R: Operations to map curved 'ergm()' parameters onto canonical...

ergm.eta {ergm} | R Documentation

## Operations to map curved `ergm()` parameters onto canonical parameters

### Description

The `ergm.eta` function calculates and returns eta, mapped from theta using the `etamap` object, usually attached as the `$etamap` element of an `ergm_model` object.

The `ergm.etagrad` function caculates and returns the gradient of eta mapped from theta using the etamap object created by `ergm.etamap`. If the gradient is only intended to be a multiplier for some vector, the more efficient `ergm.etagradmult` is recommended.

The `ergm.etagradmult` function calculates and returns the product of the gradient of eta with a vector `v`.

### Usage

    ergm.eta(theta, etamap)

    ergm.etagrad(theta, etamap)

    ergm.etagradmult(theta, v, etamap)

### Arguments

`theta` | the curved model parameters
`etamap` | the list of values that describes the theta -> eta mapping, usually attached as `$etamap` element of an `ergm_model` object. At this time, it is a list with the following elements: `canonical` a numeric vector whose `i`th entry specifies whether the `i`th component of theta is canonical (via non-negative integers) or curved (via zeroes) `offsetmap` a logical vector whose `i`th entry tells whether the ith coefficient of the canonical parameterization was "offset", i.e fixed `offset` a logical vector whose ith entry tells whether the ith model term was offset/fixed `offsettheta` a logical vector whose ith entry tells whether the ith curved theta coeffient was offset/fixed; `curved` a list with one component per curved EF term in the model containing `from` the indices of the curved theta parameter that are to be mapped from `to` the indices of the canonical eta parameters to be mapped to `map` the map provided by `InitErgmTerm` `gradient` the gradient function provided by `InitErgmTerm` `cov` optional additional covariates to be passed to the map and the gradient functions `etalength` the length of the eta vector
`v` | a vector of the same length as the vector of mapped eta parameters

### Details

These functions are mainly important in the case of curved exponential family models, i.e., those in which the parameter of interest (theta) is not a linear function of the natural parameters (eta) in the exponential-family model. In non-curved models, we may assume without loss of generality that eta(theta)=theta.

A succinct description of how eta(theta) is incorporated into an ERGM is given by equation (5) of Hunter (2007). See Hunter and Handcock (2006) and Hunter (2007) for further details about how eta and its derivatives are used in the estimation process.

### Value

For `ergm.eta`, the canonical eta parameters as mapped from theta.

For `ergm.etagrad`, a matrix of the gradient of eta with respect to theta.

For `ergm.etagradmult`, the vector that is the product of the gradient of eta and `v`.

### References

- Hunter, D. R. and M. S. Handcock (2006). Inference in curved exponential family models for networks. Journal of Computational and Graphical Statistics, 15: 565–583.

- Hunter, D. R. (2007). Curved exponential family models for social networks. Social Networks, 29: 216–230.

### See Also

`ergmTerm`

[Package ergm version 4.2.1 Index]