The spatstat Package March 26, 2004 Version 1.4-5 Date 25 March 2004 Title Analysis of spatial point patterns Author Adrian Baddeley and Rolf Turner , with contributions by Maintainer Adrian Baddeley Depends R (>= 1.8.0), mgcv, modreg, sm Description Spatial Point Pattern data analysis, modelling and simulation including multitype/marked points and spatial covariates License GPL version 2 or newer URL http://www.maths.uwa.edu.au/˜adrian/spatstat.html
R topics documented: DiggleGratton . Fest . . . . . . Gcross . . . . . Gdot . . . . . . Gest . . . . . . Geyer . . . . . Gmulti . . . . . Jcross . . . . . Jdot . . . . . . Jest . . . . . . Jmulti . . . . . Kcross . . . . . Kdot . . . . . . Kest . . . . . . Kest.fft . . . . Kinhom . . . . Kmeasure . . . Kmulti . . . . . LennardJones . MultiStrauss .
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R topics documented: MultiStraussHard Ord . . . . . . . OrdThresh . . . PairPiece . . . . Pairwise . . . . . Poisson . . . . . Saturated . . . . Softcore . . . . . Strauss . . . . . . StraussHard . . . affine . . . . . . . affine.owin . . . . affine.ppp . . . . allstats . . . . . . alltypes . . . . . amacrine . . . . . applynbd . . . . area.owin . . . . as.im . . . . . . . as.mask . . . . . as.owin . . . . . as.ppp . . . . . . as.rectangle . . . bdist.pixels . . . bdist.points . . . bounding.box . . bramblecanes . . cells . . . . . . . centroid.owin . . coef.ppm . . . . . complement.owin concatxy . . . . . corners . . . . . . cut.ppp . . . . . data.ppm . . . . default.dummy . demopat . . . . . diameter . . . . . dirichlet.weights dummy.ppm . . . erode.owin . . . . eroded.areas . . . fasp.object . . . . fitted.ppm . . . . fv.object . . . . . ganglia . . . . . . gridcentres . . . gridweights . . . hamster . . . . . harmonic . . . . identify.ppp . . . im . . . . . . . .
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R topics documented: im.object . . . . . inside.owin . . . . spatstat-internal . intersect.owin . . . is.im . . . . . . . . is.marked . . . . . is.marked.ppm . . is.marked.ppp . . . is.owin . . . . . . . is.ppm . . . . . . . is.ppp . . . . . . . is.subset.owin . . . kaplan.meier . . . km.rs . . . . . . . ksmooth.ppp . . . lansing . . . . . . . letterR . . . . . . . longleaf . . . . . . markcorr . . . . . . mpl . . . . . . . . nearest.raster.point nndist . . . . . . . nztrees . . . . . . . ord.family . . . . . owin . . . . . . . . owin.object . . . . pairdist . . . . . . pairsat.family . . . pairwise.family . . pcf . . . . . . . . . plot.fasp . . . . . . plot.fv . . . . . . . plot.owin . . . . . plot.ppm . . . . . . plot.ppp . . . . . . plot.quad . . . . . ppm.object . . . . ppp . . . . . . . . ppp.object . . . . . predict.ppm . . . . print.im . . . . . . print.owin . . . . . print.ppm . . . . . print.ppp . . . . . print.quad . . . . . quad.object . . . . quad.ppm . . . . . quadscheme . . . . rMatClust . . . . . rMaternI . . . . . . rMaternII . . . . . rNeymanScott . . .
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107 108 109 111 113 114 115 116 117 118 118 119 120 121 122 123 124 125 125 128 133 134 135 136 137 139 140 141 142 143 145 147 148 150 151 153 155 156 158 159 161 162 163 164 165 166 167 168 170 171 172 173
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R topics documented: rSSI . . . . . . rThomas . . . . raster.x . . . . reduced.sample redwood . . . . redwoodfull . . ripras . . . . . rmh . . . . . . rmh.default . . rmh.ppm . . . rmpoint . . . . rmpoispp . . . rotate . . . . . rotate.owin . . rotate.ppp . . . rpoint . . . . . rpoispp . . . . runifpoint . . . setmarks . . . . shift . . . . . . shift.im . . . . shift.owin . . . shift.ppp . . . . simdat . . . . . spatstat.options spokes . . . . . square . . . . . stratrand . . . subset.fasp . . subset.fv . . . . subset.im . . . subset.ppp . . . summary.im . . summary.owin . summary.ppm . summary.ppp . summary.quad superimpose . . swedishpines . union.quad . . unmark . . . . update.ppm . .
Index
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DiggleGratton
DiggleGratton
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Diggle-Gratton model
Description Creates an instance of the Diggle-Gratton pairwise interaction point process model, which can then be fitted to point pattern data. Usage DiggleGratton(delta, rho) Arguments delta rho
lower threshold δ upper threshold ρ
Details Diggle and Gratton (1984, pages 208-210) introduced the pairwise interaction point process with pair potential h(t) of the form κ t−δ if δ ≤ t ≤ ρ h(t) = ρ−δ with h(t) = 0 for t < δ and h(t) = 1 for t > ρ. Here δ, ρ and κ are parameters. Note that we use the symbol κ where Diggle and Gratton (1984) and Diggle, Gates and Stibbard (1987) use β, since in spatstat we reserve the symbol β for an intensity parameter. The parameters must all be nonnegative, and must satisfy δ ≤ ρ. The potential is inhibitory, i.e. this model is only appropriate for regular point patterns. The strength of inhibition increases with κ. For κ = 0 the model is a hard core process with hard core radius δ. For κ = ∞ the model is a hard core process with hard core radius ρ. The irregular parameters δ, ρ must be given in the call to DiggleGratton, while the regular parameter κ will be estimated. Value An object of class "interact" describing the interpoint interaction structure of a point process. Author(s) Adrian Baddeley
[email protected] http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner
[email protected] http://www.math.unb.ca/~rolf References Diggle, P.J., Gates, D.J. and Stibbard, A. (1987) A nonparametric estimator for pairwiseinteraction point processes. Biometrika 74, 763 – 770. Diggle, P.J. and Gratton, R.J. (1984) Monte Carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society, series B 46, 193 – 212.
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Fest
See Also mpl, ppm.object, Pairwise Examples data(cells) mpl(cells, ~1, DiggleGratton(0.05, 0.1))
Estimate the empty space function F
Fest
Description Estimates the empty space function F (r) from a point pattern in a window of arbitrary shape. Usage Fest(X, eps) Fest(X, eps, r) Fest(X, eps, breaks) Arguments X
The observed point pattern, from which an estimate of F (r) will be computed. An object of class ppp, or data in any format acceptable to as.ppp().
eps
A positive number. The resolution of the discrete approximation to Euclidean distance (see below). There is a sensible default.
r
numeric vector. The values of the argument r at which F (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 the Details section.
Details The empty space function (also called the “spherical contact distribution” or the “point-tonearest-event” distribution) of a stationary point process X is the cumulative distribution function F of the distance from a fixed point in space to the nearest point of X. An estimate of F derived from a spatial point pattern dataset can be used in exploratory data analysis and formal inference about the pattern (Cressie, 1991; Diggle, 1983; Ripley, 1988). In exploratory analyses, the estimate of F is a useful statistic summarising the sizes of gaps in the pattern. For inferential purposes, the estimate of F is usually compared to the true value of F for a completely random (Poisson) point process, which is F (r) = 1 − e−λπr
2
where λ is the intensity (expected number of points per unit area). Deviations between the empirical and theoretical F curves may suggest spatial clustering or spatial regularity.
Fest
7 This algorithm estimates the empty space function F from the point pattern X. It assumes that X can be treated as a realisation of a stationary (spatially homogeneous) random spatial point process in the plane, observed through a bounded window. The window (which is specified in X) may have arbitrary shape. The argument X is interpreted as a point pattern object (of class "ppp", see ppp.object) and can be supplied in any of the formats recognised by as.ppp. The algorithm uses two discrete approximations which are controlled by the parameter eps and by the spacing of values of r respectively. (See below for details.) First-time users are strongly advised not to specify these arguments. The estimation of F is hampered by edge effects arising from the unobservability of points of the random pattern outside the window. An edge correction is needed to reduce bias (Baddeley, 1998; Ripley, 1988). The two edge corrections implemented here are the border method or ”reduced sample” estimator, and the spatial Kaplan-Meier estimator (Baddeley and Gill, 1997). Our implementation makes essential use of the distance transform algorithm of image processing (Borgefors, 1986). A fine grid of pixels is created in the observation window. The Euclidean distance between two pixels is approximated by the length of the shortest path joining them in the grid, where a path is a sequence of steps√between adjacent pixels, and horizontal, vertical and diagonal steps have length 1, 1 and 2 respectively in pixel units. If the pixel grid is sufficiently fine then this is an accurate approximation. The parameter eps is the pixel width of the rectangular raster used to compute the distance transform (see below). It must not be too large: the absolute error in distance values due to discretisation is bounded by eps. If eps is not specified, the function checks whether the window X$window contains pixel raster information. If so, then eps is set equal to the pixel width of the raster; otherwise, eps defaults to 1/100 of the width of the observation window. The argument r is the vector of values for the distance r at which F (r) should be evaluated. It is also used to determine the breakpoints (in the sense of hist for the computation of histograms of distances. The reduced-sample and Kaplan-Meier estimators are computed from histogram counts. In the case of the Kaplan-Meier estimator this introduces a discretisation error which is controlled by the fineness of the breakpoints. First-time users would be strongly advised not to specify r. However, if it is specified, r must satisfy r[1] = 0, and max(r) must be larger than the radius of the largest disc contained in the window. Furthermore, the spacing of successive r values must be very fine (ideally not greater than eps/4). The algorithm also returns an estimate of the hazard rate function, λ(r), of F (r). The hazard rate is defined by d λ(r) = − log(1 − F (r)) dr The hazard rate of F has been proposed as a useful exploratory statistic (Baddeley and Gill, 1994). The estimate of λ(r) given here is a discrete approximation to the hazard rate of the Kaplan-Meier estimator of F . Note that F is absolutely continuous (for any stationary point process X), so the hazard function always exists (Baddeley and Gill, 1997). The naive empirical distribution of distances from each location in the window to the nearest point of the data pattern, is a biased estimate of F . However this is also returned by the algorithm, as it is sometimes useful in other contexts. Care should be taken not to use the uncorrected empirical F as if it were an unbiased estimator of F .
8
Fest
Value An object of class "fv", see fv.object, which can be plotted directly using plot.fv. Essentially a data frame containing six columns: r
the values of the argument r at which the function F (r) has been estimated
rs
the “reduced sample” or “border correction” estimator of F (r)
km
the spatial Kaplan-Meier estimator of F (r)
hazard
the hazard rate λ(r) of F (r) by the spatial Kaplan-Meier method
raw
the uncorrected estimate of F (r), i.e. the empirical distribution of the distance from a random point in the window to the nearest point of the data pattern X
theo
the theoretical value of F (r) for a stationary Poisson process of the same estimated intensity.
Warnings The reduced sample (border method) estimator of F is pointwise approximately unbiased, but need not be a valid distribution function; it may not be a nondecreasing function of r. Its range is always within [0, 1]. The spatial Kaplan-Meier estimator of F is always nondecreasing but its maximum value may be less than 1. The estimate of λ(r) returned by the algorithm is an approximately unbiased estimate for the integral of λ() over the corresponding histogram cell. It may exhibit oscillations due to discretisation effects. We recommend modest smoothing, such as kernel smoothing with kernel width equal to the width of a histogram cell. Note Sizeable amounts of memory may be needed during the calculation. Author(s) Adrian Baddeley
[email protected] http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner
[email protected] http://www.math.unb.ca/~rolf 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. 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.
Gcross
9
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. See Also Gest, Jest, Kest, km.rs, reduced.sample, kaplan.meier Examples require(spatstat) data(cells) Fc
