GaSPModel {GaSP}

Create a GaSPModel object.

Description

Return a template for a GaSPModel object.

Usage

GaSPModel(
  x,
  y,
  reg_model,
  sp_model = NULL,
  cor_family = c("PowerExponential", "Matern"),
  cor_par,
  random_error = c(FALSE, TRUE),
  sp_var,
  error_var = 0
)

Arguments

x

A data frame containing the input (explanatory variable) training data.

y

A vector or a data frame with one column containing the output (response) training data.

reg_model

The regression model, specified as a formula, but note the left-hand side of the formula is unused; see example.

sp_model

An optional stochastic process model, specified as a formula, but note the left-hand side of the formula and the intercept are unused. The default NULL uses all column names in x.

cor_family

A character string specifying the (product, anisoptropic) correlation-function family: "PowerExponential" for the power-exponential family or "Matern" for the Matern family.

cor_par

A data frame containing the correlation parameters with one row per sp_model term and two columns (see Details).

random_error

A boolean for the presence or not of a random (measurement, white-noise) error term.

sp_var

The stochastic process variance.

error_var

The random error variance, with default 0.

Details

The data frame cor_par contains one row for each term in the stochastic process model. There are two columns. The first is named Theta, and the second is either Alpha (power-exponential) or Derivatives (Matern). Let h_j be a distance between points for term j in the stochastic-process model. For power-exponential, the contribution to the product correlation from term j depends on a distance-scale parameter \theta_j from the Theta column and a smoothness parameter \alpha_j from the Alpha column; the contribution is exp(-\theta_j h_j^{2 - \alpha_j}). For example, \alpha_j = 0 gives the squared-exponential (Gaussian) correlation. The contribution to the product correlation for Matern also depends on \theta_j, and the second parameter is the number of derivatives \delta_j = 0, 1, 2, 3 from the Derivatives column. The contribution is exp(-\theta_j h_j) for \delta_j = 0 (the exponential correlation), exp(-\theta_j h_j) (\theta_j h_j + 1) for \delta_j = 1, exp(-\theta_j h_j) ((\theta_j h_j)^2 / 3 + \theta_j h_j + 1) for \delta_j = 2, and exp(-\theta_j h_j^2) for \delta_j = 3 (the squared-exponential correlation). Note that \delta_j = 3 codes for a limiting infinite number of derivatives. This is not the usual parameterization of the Matern, but it is consistent with power-exponential for the exponential and squared-exponential special cases common to both.

A value should be given to error_var if the model has a random-error term (random_error = TRUE), and a small "nugget" such as 10^{-9} may be needed for improved numerical conditioning.

Value

A GaSPModel object, which is a list with the following components:

x

The data frame containing the input training data.

y

The training output data, now as a vector.

reg_model

The regression model, now in the form of a data frame.

sp_model

The stochastic process model, now in the form of a data frame.

cor_family

The correlation family.

cor_par

The data frame containing the correlation parameters.

random_error

The boolean for the presence or not of a random error term.

sp_var

The stochastic process variance.

error_var

The random error variance.

beta

A placeholder for a data frame to hold the regression-model parameters.

objective

A placeholder for the maximum fit objective.

cond_num

A placeholder for the condition number.

CVRMSE

A placeholder for the model's cross-validated root mean squared error.

Note

This function does not excecute Fit and is intended for CrossValidate, Predict and Visualize with models trained otherwise by the user. Placeholders do not need to be specified to excecute these further functions, as they are always recomputed as needed.

References

Sacks, J., Welch, W.J., Mitchell, T.J., and Wynn, H.P. (1989) "Design and Analysis of Computer Experiments", Statistical Science, 4, pp. 409-423, doi:10.1214/ss/1177012413.

Examples

x <- borehole$x
y <- borehole$y
theta <- c(
  5.767699e+01, 0.000000e+00, 0.000000e+00, 1.433571e-06,
  0.000000e+00, 2.366557e-06, 1.695619e-07, 2.454376e-09
)
alpha <- c(
  1.110223e-16, 0.000000e+00, 0.000000e+00, 0.000000e+00,
  0.000000e+00, 0.000000e+00, 2.494862e-03, 0.000000e+00
)
cor_par <- data.frame(Theta = theta, Alpha = alpha)
rownames(cor_par) <- colnames(borehole$x)
sp_var <- 38783.7
borehole_gasp <- GaSPModel(
  x = borehole$x, y = borehole$y,
  reg_model = ~1, cor_family = "PowerExponential",
  cor_par = cor_par, random_error = FALSE,
  sp_var = sp_var
)