# vcov.ppm

##### 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.

##### Usage

```
## S3 method for class 'ppm':
vcov(object, \dots, what = "vcov", verbose = TRUE,
gam.action=c("warn", "fatal", "silent"),
matrix.action=c("warn", "fatal", "silent"),
logi.action=c("warn", "fatal", "silent"),
hessian=FALSE)
```

##### Arguments

- object
- A fitted point process model (an object of class
`"ppm"`

.) - ...
- Ignored.
- what
- 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"`

- verbose
- Logical. If
`TRUE`

, a message will be printed if various minor problems are encountered. - gam.action
- String indicating what to do if
`object`

was fitted by`gam`

. - matrix.action
- String indicating what to do if the matrix is ill-conditioned (so that its inverse cannot be calculated).
- logi.action
- String indicating what to do if
`object`

was fitted via the logistic regression approximation using a non-standard dummy point process. - hessian
- Logical. Use the negative Hessian matrix of the log pseudolikelihood instead of the Fisher information.

##### Details

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:
[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
`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))`

.

##### Value

- A square matrix.

##### Error messages

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.

##### References

Baddeley, A., Coeurjolly, J.-F., Rubak, E. and Waagepetersen, R. (2013)
*A logistic regression estimating function
for spatial Gibbs point processes.*
Research Report, Centre for Stochastic Geometry and Bioimaging,
Denmark.

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,
Denmark.

Kutoyants, Y.A. (1998)
**Statistical Inference for Spatial Poisson Processes**,
Lecture Notes in Statistics 134.
New York: Springer 1998.

##### See Also

`vcov`

for the generic,

`ppm`

for information about fitted models,
`confint`

for confidence intervals.

##### Examples

```
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)))
```

*Documentation reproduced from package spatstat, version 1.37-0, License: GPL (>= 2)*