# Kinhom

0th

Percentile

##### Inhomogeneous K-function

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

Keywords
spatial
##### Usage
Kinhom(X, lambda, r = NULL, breaks = NULL, slow = FALSE,
correction=c("border", "bord.modif", "isotropic", "translate"),
..., lambda2
)
##### 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
Values of the estimated intensity function. Either a vector giving the intensity values at the points of the pattern X, or a pixel image (object of class "im") giving the intensity values at all locations.
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
An alternative to the argument r. Not normally invoked by the user. See Details.
slow
Not normally given by the user. Logical flag which selects the algorithm used to compute the inhomogeneous $K$ function. The default (slow=FALSE) is faster than the alternative (slow=TRUE). The slow algorithm is
correction
A character vector containing any selection of the options "border", "bord.modif", "isotropic", "Ripley" or "translate". It specifies the edge correction(s) to be applied.
...
Currently ignored.
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 $x_j$.
##### 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 $K_{\rm inhom}(r)$ 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 $K_{\rm inhom}(r)$ to be the expected value, given that $u$ is a point of $x$, of the sum of all terms $1/\lambda(u)\lambda(x_j)$ 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 $K_{\rm inhom}(r) = \pi r^2$.

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

The argument lambda must supply the (estimated) values of the intensity function $\lambda$. It may be either [object Object],[object Object]

Edge corrections are used to correct bias in the estimation of $K_{\rm inhom}$. Each edge-corrected estimate of $K_{\rm inhom}(r)$ is of the form $$\widehat K_{\rm inhom}(r) = \sum_i \sum_j \frac{1{d_{ij} \le r} e(x_i,x_j,r)}{\lambda(x_i)\lambda(x_j)}$$ where $d_{ij}$ 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, $$e(x_i,x_j,r) = \frac{1(b_i > r)}{\mbox{area}(W \ominus r)}$$ 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, $$e(x_i,x_j,r) = \frac 1 {\mbox{area}(W \cap (W + (x_j - x_i)))}$$ and for the isotropic' correction, $$e(x_i,x_j,r) = \frac 1 {\mbox{area}(W) g(x_i,x_j)}$$ 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. The pair correlation function can also be applied to the result of Kinhom; see pcf.

##### Value

• An object of class "fv" (see fv.object). Essentially a data frame containing at least the following columns,
• rthe vector of values of the argument $r$ at which the pair correlation function $g(r)$ has been estimated
• theovector of values of $\pi r^2$, the theoretical value of $K_{\rm inhom}(r)$ 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 $K_{\rm inhom}(r)$ using the border correction, translation correction, and Ripley isotropic correction, respectively.

##### References

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

Kest, pcf

• Kinhom
##### Examples
data(lansing)
# inhomogeneous pattern of maples
X <- unmark(split(lansing)$maple) <testonly>sub <- sample(c(TRUE,FALSE), X$n, replace=TRUE, prob=c(0.1,0.9))
X <- X[sub , ]</testonly>

# (1) intensity function estimated by model-fitting
# Fit spatial trend: polynomial in x and y coordinates
fit <- ppm(X, ~ polynom(x,y,2), Poisson())
# predict intensity values at points themselves
lambda <- predict(fit, locations=X, type="trend")
# inhomogeneous K function
Ki <- Kinhom(X, lambda)
plot(Ki)

# (2) intensity function estimated by heavy smoothing
smo <- density.ppp(X, sigma=0.1)
# evaluate smoothed intensity at data points
lambda <- smo[X]
# inhomogeneous K function
Ki <- Kinhom(X, lambda)
plot(Ki)

# (3) simulated data: known intensity function
lamfun <- function(x,y) { 100 * x }
# inhomogeneous Poisson process
Y <- rpoispp(lamfun, 100, owin())
# evaluate intensity at points of pattern
lambda <- lamfun(Y$x, Y$y)
# inhomogeneous K function
Ki <- Kinhom(Y, lambda)
plot(Ki)

# How to make simulation envelopes:
#      Example shows method (2)
smo <- density.ppp(X, sigma=0.1)
myKfun <- function(X, ..., lam) { Kinhom(X, lambda=lam[X], ...) }
Ken <- envelope(X, myKfun, nsim=99,
simulate=expression(rpoispp(smo)),
lam=smo, correction="trans")
plot(Ken)
Documentation reproduced from package spatstat, version 1.9-0, License: GPL version 2 or newer

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