# Methods and functions to support normalizing to marginal variance of one.

### Description

The basis functions can be normalized so that the
marginal variance of the process at each level and at all locations is
one. A generic function `LKrigNormalizeBasis`

computes this for
any LatticeKrig model. However, in special cases the normalization can
be accelerated taking advantage of the geometry and the model for the
lattice weights. This alternative is the method `LKrigNormalizeBasisFast`

.

### Usage

1 2 3 4 5 6 7 | ```
LKrigNormalizeBasis( LKinfo, Level, PHI, ...)
LKrigNormalizeBasisFast(LKinfo, ...)
## Default S3 method:
LKrigNormalizeBasisFast(LKinfo, ...)
## S3 method for class 'LKRectangle'
LKrigNormalizeBasisFast(LKinfo, Level, x, ...)
LKRectangleSetupNormalization(mx, a.wght)
``` |

### Arguments

`a.wght` |
A.wght parameters. |

`LKinfo` |
An LKinfo object. NOTE: Here |

`Level` |
The multiresolution level. |

`mx` |
Matrix of lattice sizes. |

`PHI` |
Unnormalized basis functions evaluated at the locations to find the normalization weights. |

`x` |
Locations to find normalization weights. |

`...` |
Additional arguments for method. |

### Details

Normalization to unit variance is useful for reducing the artifacts of the lattice points and creates a model that is closer to being stationary. The computation must be done for every point evaluated with the basis functions The basic calculation is

1 2 3 4 | ```
tempB <- LKrigSAR(LKinfo, Level = Level)
tempB <- LKrig.spind2spam(tempB)
wght <- LKrig.quadraticform(t(tempB)
choleskyMemory = choleskyMemory)
``` |

This generic method uses the low level utility `LKrig.quadraticform`

that evaluates diag( t(PHI) solve( Q.l) PHI ) where PHI are the basis functions evaluated at the locations and Q.l is the precision matrix for the lattice coefficients at level l.

For constant a.wght and for the rectangular geometry one can use an eigendecomposition of the Kronecker product of the SAR matrix.

### Value

A vector of weights. The basis are normalized by dividing by the square root of the weights.

### Author(s)

Doug Nychka