npreg (version 1.1.0)

vcov: Calculate Variance-Covariance Matrix for a Fitted Smooth Model

Description

Returns the variance-covariance matrix for the basis function coefficients from a fit smoothing spline (fit by ss), smooth model (fit by sm), or generalized smooth model (fit by gsm).

Usage

# S3 method for ss
vcov(object, ...)
# S3 method for sm
vcov(object, ...)
# S3 method for gsm
vcov(object, ...)

Value

Returns the (symmetric) matrix such that cell (\(i,j\)) contains the covariance between the \(i\)-th and \(j\)-th elements of the coefficient vector.

Arguments

object

an object of class "gsm" output by the gsm function, "sm" output by the sm function, or "ss" output by the ss function

...

other arugments (currently ignored)

Author

Nathaniel E. Helwig <[e-mail]>

Details

The variance-covariance matrix is calculated using the Bayesian interpretation of a smoothing spline. Unlike the classic treatments (e.g., Wahba, 1983; Nychka, 1988), which interpret the smoothing spline as a Bayesian estimate of a Gaussian process, this treatment applies the Bayesian interpretation directly on the coefficient vector. More specifically, the smoothing spline basis function coefficients are interpreted as Bayesian estimates of the basis function coefficients (see Helwig, 2020).

References

Helwig, N. E. (2020). Multiple and Generalized Nonparametric Regression. In P. Atkinson, S. Delamont, A. Cernat, J. W. Sakshaug, & R. A. Williams (Eds.), SAGE Research Methods Foundations. tools:::Rd_expr_doi("10.4135/9781526421036885885")

Nychka, D. (1988). Bayesian confience intervals for smoothing splines. Journal of the American Statistical Association, 83(404), 1134-1143. tools:::Rd_expr_doi("10.2307/2290146")

Wahba, G. (1983). Bayesian "confidence intervals" for the cross-validated smoothing spline. Journal of the Royal Statistical Society. Series B, 45(1), 133-150. tools:::Rd_expr_doi("10.1111/j.2517-6161.1983.tb01239.x")

See Also

ss, sm, gsm for model fitting

boot.ss, boot.sm, boot.gsm for bootstrapping

Examples

## for 'ss' objects this function is defined as
  function(object, ...){
    Sigma <- tcrossprod(object$fit$cov.sqrt)
    rownames(Sigma) <- colnames(Sigma) <- names(object$fit$coef)
    Sigma
  }

## for 'sm' and 'gsm' objects this function is defined as
  function(object, ...){
    Sigma <- tcrossprod(object$cov.sqrt)
    rownames(Sigma) <- colnames(Sigma) <- names(object$coefficients)
    Sigma
  }