Description Usage Arguments Details Value References See Also
Fit a centered autologistic model using maximum pseudolikelihood estimation or MCMC for Bayesian inference.
1 2 |
formula |
an object of class |
data |
an optional data frame, list, or environment (or object coercible by |
A |
the adjacency matrix for the underlying graph. The matrix need not be binary, but it must be numeric and symmetric. |
method |
the method to use for inference. “ |
model |
a logical value indicating whether the model frame should be included as a component of the returned value. |
x |
a logical value indicating whether the model matrix used in the fitting process should be returned as a component of the returned value. |
y |
a logical value indicating whether the response vector used in the fitting process should be returned as a component of the returned value. |
verbose |
a logical value indicating whether to print various messages to the screen, including progress updates when |
control |
a list of the following control parameters.
|
This function fits the centered autologistic model of Caragea and Kaiser (2009) using maximum pseudolikelihood estimation or MCMC for Bayesian inference. The joint distribution for the centered autologistic model is
π(Z | θ)=c(θ)^{-1} exp(Z'Xβ - η Z'Aμ + 0.5 η Z'AZ),
where θ = (β', η)' is the parameter vector, c(θ) is an intractable normalizing function, Z is the response vector, X is the design matrix,
β is a (p-1)-vector of regression coefficients, A is the adjacency matrix for the underlying graph, μ is the vector of independence expectations,
and η is the spatial dependence parameter.
Maximum pseudolikelihood estimation sidesteps the intractability of c(θ) by maximizing the product of the conditional likelihoods.
Confidence intervals are then obtained using a parametric bootstrap or a normal approximation, i.e., sandwich estimation. The bootstrap datasets are generated by perfect sampling (rautologistic
).
The bootstrap samples can be generated in parallel using the snow package.
Bayesian inference is obtained using the auxiliary variable algorithm of Moller et al. (2006).
The auxiliary variables are generated by perfect sampling.
The prior distributions are (1) zero-mean normal with independent coordinates for β, and (2) uniform for η.
The common standard deviation for the normal prior can be supplied by the user as control parameter sigma
. The default is 1,000. The uniform prior has support [0, 2] by default, but the right endpoint can be supplied (as control parameter eta.max
) by the user.
The posterior covariance matrix of θ is estimated using samples obtained during a training run. The default number of iterations for the training run is 100,000, but this can be controlled by the user (via parameter trainit
). The estimated covariance matrix is then used as the proposal variance for a Metropolis-Hastings random walk. The proposal distribution is normal. The posterior samples obtained during the second run are used for inference. The length of the run can be controlled by the user via parameters minit
, maxit
, and tol
. The first determines the minimum number of iterations. If minit
has been reached, the sampler will terminate when maxit
is reached or all Monte Carlo standard errors are smaller than tol
, whichever happens first. The default values for minit
, maxit
, and tol
are 100,000; 1,000,000; and 0.01, respectively.
autologistic
returns an object of class “autologistic
”, which is a list containing the following components.
coefficients |
the point estimate of θ. |
fitted.values |
the fitted mean values, obtained by transforming the linear predictors by the inverse of the link function. |
linear.predictors |
the linear fit on link scale. |
residuals |
the response residuals. |
iter |
the size of the bootstrap/posterior sample. |
sample |
(where relevant) an |
mcse |
(where relevant) a p-vector of Monte Carlo standard errors. |
S |
(where relevant) the estimated sandwich matrix. |
accept |
(Bayes) the acceptance rate for the MCMC sampler. |
y |
if requested (the default), the |
X |
if requested, the model matrix. |
model |
if requested (the default), the model frame. |
call |
the matched call. |
formula |
the formula supplied. |
method |
the method used for inference. |
convergence |
the integer code returned by |
message |
a character string to go along with |
terms |
the |
data |
the |
xlevels |
(where relevant) a record of the levels of the factors used in fitting. |
control |
a list containing the names and values of the control parameters. |
value |
the value of the negative log pseudolikelihood at its minimum. |
Caragea, P. and Kaiser, M. (2009) Autologistic models with interpretable parameters. Journal of Agricultural, Biological, and Environmental Statistics, 14(3), 281–300.
Hughes, J., Haran, M. and Caragea, P. C. (2011) Autologistic models for binary data on a lattice. Environmetrics, 22(7), 857–871.
Moller, J., Pettitt, A., Berthelsen, K., and Reeves, R. (2006) An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants. Biometrika, 93(2), 451–458.
rautologistic
, residuals.autologistic
, summary.autologistic
, vcov.autologistic
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