# Jest

0th

Percentile

##### Estimate the J-function

Estimates the summary function $J(r)$ for a point pattern in a window of arbitrary shape.

Keywords
spatial, nonparametric
##### Usage
Jest(X, eps=NULL, r=NULL, breaks=NULL)
##### Arguments
X
The observed point pattern, from which an estimate of $J(r)$ will be computed. An object of class "ppp", or data in any format acceptable to as.ppp().
eps
the resolution of the discrete approximation to Euclidean distance (see below). There is a sensible default.
r
vector of values for the argument $r$ at which $J(r)$ should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. See below for important conditions on r.
breaks
An alternative to the argument r. Not normally invoked by the user. See Details section.
##### Details

The $J$ function (Van Lieshout and Baddeley ,1996) of a stationary point process is defined as $$J(r) = \frac{1-G(r)}{1-F(r)}$$ where $G(r)$ is the nearest neighbour distance distribution function of the point process (see Gest) and $F(r)$ is its empty space function (see Fest).

For a completely random (uniform Poisson) point process, the $J$-function is identically equal to $1$. Deviations $J(r) < 1$ or $J(r) > 1$ typically indicate spatial clustering or spatial regularity, respectively. The $J$-function is one of the few characteristics that can be computed explicitly for a wide range of point processes. See Van Lieshout and Baddeley (1996), Baddeley et al (2000), Thonnes and Van Lieshout (1999) for further information.

An estimate of $J$ derived from a spatial point pattern dataset can be used in exploratory data analysis and formal inference about the pattern. The estimate of $J(r)$ is compared against the constant function $1$. Deviations $J(r) < 1$ or $J(r) > 1$ may suggest spatial clustering or spatial regularity, respectively.

##### Note

Sizeable amounts of memory may be needed during the calculation.

##### References

Baddeley, A.J. Spatial sampling and censoring. In O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. van Lieshout (eds) Stochastic Geometry: Likelihood and Computation. Chapman and Hall, 1998. Chapter 2, pages 37--78. Baddeley, A.J. and Gill, R.D. The empty space hazard of a spatial pattern. Research Report 1994/3, Department of Mathematics, University of Western Australia, May 1994.

Baddeley, A.J. and Gill, R.D. Kaplan-Meier estimators of interpoint distance distributions for spatial point processes. Annals of Statistics 25 (1997) 263--292.

Baddeley, A., Kerscher, M., Schladitz, K. and Scott, B.T. Estimating the J function without edge correction. Statistica Neerlandica 54 (2000) 315--328.

Borgefors, G. Distance transformations in digital images. Computer Vision, Graphics and Image Processing 34 (1986) 344--371.

Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.

Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.

Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.

Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition. Springer Verlag, 1995.

Thonnes, E. and Van Lieshout, M.N.M, A comparative study on the power of Van Lieshout and Baddeley's J-function. Biometrical Journal 41 (1999) 721--734.

Van Lieshout, M.N.M. and Baddeley, A.J. A nonparametric measure of spatial interaction in point patterns. Statistica Neerlandica 50 (1996) 344--361.

• Jest
##### Examples
data(cells)
J <- Jest(cells, 0.01)
plot(J, main="cells data")
# values are far above J= 1, indicating regular pattern

data(redwood)
J <- Jest(redwood, 0.01)
plot(J, main="redwood data")
# values are below J= 1, indicating clustered pattern
Documentation reproduced from package spatstat, version 1.11-8, License: GPL version 2 or newer

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