pdNatural {nlme} R Documentation

## General Positive-Definite Matrix in Natural Parametrization

### Description

This function is a constructor for the `pdNatural` class, representing a general positive-definite matrix, using a natural parametrization . If the matrix associated with `object` is of dimension n, it is represented by n*(n+1)/2 parameters. Letting S(i,j) denote the ij-th element of the underlying positive definite matrix and r(i,j) = S(i,j)/sqrt(S(i,i)S(j,j)), i not equal to j denote the associated "correlations", the "natural" parameters are given by sqrt(Sii), i=1,..,n and log((1+r(i,j))/(1-r(i,j))), i not equal to j. Note that all natural parameters are individually unrestricted, but not jointly unrestricted (meaning that not all unrestricted vectors would give positive-definite matrices). Therefore, this parametrization should NOT be used for optimization. It is mostly used for deriving approximate confidence intervals on parameters following the optimization of an objective function. When `value` is `numeric(0)`, an uninitialized `pdMat` object, a one-sided formula, or a vector of character strings, `object` is returned as an uninitialized `pdSymm` object (with just some of its attributes and its class defined) and needs to have its coefficients assigned later, generally using the `coef` or `matrix` replacement functions. If `value` is an initialized `pdMat` object, `object` will be constructed from `as.matrix(value)`. Finally, if `value` is a numeric vector, it is assumed to represent the natural parameters of the underlying positive-definite matrix.

### Usage

```pdNatural(value, form, nam, data)
```

### Arguments

 `value` an optional initialization value, which can be any of the following: a `pdMat` object, a positive-definite matrix, a one-sided linear formula (with variables separated by `+`), a vector of character strings, or a numeric vector. Defaults to `numeric(0)`, corresponding to an uninitialized object. `form` an optional one-sided linear formula specifying the row/column names for the matrix represented by `object`. Because factors may be present in `form`, the formula needs to be evaluated on a data.frame to resolve the names it defines. This argument is ignored when `value` is a one-sided formula. Defaults to `NULL`. `nam` an optional vector of character strings specifying the row/column names for the matrix represented by object. It must have length equal to the dimension of the underlying positive-definite matrix and unreplicated elements. This argument is ignored when `value` is a vector of character strings. Defaults to `NULL`. `data` an optional data frame in which to evaluate the variables named in `value` and `form`. It is used to obtain the levels for `factors`, which affect the dimensions and the row/column names of the underlying matrix. If `NULL`, no attempt is made to obtain information on `factors` appearing in the formulas. Defaults to the parent frame from which the function was called.

### Value

a `pdNatural` object representing a general positive-definite matrix in natural parametrization, also inheriting from class `pdMat`.

### Author(s)

Jose Pinheiro Jose.Pinheiro@pharma.novartis.com and Douglas Bates bates@stat.wisc.edu

`as.matrix.pdMat`, `coef.pdMat`, `matrix<-.pdMat`
```pdNatural(diag(1:3))