Emulator {hmer}

Bayes Linear Emulator

Description

Creates a univariate emulator object.

The structure of the emulator is f(x) = g(x) * beta + u(x), for regression functions g(x), regression coefficients beta, and correlation structure u(x). An emulator can be created with or without data; the preferred method is to create an emulator based on prior specifications in the absence of data, then use that emulator with data to generate a new one (see examples).

Constructor

Emulator$new(basis_f, beta, u, ranges, ...)

Arguments

Required:

basis_f A list of basis functions to be used. The constant function function(x) 1 should be provided as the first element.

beta The specification for the regression parameters. This should be provided in the form list(mu, sigma), where mu are the expectations of the coefficients (aligning with the ordering of basis_f) and sigma the corresponding covariance matrix.

u The specifications for the correlation structure. This should be specified in the form list(sigma, corr), where sigma is a single-valued object, and corr is a Correlator object.

ranges A named list of ranges for the input parameters, provided as a named list of length-two numeric vectors.

Optional:

data A data.frame consisting of the data with which to adjust the emulator, consisting of input values for each parameter and the output.

out_name The name of the output variable.

a_vars A logical vector indicating which variables are active for this emulator.

discs Model discrepancies: does not include observational error. Ideally split into list(internal = ..., external = ...).

Internal:

model If a linear model, or otherwise, has been fitted to the data, it lives here.

original_em If the emulator has been adjusted, the unadjusted Emulator object is stored, for use of set_sigma or similar.

multiplier A multiplicative factor to be applied to u_sigma. Typically equal to 1, unless changes have been made by, for example, mult_sigma.

Constructor Details

The constructor must take, as a minimum: a list of vectorised basis functions, whose length is equal to the number of regression coefficients; a correlation structure, which can be non-stationary; and the parameter ranges, used to scale all inputs to the range [-1,1].

The construction of a correlation structure is detailed in the documentation for Correlator.

Accessor Methods

get_exp(x, include_c) Returns the emulator expectation at a point, or at a collection of points. If include_c = FALSE, the contribution made by the correlation structure is not included.

get_cov(x, xp = NULL, full = FALSE, include_c) Returns the covariance between collections of points x and xp. If xp is not supplied, then this is equivalent to get_cov(x, x, ...); if full = TRUE, then the full covariance matrix is calculated - this is FALSE by default due to most built-in uses requiring only the diagonal terms, and allows us to take advantage of computational tricks for efficiency.

implausibility(x, z, cutoff = NULL) Returns the implausibility for a collection of points x. The implausibility is the distance between the emulator expectation and a desired output value, weighted by the emulator variance and any external uncertainty. The target, z, should be specified as a named pair list(val, sigma), or a single numeric value. If cutoff = NULL, the output is a numeric I; if cutoff is a numeric value, then the output is boolean corresponding to I <= cutoff.

get_exp_d(x, p) Returns the expectation of the derivative of the emulated function, E[f'(x)]. Similar in structure to get_exp but for the additional parameter p, which indicates which of the input dimensions the derivative is performed with respect to.

get_cov_d(x, p1, xp = NULL, p2 = NULL, full = FALSE) Returns the variance of the derivative of the emulated function, Var[f'(x)]. The arguments are similar to that of get_cov, but for the addition of parameters p1 and p2, which indicate the derivative directions. Formally, the output of this function is equivalent to Cov[df/dp1, df/dp2].

print(...) Returns a summary of the emulator specifications.

plot(...) A wrapper for emulator_plot for a single Emulator object.

Object Methods

adjust(data, out_name) Performs Bayes Linear Adjustment, given data. The data should contain all input parameters, even inactive ones, and the single output that we wish to emulate. adjust creates a new Emulator object with the adjusted expectation and variance resulting from Bayes Linear adjustment, allowing for the requisite predictions to be made using get_exp and get_cov.

set_sigma(sigma) Modifies the (usually constant) global variance of the correlation structure, Var[u(X)]. If the emulator has been trained, the original emulator is modified and Bayes Linear adjustment is again performed.

mult_sigma(m) Modifies the global variance of the correlation structure via a multiplicative factor. As with set_sigma, this change will chain through any prior emulators if the emulator in question is Bayes Linear adjusted.

set_hyperparams(hp, nugget) Modifies the underlying correlator for u(x). Behaves in a similar way to set_sigma as regards trained emulators. See the Correlator documentation for details of hp and nugget.

References

Goldstein & Wooff (2007) <ISBN: 9780470065662>

Craig, Goldstein, Seheult & Smith (1998) <doi:10.1111/1467-9884.00115>

Examples

basis_functions <- list(function(x) 1, function(x) x[[1]], function(x) x[[2]])
beta <- list(mu = c(1,2,3),
             sigma = matrix(c(0.5, -0.1, 0.2, -0.1, 1, 0, 0.2, 0, 1.5), nrow = 3))
u <- list(mu = function(x) 0, sigma = 3, corr = Correlator$new('exp_sq', list(theta = 0.1)))
ranges <- list(a = c(-0.5, 0.5), b = c(-1, 2))
em <- Emulator$new(basis_functions, beta, u, ranges)
em
# Individual evaluations of points
# Points should still be declared in a data.frame
em$get_exp(data.frame(a = 0.1, b = 0.1)) #> 0.6
em$get_cov(data.frame(a = 0.1, b = 0.1)) #> 9.5
# 4x4 grid of points
sample_points <- expand.grid(a = seq(-0.5, 0.5, length.out = 4), b = seq(-1, 2, length.out = 4))
em$get_exp(sample_points) # Returns 16 expectations
em$get_cov(sample_points) # Returns 16 variances
sample_points_2 <- expand.grid(a = seq(-0.5, 0.5, length.out = 3),
                               b = seq(-1, 2, length.out = 4))
em$get_cov(sample_points, xp = sample_points_2, full = TRUE) # Returns a 16x12 matrix of covariances


fake_data <- data.frame(a = runif(10, -0.5, 0.5), b = runif(10, -1, 2))
fake_data$c <- fake_data$a + 2*fake_data$b
newem <- em$adjust(fake_data, 'c')
all(round(newem$get_exp(fake_data[,names(ranges)]),5) == round(fake_data$c,5)) #>TRUE

matern_em <- Emulator$new(basis_f = c(function(x) 1, function(x) x[[1]], function(x) x[[2]]),
 beta = list(mu = c(1, 0.5, 2), sigma = diag(0, nrow = 3)),
 u = list(corr = Correlator$new('matern', list(nu = 1.5, theta = 0.4))),
 ranges = list(x = c(-1, 1), y = c(0, 3)))
matern_em$get_exp(data.frame(x = 0.4, y = 2.3))

newem_data <- Emulator$new(basis_functions, beta, u, ranges, data = fake_data)
all(round(newem$get_exp(fake_data[,names(ranges)]),5)
   == round(newem_data$get_exp(fake_data[,names(ranges)]), 5)) #>TRUE
newem$get_exp_d(sample_points, 'a')
newem$get_cov_d(sample_points, 'b', p2 = 'a')