R: Constraint Matrices

constraints {VGAM} | R Documentation

## Constraint Matrices

### Description

Extractor function for the constraint matrices of objects in the VGAM package.

### Usage

    constraints(object, ...)
    constraints.vlm(object, type = c("lm", "term"), all = TRUE, which,
                    matrix.out = FALSE, colnames.arg = TRUE, ...)

### Arguments

`object` | Some VGAM object, for example, having class `vglmff-class`.
`type` | Character. Whether LM- or term-type constraints are to be returned. The number of such matrices returned is equal to `nvar(object, type = "lm")` and the number of terms, respectively.
`all, which` | If `all = FALSE` then `which` gives the integer index or a vector of logicals specifying the selection.
`matrix.out` | Logical. If `TRUE` then the constraint matrices are `cbind()ed` together. The result is usually more compact because the default is a list of constraint matrices.
`colnames.arg` | Logical. If `TRUE` then column names are assigned corresponding to the variables.
`...` | Other possible arguments such as `type`.

### Details

Constraint matrices describe the relationship of coefficients/component functions of a particular explanatory variable between the linear/additive predictors in VGLM/VGAM models. For example, they may be all different (constraint matrix is the identity matrix) or all the same (constraint matrix has one column and has unit values).

VGLMs and VGAMs have constraint matrices which are known. The class of RR-VGLMs have constraint matrices which are unknown and are to be estimated.

### Value

The extractor function `constraints()` returns a list comprising of constraint matrices—usually one for each column of the VLM model matrix, and in that order. The list is labelled with the variable names. Each constraint matrix has M rows, where M is the number of linear/additive predictors, and whose rank is equal to the number of columns. A model with no constraints at all has an order M identity matrix as each variable's constraint matrix.

For `vglm` and `vgam` objects, feeding in `type = "term"` constraint matrices back into the same model should work and give an identical model. The default are the `"lm"`-type constraint matrices; this is a list with one constraint matrix per column of the LM matrix. See the `constraints` argument of `vglm`, and the example below.

### Note

In all VGAM family functions `zero = NULL` means none of the linear/additive predictors are modelled as intercepts-only. Other arguments found in certain VGAM family functions which affect constraint matrices include `parallel` and `exchangeable`.

The `constraints` argument in `vglm` and `vgam` allows constraint matrices to be inputted. If so, then `constraints(fit, type = "lm")` can be fed into the `constraints` argument of the same object to get the same model.

The `xij` argument does not affect constraint matrices; rather, it allows each row of the constraint matrix to be multiplied by a specified vector.

### Author(s)

T. W. Yee

### References

Yee, T. W. and Wild, C. J. (1996) Vector generalized additive models. Journal of the Royal Statistical Society, Series B, Methodological, 58, 481–493.

Yee, T. W. and Hastie, T. J. (2003) Reduced-rank vector generalized linear models. Statistical Modelling, 3, 15–41.

### See Also

`is.parallel`, `is.zero`. VGLMs are described in `vglm-class`; RR-VGLMs are described in `rrvglm-class`.

Arguments such as `zero` and `parallel` found in many VGAM family functions are a way of creating/modifying constraint matrices conveniently, e.g., see `zero`. See `CommonVGAMffArguments` for more information.

### Examples

    # Fit the proportional odds model:
    pneumo <- transform(pneumo, let = log(exposure.time))
    (fit1 <- vglm(cbind(normal, mild, severe) ~ sm.bs(let, 3),
                  cumulative(parallel = TRUE, reverse = TRUE), data = pneumo))
    coef(fit1, matrix = TRUE)
    constraints(fit1)  # Parallel assumption results in this
    constraints(fit1, type = "term")  # Same as the default ("vlm"-type)
    is.parallel(fit1)

    # An equivalent model to fit1 (needs the type "term" constraints):
    clist.term <- constraints(fit1, type = "term")  # "term"-type constraints
    (fit2 <- vglm(cbind(normal, mild, severe) ~ sm.bs(let, 3), data = pneumo,
                  cumulative(reverse = TRUE), constraints = clist.term))
    abs(max(coef(fit1, matrix = TRUE) -
            coef(fit2, matrix = TRUE)))  # Should be zero

    # Fit a rank-1 stereotype (RR-multinomial logit) model:
    fit <- rrvglm(Country ~ Width + Height + HP, multinomial, data = car.all)
    constraints(fit)  # All except the first are the estimated A matrix

[Package VGAM version 1.0-3 Index]