R: Projection of a model matrix on the orthogonal complement of...

detrend {Epi} | R Documentation

## Projection of a model matrix on the orthogonal complement of a trend or curvature.

### Description

The columns of a model matrix `M` is projected on the orthogonal complement to the matrix `(1,t)`, resp. `(1,t,t^2)`.

Orthogonality is w.r.t. an inner product defined by the positive definite matrix matrix `diag(weight)`. Non-diagonal matrices defining the inner product is not supported.

### Usage

      detrend( M, t, weight = rep(1, nrow(M)) )
      decurve( M, t, weight = rep(1, nrow(M)) )

### Arguments

`M` | A model matrix.
`t` | The trend defining a subspace. A numerical vector of length `nrow(M)`.
`weight` | Weights defining the inner product of vectors `x` and `y` as `sum(x*w*y)`. A numerical vector of length `nrow(M)`, defaults to a vector of `1`s. Must be all non-negative.

### Details

The functions are intended to be used in construction of particular parametrizations of age-period-cohort models.

### Value

`detrend` returns full-rank matrix with columns orthogonal to `(1,t)`; `decurve` returns full-rank matrix with columns orthogonal to `(1,t,t^2)`.

### Author(s)

Bendix Carstensen, Steno Diabetes Center Copenhagen, http://BendixCarstensen.com, with essential help from Peter Dalgaard.

### See Also

`projection.ip`

[Package Epi version 2.40 Index]