Softcore {spatstat.model}

The Soft Core Point Process Model

Description

Creates an instance of the Soft Core point process model which can then be fitted to point pattern data.

Usage

  Softcore(kappa, sigma0=NA)

Arguments

kappa

The exponent \kappa of the Soft Core interaction

sigma0

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

Details

The (stationary) Soft Core point process with parameters \beta and \sigma and exponent \kappa is the pairwise interaction point process in which each point contributes a factor \beta to the probability density of the point pattern, and each pair of points contributes a factor

\exp \left\{ - \left( \frac{\sigma}{d} \right)^{2/\kappa} \right\}

to the density, where d is the distance between the two points. See the Examples for a plot of this interaction curve.

Thus the process has probability density

f(x_1,\ldots,x_n) = \alpha \beta^{n(x)} \exp \left\{ - \sum_{i < j} \left( \frac{\sigma}{||x_i-x_j||} \right)^{2/\kappa} \right\}

where x_1,\ldots,x_n represent the points of the pattern, n(x) is the number of points in the pattern, \alpha is the normalising constant, and the sum on the right hand side is over all unordered pairs of points of the pattern.

This model describes an “ordered” or “inhibitive” process, with the strength of inhibition decreasing smoothly with distance. The interaction is controlled by the parameters \sigma and \kappa.

• The spatial scale of interaction is controlled by the parameter \sigma, which is a positive real number interpreted as a distance, expressed in the same units of distance as the spatial data. The parameter \sigma is the distance at which the pair potential reaches the threshold value 0.37.

• The shape of the interaction function is controlled by the exponent \kappa which is a dimensionless number in the range (0,1), with larger values corresponding to a flatter shape (or a more gradual decay rate). The process is well-defined only for \kappa in (0,1). The limit of the model as \kappa \to 0 is the hard core process with hard core distance h=\sigma.

• The “strength” of the interaction is determined by both of the parameters \sigma and \kappa. The larger the value of \kappa, the wider the range of distances over which the interaction has an effect. If \sigma is very small, the interaction is very weak for all practical purposes (theoretically if \sigma = 0 the model reduces to the Poisson point process).

The nonstationary Soft Core process is similar except that the contribution of each individual point x_i is a function \beta(x_i) of location, rather than a constant beta.

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 Soft Core process pairwise interaction is yielded by the function Softcore(). See the examples below.

The main argument is the exponent kappa. When kappa is fixed, the model becomes an exponential family with canonical parameters \log \beta and

\log \gamma = \frac{2}{\kappa} \log\sigma

The canonical parameters are estimated by ppm(), not fixed in Softcore().

The optional argument sigma0 can be used to improve numerical stability. If sigma0 is given, it should be a positive number, and it should be a rough estimate of the parameter \sigma.

Value

An object of class "interact" describing the interpoint interaction structure of the Soft Core process with exponent \kappa.

Author(s)

Adrian Baddeley [e-mail], Rolf Turner [e-mail] and Ege Rubak [e-mail].

References

Ogata, Y, and Tanemura, M. (1981). Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure. Annals of the Institute of Statistical Mathematics, B 33, 315–338.

Ogata, Y, and Tanemura, M. (1984). Likelihood analysis of spatial point patterns. Journal of the Royal Statistical Society, series B 46, 496–518.

See Also

ppm, pairwise.family, ppm.object

Examples

   # fit the stationary Soft Core process to `cells' 
   fit5 <- ppm(cells ~1, Softcore(kappa=0.5), correction="isotropic")

   # study shape of interaction and explore effect of parameters
   fit2 <- update(fit5, Softcore(kappa=0.2))
   fit8 <- update(fit5, Softcore(kappa=0.8))
   plot(fitin(fit2), xlim=c(0, 0.4),
        main="Pair potential (sigma = 0.1)", 
        xlab=expression(d), ylab=expression(h(d)), legend=FALSE)
   plot(fitin(fit5), add=TRUE, col=4)
   plot(fitin(fit8), add=TRUE, col=3)
   legend("bottomright", col=c(1,4,3), lty=1,
          legend=expression(kappa==0.2, kappa==0.5, kappa==0.8))