gp.fisher: Fisher information matrix

In GPBayes: Tools for Gaussian Process Modeling in Uncertainty Quantification

Fisher information matrix

Description

This function computes the Fisher information matrix I(\sigma^2, \boldsymbol \theta) for a Gaussian process model y(\cdot) \sim \mathcal{GP}(h^\top(\mathbf{x})\mathbf{b}, \sigma^2 c(\cdot, \cdot) ), where c(\mathbf{x}_1, \mathbf{x}_2) = r(\mathbf{x}_1, \mathbf{x}_2) + \tau^2\mathbf{1}(\mathbf{x}_1=\mathbf{x}_2) with correlation function r(\cdot, \cdot) and nugget parameter \tau^2; \mathbf{b} is a vector of regression coefficients, \sigma^2 is the variance parameter (or partial sill).

Given n data points that are assumed to be realizations from the GP model, the standard likelihood is defined as

L(\mathbf{b}, \sigma^2, \boldsymbol \theta; \mathbf{y}) = \mathcal{N}_n(\mathbf{H}\mathbf{b}, \sigma^2 (\mathbf{R} + \tau^2\mathbf{I}) ),

where \mathbf{y}:=(y(\mathbf{x}_1), \ldots, y(\mathbf{x}_n))^\top is a vector of n observations. \mathbf{H} is a matrix of covariates, \boldsymbol \theta contains correlation parameters and nugget parameter, \mathbf{R} denotes the n-by-n correlation matrix.

The integrated likelihood is defined as

L^{I}(\sigma^2, \boldsymbol \theta; \mathbf{y}) = \int L(\mathbf{b}, \sigma^2, \boldsymbol \theta; \mathbf{y}) \pi^{R}(\mathbf{b} \mid \sigma^2, \boldsymbol \theta) d \mathbf{b},

where \pi^{R}(\mathbf{b} \mid \sigma^2, \boldsymbol \theta)=1 is the conditional Jeffreys-rule (or reference prior) in the model with the above standard likelihood when (\sigma^2, \boldsymbol \theta) is assumed to be known.

• For the Matérn class, current implementation only computes Fisher information matrix for variance parameter \sigma^2, range parameter \phi, and nugget variance parameter \tau^2. That is, I(\sigma^2, \boldsymbol \theta) = I(\sigma^2, \phi, \tau^2).

• For the Confluent Hypergeometric class, current implementation computes Fisher information matrix for variance parameter \sigma^2, range parameter \beta, tail decay parameter \alpha, smoothness parameter \nu and nugget variance parameter \tau^2. That is, I(\sigma^2, \boldsymbol \theta) = I(\sigma^2, \beta, \alpha, \nu, \tau^2).

Usage

gp.fisher(
  obj = NULL,
  intloglik = FALSE,
  formula = ~1,
  input = NULL,
  param = NULL,
  cov.model = NULL,
  dtype = "Euclidean"
)

Arguments

obj	a gp object. It is optional with default value NULL.
intloglik	a logical value with default value FALSE. If it is FALSE, Fisher information matrix I(\sigma^2, \boldsymbol \theta) is derived based on the standard likelihood; otherwise, Fisher information matrix I(\sigma^2, \boldsymbol \theta) is derived based on the integrated likelihood.
formula	an object of formula class that specifies regressors; see formula for details.
input	a matrix including inputs in a GaSP
param	a list including values for regression parameters, covariance parameters, and nugget variance parameter. The specification of param should depend on the covariance model. The regression parameters are denoted by coeff. Default value is \mathbf{0}. The marginal variance or partial sill is denoted by sig2. Default value is 1. The nugget variance parameter is denoted by nugget for all covariance models. Default value is 0. For the Confluent Hypergeometric class, range is used to denote the range parameter \beta. tail is used to denote the tail decay parameter \alpha. nu is used to denote the smoothness parameter \nu. For the generalized Cauchy class, range is used to denote the range parameter \phi. tail is used to denote the tail decay parameter \alpha. nu is used to denote the smoothness parameter \nu. For the Matérn class, range is used to denote the range parameter \phi. nu is used to denote the smoothness parameter \nu. When \nu=0.5, the Matérn class corresponds to the exponential covariance. For the powered-exponential class, range is used to denote the range parameter \phi. nu is used to denote the smoothness parameter. When \nu=2, the powered-exponential class corresponds to the Gaussian covariance.
cov.model	a list of two strings: family, form, where family indicates the family of covariance functions including the Confluent Hypergeometric class, the Matérn class, the Cauchy class, the powered-exponential class. form indicates the specific form of covariance structures including the isotropic form, tensor form, automatic relevance determination form. family CH The Confluent Hypergeometric correlation function is given by C(h) = \frac{\Gamma(\nu+\alpha)}{\Gamma(\nu)} \mathcal{U}\left(\alpha, 1-\nu, \left(\frac{h}{\beta}\right)^2\right), where \alpha is the tail decay parameter. \beta is the range parameter. \nu is the smoothness parameter. \mathcal{U}(\cdot) is the confluent hypergeometric function of the second kind. For details about this covariance, see Ma and Bhadra (2023; \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/01621459.2022.2027775")}). cauchy The generalized Cauchy covariance is given by C(h) = \left\{ 1 + \left( \frac{h}{\phi} \right)^{\nu} \right\}^{-\alpha/\nu}, where \phi is the range parameter. \alpha is the tail decay parameter. \nu is the smoothness parameter with default value at 2. matern The Matérn correlation function is given by C(h)=\frac{2^{1-\nu}}{\Gamma(\nu)} \left( \frac{h}{\phi} \right)^{\nu} \mathcal{K}_{\nu}\left( \frac{h}{\phi} \right), where \phi is the range parameter. \nu is the smoothness parameter. \mathcal{K}_{\nu}(\cdot) is the modified Bessel function of the second kind of order \nu. exp The exponential correlation function is given by C(h)=\exp(-h/\phi), where \phi is the range parameter. This is the Matérn correlation with \nu=0.5. matern_3_2 The Matérn correlation with \nu=1.5. matern_5_2 The Matérn correlation with \nu=2.5. powexp The powered-exponential correlation function is given by C(h)=\exp\left\{-\left(\frac{h}{\phi}\right)^{\nu}\right\}, where \phi is the range parameter. \nu is the smoothness parameter. gauss The Gaussian correlation function is given by C(h)=\exp\left(-\frac{h^2}{\phi^2}\right), where \phi is the range parameter. form isotropic This indicates the isotropic form of covariance functions. That is, C(\mathbf{h}) = C^0(\|\mathbf{h}\|; \boldsymbol \theta), where \| \mathbf{h}\| denotes the Euclidean distance or the great circle distance for data on sphere. C^0(\cdot) denotes any isotropic covariance family specified in family. tensor This indicates the tensor product of correlation functions. That is, C(\mathbf{h}) = \prod_{i=1}^d C^0(|h_i|; \boldsymbol \theta_i), where d is the dimension of input space. h_i is the distance along the ith input dimension. This type of covariance structure has been often used in Gaussian process emulation for computer experiments. ARD This indicates the automatic relevance determination form. That is, C(\mathbf{h}) = C^0\left(\sqrt{\sum_{i=1}^d\frac{h_i^2}{\phi^2_i}}; \boldsymbol \theta \right), where \phi_i denotes the range parameter along the ith input dimension.
dtype	a string indicating the type of distance: Euclidean Euclidean distance is used. This is the default choice. GCD Great circle distance is used for data on sphere.

Value

a numerical matrix of Fisher information

Author(s)

Pulong Ma [e-mail]

See Also

GPBayes-package, GaSP, gp, kernel, ikernel,

Examples

 
n=100
input = seq(0, 20, length=n)
range = 1
tail = .5
nu = 1.5
sig2 = 1
nugget = 0.01
coeff = 0
par = list(range=range, tail=tail, nu=nu, sig2=sig2, nugget=nugget, coeff=coeff)
I = gp.fisher(formula=~1, input=input, 
        param=list(range=4, nugget=0.1,nu=2.5),
        cov.model=list(family="CH", form="isotropic"))

GPBayes documentation built on May 29, 2024, 7:30 a.m.