nsgpr: Estimation of a nonseparable and/or nonstationary covariance structure (NSGPR model)

Description

Estimate the covariance structure of a zero-mean Gaussian Process with Q-dimensional input coordinates (covariates).

Multiple realisations for the response variable can be used, provided they are observed on the same grid of dimension n_1 x n_2 x ... x n_Q.

Let n = n_1 x n_2 x ... x n_Q and let nSamples be the number of realisations.

Usage

nsgpr(
  response,
  input,
  corrModel = "pow.ex",
  gamma = 2,
  nu = 1.5,
  whichTau = NULL,
  nBasis = 5,
  cyclic = NULL,
  unitSignalVariance = F,
  zeroNoiseVariance = F,
  sepCov = F,
  nInitCandidates = 300,
  absBounds = 6,
  inputSubsetIdx = NULL
)

Value

A list containing:

MLEsts: Maximum likelihood estimates of B-spline coefficients and noise variance.
response: Matrix of response.
inputMat: Input coordinates in a matrix form
corrModel: Correlation function specification used for g(.)

Arguments

response: Response variable. This should be a (n x nSamples) matrix where each column is a realisation
input: List of Q input variables (see Details).
corrModel: Correlation function specification used for g(.). It can be either "pow.ex" or "matern".
gamma: Power parameter used in powered exponential kernel function. It must be 0<gamma<=2.
nu: Smoothness parameter of the Matern class. It must be a positive value.
whichTau: Logical vector of dimension Q identifying which input coordinates the parameters are function of. For example, if Q=2 and parameters change only with respect to the first coordinate, then we set whichTau=c(T,F).
nBasis: Number of B-spline basis functions in each coordinate direction along which parameters change.
cyclic: Logical vector of dimension Q which defines which covariates are cyclic (periodic). For example, if basis functions should be cyclic only in the first coordinate direction, then cyclic=c(T,F). cyclic must have the same dimension of whichTau. If cyclic is TRUE for some coordinate direction, then cyclic B-spline functions will be used and the varying parameters (and their first two derivatives) will match at the boundaries of that coordinate direction.
unitSignalVariance: Logical. TRUE if we assume realisations have variance 1. This is useful when we want to estimate an NSGP correlation function.
zeroNoiseVariance: Logical. TRUE if we assume the realisations are noise-free.
sepCov: Logical. TRUE only if we fix to zero all off-diagonal elements of the varying anisotropy matrix. Default to FALSE, allowing for a separable covariance function.
nInitCandidates: number of initial hyperparameter vectors which are used to evaluate the log-likelihood function at a first step. After evaluating the log-likelihood using these 'nInitCandidates' vectors, the optimisation via nlminb() begins with the best of these vectors.
absBounds: lower and upper boundaries for B-spline coefficients (if wanted).
inputSubsetIdx: A list identifying a subset of the input values to be used in the estimation (see Details).

Details

The input argument for Q=2 can be constructed as follows:


  n1 <- 10
  n2 <- 1000
  input <- list()
  input[[1]] <- seq(0,1,length.out = n1)
  input[[2]] <- seq(0,1,length.out = n2)

If we want to use every third lattice point in the second input variable (using Subset of Data), then we can set


  inputSubsetIdx <- list()
  inputSubsetIdx[[1]] <- 1:n1
  inputSubsetIdx[[2]] <- seq(1,n2, by=3)

References

Konzen, E., Shi, J. Q. and Wang, Z. (2020) "Modeling Function-Valued Processes with Nonseparable and/or Nonstationary Covariance Structure" <arXiv:1903.09981>.

Examples

Run this code

## See examples in vignette:
# vignette("nsgpr", package = "GPFDA")

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