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

## 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

 `1` ```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

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.Ta`
 ```1 2 3 4 5``` ```model_Ta <- Ta(sample_data = stackloss[-c(2, 12, 19), ], subtracts_V_e = TRUE, includes_transformed_data = TRUE) (model_Ta\$M_hat) ```