MARSShessian {MARSS}

Parameter Variance-Covariance Matrix from the Hessian Matrix

Description

Calculates an approximate parameter variance-covariance matrix for the parameters using an inverse of the Hessian of the negative log-likelihood function at the MLEs (the observed Fisher Information matrix). It appends $Hessian, $parMean, $parSigma to the marssMLE object.

Usage

MARSShessian(MLEobj, method=c("Harvey1989", "fdHess", "optim"))

Arguments

MLEobj

An object of class marssMLE. This object must have a $par element containing MLE parameter estimates from e.g. MARSSkem.

method

The method to use for computing the Hessian. Options are Harvey1989 to use the Harvey (1989) recursion, which is an analytical solution, fdHess or optim which are two numerical methods. Although optim can be passed to this function, in the internal functions which call this function, fdHess will be used if a numerical estimate is requested.

Details

See MARSSFisherI for a discussion of the observed Fisher Information matrix and references.

Method fdHess uses fdHess from package nlme to numerically estimate the Hessian matrix (the matrix of partial 2nd derivatives of the negative log-likelihood function at the MLE). Method optim uses optim with hessian=TRUE and list(maxit=0) to ensure that the Hessian is computed at the values in the par element of the MLE object. Method Harvey1989 (the default) uses the recursion in Harvey (1989) to compute the observed Fisher Information of a MARSS model analytically.

Note that the parameter confidence intervals computed with the observed Fisher Information matrix are based on the asymptotic normality of maximum-likelihood estimates under a large-sample approximation.

Value

MARSShessian() attaches Hessian, parMean and parSigma to the marssMLE object that is passed into the function.

Author(s)

Eli Holmes, NOAA, Seattle, USA.

References

Harvey, A. C. (1989) Section 3.4.5 (Information matrix) in Forecasting, structural time series models and the Kalman filter. Cambridge University Press, Cambridge, UK.

See also J. E. Cavanaugh and R. H. Shumway (1996) On computing the expected Fisher information matrix for state-space model parameters. Statistics & Probability Letters 26: 347-355. This paper discusses the Harvey (1989) recursion (and proposes an alternative).

See Also

MARSSFisherI(), MARSSharveyobsFI(), MARSShessian.numerical(), MARSSparamCIs(), marssMLE

Examples

dat <- t(harborSeal)
dat <- dat[c(2, 11), ]
MLEobj <- MARSS(dat)
MLEobj.hessian <- MARSShessian(MLEobj)

# show the approx Hessian
MLEobj.hessian$Hessian

# generate a parameter sample using the Hessian
# this uses the rmvnorm function in the mvtnorm package
hess.params <- mvtnorm::rmvnorm(1,
  mean = MLEobj.hessian$parMean,
  sigma = MLEobj.hessian$parSigma
)