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The covariance of a curve's derivative that is not coefficient uncertainty

Source: R/extra-deriv.R
frm_extra_cov_deriv.Rd

The derivative counterpart of extra_cov from frm_lp_basis(): the n x n covariance, over the rows of newdata, of the order-th derivative in var of the part of the linear predictor that is not a function of the coefficients. Two sources contribute, and each is differentiated before its covariance is formed:

Usage

frm_extra_cov_deriv(
  object,
  newdata,
  var,
  order = 1L,
  eps = NULL,
  dpar = NULL,
  resp = NULL,
  re_formula = NULL,
  allow_new_levels = FALSE
)

Arguments

object

A fitted frmtmb_fit.

newdata

The grid, a data frame.

var

The variable to differentiate in, a column of newdata.

order

1 or 2.

eps

The stencil's step for a new level's design rows.

dpar, resp

The distributional parameter and response to take the linear predictor of. Both default the way predict() defaults them.

re_formula

NULL keeps every random effect, NA drops the group-level terms and keeps every smooth, a one-sided formula keeps the terms it names. See frm_linpred() for the full rule.

allow_new_levels

Whether a grouping level the fit never saw is allowed.

Value

An n x n matrix, as extra_cov is: a sparse Matrix::sparseMatrix() where the sources cover few of its entries (an empty one when there is no source), dense otherwise.

Details

  • An exact gp() term's kriging residual. Its derivative's covariance is the kernel's closed form, d^o_x d^o_x' k(x, x') - d^o_x k(x, X) K^-1 d^o_x' k(X, x'), from the squared exponential kernel's Hermite-polynomial derivatives. It is not zero at an observed position: the field's slope there is not fixed by its value. The nugget is white noise and has no derivative, so it is not in this covariance.

  • A grouping level the fit never saw (allow_new_levels = TRUE). The design rows of its draw are differenced with the central stencil of step eps, as a design is, and the covariance is then (L M) S (L M)', which carries no differenced rounding.

A Hilbert-space gp(k = ) and a smooth have no such part: their uncertainty is all in their coefficients.

See also

frm_lp_basis(), whose extra_cov this differentiates.

Examples

set.seed(5)
d <- data.frame(x = sort(runif(40, 0, 4)))
d$y <- sin(d$x) + rnorm(40, 0, 0.2)
fit <- frm(bf(y ~ gp(x)), data = d)
nd <- data.frame(x = c(4.5, 5, 6))
# the slope's and the curvature's variance past the data
diag(frm_extra_cov_deriv(fit, nd, var = "x", order = 1))
#> [1] 3.060007e-06 1.122850e-04 8.706461e-03
diag(frm_extra_cov_deriv(fit, nd, var = "x", order = 2))
#> [1] 5.940085e-05 1.007320e-03 2.142198e-02