`NestedKriging`

Description

Create a NestedKriging object: a divide-and-conquer Kriging for large designs (\(n \sim 10^4\)–\(10^6\)). The design is partitioned into nb_groups groups (k-means on the inputs, or random), one Kriging submodel is fitted per group — all sharing a common prior after hyper-parameter unification — and predictions are aggregated with:

aggregation = "NK" (default): the optimal nested-kriging aggregation of Rullière, Durrande, Bachoc & Chevalier (Statistics & Computing, 2018). It is itself a kriging predictor: it interpolates the design and provides consistent variances;
aggregation = "PoE", "gPoE", "BCM" or "rBCM": precision-weighted products of experts (cheaper, no interpolation guarantee).

The fit cost drops from \(O(n^3)\) to \(O(n^3/p^2)\) per likelihood evaluation for \(p\) groups.

Usage

Python

k = NestedKriging(y, X, kernel="matern5_2", nb_groups=8,
                  aggregation="NK", partition="kmeans", seed=123,
                  regmodel="constant", optim="BFGS", objective="LL",
                  parameters=None, warping=[])

R

k <- NestedKriging(y, X, kernel = "matern5_2", nb_groups = 8,
                   aggregation = "NK", partition = "kmeans", seed = 123,
                   regmodel = "constant", optim = "BFGS", objective = "LL",
                   parameters = NULL, warping = NULL)

Matlab/Octave

k = NestedKriging(y, X, "matern5_2", 8, "NK", "kmeans", 123, ...
                  "constant", "BFGS", "LL", [], {})

Julia

k = NestedKriging(y, X, "matern5_2", 8; aggregation="NK",
                  partition="kmeans", seed=123, regmodel="constant",
                  optim="BFGS", objective="LL", warping=String[])

Arguments

Argument	Description
`y`	Numeric vector of response values.
`X`	Numeric matrix of input design.
`kernel`	Character defining the covariance model: `"gauss"`, `"exp"`, `"matern3_2"`, `"matern5_2"`.
`nb_groups`	Number of submodels; each group holds about `nrow(X)/nb_groups` points (groups of ~100–1000 points are typical).
`aggregation`	`"NK"` (optimal nested-kriging aggregation, interpolating, default), `"PoE"`, `"gPoE"`, `"BCM"` or `"rBCM"`.
`partition`	`"kmeans"` (default) or `"random"`.
`seed`	Integer seed for the partition (and hyper-parameter subsampling), for reproducibility.
`regmodel`	Universal Kriging linear trend. `"NK"` aggregation requires `"constant"`; the PoE family accepts any trend.
`optim`	Optimization method for the submodel hyper-parameters: `"BFGS"` (default) or `"none"`.
`objective`	`"LL"` (default), `"LOO"`, `"LMP"` — or `"VLL(m)"`: the common prior \((\theta, \sigma^2, \beta)\) is then estimated by one global Vecchia fit in \(O(n\,m^3)\) (cross-group information, one optimization instead of \(p\)) and every submodel is fitted in closed form on the seeded prior.
`parameters`	Initial or fixed values for the hyper-parameters (named list with `"sigma2"`, `"theta"`, `"beta"`).
`warping`	Optional per-dimension warp specs (see `WarpKriging`); submodels are then `WarpKriging` sharing a common warped prior \(\sigma^2 k(\Phi(x), \Phi(x'))\), with \((\theta, \text{warp})\) estimated by a single reference fit on a global subsample. Not compatible with `objective="VLL(m)"`.

Details

The hyper-parameter unification builds the common GP prior required by all aggregations: with objective="LL" (default), \(\theta\) is the group-size weighted geometric mean of the per-group estimates and \(\sigma^2, \beta_0\) weighted means, then every submodel is refitted with optim="none". The NK aggregation then kriges \(Y(x)\) on the submodel predictors \(M_i(x)\), using their exact cross-covariances under the common prior.

Value

A NestedKriging object, to be used with its predict method. Accessors: kernel(), aggregation(), nb_groups(), groups(), theta(), sigma2(), beta0(), warping().

Examples

f <- function(X) apply(X, 1, function(x) sin(3 * x[1]) + cos(5 * x[2]))
set.seed(123)
X <- matrix(runif(2 * 1000), ncol = 2)
y <- f(X)

k <- NestedKriging(y, X, kernel = "matern5_2", nb_groups = 10)
print(k)

x <- matrix(runif(2 * 100), ncol = 2)
p <- predict(k, x)

NestedKriging