Returns the variance-covariance matrix of the estimates of the parameters of a fitted point process model.
# S3 method for ppm
vcov(object, …, what = "vcov", verbose = TRUE,
fine=FALSE,
gam.action=c("warn", "fatal", "silent"),
matrix.action=c("warn", "fatal", "silent"),
logi.action=c("warn", "fatal", "silent"),
nacoef.action=c("warn", "fatal", "silent"),
hessian=FALSE)A fitted point process model (an object of class "ppm".)
Ignored.
Character string (partially-matched)
that specifies what matrix is returned.
Options are "vcov" for the variance-covariance matrix,
"corr" for the correlation matrix, and
"fisher" or "Fisher"
for the Fisher information matrix.
Logical value indicating whether to use a quick estimate
(fine=FALSE, the default) or a slower, more accurate
estimate (fine=TRUE).
Logical. If TRUE, a message will be printed
if various minor problems are encountered.
String indicating what to do if object was
fitted by gam.
String indicating what to do if the matrix is ill-conditioned (so that its inverse cannot be calculated).
String indicating what to do if object was
fitted via the logistic regression approximation using a
non-standard dummy point process.
String indicating what to do if some of the fitted coefficients
are NA (so that variance cannot be calculated).
Logical. Use the negative Hessian matrix of the log pseudolikelihood instead of the Fisher information.
A square matrix.
An error message that reports system is computationally singular indicates that the determinant of the Fisher information matrix was either too large or too small for reliable numerical calculation.
If this message occurs, try repeating the calculation
using fine=TRUE.
Singularity can occur because of numerical overflow or collinearity in the covariates. To check this, rescale the coordinates of the data points and refit the model. See the Examples.
In a Gibbs model, a singular matrix may also occur if the fitted model is a hard core process: this is a feature of the variance estimator.
This function computes the asymptotic variance-covariance
matrix of the estimates of the canonical parameters in the
point process model object. It is a method for the
generic function vcov.
object should be an object of class "ppm", typically
produced by ppm.
The canonical parameters of the fitted model object
are the quantities returned by coef.ppm(object).
The function vcov calculates the variance-covariance matrix
for these parameters.
The argument what provides three options:
what="vcov"return the variance-covariance matrix of the parameter estimates
what="corr"return the correlation matrix of the parameter estimates
what="fisher"return the observed Fisher information matrix.
In all three cases, the result is a square matrix.
The rows and columns of the matrix correspond to the canonical
parameters given by coef.ppm(object). The row and column
names of the matrix are also identical to the names in
coef.ppm(object).
For models fitted by the Berman-Turner approximation (Berman and Turner, 1992;
Baddeley and Turner, 2000) to the maximum pseudolikelihood (using the
default method="mpl" in the call to ppm), the implementation works
as follows.
If the fitted model object is a Poisson process,
the calculations are based on standard asymptotic theory for the maximum
likelihood estimator (Kutoyants, 1998).
The observed Fisher information matrix of the fitted model
object is first computed, by
summing over the Berman-Turner quadrature points in the fitted model.
The asymptotic variance-covariance matrix is calculated as the
inverse of the
observed Fisher information. The correlation matrix is then obtained
by normalising.
If the fitted model is not a Poisson process (i.e. it is some other Gibbs point process) then the calculations are based on Coeurjolly and Rubak (2012). A consistent estimator of the variance-covariance matrix is computed by summing terms over all pairs of data points. If required, the Fisher information is calculated as the inverse of the variance-covariance matrix.
For models fitted by the Huang-Ogata method (method="ho" in
the call to ppm), the implementation uses the
Monte Carlo estimate of the Fisher information matrix that was
computed when the original model was fitted.
For models fitted by the logistic regression approximation to the
maximum pseudolikelihood (method="logi" in the call to
ppm), calculations are based on (Baddeley et al.,
2013). A consistent estimator of the variance-covariance matrix is
computed by summing terms over all pairs of data points. If required,
the Fisher information is calculated as the inverse of the
variance-covariance matrix. In this case the calculations depend on
the type of dummy pattern used, and currently only the types
"stratrand", "binomial" and "poisson" as
generated by quadscheme.logi are implemented. For other
types the behavior depends on the argument logi.action. If
logi.action="fatal" an error is produced. Otherwise, for types
"grid" and "transgrid" the formulas for
"stratrand" are used which in many cases should be
conservative. For an arbitrary user specified dummy pattern (type
"given") the formulas for "poisson" are used which in
many cases should be conservative. If logi.action="warn" a
warning is issued otherwise the calculation proceeds without a
warning.
The argument verbose makes it possible to suppress some
diagnostic messages.
The asymptotic theory is not correct if the model was fitted using
gam (by calling ppm with use.gam=TRUE).
The argument gam.action determines what to do in this case.
If gam.action="fatal", an error is generated.
If gam.action="warn", a warning is issued and the calculation
proceeds using the incorrect theory for the parametric case, which is
probably a reasonable approximation in many applications.
If gam.action="silent", the calculation proceeds without a
warning.
If hessian=TRUE then the negative Hessian (second derivative)
matrix of the log pseudolikelihood, and its inverse, will be computed.
For non-Poisson models, this is not a valid estimate of variance,
but is useful for other calculations.
Note that standard errors and 95% confidence intervals for
the coefficients can also be obtained using
confint(object) or coef(summary(object)).
Baddeley, A., Coeurjolly, J.-F., Rubak, E. and Waagepetersen, R. (2014) Logistic regression for spatial Gibbs point processes. Biometrika 101 (2) 377--392.
Coeurjolly, J.-F. and Rubak, E. (2013) Fast covariance estimation for innovations computed from a spatial Gibbs point process. Scandinavian Journal of Statistics 40 669--684.
Kutoyants, Y.A. (1998) Statistical Inference for Spatial Poisson Processes, Lecture Notes in Statistics 134. New York: Springer 1998.
vcov for the generic,
ppm for information about fitted models,
confint for confidence intervals.
# NOT RUN {
X <- rpoispp(42)
fit <- ppm(X, ~ x + y)
vcov(fit)
vcov(fit, what="Fish")
# example of singular system
m <- ppm(demopat ~polynom(x,y,2))
# }
# NOT RUN {
try(v <- vcov(m))
# }
# NOT RUN {
# rescale x, y coordinates to range [0,1] x [0,1] approximately
demopatScale <- rescale(demopat, 10000)
m <- ppm(demopatScale ~ polynom(x,y,2))
v <- vcov(m)
# Gibbs example
fitS <- ppm(swedishpines ~1, Strauss(9))
coef(fitS)
sqrt(diag(vcov(fitS)))
# }
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