LennardJones
The Lennard-Jones Potential
Creates the Lennard-Jones pairwise interaction structure which can then be fitted to point pattern data.
Usage
LennardJones()
Details
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 $$\exp \left{ - 4\epsilon \left[ \left( \frac{\sigma}{d} \right)^{12} - \left( \frac{\sigma}{d} \right)^6 \right] \right}$$ 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.
The ``canonical regular parameters'' estimated by ppm
are
$\theta_1 = 4 \epsilon \sigma^{12}$
and
$\theta_2 = 4 \epsilon \sigma^6$.
Value
- An object of class
"interact"
describing the Lennard-Jones interpoint interaction structure.
Warnings
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.
The fitting algorithm may also fail to converge
when the spatial window of the point pattern dataset has dimensions
that are much greater than 1 unit or much less than 1 unit,
leading to overflow or underflow in the 12th power of the coordinates.
To fix this problem, rescale the coordinates so that the window is
close to the unit square (see Examples).
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.
References
Lennard-Jones, J.E. (1924) On the determination of molecular fields. Proc Royal Soc London A 106, 463--477.
See Also
Examples
data(demopat)
demopat
X <- rescale(unmark(demopat), 5000)
X
fit <- ppm(X, ~1, LennardJones(), rbord=0.1)
fit
plot(fitin(fit), xlim=c(0,0.01))