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

For a marked point pattern,
estimate the multitype *K* function
which counts the expected number of points of subset *J*
within a given distance from a typical point in subset `I`

.

1 |

`X` |
The observed point pattern,
from which an estimate of the multitype |

`I` |
Subset index specifying the points of |

`J` |
Subset index specifying the points in |

`r` |
numeric vector. The values of the argument |

`breaks` |
This argument is for internal use only. |

`correction` |
A character vector containing any selection of the
options |

`...` |
Ignored. |

`ratio` |
Logical.
If |

The function `Kmulti`

generalises `Kest`

(for unmarked point
patterns) and `Kdot`

and `Kcross`

(for
multitype point patterns) to arbitrary marked point patterns.

Suppose *X[I]*, *X[J]* are subsets, possibly
overlapping, of a marked point process.
The multitype *K* function
is defined so that
*lambda[J] KIJ(r)* equals the expected number of
additional random points of *X[J]*
within a distance *r* of a
typical point of *X[I]*.
Here *lambda[J]*
is the intensity of *X[J]*
i.e. the expected number of points of *X[J]* per unit area.
The function *KIJ* is determined by the
second order moment properties of *X*.

The argument `X`

must be a point pattern (object of class
`"ppp"`

) or any data that are acceptable to `as.ppp`

.

The arguments `I`

and `J`

specify two subsets of the
point pattern. They may be any type of subset indices, for example,
logical vectors of length equal to `npoints(X)`

,
or integer vectors with entries in the range 1 to
`npoints(X)`

, or negative integer vectors.

Alternatively, `I`

and `J`

may be **functions**
that will be applied to the point pattern `X`

to obtain
index vectors. If `I`

is a function, then evaluating
`I(X)`

should yield a valid subset index. This option
is useful when generating simulation envelopes using
`envelope`

.

The argument `r`

is the vector of values for the
distance *r* at which *KIJ(r)* should be evaluated.
It is also used to determine the breakpoints
(in the sense of `hist`

)
for the computation of histograms of distances.

First-time users would be strongly advised not to specify `r`

.
However, if it is specified, `r`

must satisfy `r[1] = 0`

,
and `max(r)`

must be larger than the radius of the largest disc
contained in the window.

This algorithm assumes that `X`

can be treated
as a realisation of a stationary (spatially homogeneous)
random spatial point process in the plane, observed through
a bounded window.
The window (which is specified in `X`

as `Window(X)`

)
may have arbitrary shape.

Biases due to edge effects are
treated in the same manner as in `Kest`

.
The edge corrections implemented here are

- border
the border method or “reduced sample” estimator (see Ripley, 1988). This is the least efficient (statistically) and the fastest to compute. It can be computed for a window of arbitrary shape.

- isotropic/Ripley
Ripley's isotropic correction (see Ripley, 1988; Ohser, 1983). This is currently implemented only for rectangular and polygonal windows.

- translate
Translation correction (Ohser, 1983). Implemented for all window geometries.

The pair correlation function `pcf`

can also be applied to the
result of `Kmulti`

.

An object of class `"fv"`

(see `fv.object`

).

Essentially a data frame containing numeric columns

`r` |
the values of the argument |

`theo` |
the theoretical value of |

together with a column or columns named
`"border"`

, `"bord.modif"`

,
`"iso"`

and/or `"trans"`

,
according to the selected edge corrections. These columns contain
estimates of the function *KIJ(r)*
obtained by the edge corrections named.

If `ratio=TRUE`

then the return value also has two
attributes called `"numerator"`

and `"denominator"`

which are `"fv"`

objects
containing the numerators and denominators of each
estimate of *K(r)*.

The function *KIJ* is not necessarily differentiable.

The border correction (reduced sample) estimator of
*KIJ* used here is pointwise approximately
unbiased, but need not be a nondecreasing function of *r*,
while the true *KIJ* must be nondecreasing.

.

Cressie, N.A.C. *Statistics for spatial data*.
John Wiley and Sons, 1991.

Diggle, P.J. *Statistical analysis of spatial point patterns*.
Academic Press, 1983.

Diggle, P. J. (1986).
Displaced amacrine cells in the retina of a
rabbit : analysis of a bivariate spatial point pattern.
*J. Neurosci. Meth.* **18**, 115–125.

Harkness, R.D and Isham, V. (1983)
A bivariate spatial point pattern of ants' nests.
*Applied Statistics* **32**, 293–303

Lotwick, H. W. and Silverman, B. W. (1982).
Methods for analysing spatial processes of several types of points.
*J. Royal Statist. Soc. Ser. B* **44**, 406–413.

Ripley, B.D. *Statistical inference for spatial processes*.
Cambridge University Press, 1988.

Stoyan, D, Kendall, W.S. and Mecke, J.
*Stochastic geometry and its applications*.
2nd edition. Springer Verlag, 1995.

Van Lieshout, M.N.M. and Baddeley, A.J. (1999)
Indices of dependence between types in multivariate point patterns.
*Scandinavian Journal of Statistics* **26**, 511–532.

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