# lurking

##### Lurking Variable Plot

Plot spatial point process residuals against a covariate

##### Usage

`lurking(object, …)`# S3 method for ppm
lurking(object, covariate,
type="eem",
cumulative=TRUE,
…,
plot.it = TRUE,
plot.sd = is.poisson(object),
clipwindow=default.clipwindow(object),
rv = NULL,
envelope=FALSE, nsim=39, nrank=1,
typename,
covname,
oldstyle=FALSE,
check=TRUE,
verbose=TRUE,
nx=128,
splineargs=list(spar=0.5),
internal=NULL)

# S3 method for ppp
lurking(object, covariate,
type="eem",
cumulative=TRUE,
…,
plot.it = TRUE,
plot.sd = is.poisson(object),
clipwindow=default.clipwindow(object),
rv = NULL,
envelope=FALSE, nsim=39, nrank=1,
typename,
covname,
oldstyle=FALSE,
check=TRUE,
verbose=TRUE,
nx=128,
splineargs=list(spar=0.5),
internal=NULL)

##### Arguments

- object
The fitted point process model (an object of class

`"ppm"`

) for which diagnostics should be produced. This object is usually obtained from`ppm`

. Alternatively,`object`

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

).- 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"`

. See`diagnose.ppm`

for all possible choices.- 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`

).- …
Arguments passed to

`plot.default`

and`lines`

to control the plot behaviour.- 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*.- plot.sd
Logical value indicating whether error bounds should be added to plot. The default is

`TRUE`

for Poisson models and`FALSE`

for non-Poisson models. See Details.- 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. Then`clipwindow`

should be a window object of class`"owin"`

.- rv
Usually absent. If this argument is present, the point process residuals will not be calculated from the fitted model

`object`

, but will instead be taken directly from`rv`

.- envelope
Logical value indicating whether to compute simulation envelopes for the plot. Alternatively

`envelope`

may be a list of point patterns to use for computing the simulation envelopes, or an object of class`"envelope"`

containing simulated point patterns.- nsim
Number of simulated point patterns to be generated to produce the simulation envelope, if

`envelope=TRUE`

.- nrank
Integer. Rank of the envelope value amongst the

`nsim`

simulated values. A rank of 1 means that the minimum and maximum simulated values will be used.- 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.

- oldstyle
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 (`oldstyle=FALSE`

).- check
Logical flag indicating whether the integrity of the data structure in

`object`

should be checked.- verbose
Logical value indicating whether to print progress reports during Monte Carlo simulation.

- nx
Integer. Number of covariate values to be used in the plot.

- splineargs
A list of arguments passed to

`smooth.spline`

for the estimation of the derivatives in the case`cumulative=FALSE`

.- internal
Internal use only.

##### 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.

The function `lurking`

is generic, with methods for
`ppm`

and `ppp`

documented here, and possibly other methods.

The argument `object`

would usually be a fitted point process model
(object of class `"ppm"`

) produced by the
model-fitting algorithm `ppm`

). If
`object`

is a point pattern (object of class `"ppp"`

) then
the model is taken to be the uniform Poisson process (Complete
Spatial Randomness) fitted to this point pattern.

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 (grey shading).

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.

For the point process residuals \(E(H(z)) = 0\), while for the exponential energy weights \(E(H(z)) = \) area of the subset of the window satisfying \(Z(u) <= z\).

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.

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 `oldstyle=TRUE`

,
the two-standard-deviation limits are calculated
using the innovation variance, an over-estimate of the true
variance of the residuals.

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.

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 the object `rv`

.
If `type = "eem"`

then `rv`

should be similar to the
return value of `eem`

, namely, a numeric vector
with length equal to the number of data points in the original point
pattern. Otherwise, `rv`

should be
similar to the return value of `residuals.ppm`

,
that is, `rv`

should be an object of class
`"msr"`

(see `msr`

) representing a signed measure.

##### Value

The (invisible) return value is an object
belonging to the class `"lurk"`

, for which there
are methods for `plot`

and `print`

.

This object is a list containing two dataframes
`empirical`

and `theoretical`

.
The first dataframe `empirical`

contains columns
`covariate`

and `value`

giving the coordinates of the
lurking variable plot. The second dataframe `theoretical`

contains columns `covariate`

, `mean`

and `sd`

giving the coordinates of the plot of the theoretical mean
and standard deviation.

##### References

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.
*Annals of the Institute of Statistical Mathematics*
**60**, 627--649.

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

```
# NOT RUN {
(a <- lurking(nztrees, expression(x), type="raw"))
fit <- ppm(nztrees ~x, Poisson(), nd=128)
(b <- lurking(fit, expression(x), type="raw"))
lurking(fit, expression(x), type="raw", cumulative=FALSE)
# }
```

*Documentation reproduced from package spatstat, version 1.56-1, License: GPL (>= 2)*