# linearpcfinhom: Inhomogeneous Linear Pair Correlation Function In spatstat.linnet: Linear Networks Functionality of the 'spatstat' Family

## Description

Computes an estimate of the inhomogeneous linear pair correlation function for a point pattern on a linear network.

## Usage

 ```1 2 3 4``` ```linearpcfinhom(X, lambda=NULL, r=NULL, ..., correction="Ang", normalise=TRUE, normpower=1, update = TRUE, leaveoneout = TRUE, ratio = FALSE) ```

## Arguments

 `X` Point pattern on linear network (object of class `"lpp"`). `lambda` Intensity values for the point pattern. Either a numeric vector, a `function`, a pixel image (object of class `"im"`) or a fitted point process model (object of class `"ppm"` or `"lppm"`). `r` Optional. Numeric vector of values of the function argument r. There is a sensible default. `...` Arguments passed to `density.default` to control the smoothing. `correction` Geometry correction. Either `"none"` or `"Ang"`. See Details. `normalise` Logical. If `TRUE` (the default), the denominator of the estimator is data-dependent (equal to the sum of the reciprocal intensities at the data points, raised to `normpower`), which reduces the sampling variability. If `FALSE`, the denominator is the length of the network. `normpower` Integer (usually either 1 or 2). Normalisation power. See explanation in `linearKinhom`. `update` Logical value indicating what to do when `lambda` is a fitted model (class `"lppm"` or `"ppm"`). If `update=TRUE` (the default), the model will first be refitted to the data `X` (using `update.lppm` or `update.ppm`) 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 `fitted.lppm` or `fitted.ppm`) specifying whether to use a leave-one-out rule when calculating the intensity, when `lambda` is a fitted model. Supported only when `update=TRUE`. `ratio` Logical. If `TRUE`, the numerator and denominator of each estimate will also be saved, for use in analysing replicated point patterns.

## Details

This command computes the inhomogeneous version of the linear pair correlation function from point pattern data on a linear network.

If `lambda = NULL` the result is equivalent to the homogeneous pair correlation function `linearpcf`. If `lambda` is given, then it is expected to provide estimated values of the intensity of the point process at each point of `X`. The argument `lambda` may be a numeric vector (of length equal to the number of points in `X`), or a `function(x,y)` that will be evaluated at the points of `X` to yield numeric values, or a pixel image (object of class `"im"`) or a fitted point process model (object of class `"ppm"` or `"lppm"`).

If `lambda` is a fitted point process model, the default behaviour is to update the model by re-fitting it to the data, before computing the fitted intensity. This can be disabled by setting `update=FALSE`.

If `correction="none"`, the calculations do not include any correction for the geometry of the linear network. If `correction="Ang"`, the pair counts are weighted using Ang's correction (Ang, 2010).

The bandwidth for smoothing the pairwise distances is determined by arguments `...` passed to `density.default`, mainly the arguments `bw` and `adjust`. The default is to choose the bandwidth by Silverman's rule of thumb `bw="nrd0"` explained in `density.default`.

## Value

Function value table (object of class `"fv"`).

If `ratio=TRUE` then the return value also has two attributes called `"numerator"` and `"denominator"` which are `"fv"` objects containing the numerators and denominators of each estimate of g(r).

## Author(s)

Ang Qi Wei aqw07398@hotmail.com and \adrian.

## References

Ang, Q.W. (2010) Statistical methodology for spatial point patterns on a linear network. MSc thesis, University of Western Australia.

Ang, Q.W., Baddeley, A. and Nair, G. (2012) Geometrically corrected second-order analysis of events on a linear network, with applications to ecology and criminology. Scandinavian Journal of Statistics 39, 591–617.

Okabe, A. and Yamada, I. (2001) The K-function method on a network and its computational implementation. Geographical Analysis 33, 271-290.

`linearpcf`, `linearKinhom`, `lpp`
 ```1 2 3 4 5``` ``` data(simplenet) X <- rpoislpp(5, simplenet) fit <- lppm(X ~x) K <- linearpcfinhom(X, lambda=fit) plot(K) ```