R: Maximum likelihood estimates

mle {spam} | R Documentation

## Maximum likelihood estimates

### Description

Maximum likelihood estimates of a simple spatial model

### Usage

    neg2loglikelihood.spam(y, X, distmat, Covariance,
                     beta, theta, Rstruct = NULL, ...)

    neg2loglikelihood(y, X, distmat, Covariance,
                     beta, theta, ...)

    mle.spam(y, X, distmat, Covariance,
         beta0, theta0,
         thetalower, thetaupper, optim.control=NULL,
         Rstruct = NULL, hessian = FALSE,...)

    mle(y, X, distmat, Covariance,
         beta0, theta0,
         thetalower, thetaupper, optim.control=NULL,
         hessian = FALSE, ...)

    mle.nomean.spam(y, distmat, Covariance,
         theta0,
         thetalower, thetaupper, optim.control=NULL,
         Rstruct = NULL, hessian = FALSE, ...)

    mle.nomean(y, distmat, Covariance,
         theta0,
         thetalower, thetaupper, optim.control=NULL,
         hessian = FALSE, ...)

### Arguments

`y` | data vector of length n.
`X` | the design matrix of dimension n x p.
`distmat` | a distance matrix. Usually the result of a call to `nearest.dist`.
`Covariance` | function defining the covariance. See example.
`beta` | parameters of the trend (fixed effects).
`theta` | parameters of the covariance structure.
`Rstruct` | the Cholesky structure of the covariance matrix.
`beta0,theta0` | inital values.
`thetalower,thetaupper` | lower and upper bounds of the parameter `theta`.
`optim.control` | arguments passed to `optim`.
`hessian` | Logical. Should a numerically differentiated Hessian matrix be returned?
`...` | additional arguments passed to `chol`.

### Details

We provide functions to calculate the negative-2-log-likelihood and maximum likelihood estimates for the model

y ~ N_n( X beta, Sigma(h;theta) )

in the case of a sparse or ordinary covariance matrices.

In the case of the `*.spam` versions, the covariance function has to return a `spam` object. In the other case, the methods are correctly overloaded and work either way, slightly slower than the `*.spam` counterparts though.

When working on the sphere, the distance matrix has to be transformed by

h -> R / 2 sin(h/2)

where R is the radius of the sphere.

The covariance function requires that the first argument is the distance matrix and the second the parameters. One can image cases in which the covariance function does not take the entire distance matrix but only some partial information thereof. (An example is the use of a kronecker type covariance structure.) In case of a sparse covariance construction where the argument `Rstruct` is not given, the first parameter element needs to be the range parameter. (This results from the fact, that a sparse structure is constructed that is independent of the parameter values to exploit the fast Choleski decomposition.)

In the zero-mean case, the `neg2loglikelihood` is calculated by setting the parameters `X` or `beta` to zero.

### Value

The negative-2-loglikelihood or the output from the function `optim`.

### Author(s)

Reinhard Furrer

### See Also

`covmat`

### Examples

    # True parameter values:
    truebeta <- c(1,2,.2)    # beta = (intercept, linear in x, linear in y)
    truetheta <- c(.5,2,.02) # theta = (range, sill, nugget)

    # We now define a grid, distance matrix, and a sample:
    x <- seq(0,1,l=5)
    locs <- expand.grid( x, x)
    X <- as.matrix( cbind(1,locs))  # design matrix

    distmat <- nearest.dist( locs, upper=NULL) # distance matrix
    Sigma <- cov.sph( distmat, truetheta)    # true covariance matrix

    set.seed(15)
    y <- c(rmvnorm.spam(1,X %*% truebeta,Sigma)) # construct sample

    # Here is the negative 2 log likelihood:
    neg2loglikelihood.spam( y, X, distmat, cov.sph,
                           truebeta, truetheta)

    # We pass now to the mle:
    res <- mle.spam(y, X, distmat, cov.sph,
             truebeta, truetheta,thetalower=c(0,0,0),thetaupper=c(1,Inf,Inf))

    # Similar parameter estimates here, of course:
    mle.nomean.spam(y-X%*%res$par[1:3], distmat, cov.sph,
             truetheta, thetalower=c(0,0,0), thetaupper=c(1,Inf,Inf))

[Package spam version 2.2-2 Index]