensionality <-
                 generates_dimensionality_reduction_function(iris_versicolor)

is.function(reduces_dimensionality) # TRUE

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

reduces_dimensionality <-
                 generates_dimensionality_reduction_function(iris_versicolor)

is.function(reduces_dimensionality) # TRUE

Wrapper function to generate a model for a family of Taguchi (T) methods

Description

generates_model generates a model for a family of Taguchi (MT) methods. The model of T1 method, Ta method or the Tb method can be generated by passing a method name (character) into a parameter method.

Usage

generates_model(unit_space_data, signal_space_data, sample_data,
  method = c("T1", "Ta", "Tb"), subtracts_V_e = TRUE,
  includes_transformed_data = FALSE)
generates_model(unit_space_data, signal_space_data, sample_data,
  method = c("T1", "Ta", "Tb"), subtracts_V_e = TRUE,
  includes_transformed_data = FALSE)

Arguments

`unit_space_data`	Used only for the T1 method. Matrix with n rows (samples) and (p + 1) columns (variables). The 1 ~ p th columns are independent variables and the (p + 1) th column is a dependent variable. Underlying data to obtain a representative point for the normalization of `signal_space_data`. All data should be continuous values and should not have missing values.
`signal_space_data`	Used only for the T1 method. Matrix with m rows (samples) and (p + 1) columns (variables). The 1 ~ p th columns are independent variables and the (p + 1) th column is a dependent variable. Underlying data to generate a prediction expression. All data should be continuous values and should not have missing values.
`sample_data`	Used for the Ta and the Tb methods. Matrix with n rows (samples) and (p + 1) columns (variables). The 1 ~ p th columns are independent variables and the (p + 1) th column is a dependent variable. All data should be continuous values and should not have missing values.
`method`	Character to designate a method. Currently, "MT", "MTA", and "RT" are available.
`subtracts_V_e`	If `TRUE`, then the error variance is subtracted in the numerator when calculating `eta_hat`.
`includes_transformed_data`	If `TRUE`, then the transformed data are included in a return object.

Value

A returned object depends on the selected method. See T1, Ta or Tb.

Examples

# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

# The following samples are data other than the unit space data and the test
# data.
stackloss_signal <- stackloss[-c(2, 9, 10, 11, 12, 19, 20, 21), ]

# The following test samples are chosen casually.
stackloss_test <- stackloss[c(2, 12, 19), -4]

# T1 method
model_T1 <- generates_model(unit_space_data = stackloss_center,
                            signal_space_data = stackloss_signal,
                            method = "T1",
                            subtracts_V_e = TRUE)

forecasting_T1 <- forecasting(model = model_T1,
                              newdata = stackloss_test)

(forecasting_T1$y_hat)

# Ta method
model_Ta <- generates_model(sample_data =
                                   rbind(stackloss_center, stackloss_signal),
                            method = "Ta",
                            subtracts_V_e = TRUE)

forecasting_Ta <- forecasting(model = model_Ta,
                              newdata = stackloss_test)

(forecasting_Ta$y_hat)

# Tb method
model_Tb <- generates_model(sample_data =
                                   rbind(stackloss_center, stackloss_signal),
                            method = "Tb",
                            subtracts_V_e = TRUE)

forecasting_Tb <- forecasting(model = model_Tb,
                              newdata = stackloss_test)

(forecasting_Tb$y_hat)

# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

# The following samples are data other than the unit space data and the test
# data.
stackloss_signal <- stackloss[-c(2, 9, 10, 11, 12, 19, 20, 21), ]

# The following test samples are chosen casually.
stackloss_test <- stackloss[c(2, 12, 19), -4]

# T1 method
model_T1 <- generates_model(unit_space_data = stackloss_center,
                            signal_space_data = stackloss_signal,
                            method = "T1",
                            subtracts_V_e = TRUE)

forecasting_T1 <- forecasting(model = model_T1,
                              newdata = stackloss_test)

(forecasting_T1$y_hat)

# Ta method
model_Ta <- generates_model(sample_data =
                                   rbind(stackloss_center, stackloss_signal),
                            method = "Ta",
                            subtracts_V_e = TRUE)

forecasting_Ta <- forecasting(model = model_Ta,
                              newdata = stackloss_test)

(forecasting_Ta$y_hat)

# Tb method
model_Tb <- generates_model(sample_data =
                                   rbind(stackloss_center, stackloss_signal),
                            method = "Tb",
                            subtracts_V_e = TRUE)

forecasting_Tb <- forecasting(model = model_Tb,
                              newdata = stackloss_test)

(forecasting_Tb$y_hat)

Function to generate the data normalization function

Description

generates_normalization_function returns the data normalization function. The data normalization function is generated based on unit_space_data.

Usage

generates_normalization_function(unit_space_data, unit_space_center,
  unit_space_scale, is_scaled = TRUE)
generates_normalization_function(unit_space_data, unit_space_center,
  unit_space_scale, is_scaled = TRUE)

Arguments

`unit_space_data`	Matrix with n rows (samples) and p columns (variables). Data to generate the unit space. All data should be continuous values and should not have missing values.
`unit_space_center`	Vector with length p. The values are subtrahends in normalization. If missing, the mean for each column of `unit_space_data` is used for normalization.
`unit_space_scale`	Vector with length p. The values are divisors in normalization. If missing and `is_scaled` is `TRUE`, then the unbiased standard deviation for each column of `unit_space_data` is used for normalization.
`is_scaled`	Logical. If `TRUE` (default value), normalization is conducted by subtracting `unit_space_center` and dividing by `unit_space_scale`. If `FALSE`, normalization is conducted by subtracting `unit_space_center` only.

Value

Function is returned which takes an n x p matrix as an (only) argument and returns a normalized n x p matrix. The normalization is conducted based on unit_space_data.

Examples

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

normalizes_data <- generates_normalization_function(iris_versicolor)

is.function(normalizes_data) # TRUE

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

normalizes_data <- generates_normalization_function(iris_versicolor)

is.function(normalizes_data) # TRUE

Function to generate data transformation functions for the T1 methods

Description

generates_transformation_functions_T1 is the argument for the parameter generates_transform_functions in genera_T, which is used in the T1 method. In addtion, the Ta method also uses this function for the argument.

Usage

generates_transformation_functions_T1(unit_space_data)
generates_transformation_functions_T1(unit_space_data)

Arguments

unit_space_data

Matrix with n rows (samples) and (p + 1) columns (variables). Data to generate the unit space. All data should be continuous values and should not have missing values.

Value

generates_transformation_functions_T1 returns a list containing three functions. For the first component, the data transformation function for independent variables is a function that subtracts the mean of each independent variable. For the second component, the data transformation function for a dependent variable is a function that subtracts the mean of a dependent variable. For the third component, the inverse function of the data transformation function for a dependent variable is a function that adds the mean of a dependent variable. The mean used is the mean of the unit_space_data.

Examples

# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

tmp <- generates_transformation_functions_T1(stackloss_center)
mean_subtraction_function <- tmp[[1]]
subtracts_M_0 <- tmp[[2]]
adds_M_0 <- tmp[[3]]

is.function(mean_subtraction_function) # TRUE
is.function(subtracts_M_0) # TRUE
is.function(adds_M_0) # TRUE

# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

tmp <- generates_transformation_functions_T1(stackloss_center)
mean_subtraction_function <- tmp[[1]]
subtracts_M_0 <- tmp[[2]]
adds_M_0 <- tmp[[3]]

is.function(mean_subtraction_function) # TRUE
is.function(subtracts_M_0) # TRUE
is.function(adds_M_0) # TRUE

Function to generate data transformation functions for the Tb methods

Description

generates_transformation_functions_Tb is the argument for the parameter generates_transform_functions in genera_T, which is used in the Tb method.

Usage

generates_transformation_functions_Tb(sample_data, subtracts_V_e)
generates_transformation_functions_Tb(sample_data, subtracts_V_e)

Arguments

`sample_data`	Matrix with n rows (samples) and (p + 1) columns (variables). The Tb method uses all data to generate the unit space. All data should be continuous values and should not have missing values.
`subtracts_V_e`	If `TRUE`, then the error variance is subtracted in the numerator when calculating `eta_hat`.

Value

generates_transformation_functions_Tb returns a list containing three functions. For the first component, the data transformation function for independent variables is a function that subtracts the center of each independent variable. The center is determined in a specific manner for the Tb method. The center consists of each sample value which maximizes the signal-to-noise ratio (S/N) per independent variable. The values are determined independently so that different samples may be selected for different variables. For the second component, the data transformation function for a dependent variable is a function that subtracts the dependent variable of the sample which maximizes the S/N per independent variable. For the third component, the inverse function of the data transformation function for a dependent variable is a function that adds the weighted mean of a dependent variable. The weighted mean is calculated based on the S/N and the frequency of being selected in independent variables.

References

Kawada, H., & Nagata, Y. (2015). An application of a generalized inverse regression estimator to Taguchi's T-Method. Total Quality Science, 1(1), 12-21.

Examples

# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

tmp <- generates_transformation_functions_Tb(stackloss_center, TRUE)
center_subtraction_function <- tmp[[1]]
subtracts_ys <- tmp[[2]]
adds_M_0 <- tmp[[3]]

is.function(center_subtraction_function) # TRUE
is.function(subtracts_ys) # TRUE
is.function(adds_M_0) # TRUE

# Note that dynamic scope is used when the parameter "subtracts_V_e" is not
# set.
subtracts_V_e <- FALSE
tmp <- generates_transformation_functions_Tb(stackloss_center)
center_subtraction_function <- tmp[[1]]
subtracts_ys <- tmp[[2]]
adds_M_0 <- tmp[[3]]

is.function(center_subtraction_function) # TRUE
is.function(subtracts_ys) # TRUE
is.function(adds_M_0) # TRUE

# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

tmp <- generates_transformation_functions_Tb(stackloss_center, TRUE)
center_subtraction_function <- tmp[[1]]
subtracts_ys <- tmp[[2]]
adds_M_0 <- tmp[[3]]

is.function(center_subtraction_function) # TRUE
is.function(subtracts_ys) # TRUE
is.function(adds_M_0) # TRUE

# Note that dynamic scope is used when the parameter "subtracts_V_e" is not
# set.
subtracts_V_e <- FALSE
tmp <- generates_transformation_functions_Tb(stackloss_center)
center_subtraction_function <- tmp[[1]]
subtracts_ys <- tmp[[2]]
adds_M_0 <- tmp[[3]]

is.function(center_subtraction_function) # TRUE
is.function(subtracts_ys) # TRUE
is.function(adds_M_0) # TRUE

Wrapper function to generate a unit space for a family of Mahalanobis-Taguchi (MT) methods

Description

generates_unit_space generates a unit space for a family of Mahalanobis-Taguchi (MT) methods. The unit space of MT method, MTA method or RT method can be generated by passing a method name (character) into a parameter method.

Usage

generates_unit_space(unit_space_data, method = c("MT", "MTA", "RT"),
  includes_transformed_data = FALSE, ...)
generates_unit_space(unit_space_data, method = c("MT", "MTA", "RT"),
  includes_transformed_data = FALSE, ...)

Arguments

`unit_space_data`	Matrix with n rows (samples) and p columns (variables). Data to generate the unit space. All data should be continuous values and should not have missing values.
`method`	Character to designate a method. Currently, "MT", "MTA", and "RT" are available.
`includes_transformed_data`	If `TRUE`, then the transformed data are included in a return object.
`...`	Passed to `solve` for computing the inverse of the correlation matrix in `MT` and `RT` method.

Value

A returned object depends on the selected method. See MT, MTA or RT.

Examples

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

# 10 data for each kind (setosa, versicolor, virginica) in the iris dataset
iris_test <- iris[c(1:10, 51:60, 101:111), -5]

# MT method
unit_space_MT <- generates_unit_space(unit_space_data = iris_versicolor,
                                      method = "MT")

diagnosis_MT <- diagnosis(unit_space = unit_space_MT,
                          newdata = iris_test,
                          threshold = 4)

(diagnosis_MT$distance)
(diagnosis_MT$le_threshold)

# MTA method
unit_space_MTA <- generates_unit_space(unit_space_data = iris_versicolor,
                                       method = "MTA")

diagnosis_MTA <- diagnosis(unit_space = unit_space_MTA,
                           newdata = iris_test,
                           threshold = 0.5)

(diagnosis_MTA$distance)
(diagnosis_MTA$le_threshold)

# RT method
unit_space_RT <- generates_unit_space(unit_space_data = iris_versicolor,
                                      method = "RT")

diagnosis_RT <- diagnosis(unit_space = unit_space_RT,
                          newdata = iris_test,
                          threshold = 0.2)

(diagnosis_RT$distance)
(diagnosis_RT$le_threshold)

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

# 10 data for each kind (setosa, versicolor, virginica) in the iris dataset
iris_test <- iris[c(1:10, 51:60, 101:111), -5]

# MT method
unit_space_MT <- generates_unit_space(unit_space_data = iris_versicolor,
                                      method = "MT")

diagnosis_MT <- diagnosis(unit_space = unit_space_MT,
                          newdata = iris_test,
                          threshold = 4)

(diagnosis_MT$distance)
(diagnosis_MT$le_threshold)

# MTA method
unit_space_MTA <- generates_unit_space(unit_space_data = iris_versicolor,
                                       method = "MTA")

diagnosis_MTA <- diagnosis(unit_space = unit_space_MTA,
                           newdata = iris_test,
                           threshold = 0.5)

(diagnosis_MTA$distance)
(diagnosis_MTA$le_threshold)

# RT method
unit_space_RT <- generates_unit_space(unit_space_data = iris_versicolor,
                                      method = "RT")

diagnosis_RT <- diagnosis(unit_space = unit_space_RT,
                          newdata = iris_test,
                          threshold = 0.2)

(diagnosis_RT$distance)
(diagnosis_RT$le_threshold)

Function to generate a unit space for the Mahalanobis-Taguchi (MT) method

Description

MT generates a unit space for the Mahalanobis-Taguchi (MT) method. In general_MT, the inversed correlation matrix is used for A and the data are normalized based on unit_space_data.

Usage

MT(unit_space_data, includes_transformed_data = FALSE, ...)
MT(unit_space_data, includes_transformed_data = FALSE, ...)

Arguments

`unit_space_data`	Matrix with n rows (samples) and p columns (variables). Data to generate the unit space. All data should be continuous values and should not have missing values.
`includes_transformed_data`	If `TRUE`, then the transformed data are included in a return object.
`...`	Passed to `solve` for computing the inverse of the correlation matrix.

Value

MT returns an object of S3 class "MT". An object of class "MT" is a list containing the following components:

`A`	p x p (q x q) matrix. Inversed correlation matrix of `unit_space_data` (the transformed data).
`calc_A`	`function(x) solve(cor(x), ...)`.
`transforms_data`	Function to be generated from `generates_normalization_function` based on `unit_space_data`.
`distance`	Vector with length n. Distances from the unit space to each sample.
`n`	The number of samples.
`q`	The number of variables after the data transformation. q is equal to p.
`x`	If `includes_transformed_data` is `TRUE`, then the transformed data are included.

References

Taguchi, G. (1995). Pattern Recognition and Quality Engineering (1). Journal of Quality Engineering Society, 3(2), 2-5. (In Japanese)

Taguchi, G., Wu, Y., & Chodhury, S. (2000). Mahalanobis-Taguchi System. McGraw-Hill Professional.

Taguchi, G., & Jugulum, R. (2002). The Mahalanobis-Taguchi strategy: A pattern technology system. John Wiley & Sons.

Woodall, W. H., Koudelik, R., Tsui, K. L., Kim, S. B., Stoumbos, Z. G., & Carvounis, C. P. (2003). A review and analysis of the Mahalanobis-Taguchi system. Technometrics, 45(1), 1-15.

Examples

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

unit_space_MT <- MT(unit_space_data = iris_versicolor,
                    includes_transformed_data = TRUE)

# The following tol is a parameter passed to solve function.
unit_space_MT <- MT(unit_space_data = iris_versicolor,
                    includes_transformed_data = TRUE,
                    tol = 1e-9)

(unit_space_MT$distance)

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

unit_space_MT <- MT(unit_space_data = iris_versicolor,
                    includes_transformed_data = TRUE)

# The following tol is a parameter passed to solve function.
unit_space_MT <- MT(unit_space_data = iris_versicolor,
                    includes_transformed_data = TRUE,
                    tol = 1e-9)

(unit_space_MT$distance)

Function to generate a unit space for the Mahalanobis-Taguchi Adjoint (MTA) method

Description

MTA generates a unit space for the Mahalanobis-Taguchi Adjoint (MTA) method. In general_MT, cofactor matrix is used for A and the data are normalized based on unit_space_data.

Usage

MTA(unit_space_data, includes_transformed_data = FALSE)
MTA(unit_space_data, includes_transformed_data = FALSE)

Arguments

`unit_space_data`	Matrix with n rows (samples) and p columns (variables). Data to generate the unit space. All data should be continuous values and should not have missing values.
`includes_transformed_data`	If `TRUE`, then the transformed data are included in a return object.

Value

MTA returns an object of S3 class "MTA". An object of class "MTA" is a list containing the following components:

`A`	p x p (q x q) matrix. Cofactor matrix of `unit_space_data` (the transformed data).
`calc_A`	`calc_cofactor`.
`transforms_data`	Function to be generated from the `generates_normalization_function` based on the `unit_space_data`.
`distance`	Vector with length n. Distances from the unit space to each sample.
`n`	The number of samples.
`q`	The number of variables after the data transformation. q equals p.
`x`	If `includes_transformed_data` is `TRUE`, then the transformed data are included.

References

Taguchi, G. & Kanetaka, T. (2002). Engineering Technical Development in MT System - Lecture on Applied Quality. Japanese Standards Association. (In Japanese)

Taguchi, G., & Jugulum, R. (2002). The Mahalanobis-Taguchi strategy: A pattern technology system. John Wiley & Sons.

Examples

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

unit_space_MTA <- MTA(unit_space_data = iris_versicolor,
                      includes_transformed_data = TRUE)

(unit_space_MTA$distance)

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

unit_space_MTA <- MTA(unit_space_data = iris_versicolor,
                      includes_transformed_data = TRUE)

(unit_space_MTA$distance)

Function to generate a unit space for the Recognition-Taguchi (RT) method

Description

RT generates a unit space for the Recognition-Taguchi (RT) method. In general_MT, the inversed correlation matrix is used for A and the data are transformed by the function to be generated by generates_dimensionality_reduction_function based on unit_space_data. In the transformation, the p variables in unit_space_data are reduced into 2 synthetic variables.

Usage

RT(unit_space_data, includes_transformed_data = FALSE, ...)
RT(unit_space_data, includes_transformed_data = FALSE, ...)

Arguments

`unit_space_data`	Matrix with n rows (samples) and p columns (variables). Data to generate the unit space. All data should be continuous values and should not have missing values.
`includes_transformed_data`	If `TRUE`, then the transformed data are included in a return object.
`...`	Passed to `solve` for computing the inverse of the correlation matrix.

Value

RT returns an object of S3 class "RT". An object of class "RT" is a list containing the following components:

`A`	2 x 2 matrix. Inversed correlation matrix of the transformed `unit_space_data`.
`calc_A`	`function(x) solve(cor(x), ...)`.
`transforms_data`	Function to be generated from `generates_dimensionality_reduction_function` based on `unit_space_data`.
`distance`	Vector with length n. Distances from the unit space to each sample.
`n`	The number of samples.
`q`	The number of variables after the data transformation. q is always 2.
`x`	If `includes_transformed_data` is `TRUE`, then the transformed data are included.

References

Taguchi, G. (2006). Objective Function and Generic Function (11). Journal of Quality Engineering Society, 14(2), 5-9. (In Japanese)

Examples

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

unit_space_RT <- RT(unit_space_data = iris_versicolor,
                    includes_transformed_data = TRUE)

# The following "tol" is a parameter passed to the solve function.
unit_space_RT <- RT(unit_space_data = iris_versicolor,
                    includes_transformed_data = TRUE,
                    tol = 1e-9)

(unit_space_RT$distance)

# 40 data for versicolor in the iris dataset
iris_versicolor <- iris[61:100, -5]

unit_space_RT <- RT(unit_space_data = iris_versicolor,
                    includes_transformed_data = TRUE)

# The following "tol" is a parameter passed to the solve function.
unit_space_RT <- RT(unit_space_data = iris_versicolor,
                    includes_transformed_data = TRUE,
                    tol = 1e-9)

(unit_space_RT$distance)

Function to generate a prediction expression for the two-sided Taguchi (T1) method

Description

T1 generates a prediction expression for the two-sided Taguchi (T1) method. In general_T, the data are normalized by subtracting the mean and without scaling based on unit_space_data. The sample data should be divided into 2 datasets in advance. One is for the unit space and the other is for the signal space.

Usage

T1(unit_space_data, signal_space_data, subtracts_V_e = TRUE,
  includes_transformed_data = FALSE)
T1(unit_space_data, signal_space_data, subtracts_V_e = TRUE,
  includes_transformed_data = FALSE)

Arguments

`unit_space_data`	Matrix with n rows (samples) and (p + 1) columns (variables). The 1 ~ p th columns are independent variables and the (p + 1) th column is a dependent variable. Underlying data to obtain a representative point for the normalization of the `signal_space_data`. All data should be continuous values and should not have missing values.
`signal_space_data`	Matrix with m rows (samples) and (p + 1) columns (variables). The 1 ~ p th columns are independent variables and the (p + 1) th column is a dependent variable. Underlying data to generate a prediction expression. All data should be continuous values and should not have missing values.
`subtracts_V_e`	If `TRUE`, then the error variance is subtracted in the numerator when calculating `eta_hat`.
`includes_transformed_data`	If `TRUE`, then the transformed data are included in a return object.

Value

A list containing the following components is returned.

`beta_hat`	Vector with length q. Estimated proportionality constants between each independent variable and the dependent variable.
`subtracts_V_e`	Logical. If `TRUE`, then `eta_hat` was calculated without subtracting the error variance in the numerator.
`eta_hat`	Vector with length q. Estimated squared signal-to-noise ratios (S/N) coresponding to `beta_hat`.
`M_hat`	Vector with length n. The estimated values of the dependent variable after the data transformation for `signal_space_data`.
`overall_prediction_eta`	Numeric. The overall squared signal-to-noise ratio (S/N).
`transforms_independent_data`	Data transformation function generated from `generates_transform_functions` based on the `unit_space_data`. The function for independent variables takes independent variable data (a matrix of p columns) as an (only) argument and returns the transformed independent variable data.
`transforms_dependent_data`	Data transformation function generated from `generates_transform_functions` based on the `unit_space_data`. The function for a dependent variable takes dependent variable data (a vector) as an (only) argument and returns the transformed dependent variable data.
`inverses_dependent_data`	Data transformation function generated from `generates_transform_functions` based on the `unit_space_data`. The function of the takes the transformed dependent variable data (a vector) as an (only) argument and returns the dependent variable data inversed from the transformed dependent variable data.
`m`	The number of samples for `signal_space_data`.
`q`	The number of independent variables after the data transformation. q equals p.
`X`	If `includes_transformed_data` is `TRUE`, then the independent variable data after the data transformation for the `signal_space_data` are included.
`M`	If `includes_transformed_data` is `TRUE`, then the (true) value of the dependent variable after the data transformation for the `signal_space_data` are included.

References

Taguchi, G. (2006). Objective Function and Generic Function (12). Journal of Quality Engineering Society, 14(3), 5-9. (In Japanese)

Kawada, H., & Nagata, Y. (2015). An application of a generalized inverse regression estimator to Taguchi's T-Method. Total Quality Science, 1(1), 12-21.

Examples

# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

# The following samples are data other than the unit space data and the test
# data.
stackloss_signal <- stackloss[-c(2, 9, 10, 11, 12, 19, 20, 21), ]

model_T1 <- T1(unit_space_data = stackloss_center,
               signal_space_data = stackloss_signal,
               subtracts_V_e = TRUE,
               includes_transformed_data = TRUE)

(model_T1$M_hat)

# The value of the dependent variable of the following samples mediates
# in the stackloss dataset.
stackloss_center <- stackloss[c(9, 10, 11, 20, 21), ]

# The following samples are data other than the unit space data and the test
# data.
stackloss_signal <- stackloss[-c(2, 9, 10, 11, 12, 19, 20, 21), ]

model_T1 <- T1(unit_space_data = stackloss_center,
               signal_space_data = stackloss_signal,
               subtracts_V_e = TRUE,
               includes_transformed_data = TRUE)

(model_T1$M_hat)

Function to generate a prediction expression for the Ta method

Description

Ta generates a prediction expression for the Ta method. In general_T, the data are normalized by subtracting the mean and without scaling based on sample_data. The sample data are not divided into 2 datasets. All the sample data are used for both unit space and signal space.

Usage

Ta(sample_data, subtracts_V_e = TRUE, includes_transformed_data = FALSE)
Ta(sample_data, subtracts_V_e = TRUE, includes_transformed_data = FALSE)

Arguments

`sample_data`	Matrix with n rows (samples) and (p + 1) columns (variables). The 1 ~ p th columns are independent variables and the (p + 1) th column is a dependent variable. All data should be continuous values and should not have missing values.
`subtracts_V_e`	If `TRUE`, then the error variance is subtracted in the numerator when calculating `eta_hat`.
`includes_transformed_data`	If `TRUE`, then the transformed data are included in a return object.

Value

A list containing the following components is returned.

`beta_hat`	Vector with length q. Estimated proportionality constants between each independent variable and the dependent variable.
`subtracts_V_e`	Logical. If `TRUE`, then `eta_hat` was calculated without subtracting the error variance in the numerator.
`eta_hat`	Vector with length q. Estimated squared signal-to-noise ratios (S/N) coresponding to `beta_hat`.
`M_hat`	Vector with length n. The estimated values of the dependent variable after the data transformation for `sample_data`.
`overall_prediction_eta`	Numeric. The overall squared signal-to-noise ratio (S/N).
`transforms_independent_data`	Data transformation function generated from `generates_transform_functions` based on the `unit_space_data`. The function for independent variables takes independent variable data (a matrix of p columns) as an (only) argument and returns the transformed independent variable data.
`transforms_dependent_data`	Data transformation function generated from `generates_transform_functions` based on the `unit_space_data`. The function for a dependent variable takes dependent variable data (a vector) as an (only) argument and returns the transformed dependent variable data.
`inverses_dependent_data`	Data transformation function generated from `generates_transform_functions` based on the `unit_space_data`. The function of the takes the transformed dependent variable data (a vector) as an (only) argument and returns the dependent variable data inversed from the transformed dependent variable data.
`m`	The number of samples for `sample_data`.
`q`	The number of independent variables after the data transformation. q equals p.
`X`	If `includes_transformed_data` is `TRUE`, then the independent variable data after the data transformation for the `sample_data` are included.
`M`	If `includes_transformed_data` is `TRUE`, then the (true) value of the dependent variable after the data transformation for the `sample_data` are included.

References

Kawada, H., & Nagata, Y. (2015). An application of a generalized inverse regression estimator to Taguchi's T-Method. Total Quality Science, 1(1), 12-21.

Examples

model_Ta <- Ta(sample_data = stackloss[-c(2, 12, 19), ],
               subtracts_V_e = TRUE,
               includes_transformed_data = TRUE)

(model_Ta$M_hat)

model_Ta <- Ta(sample_data = stackloss[-c(2, 12, 19), ],
               subtracts_V_e = TRUE,
               includes_transformed_data = TRUE)

(model_Ta$M_hat)

Function to generate a prediction expression for the Tb method

Description

Tb generates a prediction expression for the Tb method. In general_T, the data are normalized by subtracting the center and without scaling based on sample_data. The center is determined by the specific way for the Tb method. For details, please see generates_transformation_functions_Tb. All the sample data are used for both unit space and signal space.

Usage

Tb(sample_data, subtracts_V_e = TRUE, includes_transformed_data = FALSE)
Tb(sample_data, subtracts_V_e = TRUE, includes_transformed_data = FALSE)

Arguments

`sample_data`	Matrix with n rows (samples) and (p + 1) columns (variables). The 1 ~ p th columns are independent variables and the (p + 1) th column is a dependent variable. All data should be continuous values and should not have missing values.
`subtracts_V_e`	If `TRUE`, then the error variance is subtracted in the numerator when calculating `eta_hat`.
`includes_transformed_data`	If `TRUE`, then the transformed data are included in a return object.

Value

A list containing the following components is returned.

`beta_hat`	Vector with length q. Estimated proportionality constants between each independent variable and the dependent variable.
`subtracts_V_e`	Logical. If `TRUE`, then `eta_hat` was calculated without subtracting the error variance in the numerator.
`eta_hat`	Vector with length q. Estimated squared signal-to-noise ratios (S/N) coresponding to `beta_hat`.
`M_hat`	Vector with length n. The estimated values of the dependent variable after the data transformation for `sample_data`.
`overall_prediction_eta`	Numeric. The overall squared signal-to-noise ratio (S/N).
`transforms_independent_data`	Data transformation function generated from `generates_transform_functions` based on the `unit_space_data`. The function for independent variables takes independent variable data (a matrix of p columns) as an (only) argument and returns the transformed independent variable data.
`transforms_dependent_data`	Data transformation function generated from `generates_transform_functions` based on the `unit_space_data`. The function for a dependent variable takes dependent variable data (a vector) as an (only) argument and returns the transformed dependent variable data.
`inverses_dependent_data`	Data transformation function generated from `generates_transform_functions` based on the `unit_space_data`. The function of the takes the transformed dependent variable data (a vector) as an (only) argument and returns the dependent variable data inversed from the transformed dependent variable data.
`m`	The number of samples for `sample_data`.
`q`	The number of independent variables after the data transformation. q equals p.
`X`	If `includes_transformed_data` is `TRUE`, then the independent variable data after the data transformation for the `sample_data` are included.
`M`	If `includes_transformed_data` is `TRUE`, then the (true) value of the dependent variable after the data transformation for the `sample_data` are included.

References

Kawada, H., & Nagata, Y. (2015). An application of a generalized inverse regression estimator to Taguchi's T-Method. Total Quality Science, 1(1), 12-21.

Examples

model_Tb <- Tb(sample_data = stackloss[-c(2, 12, 19), ],
               subtracts_V_e = TRUE,
               includes_transformed_data = TRUE)

(model_Tb$M_hat)

model_Tb <- Tb(sample_data = stackloss[-c(2, 12, 19), ],
               subtracts_V_e = TRUE,
               includes_transformed_data = TRUE)

(model_Tb$M_hat)

Package 'MTSYS'

Help Index

Function to calculate a cofactor matrix

Description

Usage

Arguments

Value

See Also

Examples

Function to estimate M value (M hat) for a family of T methods.

Description

Usage

Arguments

Value

See Also

Examples

Function to calculate overall prediction eta for the T method

Description

Usage

Arguments

Value

See Also

Examples

Function to predict a diagnosis for a family of Mahalanobis-Taguchi (MT) methods

Description

Usage

Arguments

Value

See Also

Diagnosis method for the Mahalanobis-Taguchi (MT) method

Description

Usage

Arguments

Value

References

See Also

Examples

Diagnosis method for the Mahalanobis-Taguchi Adjoint (MTA) method

Description

Usage

Arguments

Value

References

See Also

Examples

Diagnosis method for the Recognition-Taguchi (RT) method

Description

Usage

Arguments

Value

References

See Also

Examples

Function to predict a forecasting for a family of Taguchi (T) methods

Description

Usage

Arguments

Value

See Also

Forecasting method for the T1 method

Description

Usage

Arguments

Value

References

See Also

Examples

Forecasting method for the Ta method

Description

Usage

Arguments

Value

References

See Also

Examples

Forecasting method for the Tb method

Description

Usage

Arguments

Value