Fractional Brownian motion prior

fBm has E(W_s W_t) = (s^(2H) + t^(2H) - |t - s|^(2H)) / 2. The Hurst index H sets the path smoothness directly, so fBm is the natural prior when the smoothness is to be chosen rather than inherited; the kernel is evaluated on a grid and checked for the two properties the theory needs, the diagonal E W_t^2 = t^(2H) and positive definiteness (all leading principal minors positive, by Gaussian elimination).

Usage

Ghosalfbmprior(H = 0.7, ts = c(0.25, 0.5, 0.75))

Arguments

H

Hurst index, in (0, 1).

ts

Grid of time points, all positive.

Value

List with estimate (K(t1, t1)), kernel, var_gap, positive_definite, method.

Details

Formula: K(s, t) = (s^(2H) + t^(2H) - |t - s|^(2H)) / 2.

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, CUP, Example 11.9, eq. (11.6).

Examples

Ghosalfbmprior()
#> $estimate
#> [1] 0.1435873
#> 
#> $kernel
#>           [,1]      [,2]      [,3]
#> [1,] 0.1435873 0.1894646 0.2165670
#> [2,] 0.1894646 0.3789291 0.4519089
#> [3,] 0.2165670 0.4519089 0.6684759
#> 
#> $var_gap
#> [1] 0
#> 
#> $positive_definite
#> [1] TRUE
#> 
#> $method
#> [1] "fBm covariance (GvdV 2017 eq. 11.6)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"