Creates the Lennard-Jones pairwise interaction structure which can then be fitted to point pattern data.

`LennardJones(sigma0=NA)`

sigma0

Optional. Initial estimate of the parameter \(\sigma\). A positive number.

An object of class `"interact"`

describing the Lennard-Jones interpoint interaction
structure.

To avoid numerical instability,
the interpoint distances `d`

are rescaled
when fitting the model.

Distances are rescaled by dividing by `sigma0`

.
In the formula for \(v(d)\) above,
the interpoint distance \(d\) will be replaced by `d/sigma0`

.

The rescaling happens automatically by default.
If the argument `sigma0`

is missing or `NA`

(the default),
then `sigma0`

is taken to be the minimum
nearest-neighbour distance in the data point pattern (in the
call to `ppm`

).

If the argument `sigma0`

is given, it should be a positive
number, and it should be a rough estimate of the
parameter \(\sigma\).

The ``canonical regular parameters'' estimated by `ppm`

are
\(\theta_1 = 4 \epsilon (\sigma/\sigma_0)^{12}\)
and
\(\theta_2 = 4 \epsilon (\sigma/\sigma_0)^6\).

Fitting the Lennard-Jones model is extremely unstable, because
of the strong dependence between the functions \(d^{-12}\)
and \(d^{-6}\). The fitting algorithm often fails to
converge. Try increasing the number of
iterations of the GLM fitting algorithm, by setting
`gcontrol=list(maxit=1e3)`

in the call to `ppm`

.

Errors are likely to occur if this model is fitted to a point pattern dataset
which does not exhibit both short-range inhibition and
medium-range attraction between points. The values of the parameters
\(\sigma\) and \(\epsilon\) may be `NA`

(because the fitted canonical parameters have opposite sign, which
usually occurs when the pattern is completely random).

An absence of warnings does not mean that the fitted model is sensible. A negative value of \(\epsilon\) may be obtained (usually when the pattern is strongly clustered); this does not correspond to a valid point process model, but the software does not issue a warning.

In a pairwise interaction point process with the Lennard-Jones pair potential (Lennard-Jones, 1924) each pair of points in the point pattern, a distance \(d\) apart, contributes a factor $$ v(d) = \exp \left\{ - 4\epsilon \left[ \left( \frac{\sigma}{d} \right)^{12} - \left( \frac{\sigma}{d} \right)^6 \right] \right\} $$ to the probability density, where \(\sigma\) and \(\epsilon\) are positive parameters to be estimated.

See **Examples** for a plot of this expression.

This potential causes very strong inhibition between points at short
range, and attraction between points at medium range.
The parameter \(\sigma\) is called the
*characteristic diameter* and controls the scale of interaction.
The parameter \(\epsilon\) is called the *well depth*
and determines the strength of attraction.
The potential switches from inhibition to attraction at
\(d=\sigma\).
The maximum value of the pair potential is
\(\exp(\epsilon)\)
occuring at distance
\(d = 2^{1/6} \sigma\).
Interaction is usually considered to be negligible for distances
\(d > 2.5 \sigma \max\{1,\epsilon^{1/6}\}\).

This potential is used to model interactions between uncharged molecules in statistical physics.

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 Lennard-Jones pairwise interaction is
yielded by the function `LennardJones()`

.
See the examples below.

Lennard-Jones, J.E. (1924) On the determination of molecular fields.
*Proc Royal Soc London A* **106**, 463--477.

# NOT RUN { badfit <- ppm(cells ~1, LennardJones(), rbord=0.1) badfit fit <- ppm(unmark(longleaf) ~1, LennardJones(), rbord=1) fit plot(fitin(fit)) # Note the Longleaf Pines coordinates are rounded to the nearest decimetre # (multiple of 0.1 metres) so the apparent inhibition may be an artefact # }