mpp.characteristics {MarkedPointProcess} | R Documentation |
mpp.characteristics
returns summary statistics for the marks of
a marked point process
mpp.characteristics(...,bin=NULL, rep=1, p=0.8, name="", normalize=TRUE, show=FALSE, model=NULL, param=NULL, summarize=TRUE, PrintLevel=RFparameters()$Print, dev=if (name=="") 2 else FALSE, rdline=if (is.logical(dev)) NULL else readline, staticchoice=FALSE)
... |
coordinates and the data for the species, see Details |
bin |
sequence of increasing bin margins for the functions E, V, and S; see Details |
rep |
number of independent measurements of the marks (at each point); usually 1 |
p |
in [0,1]; outlier threshold for the robustified distance function of the test statistics |
name |
character; if show=FALSE this parameter is
ignored. Otherwise,
if name="" then plots are printed on the screen else name
is the main name of the postscript files |
normalize |
logical; if TRUE the data are (marginally)
transformed to Gaussian variables (for each species, each kind of
mark, and each realisation, seperately)
before being analysed |
show |
logical; if TRUE the results are also shown
graphically |
model |
the variogram model to compare with the mark variogram; see
CovarianceFct |
param |
the parameters for the variogram model; see
CovarianceFct |
summarize |
logical; if FALSE results are for each
realisation seperately (instead of being averaged over the realisations). |
PrintLevel |
0,1 or 2. The function gives some short messages if
PrintLevel is 1 or 2. |
rdline |
NULL or function.
if not NULL then the function is called after each plot with
a string parameter that gives file or image information
|
dev |
the graphical device for the output, see
Dev |
staticchoice |
logical. The calculation of some variances requires
the splitting of the data into group. If staticchoice=FALSE
this is done in a random way. staticchoice=TRUE is only
used for internal testing.
|
bin
: analogously to
the variogram in geostatistics, the characteristics E, V,
and S of a stationary and isotropic
marked point process depend on the distance
r. Instead of returning a cloud of values, binned values are
calculated in the same way the binned variogram is calculated.
bin
gives
the margins of the bins (left open, right closed ones) as an
increasing sequence. The first bin must include the zero, i.e.,
bin=c(-1, 0, ...)
.
...
: data for mark point processes typically split up into
different species (ill/healthy cells; beaches/oaks/pines);
furthermore,
multivariate data are measured for each individual (size of the cell;
diameter of the stem, height of the tree). The function
calculates many cross-statistics; for example the cross variogram
of mark A of species B and mark C of
species D given species B and species D
are a distance r apart.
Denote by S_i species i, i=1,...,s.
Due to the potential complexity of the data,
the data are passed to mpp.characteristics
in the following way:
(coordinates of species S_i), (marks of species A),...,
(coordinates of species Z), (marks of species Z)
The coordinates are n_i x 2 matrices; the data
are n_i x m x rep
matrices,
if the data are m-variate, and rep
independent observations
of the data exist. In case rep
>1 the sequence for the
data is:
columns 1:m : first set of the m-variate data,...,
columns (m * rep
- m + 1) : (m * rep
) : last set of the
m-variate data.
Note that m and rep
must be identical for all species.
The function returns the following values if summarize=TRUE
.
Denote by M_k(S_i) the
kth mark of species S_i, and by d_ij
the distance of two indviduals of species S_i and S_j.
Denote by s the number of species.
E
: (length(bin)-1
)x (ms^2) matrix.
function E in the following ordering of the
columns.
The index for the marks runs fastest, then the index of the
conditioning species, then the index for the species the marks
belong to.
That is,
ETest
: matrix of 66 rows and the same structure of columns as
E
.
The rows correspond to different algorithms for calculating the
deviance of E
from a horizontal line. Let denote
by E(i) the value of the ith bin, i=0,...,b,
where E(0)
is the bin that includes 0.
p
th quantile of
{ |E(i) - E(0)| : i=0,...,b }.
The functions s first increase linearly,
then quadratically: s(x) = x if x<a
and, b (x+a)^2 otherwise.
Here, a = 1/10 and b_k = 0.25 / a.
Weights
a : w_i ~ 1
b : w_i ~ 1 /
sum_{j=0}^{i-1} Ebin
(j)
c : w_i ~ sqrt(1 / sum_{j=0}^{i-1} Ebin
(j))
d : w_i ~ 1 /
sum_{j=0}^{i-1} sqrt(Ebin
(j))
e : w_i ~ Ebin
(i)
f : w_i ~ sqrt(Ebin
(i))
g : w_i ~ 1 / sqrt(var(E_j,unbinned; distance in ith bin))
The sequence is 1a, 2a, ..., 9a, 1b, ..., 9f.
VAR
: (length(bin)-1
)x (m(m-1)s^2/2)
matrix.
function V if m=1
. Otherwise, the
covariances are returned in the following ordering.
The index for
the lower triangle of the covariance matrix (including the
diagonal) runs fastest, then the index of the
conditioning species, then the index for the species the marks
belong to.
That is,
VARTest
: matrix of 66 rows and the same structure of columns as
VAR
. See VAR
and ETest
for details.
SQ
: (length(bin)-1
)x (m(m-1)s^2/2)
matrix. Equals sign(V) * sqrt(|V|).
SQTest
: Test results for SQ. Same matrix
dimensions as VARTest
.
KMM
: Stoyan's kmm functionlength(bin)-1
)x (ms(ms+1)/2) matrix.
The columns of KMM
are returned in the following ordering:
It is the lower triangle (including the diagonal) of
expectation of the matrix v v^T. Here v is
vector where first m components are the marks of species
S_1 the next components are the marks of S_2 and so on.
That is,
GAM
: mark variogrammlength(bin)-1
)x (ms(ms+1)/2) matrix;
see KMM for details.
Ebin
: number of values a binned value of E is based on.
VARbin
: number of values a binned value of V or
SQ is based on.
KMMbin
: number of values a binned value of
Stoyan's kmm function is based on.
GAMbin
: number of values a binned value of the mark variogram
is based on.
midbin
: centers of the bins
call
: match.call()
of mpp.characteristics
summarize=FALSE
: the column structure of the above
matrices is rep
times repeated.
mpp.characteristics
returns
list(E, ETest, VAR, VARTest, SQ, SQTest, KMM, GAM, Ebin,
VARbin, KMMbin, GAMbin, midbin, call = match.call())
;
see Details. The return is invisible
if show=TRUE
.
Martin Schlather, martin.schlather@math.uni-goettingen.de http://www.stochastik.math.uni-goettingen.de/institute
Schlather, M., Ribeiro, P., and Diggle, P. (2004) Detecting Dependence Between Marks and Locations of Marked Point Processes J. R. Statist. Soc., Ser. B , .
Schlather, M. (2001) On the second order characteristics of marked point processes, Bernoulli 7, 99-117.
data(BITOEK) bin <- c(-1, seq(0, 50, 2)) normalize <- TRUE mpp <- mpp.characteristics(bin=bin, normalize=normalize, show=TRUE, coord=steigerwald$coord, diam=steigerwald$diam) str(mpp)