R: Parameter restriction families

mrf2d-family {mrf2d}

R Documentation

Parameter restriction families

Description

Different parameter restrictions can be included in estimation processes to make sure mrf2d can successfully include a wide range of model types in its inference functions.

For model identifiability, at least one linear restriction is necessary. mrf2d always assume \theta_{0,0,r} = 0 for all relative positions r.

Additionally, each family of restrictions may introduce other restrictions:

'onepar'

This family assumes the model is defined by a single parameter by adding the restriction

\theta_{a,b,r} = \phi * 1(a != b).

Here 1() denotes de indicator function. In words, the parameter must be the same value for any pair with different values and 0 for any equal-valued pair.

'oneeach'

Similar to 'onepar', parameters are 0 for equal-valued pairs and a constant for pairs with different values, but the constant may differ between different relative positions r:

\theta{a,b,r} = \phi_r * 1(a != b).

'absdif'

All parameters \theta_{a,b,r} with the same absolute difference between a and b must be equal within each relative position r. (Note that 'absdif' is equal to 'oneeach' for binary images).

\theta_{a,b,r} = \sum_d \phi_{d,r} * 1(|a-b| == d)

'dif'

The same as 'absdif', but parameters may differ between positive and negative differences.

\theta_{a,b,r} = \sum_d \phi_{d,r} * 1(a-b == d)

'free'

No additional restriction, all parameters other than \theta_{0,0,r} vary freely.

Author(s)

Victor Freguglia