Signed or Vector-Valued Measure
Defines an object representing a signed measure or vector-valued measure on a spatial domain.
msr(qscheme, discrete, density, check=TRUE)
qscheme |
A quadrature scheme (object of class |
discrete |
Vector or matrix containing the values (masses) of the discrete component
of the measure, for each of the data points in |
density |
Vector or matrix containing values of the density of the
diffuse component of the measure, for each of the
quadrature points in |
check |
Logical. Whether to check validity of the arguments. |
This function creates an object that represents a signed or vector valued measure on the two-dimensional plane. It is not normally called directly by the user.
A signed measure is a classical mathematical object (Diestel and Uhl, 1977) which can be visualised as a collection of electric charges, positive and/or negative, spread over the plane. Electric charges may be concentrated at specific points (atoms), or spread diffusely over a region.
An object of class "msr" represents a signed (i.e. real-valued)
or vector-valued measure in the spatstat package.
Spatial residuals for point process models
(Baddeley et al, 2005, 2008) take the form of a real-valued
or vector-valued measure. The function
residuals.ppm returns an object of
class "msr" representing the residual measure.
Various other diagnostic tools such as dfbetas.ppm and
dffit.ppm also return an object of class "msr".
The function msr would not normally be called directly by the
user. It is the low-level creator function that
makes an object of class "msr" from raw data.
The first argument qscheme is a quadrature scheme (object of
class "quad"). It is typically created by quadscheme or
extracted from a fitted point process model using
quad.ppm. A quadrature scheme contains both data points
and dummy points. The data points of qscheme are used as the locations
of the atoms of the measure. All quadrature points
(i.e. both data points and dummy points)
of qscheme are used as sampling points for the density
of the continuous component of the measure.
The argument discrete gives the values of the
atomic component of the measure for each data point in qscheme.
It should be either a numeric vector with one entry for each
data point, or a numeric matrix with one row
for each data point.
The argument density gives the values of the density
of the diffuse component of the measure, at each
quadrature point in qscheme.
It should be either a numeric vector with one entry for each
quadrature point, or a numeric matrix with one row
for each quadrature point.
If both discrete and density are vectors
(or one-column matrices) then the result is a signed (real-valued) measure.
Otherwise, the result is a vector-valued measure, with the dimension
of the vector space being determined by the number of columns
in the matrices discrete and/or density.
(If one of these is a k-column matrix and the other
is a 1-column matrix, then the latter is replicated to k columns).
The class "msr" has methods for print,
plot and [.
There is also a function Smooth.msr for smoothing a measure.
An object of class "msr".
Objects of class "msr", representing measures, are returned by the
functions residuals.ppm, dfbetas.ppm,
dffit.ppm and possibly by other functions.
There are methods for printing and plotting a measure, along with
many other operations, which can be listed by
typing methods(class="msr").
The print and summary methods report basic information
about a measure, such as the total value of the measure, and the
spatial domain on which it is defined.
The plot method displays the measure. It is documented separately in
plot.msr.
A measure can be smoothed using Smooth.msr, yielding a
pixel image which is sometimes easier to interpret than the plot
of the measure itself.
The subset operator [ can be used to restrict the measure to
a subregion of space, or to extract one of the scalar components of a
vector-valued measure. It is documented separately in
[.msr.
The total value of a measure, or the value on a subregion,
can be obtained using integral.msr.
The value of a measure m on a subregion
B can be obtained by integral(m, domain=B) or
integral(m[B]). The values of a measure m on each tile
of a tessellation A can be obtained by
integral(m, domain=A).
Some mathematical operations on measures are supported, such as multiplying a measure by a single number, or adding two measures.
Measures can be separated into components in different ways using
as.layered.msr, unstack.msr
and split.msr.
Internal components of the data structure of an "msr" object
can be extracted using with.msr.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Baddeley, A., Moller, J. and Pakes, A.G. (2008) Properties of residuals for spatial point processes. Annals of the Institute of Statistical Mathematics 60, 627–649.
Diestel, J. and Uhl, J.J. Jr (1977) Vector measures. Providence, RI, USA: American Mathematical Society.
Halmos, P.R. (1950) Measure Theory. Van Nostrand.
X <- rpoispp(function(x,y) { exp(3+3*x) })
fit <- ppm(X, ~x+y)
rp <- residuals(fit, type="pearson")
rp
rs <- residuals(fit, type="score")
rs
colnames(rs)
# An equivalent way to construct the Pearson residual measure by hand
Q <- quad.ppm(fit)
lambda <- fitted(fit)
slam <- sqrt(lambda)
Z <- is.data(Q)
m <- msr(Q, discrete=1/slam[Z], density = -slam)
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