Lurking variable plot
Plot spatial point process residuals against a covariate
lurking(object, covariate, type="eem", cumulative=TRUE, clipwindow=default.clipwindow(object), rv, plot.sd, plot.it=TRUE, typename, covname, oldstyle=FALSE, check=TRUE, ..., splineargs=list(spar=0.5))
- The fitted point process model (an object of class
"ppm") for which diagnostics should be produced. This object is usually obtained from
objectmay be a
- The covariate against which residuals should be plotted.
Either a numeric vector, a pixel image, or an
expression. See Details below.
- String indicating the type of residuals or weights to be computed.
- 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 (
- If not
NULLthis argument indicates that residuals shall only be computed inside a subregion of the window containing the original point pattern data. Then
clipwindowshould be a window object of class
- Usually absent.
If this argument is present, the values of the residuals will not be
calculated from the fitted model
objectbut will instead be taken directly from this vector.
- Logical value indicating whether
error bounds should be added to plot.
The default is
TRUEfor Poisson models and
FALSEfor non-Poisson models. See Details.
- Logical value indicating whether
plots should be shown. If
plot.it=FALSE, only the computed coordinates for the plots are returned. See Value.
- 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.
- A string name for the covariate, to be used in axis labels of plots.
- Logical flag indicating whether error bounds should be plotted
using the approximation given in the original paper
oldstyle=TRUE), or using the correct asymptotic formula (
- Logical flag indicating whether the integrity of the data structure
objectshould be checked.
- Arguments passed to
linesto control the plot behaviour.
- A list of arguments passed to
smooth.splinefor the estimation of the derivatives in the case
This function generates a `lurking variable' plot for a
fitted point process model.
Residuals from the model represented by
are plotted against the covariate specified by
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.
type selects the type of
residual or weight. See
diagnose.ppm for options
A lurking variable plot for point processes (Baddeley et al, 2004)
displays either the cumulative sum of residuals/weights
cumulative = TRUE) or a kernel-weighted average of the
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 $u$.
cumulative=TRUEthen 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$z$.
cumulative=FALSEthen 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.
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
By default, the two-standard-deviation limits are calculated
from the exact formula for the asymptotic variance
of the residuals under the asymptotic normal approximation,
equation (37) of Baddeley et al (2006).
However, for compatibility with the original paper
of Baddeley et al (2005), if
the two-standard-deviation limits are calculated
using the innovation variance, an over-estimate of the true
variance of the residuals.
object must be a fitted point process model
(object of class
"ppm") typically produced by the maximum
pseudolikelihood fitting algorithm
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
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.
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
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
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.
For advanced use, even the values of the residuals/weights
can be altered. If the argument
rv is present,
the residuals will not be calculated from the fitted model
object but will instead be taken directly from
rv should be numeric, and its length should equal
the number of data points in the original point pattern
type="eem") or the number of
quadrature points used to fit the model (otherwise).
- A list containing two dataframes
theoretical. The first dataframe
valuegiving the coordinates of the lurking variable plot. The second dataframe
sdgiving the coordinates of the plot of the theoretical mean and standard deviation.
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. (2006) Properties of residuals for spatial point processes. To appear. Stoyan, D. and Grabarnik, P. (1991) Second-order characteristics for stochastic structures connected with Gibbs point processes. Mathematische Nachrichten, 151:95--100.
data(nztrees) fit <- ppm(nztrees, ~x, Poisson()) lurking(fit, expression(x)) lurking(fit, expression(x), cumulative=FALSE)