# 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).

`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

```
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 1.41-1, License: GPL (>= 2)*