# Snorm.hess: Hessian of the loglikelihood of a multivariate normal with... In SuperGauss: Superfast Likelihood Inference for Stationary Gaussian Time Series

## Description

Superfast evaluation of loglikelihood Hessian.

## Usage

 `1` ```Snorm.hess(X, mu, acf, dmu, dacf, d2mu, d2acf) ```

## Arguments

 `X` A length-`N` vector of multivariate normal observations. `mu` A scalar or length-`N` vector of means. If missing defaults to the vector of zeros. `acf` A `Toeplitz` object or length-`N` vector containing the first column of the Toeplitz variance matrix. `dmu` A length-`p` vector or `N x p` matrix of partial derivatives of `mu` along the columns. If missing defaults to a matrix of zeros. `dacf` An `N x p` matrix with the partial derivatives of `acf` along the columns. `d2mu` A `p x p` matrix or `N x p x p` array of second partial derivatives of `mu`. If missing defaults to zeros. `d2acf` A `N x p x p` array of second partial derivatives of `acf`.

## Value

The `p x p` Hessian matrix of the loglikelihood.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31``` ```# two parameter inference acf.fun <- function(theta) theta[2]^2 * exp(-(1:N-1)) mu.fun <- function(theta) theta[1] * (1:N) + log(theta[2] + 1:N) # partial derivatives dacf.fun <- function(theta) { cbind(0, 2*theta[2] * exp(-(1:N-1))) } dmu.fun <- function(theta) cbind(1:N, 1/(theta[2] + 1:N)) # 2nd order partials d2acf.fun <- function(theta) { H <- array(0, dim = c(N, 2, 2)) H[,2,2] <- 2*exp(-(1:N-1)) H } d2mu.fun <- function(theta) { H <- array(0, dim = c(N, 2, 2)) H[,2,2] <- -1/(theta[2] + 1:N)^2 H } # generate data N <- 300 theta <- rexp(2) X <- rSnorm(n = 1, acf = acf.fun(theta)) + mu.fun(theta) # likelihood Hessian Snorm.hess(X = X, mu = mu.fun(theta), acf = acf.fun(theta), dmu = dmu.fun(theta), dacf = dacf.fun(theta), d2mu = d2mu.fun(theta), d2acf = d2acf.fun(theta)) ```

### Example output

```            [,1]       [,2]
[1,] -33163206.8 -27915.299
[2,]    -27915.3  -5079.058
```

SuperGauss documentation built on May 1, 2019, 7:58 p.m.