# T1: Function to generate a prediction expression for the... In okayaa/MT: Methods in Mahalanobis-Taguchi (MT) System

## 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

 ```1 2``` ```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)

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.

`general_T`, `generates_transformation_functions_T1`, and `forecasting.T1`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```# 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) ```