Rescaled Gaussian process

f_l(x) = f(x / l) changes the kernel to K(s/l, t/l): shrinking the length scale l roughens the sample paths and lengthening it smooths them. Because the smoothness of a Gaussian prior is what fixes its contraction rate, putting a prior on l is exactly how a single process adapts to an unknown smoothness – the mechanism of section 11.5. For the squared-exponential kernel the correlation at lag h is exp(-(h/l)^2), which falls as l shrinks.

Usage

Ghosalrescalgp(lengths = c(2, 1, 0.25), h = 0.3)

Arguments

lengths

Vector of length scales, all positive.

h

Lag.

Value

List with estimate (correlation at the last length), correlation_by_length, roughens_as_l_shrinks, method.

Details

Formula: rho_l(h) = exp(-(h / l)^2).

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, CUP, Example 11.11 and section 11.5.

Examples

Ghosalrescalgp()
#> $estimate
#> [1] 0.2369278
#> 
#> $correlation_by_length
#> [1] 0.9777512 0.9139312 0.2369278
#> 
#> $roughens_as_l_shrinks
#> [1] TRUE
#> 
#> $method
#> [1] "rescaled GP (GvdV 2017 Ex 11.11, sec. 11.5)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"