Variance-Covariance Matrix for a Fitted Point Process Model
Returns the variance-covariance matrix of the estimates of the parameters of a fitted point process model.
## S3 method for class 'ppm': vcov(object, \dots, what = "vcov", verbose = TRUE, gam.action=c("warn", "fatal", "silent"), matrix.action=c("warn", "fatal", "silent"), hessian=FALSE)
- A fitted point process model (an object of class
- Character string (partially-matched)
that specifies what matrix is returned.
"vcov"for the variance-covariance matrix,
"corr"for the correlation matrix, and
- Logical. If
TRUE, a message will be printed if various minor problems are encountered.
- String indicating what to do if
objectwas fitted by
- String indicating what to do if the matrix is ill-conditioned (so that its inverse cannot be calculated).
- Logical. Use the negative Hessian matrix of the log pseudolikelihood instead of the Fisher information.
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
object should be an object of class
The canonical parameters of the fitted model
are the quantities returned by
vcov calculates the variance-covariance matrix
for these parameters.
what provides three options:
[object Object],[object Object],[object Object]
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
For models fitted by maximum pseudolikelihood (which is the
ppm), the implementation works
- If the fitted model
objectis 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
objectis 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 (
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.
verbose makes it possible to suppress some
The asymptotic theory is not correct if the model was fitted using
gam (by calling
gamaction determines what to do in this case.
gamaction="fatal", an error is generated.
gamaction="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.
gamaction="silent", the calculation proceeds without a
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
- 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. This 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.
Coeurjolly, J.-F. and Rubak, E. (2012)
Fast covariance estimation for innovations
computed from a spatial Gibbs point process.
Research Report, Centre for Stochastic Geometry and Bioimaging,
Kutoyants, Y.A. (1998) Statistical Inference for Spatial Poisson Processes, Lecture Notes in Statistics 134. New York: Springer 1998.
vcov for the generic,
X <- rpoispp(42) fit <- ppm(X, ~ x + y) vcov(fit) vcov(fit, what="Fish") # example of singular system data(demopat) m <- ppm(demopat, ~polynom(x,y,2)) try(v <- vcov(m)) # rescale x, y coordinates to range [0,1] x [0,1] approximately demopat <- rescale(demopat, 10000) m <- ppm(demopat, ~polynom(x,y,2)) v <- vcov(m) # Gibbs example fitS <- ppm(swedishpines, ~1, Strauss(9)) coef(fitS) sqrt(diag(vcov(fitS)))