spatstat.core (version 2.2-0)

Kinhom: Inhomogeneous K-function

Description

Estimates the inhomogeneous $K$ function of a non-stationary point pattern.

Usage

Kinhom(X, lambda=NULL, …, r = NULL, breaks = NULL,
    correction=c("border", "bord.modif", "isotropic", "translate"),
    renormalise=TRUE,
    normpower=1,
    update=TRUE,
    leaveoneout=TRUE,
    nlarge = 1000,
    lambda2=NULL, reciplambda=NULL, reciplambda2=NULL,
    diagonal=TRUE,
    sigma=NULL, varcov=NULL,
    ratio=FALSE)

Arguments

X

The observed data point pattern, from which an estimate of the inhomogeneous $K$ function will be computed. An object of class "ppp" or in a format recognised by as.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" or "kppm") or a function(x,y) which can be evaluated to give the intensity value at any location.

…

Extra arguments. Ignored if lambda is present. Passed to density.ppp if lambda is omitted.

r

vector of values for the argument $r$ at which the inhomogeneous $K$ function should be evaluated. Not normally given by the user; there is a sensible default.

breaks

This argument is for internal use only.

correction

A character vector containing any selection of the options "border", "bord.modif", "isotropic", "Ripley", "translate", "translation", "none" or "best". It specifies the edge correction(s) to be applied. Alternatively correction="all" selects all options.

renormalise

Logical. Whether to renormalise the estimate. See Details.

normpower

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

update

Logical value indicating what to do when lambda is a fitted model (class "ppm", "kppm" or "dppm"). If 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.

nlarge

Optional. Efficiency threshold. If the number of points exceeds nlarge, then only the border correction will be computed, using a fast algorithm.

lambda2

Advanced use only. Matrix containing estimates of the products $\lambda (x_i)\lambda (x_j)$ of the intensities at each pair of data points $x_i$ and $x_j$.

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.

reciplambda2

Advanced use only. Alternative to lambda2. A matrix giving values of the estimated reciprocal products $1/\lambda (x_i)\lambda (x_j)$ of the intensities at each pair of data points $x_i$ and $x_j$.

diagonal

Do not use this argument.

sigma,varcov

Optional arguments passed to density.ppp to control the smoothing bandwidth, when lambda is estimated by kernel smoothing.

ratio

Logical. If TRUE, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.

Value

An object of class "fv" (see fv.object).

Essentially a data frame containing at least the following columns,

r

the vector of values of the argument $r$ at which $K_{\text{scriptsizerm inhom}}(r)$ has been estimated

theo

vector of values of $\pi r^2$, the theoretical value of $K_{\text{scriptsizerm inhom}}(r)$ for an inhomogeneous Poisson process

and containing additional columns according to the choice specified in the correction argument. The additional columns are named border, trans and iso and give the estimated values of K_{\mbox{\scriptsize\rm inhom}}(r)Kinhom(r) using the border correction, translation correction, and Ripley isotropic correction, respectively.

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 K_{\mbox{\scriptsize\rm inhom}}(r)Kinhom(r).

Details

This computes a generalisation of the $K$ function for inhomogeneous point patterns, proposed by Baddeley, Moller and Waagepetersen (2000).

The ``ordinary'' $K$ function (variously known as the reduced second order moment function and Ripley's $K$ function), is described under Kest. It is defined only for stationary point processes.

The inhomogeneous $K$ function $K_{\text{scriptsizerm inhom}}(r)$ is a direct generalisation to nonstationary point processes. Suppose $x$ is a point process with non-constant intensity $\lambda (u)$ at each location $u$. Define $K_{\text{scriptsizerm inhom}}(r)$ to be the expected value, given that $u$ is a point of $x$, of the sum of all terms $1/\lambda (x_j)$ over all points $x_j$ in the process separated from $u$ by a distance less than $r$. This reduces to the ordinary $K$ function if $\lambda ()$ is constant. If $x$ is an inhomogeneous Poisson process with intensity function $\lambda (u)$, then $K_{\text{scriptsizerm inhom}}(r)=\pi r^2$.

Given a point pattern dataset, the inhomogeneous $K$ function can be estimated essentially by summing the values $1/(\lambda (x_i)\lambda (x_j))$ for all pairs of points $x_i,x_j$ separated by a distance less than $r$.

This allows us to inspect a point pattern for evidence of interpoint interactions after allowing for spatial inhomogeneity of the pattern. Values $K_{\text{scriptsizerm inhom}}(r)>\pi r^2$ are suggestive of clustering.

The argument lambda should supply the (estimated) values of the intensity function $\lambda$. It may be either

a numeric vector

containing the values of the intensity function at the points of the pattern X.

a pixel image

(object of class "im") assumed to contain the values of the intensity function at all locations in the window.

a fitted point process model

(object of class "ppm", "kppm" or "dppm") whose fitted trend can be used as the fitted intensity. (If update=TRUE the model will first be refitted to the data X before the trend is computed.)

a function

which can be evaluated to give values of the intensity at any locations.

omitted:

if lambda is omitted, then it will be estimated using a `leave-one-out' kernel smoother.

If lambda is a numeric vector, then its length should be equal to the number of points in the pattern X. The value lambda[i] is assumed to be the the (estimated) value of the intensity $\lambda (x_i)$ for the point $x_i$ of the pattern $X$. Each value must be a positive number; NA's are not allowed.

If lambda is a pixel image, the domain of the image should cover the entire window of the point pattern. If it does not (which may occur near the boundary because of discretisation error), then the missing pixel values will be obtained by applying a Gaussian blur to lambda using blur, then looking up the values of this blurred image for the missing locations. (A warning will be issued in this case.)

If lambda is a function, then it will be evaluated in the form lambda(x,y) where x and y are vectors of coordinates of the points of X. It should return a numeric vector with length equal to the number of points in X.

If lambda is omitted, then it will be estimated using a `leave-one-out' kernel smoother, as described in Baddeley, Moller and Waagepetersen (2000). The estimate lambda[i] for the point X[i] is computed by removing X[i] from the point pattern, applying kernel smoothing to the remaining points using density.ppp, and evaluating the smoothed intensity at the point X[i]. The smoothing kernel bandwidth is controlled by the arguments sigma and varcov, which are passed to density.ppp along with any extra arguments.

Edge corrections are used to correct bias in the estimation of $K_{\text{scriptsizerm inhom}}$. Each edge-corrected estimate of $K_{\text{scriptsizerm inhom}}(r)$ is of the form $${\hat{K}}_{\text{scriptsizerm inhom}}(r)=(1/A)\sum _i\sum _j\frac{1\{d_{ij}\le r\}e(x_i,x_j,r)}{\lambda (x_i)\lambda (x_j)}$$ where A is a constant denominator, $d_{ij}$ is the distance between points $x_i$ and $x_j$, and $e(x_i,x_j,r)$ is an edge correction factor. For the `border' correction, $$e(x_i,x_j,r)=\frac{1(b_i>r)}{\sum _{j}1(b_j>r)/\lambda (x_j)}$$ where $b_i$ is the distance from $x_i$ to the boundary of the window. For the `modified border' correction, $$e(x_i,x_j,r)=\frac{1(b_i>r)}{\text{area}(W\ominus r)}$$ where $W\ominus r$ is the eroded window obtained by trimming a margin of width $r$ from the border of the original window. For the `translation' correction, $$e(x_i,x_j,r)=\frac{1}{\text{area}(W\cap (W+(x_j-x_i)))}$$ and for the `isotropic' correction, $$e(x_i,x_j,r)=\frac{1}{\text{area}(W)g(x_i,x_j)}$$ where $g(x_i,x_j)$ is the fraction of the circumference of the circle with centre $x_i$ and radius $||x_i-x_j||$ which lies inside the window.

If renormalise=TRUE (the default), then the estimates described above are multiplied by $c^{\text{normpower}}$ where $c=\text{area}(W)/\sum (1/\lambda (x_i)).$ 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 (for consistency with previous versions of spatstat) but the most sensible value is 2, which would correspond to rescaling the lambda values so that $\sum (1/\lambda (x_i))=\text{area}(W).$

If the point pattern X contains more than about 1000 points, the isotropic and translation edge corrections can be computationally prohibitive. The computations for the border method are much faster, and are statistically efficient when there are large numbers of points. Accordingly, if the number of points in X exceeds the threshold nlarge, then only the border correction will be computed. Setting nlarge=Inf or correction="best" will prevent this from happening. Setting nlarge=0 is equivalent to selecting only the border correction with correction="border".

The pair correlation function can also be applied to the result of Kinhom; see pcf.

References

Baddeley, A., Moller, J. and Waagepetersen, R. (2000) Non- and semiparametric estimation of interaction in inhomogeneous point patterns. Statistica Neerlandica 54, 329--350.

See Also

Kest, pcf

Examples

# NOT RUN {
  # inhomogeneous pattern of maples
  X <- unmark(split(lansing)$maple)
  
# }
# NOT RUN {
  # (1) intensity function estimated by model-fitting
  # Fit spatial trend: polynomial in x and y coordinates
  fit <- ppm(X, ~ polynom(x,y,2), Poisson())
  # (a) predict intensity values at points themselves,
  #     obtaining a vector of lambda values
  lambda <- predict(fit, locations=X, type="trend")
  # inhomogeneous K function
  Ki <- Kinhom(X, lambda)
  plot(Ki)
  # (b) predict intensity at all locations,
  #     obtaining a pixel image
  lambda <- predict(fit, type="trend")
  Ki <- Kinhom(X, lambda)
  plot(Ki)

  # (2) intensity function estimated by heavy smoothing
  Ki <- Kinhom(X, sigma=0.1)
  plot(Ki)

  # (3) simulated data: known intensity function
  lamfun <- function(x,y) { 50 + 100 * x }
  # inhomogeneous Poisson process
  Y <- rpoispp(lamfun, 150, owin())
  # inhomogeneous K function
  Ki <- Kinhom(Y, lamfun)
  plot(Ki)

  # How to make simulation envelopes:
  #      Example shows method (2)
  if(interactive()) {
  smo <- density.ppp(X, sigma=0.1)
  Ken <- envelope(X, Kinhom, nsim=99,
                  simulate=expression(rpoispp(smo)),
                  sigma=0.1, correction="trans")
  plot(Ken)
  }
# }