MVgauss {RTMB}

Multivariate Gaussian densities

Description

Multivariate Gaussian densities

Usage

dmvnorm(x, mu = 0, Sigma, log = FALSE, scale = 1)

dgmrf(x, mu = 0, Q, log = FALSE, scale = 1)

dautoreg(x, mu = 0, phi, log = FALSE, scale = 1)

dseparable(...)

unstructured(k)

Arguments

x

Density evaluation point

mu

Mean parameter vector

Sigma

Covariance matrix

log

Logical; Return log density?

scale

Extra scale parameter - see section 'Scaling'.

Q

Sparse precision matrix

phi

Autoregressive parameters

...

Log densities

k

Dimension

Details

Multivariate normal density evaluation is done using dmvnorm(). This is meant for dense covariance matrices. If many evaluations are needed for the same covariance matrix please note that you can pass matrix arguments: When x is a matrix the density is applied to each row of x and the return value will be a vector (length = nrow(x)) of densities.

The function dgmrf() is essentially identical to dmvnorm() with the only difference that dgmrf() is specified via the precision matrix (inverse covariance) assuming that this matrix is sparse.

Autoregressive density evaluation is implemented for all orders via dautoreg() (including the simplest AR1). We note that this variant is for a stationary, mean zero and variance one process. FIXME: Provide parameterization via partial correlations.

Separable extension can be constructed for an unlimited number of inputs. Each input must be a function returning a gaussian mean zero log density. The output of dseparable is another log density which can be evaluated for array arguments. For example dseparable(f1,f2,f3) takes as input a 3D array x. f1 acts in 1st array dimension of x, f2 in 2nd dimension and so on. In addition to x, parameters mu and scale can be supplied - see below.

Value

Vector of densities.

Functions

• dmvnorm(): Multivariate normal distribution. OSA-residuals can be used for argument x.

• dgmrf(): Multivariate normal distribution. OSA is not implemented.

• dautoreg(): Gaussian stationary mean zero AR(k) density

• dseparable(): Separable extension of Gaussian log-densities

• unstructured(): Helper to generate an unstructured correlation matrix to use with dmvnorm

Scaling

All the densities accept a scale argument which replaces SCALE and VECSCALE functionality of TMB. Scaling is applied elementwise on the residual x-mu. This works as expected when scale is a scalar or a vector object of the same length as x. In addition, dmvnorm and dgmrf can be scaled by a vector of length equal to the covariance/precision dimension. In this case the scale parameter is recycled by row to meet the special row-wise vectorization of these densities.

Unstructured correlation

Replacement of UNSTRUCTURED_CORR functionality of TMB. Constuct object using us <- unstructured(k). Now us has two methods: x <- us$parms() gives the parameter vector used as input to the objective function, and us$corr(x) turns the parameter vector into an unstructured correlation matrix.

Examples

func <- function(x, sd, parm, phi) {
   ## IID N(0, sd^2)
   f1 <- function(x)sum(dnorm(x, sd=sd, log=TRUE))
   Sigma <- diag(2) + parm
   ## MVNORM(0, Sigma)
   f2 <- function(x)dmvnorm(x, Sigma=Sigma, log=TRUE)
   ## AR(2) process
   f3 <- function(x)dautoreg(x, phi=phi, log=TRUE)
   ## Separable extension (implicit log=TRUE)
   -dseparable(f1, f2, f3)(x)
}
parameters <- list(x = array(0, c(10, 2, 10)), sd=2, parm=1, phi=c(.9, -.2))
obj <- MakeADFun(function(p)do.call(func, p), parameters, random="x")
## Check that density integrates to 1
obj$fn()
## Check that integral is independent of the outer parameters
obj$gr()
## Check that we can simulate from this density
s <- obj$simulate()