# Fiksel

##### The Fiksel Interaction

Creates an instance of Fiksel's double exponential pairwise interaction point process model, which can then be fitted to point pattern data.

##### Usage

`Fiksel(r, hc=NA, kappa)`

##### Arguments

- r
The interaction radius of the Fiksel model

- hc
The hard core distance

- kappa
The rate parameter

##### Details

Fiksel (1984) introduced a pairwise interaction point process with the following interaction function \(c\). For two points \(u\) and \(v\) separated by a distance \(d=||u-v||\), the interaction \(c(u,v)\) is equal to \(0\) if \(d < h\), equal to \(1\) if \(d > r\), and equal to $$ \exp(a \exp(-\kappa d))$$ if \(h \le d \le r\), where \(h,r,\kappa,a\) are parameters.

A graph of this interaction function is shown in the Examples. The interpretation of the parameters is as follows.

\(h\) is the hard core distance: distinct points are not permitted to come closer than a distance \(h\) apart.

\(r\) is the interaction range: points further than this distance do not interact.

\(\kappa\) is the rate or slope parameter, controlling the decay of the interaction as distance increases.

\(a\) is the interaction strength parameter, controlling the strength and type of interaction. If \(a\) is zero, the process is Poisson. If

`a`

is positive, the process is clustered. If`a`

is negative, the process is inhibited (regular).

The function `ppm()`

, which fits point process models to
point pattern data, requires an argument
of class `"interact"`

describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the Fiksel
pairwise interaction is
yielded by the function `Fiksel()`

. See the examples below.

The parameters \(h\), \(r\) and \(\kappa\) must be
fixed and given in the call to `Fiksel`

, while the canonical
parameter \(a\) is estimated by `ppm()`

.

To estimate \(h\), \(r\) and\(\kappa\)
it is possible to use `profilepl`

. The maximum likelihood
estimator of\(h\) is the minimum interpoint distance.

If the hard core distance argument `hc`

is missing or `NA`

,
it will be estimated from the data when `ppm`

is called.
The estimated value of `hc`

is the minimum nearest neighbour distance
multiplied by \(n/(n+1)\), where \(n\) is the
number of data points.

See also Stoyan, Kendall and Mecke (1987) page 161.

##### Value

An object of class `"interact"`

describing the interpoint interaction
structure of the Fiksel
process with interaction radius \(r\),
hard core distance `hc`

and
rate parameter `kappa`

.

##### References

Baddeley, A. and Turner, R. (2000)
Practical maximum pseudolikelihood for spatial point patterns.
*Australian and New Zealand Journal of Statistics*
**42**, 283--322.

Fiksel, T. (1984)
Estimation of parameterized pair potentials
of marked and non-marked Gibbsian point processes.
*Electronische Informationsverabeitung und Kybernetika*
**20**, 270--278.

Stoyan, D, Kendall, W.S. and Mecke, J. (1987)
*Stochastic geometry and its applications*. Wiley.

##### See Also

##### Examples

```
# NOT RUN {
Fiksel(r=1,hc=0.02, kappa=2)
# prints a sensible description of itself
data(spruces)
X <- unmark(spruces)
fit <- ppm(X ~ 1, Fiksel(r=3.5, kappa=1))
plot(fitin(fit))
# }
```

*Documentation reproduced from package spatstat, version 1.59-0, License: GPL (>= 2)*