Description Usage Arguments Details Value Author(s) See Also Examples

Creates an instance of a saturated pairwise interaction point process model with piecewise constant potential function. The model can then be fitted to point pattern data.

1 | ```
SatPiece(r, sat)
``` |

`r` |
vector of jump points for the potential function |

`sat` |
vector of saturation values, or a single saturation value |

This is a generalisation of the Geyer saturation point process model,
described in `Geyer`

, to the case of multiple interaction
distances. It can also be described as the saturated analogue of a
pairwise interaction process with piecewise-constant pair potential,
described in `PairPiece`

.

The saturated point process with interaction radii
*r[1], ..., r[k]*,
saturation thresholds *s[1],...,s[k]*,
intensity parameter *beta* and
interaction parameters
*gamma[1], ..., gamma[k]*,
is the point process
in which each point
*x[i]* in the pattern *X*
contributes a factor

*
beta gamma[1]^v(1, x_i, X) ... gamma[k]^v(k, x_i, X)
*

to the probability density of the point pattern, where

*
v(j, x_i, X) = min(s[j], t(j, x_i, X))
*

where *t(j,x[i],X)* denotes the
number of points in the pattern *X* which lie
at a distance between *r[j-1]* and *r[j]*
from the point *x[i]*. We take *r[0] = 0*
so that *t(1, x[i], X)* is the number of points of
*X* that lie within a distance *r[1]* of the point
*x[i]*.

`SatPiece`

is used to fit this model to data.
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 piecewise constant Saturated pairwise
interaction is yielded by the function `SatPiece()`

.
See the examples below.

Simulation of this point process model is not yet implemented. This model is not locally stable (the conditional intensity is unbounded).

The argument `r`

specifies the vector of interaction distances.
The entries of `r`

must be strictly increasing, positive numbers.

The argument `sat`

specifies the vector of saturation parameters.
It should be a vector of the same length as `r`

, and its entries
should be nonnegative numbers. Thus `sat[1]`

corresponds to the
distance range from `0`

to `r[1]`

, and `sat[2]`

to the
distance range from `r[1]`

to `r[2]`

, etc.
Alternatively `sat`

may be a single number, and this saturation
value will be applied to every distance range.

Infinite values of the
saturation parameters are also permitted; in this case
*v(j, x_i, X) = t(j, x_i, X)*
and there is effectively no ‘saturation’ for the distance range in
question. If all the saturation parameters are set to `Inf`

then
the model is effectively a pairwise interaction process, equivalent to
`PairPiece`

(however the interaction parameters
*gamma* obtained from `SatPiece`

are the
square roots of the parameters *gamma*
obtained from `PairPiece`

).

If `r`

is a single number, this model is virtually equivalent to the
Geyer process, see `Geyer`

.

An object of class `"interact"`

describing the interpoint interaction
structure of a point process.

and \rolf

in collaboration with Hao Wang and Jeff Picka

`ppm`

,
`pairsat.family`

,
`Geyer`

,
`PairPiece`

,
`BadGey`

.

1 2 3 4 5 6 7 8 9 10 11 12 | ```
SatPiece(c(0.1,0.2), c(1,1))
# prints a sensible description of itself
SatPiece(c(0.1,0.2), 1)
data(cells)
ppm(cells, ~1, SatPiece(c(0.07, 0.1, 0.13), 2))
# fit a stationary piecewise constant Saturated pairwise interaction process
## Not run:
ppm(cells, ~polynom(x,y,3), SatPiece(c(0.07, 0.1, 0.13), 2))
# nonstationary process with log-cubic polynomial trend
## End(Not run)
``` |

spatstat documentation built on Oct. 9, 2017, 1:05 a.m.

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