lurking
Lurking variable plot
Plot spatial point process residuals against a covariate
Usage
lurking(object, covariate, type="eem",
cumulative=TRUE,
clipwindow=default.clipwindow(object),
rv,
plot.sd, plot.it=TRUE,
typename,
covname, ...)
Arguments
- object
- The fitted point process model (an object of class
"ppm"
) for which diagnostics should be produced. This object is usually obtained fromppm
. - covariate
- The covariate against which residuals should be plotted.
Either a numeric vector, a pixel image, or an
expression
. See Details below. - type
- String indicating the type of residuals or weights to be computed.
Choices include
"eem"
,"raw"
,"inverse"
and"pearson"
. Seediagnose.ppm
- cumulative
- Logical flag indicating whether to plot a
cumulative sum of marks (
cumulative=TRUE
) or the derivative of this sum, a marginal density of the smoothed residual field (cumulative=FALSE
). - clipwindow
- If not
NULL
this argument indicates that residuals shall only be computed inside a subregion of the window containing the original point pattern data. Thenclipwindow
should be a window object of class"owin"
- rv
- Usually absent.
If this argument is present, the values of the residuals will not be
calculated from the fitted model
object
but will instead be taken directly from this vector. - plot.sd
- Logical value indicating whether
error bounds should be added to plot.
The default is
TRUE
for Poisson models andFALSE
for non-Poisson models. See Details. - plot.it
- Logical value indicating whether
plots should be shown. If
plot.it=FALSE
, only the computed coordinates for the plots are returned. See Value. - typename
- Usually absent. If this argument is present, it should be a string, and will be used (in the axis labels of plots) to describe the type of residuals.
- covname
- A string name for the covariate, to be used in axis labels of plots.
- ...
- Arguments passed to
smooth.spline
for the estimation of the derivatives in the casecumulative=FALSE
.
Details
This function generates a `lurking variable' plot for a
fitted point process model.
Residuals from the model represented by object
are plotted against the covariate specified by covariate
.
This plot can be used to reveal departures from the fitted model,
in particular, to reveal that the point pattern depends on the covariate.
First the residuals from the fitted model (Baddeley et al, 2004)
are computed at each quadrature point,
or alternatively the `exponential energy marks' (Stoyan and Grabarnik,
1991) are computed at each data point.
The argument type
selects the type of
residual or weight. See diagnose.ppm
for options
and explanation.
A lurking variable plot for point processes (Baddeley et al, 2004)
displays either the cumulative sum of residuals/weights
(if cumulative = TRUE
) or a kernel-weighted average of the
residuals/weights (if cumulative = FALSE
) plotted against
the covariate. The empirical plot (solid lines) is shown
together with its expected value assuming the model is true
(dashed lines) and optionally also the pointwise
two-standard-deviation limits (dotted lines).
To be more precise, let $Z(u)$ denote the value of the covariate
at a spatial location $u$.
- If
cumulative=TRUE
then we plot$H(z)$against$z$, where$H(z)$is the sum of the residuals over all quadrature points where the covariate takes a value less than or equal to$z$, or the sum of the exponential energy weights over all data points where the covariate takes a value less than or equal to$z$. - If
cumulative=FALSE
then we plot$h(z)$against$z$, where$h(z)$is the derivative of$H(z)$, computed approximately by spline smoothing.
If the empirical and theoretical curves deviate substantially
from one another, the interpretation is that the fitted model does
not correctly account for dependence on the covariate.
The correct form (of the spatial trend part of the model)
may be suggested by the shape of the plot.
If plot.sd = TRUE
, then superimposed on the lurking variable
plot are the pointwise
two-standard-deviation error limits for $H(x)$ calculated for the
inhomogeneous Poisson process. The default is plot.sd = TRUE
for Poisson models and plot.sd = FALSE
for non-Poisson
models.
The argument object
must be a fitted point process model
(object of class "ppm"
) typically produced by the maximum
pseudolikelihood fitting algorithm ppm
).
The argument covariate
is either a numeric vector, a pixel
image, or an R language expression.
If it is a numeric vector, it is assumed to contain
the values of the covariate for each of the quadrature points
in the fitted model. The quadrature points can be extracted by
quad.ppm(object)
.
If covariate
is a pixel image, it is assumed to contain the
values of the covariate at each location in the window. The values of
this image at the quadrature points will be extracted.
Alternatively, if covariate
is an expression
, it will be evaluated in the same environment
as the model formula used in fitting the model object
. It must
yield a vector of the same length as the number of quadrature points.
The expression may contain the terms x
and y
representing the
cartesian coordinates, and may also contain other variables that were
available when the model was fitted. Certain variable names are
reserved words; see ppm
.
Note that lurking variable plots for the $x$ and $y$ coordinates
are also generated by diagnose.ppm
, amongst other
types of diagnostic plots. This function is more general in that it
enables the user to plot the residuals against any chosen covariate
that may have been present.
Value
- A list containing two dataframes
empirical
andtheoretical
. The first dataframeempirical
contains columnscovariate
andvalue
giving the coordinates of the lurking variable plot. The second dataframetheoretical
contains columnscovariate
,mean
andsd
giving the coordinates of the plot of the theoretical mean and standard deviation.
References
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2004) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617--666.
Stoyan, D. and Grabarnik, P. (1991) Second-order characteristics for stochastic structures connected with Gibbs point processes. Mathematische Nachrichten, 151:95--100.
See Also
residuals.ppm
,
diagnose.ppm
,
residuals.ppm
,
qqplot.ppm
,
eem
,
ppm
Examples
data(nztrees)
fit <- ppm(nztrees, ~x, Poisson())
lurking(fit, expression(x))
lurking(fit, expression(x), cumulative=FALSE)