GaSP (version 1.0.6)

GaSPModel: 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
)

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.

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.

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
)