fields-stuff {fields} R Documentation

## Fields supporting functions

### Description

Some supporting functions that are internal to fields top level methods. Variants of these might be found in the R base but these have been written for cleaner code or efficiency.

### Usage

```fields.diagonalize2(A,B, verbose=FALSE)
fields.diagonalize(A,B)
fields.duplicated.matrix(mat, digits = 8)

fields.mkpoly(x, m = 2)

fields.derivative.poly(x, m,dcoef)

fields.evlpoly( x, coef)

fields.evlpoly2( x, coef, ptab)

```

### Arguments

 `A` A positive definite matrix `B` A positive definite matrix `mat` Arbitrary matrix for examining rows `digits` Number of significant digits to use for comparing elements to determine duplciate values. `x` Arbitrary matrix where rows are components of a multidimensional vector `m` The null space degree – results in a polynomial of degree (m-1) `dcoef` Coefficients of a multidimensional polynomial `coef` Polynomial coefficients. `ptab` Table of powers of different polnomial terms. `verbose` If TRUE prints condition number of A+B

### Details

Given two matrices A (positive definite) and B (nonnegative definite) `fields.diagonalize` and `fields.diagonalize2` finds the matrix transformation G that will convert A to a identity matrix and B to a diagonal matrix:

G\^T A G= I G\^T B G= D.

`fields.diagonalize2` is not as easy to follow as `fields.diagonalize` but may be more stable and is the version used in the Krig engine.

`fields.duplicated` finds duplicate rows in a matrix. The digits arguments is the number of digits that are considered in the comparison. The returned value is an array of integers from 1:M where M is the number of unique rows and duplicate rows are referenced in the same order that they appear as the rows of `mat`.

`fields.mkpoly` computes the complete matrix of all monomial terms up to degree (m-1). Each row of `x` is are the componets of a vector. (The fields function mkpoly returns the number of these terms.) In 2 dimensions with m=3 there 6 polynomial terms up to quadratic ( 3-1 =2) order and will be returned as the matrix:

cbind( 1 , x[,1], x[,2], x[,1]**2, x[,1]*x[,2], x[,2]**2 )

This function is used for the fixed effects polynomial or spatial drift used in spatial estimating functions Krig, Tps and mKrig. The matrix ptab is a table of the powers in each term for each variable and is included as an attribute to the matrix returned by this function. See the `attr` function for extracting an attribute from an object.

`ptab` for the example above is

```    [,1] [,2]
[1,]    0    0
[2,]    1    0
[3,]    0    1
[4,]    2    0
[5,]    1    1
[6,]    0    2
```

This information is used in finding derivatives of the polynomial.

`fields.deriviative.poly` finds the partial derivative matrix of a multidimensional polynomial of degree (m-1) at different vector values and with coefficients `dcoef`. This function has been orgainzed to be a clean utility for the predicting the derivative of the estimated function from Krig or mKrig. Within the fields context the polynomial itself would be evaluated as fields.mkpoly( x,m)%*%dcoef. If x has d columns ( also the dimension of the polynomial) and n rows the partial derivatives of this polynomial at the locations x can be organized in a nXd matrix. This is the object returned by ths function.

`evlpoly` and `evlpoly2` are FORTRAN based functions for evaluating univariate polynomials and multivariate polynomials. The table of powers (ptab) needed for evlpoly2 is the same format as that returned my the fields.mkpoly function.

Doug Nychka