Bochner: every continuous positive-definite kernel of a stationary
process is the Fourier transform of a finite spectral measure,
K(s - t) = int exp(-i <s - t, lambda>) dmu(lambda). In one dimension
the squared-exponential process has spectral density
(2 sqrt(pi))^(-1) exp(-lambda^2 / 4), and inverting that integral must
return K(h) = exp(-h^2). Doing the inversion numerically is the check
that the spectral representation used throughout the chapter is the
right one.
Usage
Ghosalstatgpspec(h = 0.6, n_grid = 4000, lam_max = 30)
Arguments
- h
Lag at which the kernel is evaluated.
- n_grid
Number of midpoint cells.
- lam_max
Truncation of the spectral integral.
Value
List with estimate (numeric kernel),
kernel_exact, bochner_gap, method.
Details
Formula: K(h) = (2 sqrt(pi))^(-1) int cos(h lambda) exp(-lambda^2/4)
dlambda, by midpoint rule over [-lam_max, lam_max].
References
Ghosal & van der Vaart (2017), Fundamentals of
Nonparametric Bayesian Inference, CUP, Example 11.8,
eqs. (11.3)-(11.4).
Examples
Ghosalstatgpspec()
#> $estimate
#> [1] 0.6976763
#>
#> $kernel_exact
#> [1] 0.6976763
#>
#> $bochner_gap
#> [1] 1.110223e-16
#>
#> $method
#> [1] "Bochner spectral kernel (GvdV 2017 eq. 11.3-11.4)"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"
