Estimates the inhomogeneous pair correlation function of a point pattern using kernel methods.

```
pcfinhom(X, lambda = NULL, ..., r = NULL,
kernel = "epanechnikov", bw = NULL, stoyan = 0.15,
correction = c("translate", "Ripley"),
divisor = c("r", "d"),
renormalise = TRUE, normpower=1,
update = TRUE, leaveoneout = TRUE,
reciplambda = NULL,
sigma = NULL, varcov = NULL, close=NULL)
```

X

A point pattern (object of class `"ppp"`

).

lambda

Optional.
Values of the estimated intensity function.
Either a vector giving the intensity values
at the points of the pattern `X`

,
a pixel image (object of class `"im"`

) giving the
intensity values at all locations, a fitted point process model
(object of class `"ppm"`

, `"kppm"`

or `"dppm"`

)
or a `function(x,y)`

which
can be evaluated to give the intensity value at any location.

r

Vector of values for the argument \(r\) at which \(g(r)\) should be evaluated. There is a sensible default.

kernel

Choice of smoothing kernel, passed to `density.default`

.

bw

Bandwidth for smoothing kernel,
passed to `density.default`

.
Either a single numeric value,
or a character string specifying a bandwidth selection rule
recognised by `density.default`

.
If `bw`

is missing or `NULL`

,
the default value is computed using
Stoyan's rule of thumb: see `bw.stoyan`

.

…

Other arguments passed to the kernel density estimation
function `density.default`

.

stoyan

Coefficient for Stoyan's bandwidth selection rule;
see `bw.stoyan`

.

correction

Character string or character vector
specifying the choice of edge correction.
See `Kest`

for explanation and options.

divisor

Choice of divisor in the estimation formula:
either `"r"`

(the default) or `"d"`

.
See `pcf.ppp`

.

renormalise

Logical. Whether to renormalise the estimate. See Details.

normpower

Integer (usually either 1 or 2). Normalisation power. See Details.

update

Logical. If `lambda`

is a fitted model
(class `"ppm"`

, `"kppm"`

or `"dppm"`

)
and `update=TRUE`

(the default),
the model will first be refitted to the data `X`

(using `update.ppm`

or `update.kppm`

)
before the fitted intensity is computed.
If `update=FALSE`

, the fitted intensity of the
model will be computed without re-fitting it to `X`

.

leaveoneout

Logical value (passed to `density.ppp`

or
`fitted.ppm`

) specifying whether to use a
leave-one-out rule when calculating the intensity.

reciplambda

Alternative to `lambda`

.
Values of the estimated *reciprocal* \(1/\lambda\)
of the intensity function.
Either a vector giving the reciprocal intensity values
at the points of the pattern `X`

,
a pixel image (object of class `"im"`

) giving the
reciprocal intensity values at all locations,
or a `function(x,y)`

which can be evaluated to give the
reciprocal intensity value at any location.

sigma,varcov

Optional arguments passed to `density.ppp`

to control the smoothing bandwidth, when `lambda`

is
estimated by kernel smoothing.

close

Advanced use only. Precomputed data. See section on Advanced Use.

A function value table (object of class `"fv"`

).
Essentially a data frame containing the variables

the vector of values of the argument \(r\) at which the inhomogeneous pair correlation function \(g_{\rm inhom}(r)\) has been estimated

vector of values equal to 1, the theoretical value of \(g_{\rm inhom}(r)\) for the Poisson process

vector of values of \(g_{\rm inhom}(r)\) estimated by translation correction

vector of values of \(g_{\rm inhom}(r)\) estimated by Ripley isotropic correction

To perform the same computation using several different bandwidths `bw`

,
it is efficient to use the argument `close`

.
This should be the result of `closepairs(X, rmax)`

for a suitably large value of `rmax`

, namely
`rmax >= max(r) + 3 * bw`

.

The inhomogeneous pair correlation function \(g_{\rm inhom}(r)\) is a summary of the dependence between points in a spatial point process that does not have a uniform density of points.

The best intuitive interpretation is the following: the probability \(p(r)\) of finding two points at locations \(x\) and \(y\) separated by a distance \(r\) is equal to $$ p(r) = \lambda(x) lambda(y) g(r) \,{\rm d}x \, {\rm d}y $$ where \(\lambda\) is the intensity function of the point process. For a Poisson point process with intensity function \(\lambda\), this probability is \(p(r) = \lambda(x) \lambda(y)\) so \(g_{\rm inhom}(r) = 1\).

The inhomogeneous pair correlation function
is related to the inhomogeneous \(K\) function through
$$
g_{\rm inhom}(r) = \frac{K'_{\rm inhom}(r)}{2\pi r}
$$
where \(K'_{\rm inhom}(r)\)
is the derivative of \(K_{\rm inhom}(r)\), the
inhomogeneous \(K\) function. See `Kinhom`

for information
about \(K_{\rm inhom}(r)\).

The command `pcfinhom`

estimates the inhomogeneous
pair correlation using a modified version of
the algorithm in `pcf.ppp`

.

If `renormalise=TRUE`

(the default), then the estimates
are multiplied by \(c^{\mbox{normpower}}\) where
\(
c = \mbox{area}(W)/\sum (1/\lambda(x_i)).
\)
This rescaling reduces the variability and bias of the estimate
in small samples and in cases of very strong inhomogeneity.
The default value of `normpower`

is 1
but the most sensible value is 2, which would correspond to rescaling
the `lambda`

values so that
\(
\sum (1/\lambda(x_i)) = \mbox{area}(W).
\)

# NOT RUN { data(residualspaper) X <- residualspaper$Fig4b plot(pcfinhom(X, stoyan=0.2, sigma=0.1)) fit <- ppm(X, ~polynom(x,y,2)) plot(pcfinhom(X, lambda=fit, normpower=2)) # }