| Title: | Latent Variable Gaussian Process Modeling with Qualitative and Quantitative Input Variables |
|---|---|
| Description: | Fit response surfaces for datasets with latent-variable Gaussian process modeling, predict responses for new inputs, and plot latent variables locations in the latent space (only 1D or 2D). The input variables of the datasets can be quantitative, qualitative/categorical or mixed. The output variable of the datasets is a scalar (quantitative). The optimization of the likelihood function is done using a successive approximation/relaxation algorithm similar to another GP modeling package "GPM". The modeling method is published in "A Latent Variable Approach to Gaussian Process Modeling with Qualitative and Quantitative Factors" by Yichi Zhang, Siyu Tao, Wei Chen, and Daniel W. Apley (2018) <arXiv:1806.07504>. The package is developed in IDEAL of Northwestern University. |
| Authors: | Siyu Tao, Yichi Zhang, Daniel W. Apley, Wei Chen |
| Maintainer: | Siyu Tao <[email protected]> |
| License: | GPL-2 |
| Version: | 2.1.5 |
| Built: | 2024-08-30 03:20:19 UTC |
| Source: | https://github.com/cran/LVGP |
LVGP PackageBuilds the correlation matrix given two datasets, and the type and parameters of the correlation function.
corr_mat(X1, X2, phi_full)corr_mat(X1, X2, phi_full)
X1, X2
|
Matrices containing the data points. The rows and columns of both |
phi_full |
The vector storing all the scale (aka roughness) parameters of the correlation function. See |
R The Correlation matrix with size nrow(X1)-by-nrow(X2). See here.
This function is NOT exported once the LVGP package is loaded.
"A Latent Variable Approach to Gaussian Process Modeling with Qualitative and Quantitative Factors", Yichi Zhang, Siyu Tao, Wei Chen, and Daniel W. Apley (arXiv)
LVGP_fit to see how a GP model can be fitted to a training dataset.LVGP_predict to use the fitted LVGP model for prediction.LVGP_plot to plot the features of the fitted model.
# see the examples in the documentation of the function LVGP_fit.# see the examples in the documentation of the function LVGP_fit.
LVGP PackageFits a latent-variable Gaussian process (LVGP) model to a dataset as described in reference 1.
The input variables can be quantitative or qualitative/categorical or mixed.
The output variable is quantitative and scalar.
LVGP_fit(X, Y, ind_qual = NULL, dim_z = 2, eps = 10^(seq(-1, -8, length.out = 15)), lb_phi_ini = -2, ub_phi_ini = 2, lb_phi_lat = -8, ub_phi_lat = 3, lb_z = -3, ub_z = 3, n_opt = 8, max_iter_ini = 100, max_iter_lat = 20, seed = 123, progress = FALSE, parallel = FALSE, noise = FALSE)LVGP_fit(X, Y, ind_qual = NULL, dim_z = 2, eps = 10^(seq(-1, -8, length.out = 15)), lb_phi_ini = -2, ub_phi_ini = 2, lb_phi_lat = -8, ub_phi_lat = 3, lb_z = -3, ub_z = 3, n_opt = 8, max_iter_ini = 100, max_iter_lat = 20, seed = 123, progress = FALSE, parallel = FALSE, noise = FALSE)
X |
Matrix or data frame containing the inputs of training data points. Each row is a data point. |
Y |
Vector containing the outputs of training data points |
ind_qual |
Vector containing the indices of columns of qualitative/categorical variables |
dim_z |
Dimensionality of latent space, usually 1 or 2 but can be higher |
eps |
The vector of smallest eigen values that the correlation matrix is allowed to have, which determines the nugget added to the correlation matrix. |
lb_phi_ini, ub_phi_ini
|
The initial lower and upper search bounds of the scale/roughness parameters ( |
lb_phi_lat, ub_phi_lat
|
The later lower and upper search bounds of the scale/roughness parameters ( |
lb_z, ub_z
|
The lower and upper search bounds of the latent parameters ( |
n_opt |
The number of times the log-likelihood function is optimized |
max_iter_ini |
The maximum number of iterations for each optimization run for largest (first) eps case |
max_iter_lat |
The maximum number of iterations for each optimization run for after first eps cases |
seed |
An integer for the random number generator. Use this to make the results reproducible. |
progress |
The switch determining whether to print function run details |
parallel |
The switch determining whether to use parallel computing |
noise |
The switch for whether the data are assumed noisy |
A model of class "LVGP model" list of the following items:
quant_param A list containing the estimated parameter phi and its search bounds for quantitative variables
qual_param A list containing the estimated parameter z and its dimensionality, vectorized form and search bounds for qualitative variables
data A list containing the fitted dataset in verbose format
fit_detail A list of more detailed variables for fitting and prediction process
optim_hist Optimization history
setting Settings for the optimization and fitting process
"A Latent Variable Approach to Gaussian Process Modeling with Qualitative and Quantitative Factors", Yichi Zhang, Siyu Tao, Wei Chen, and Daniel W. Apley (arXiv)
optim for the details on L-BFGS-B algorithm used in optimization.LVGP_predict to use the fitted LVGP model for prediction.LVGP_plot to plot the features of the fitted model.
# Math example with 2 quantitative and 1 qualitative variables (dataset included in the package): # Fit a model (with default settings) and evaluate the performance # by computing the root mean squared error (RMSE) in prediction. # Also, plot the latent variable parameters. X_tr <- math_example$X_tr Y_tr <- math_example$Y_tr X_te <- math_example$X_te Y_te <- math_example$Y_te n_te <- nrow(X_te) model <- LVGP_fit(X_tr, Y_tr, ind_qual = c(3)) output <- LVGP_predict(X_te, model) Y_hat <- output$Y_hat RRMSE <- sqrt(sum((Y_hat-Y_te)^2)/n_te)/(max(Y_te)-min(Y_te)) LVGP_plot(model)# Math example with 2 quantitative and 1 qualitative variables (dataset included in the package): # Fit a model (with default settings) and evaluate the performance # by computing the root mean squared error (RMSE) in prediction. # Also, plot the latent variable parameters. X_tr <- math_example$X_tr Y_tr <- math_example$Y_tr X_te <- math_example$X_te Y_te <- math_example$Y_te n_te <- nrow(X_te) model <- LVGP_fit(X_tr, Y_tr, ind_qual = c(3)) output <- LVGP_predict(X_te, model) Y_hat <- output$Y_hat RRMSE <- sqrt(sum((Y_hat-Y_te)^2)/n_te)/(max(Y_te)-min(Y_te)) LVGP_plot(model)
LVGP PackagePlots the qualitative/categorical variable levels in the latent space (only for 1D or 2D cases).
If the qualitative/categorical variables are not specified, all the qualified variables will be plotted.
See Arguments for more details on the options.
LVGP_plot(model, ind_qual_plot = NULL)LVGP_plot(model, ind_qual_plot = NULL)
model |
The LVGP model fitted by |
ind_qual_plot |
An array of index (indices) of the qualitative/categorical variable(s) to be plotted. Default is NULL, in which case all the qualitative/categorical variables will be plotted. |
This plot function only works for 1D or 2D latent spaces.
"A Latent Variable Approach to Gaussian Process Modeling with Qualitative and Quantitative Factors", Yichi Zhang, Siyu Tao, Wei Chen, and Daniel W. Apley (arXiv)
LVGP_fit to fit LVGP model to the datasets.LVGP_predict to use the fitted LVGP model for prediction.
# see the examples in the documentation of the function LVGP_fit.# see the examples in the documentation of the function LVGP_fit.
LVGP PackagePredicts the output and associated uncertainties of the GP model fitted by LVGP_fit.
LVGP_predict(X_new, model, MSE_on = 0)LVGP_predict(X_new, model, MSE_on = 0)
X_new |
Matrix or vector containing the input(s) where the predictions are to be made. Each row is an input vector. |
model |
The LVGP model fitted by |
MSE_on |
A scalar indicating whether the uncertainty (i.e., mean squared error |
A prediction list containing the following components:
Y_hat A vector containing the mean prediction values
MSE A vector containing the prediction uncertainty (i.e., the covariance or covariance matrix for the output(s) at prediction location(s))
"A Latent Variable Approach to Gaussian Process Modeling with Qualitative and Quantitative Factors", Yichi Zhang, Siyu Tao, Wei Chen, and Daniel W. Apley (arXiv)
LVGP_fit to fit LVGP model to the datasets.LVGP_plot to plot the features of the fitted model.
# see the examples in the documentation of the function LVGP_fit.# see the examples in the documentation of the function LVGP_fit.
Data are sampled from the modified math function based on the first example in the paper listed in codereferences. There are still 2 quantitative and 1 qualitative variables, but the qualitative variable has only 3 levels. For each level, there are 8 training data points and 30 testing data points, all generated with Latin hypercube sampling. In total, there are 24 training data points and 90 testing data points.
data(math_example)data(math_example)
A named list containing training and test data:
24-by-3 matrix for 24 training data inputs, 3rd column being the qualitative variable
24-by-1 matrix for 24 training data outputs
90-by-3 matrix for 90 testing data inputs, 3rd column being the qualitative variable
90-by-1 matrix for 90 testing data outputs
The dataset can be generated with the code at the end of this description file.
"A Latent Variable Approach to Gaussian Process Modeling with Qualitative and Quantitative Factors", Yichi Zhang, Siyu Tao, Wei Chen, and Daniel W. Apley (arXiv)
data(math_example) X_tr <- math_example$X_tr Y_tr <- math_example$Y_tr X_te <- math_example$X_te Y_te <- math_example$Y_tedata(math_example) X_tr <- math_example$X_tr Y_tr <- math_example$Y_tr X_te <- math_example$X_te Y_te <- math_example$Y_te
LVGP PackageCalculates the negative log-likelihood (excluding all the constant terms) as described in reference 1.
neg_log_l(hyperparam, p_quant, p_qual, lvs_qual, n_lvs_qual, dim_z, X_quant, X_qual, Y, min_eig, k, M)neg_log_l(hyperparam, p_quant, p_qual, lvs_qual, n_lvs_qual, dim_z, X_quant, X_qual, Y, min_eig, k, M)
hyperparam |
Hyperparameters of the LVGP model |
p_quant |
Number of quantative variables |
p_qual |
Number of qualitative variables |
lvs_qual |
Levels of each qualitative variable |
n_lvs_qual |
Number of levels of each qualitative variable |
dim_z |
Dimensionality of latent variables, usually 1 or 2 |
X_quant |
Input data of quantative variables |
X_qual |
Input data of qualitative variables |
Y |
Vector containing the outputs of data points |
min_eig |
The smallest eigen value that the correlation matrix is allowed to have, which determines the nugget added to the correlation matrix. |
k |
Number of data points, |
M |
Vector of ones with length |
LVGP_fit calls this function as its optimization objective function.
The negative log-likelihood (excluding all the constant terms) value.
This function is NOT exported once the package is loaded.
"A Latent Variable Approach to Gaussian Process Modeling with Qualitative and Quantitative Factors", Yichi Zhang, Siyu Tao, Wei Chen, and Daniel W. Apley (arXiv)
LVGP_fit to see how a GP model can be fitted to a training dataset.LVGP_predict to use the fitted LVGP model for prediction.LVGP_plot to plot the features of the fitted model.
# see the examples in the documentation of the function LVGP_fit.# see the examples in the documentation of the function LVGP_fit.
LVGP PackageTransforms qualitative/categorical variables into latent variables.
to_latent(X_qual, lvs_qual, n_lvs_qual, p_qual, z_vec, dim_z, k)to_latent(X_qual, lvs_qual, n_lvs_qual, p_qual, z_vec, dim_z, k)
X_qual |
Matrix or data frame containing (only) the qualitative/categorical data. |
lvs_qual |
List containing levels of each qualitative variable |
n_lvs_qual |
Number of levels of each qualitative variable |
p_qual |
Number of qualitative variables |
z_vec |
Latent variable parameters, i.e., latent variable values for each level of qualitative/categorical variables |
dim_z |
Dimensionality of latent variables, usually 1 or 2 |
k |
Number of data points, equal to |
Matrix containing transformed data
This function is NOT exported once the LVGP package is loaded.
"A Latent Variable Approach to Gaussian Process Modeling with Qualitative and Quantitative Factors", Yichi Zhang, Siyu Tao, Wei Chen, and Daniel W. Apley (arXiv)
LVGP_fit to see how a GP model can be fitted to a training dataset.LVGP_predict to use the fitted LVGP model for prediction.LVGP_plot to plot the features of the fitted model.
# see the examples in the documentation of the function LVGP_fit.# see the examples in the documentation of the function LVGP_fit.