rhohat

0th

Percentile

Smoothing Estimate of Intensity as Function of a Covariate

Computes a smoothing estimate of the intensity of a point process, as a function of a (continuous) spatial covariate.

Keywords
models, spatial
Usage
rhohat(object, covariate, ...)

# S3 method for ppp rhohat(object, covariate, ..., baseline=NULL, weights=NULL, method=c("ratio", "reweight", "transform"), horvitz=FALSE, smoother=c("kernel", "local"), subset=NULL, dimyx=NULL, eps=NULL, n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL, bwref=bw, covname, confidence=0.95)

# S3 method for quad rhohat(object, covariate, ..., baseline=NULL, weights=NULL, method=c("ratio", "reweight", "transform"), horvitz=FALSE, smoother=c("kernel", "local"), subset=NULL, dimyx=NULL, eps=NULL, n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL, bwref=bw, covname, confidence=0.95)

# S3 method for ppm rhohat(object, covariate, ..., weights=NULL, method=c("ratio", "reweight", "transform"), horvitz=FALSE, smoother=c("kernel", "local"), subset=NULL, dimyx=NULL, eps=NULL, n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL, bwref=bw, covname, confidence=0.95)

# S3 method for lpp rhohat(object, covariate, ..., weights=NULL, method=c("ratio", "reweight", "transform"), horvitz=FALSE, smoother=c("kernel", "local"), subset=NULL, nd=1000, eps=NULL, random=TRUE, n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL, bwref=bw, covname, confidence=0.95)

# S3 method for lppm rhohat(object, covariate, ..., weights=NULL, method=c("ratio", "reweight", "transform"), horvitz=FALSE, smoother=c("kernel", "local"), subset=NULL, nd=1000, eps=NULL, random=TRUE, n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL, bwref=bw, covname, confidence=0.95)

Arguments
object

A point pattern (object of class "ppp" or "lpp"), a quadrature scheme (object of class "quad") or a fitted point process model (object of class "ppm" or "lppm").

covariate

Either a function(x,y) or a pixel image (object of class "im") providing the values of the covariate at any location. Alternatively one of the strings "x" or "y" signifying the Cartesian coordinates.

weights

Optional weights attached to the data points. Either a numeric vector of weights for each data point, or a pixel image (object of class "im") or a function(x,y) providing the weights.

baseline

Optional baseline for intensity function. A function(x,y) or a pixel image (object of class "im") providing the values of the baseline at any location.

method

Character string determining the smoothing method. See Details.

horvitz

Logical value indicating whether to use Horvitz-Thompson weights. See Details.

smoother

Character string determining the smoothing algorithm. See Details.

subset

Optional. A spatial window (object of class "owin") specifying a subset of the data, from which the estimate should be calculated.

dimyx,eps,nd,random

Arguments controlling the pixel resolution at which the covariate will be evaluated. See Details.

bw

Smoothing bandwidth or bandwidth rule (passed to density.default).

adjust

Smoothing bandwidth adjustment factor (passed to density.default).

n, from, to

Arguments passed to density.default to control the number and range of values at which the function will be estimated.

bwref

Optional. An alternative value of bw to use when smoothing the reference density (the density of the covariate values observed at all locations in the window).

Additional arguments passed to density.default or locfit.

covname

Optional. Character string to use as the name of the covariate.

confidence

Confidence level for confidence intervals. A number between 0 and 1.

Details

This command estimates the relationship between point process intensity and a given spatial covariate. Such a relationship is sometimes called a resource selection function (if the points are organisms and the covariate is a descriptor of habitat) or a prospectivity index (if the points are mineral deposits and the covariate is a geological variable). This command uses a nonparametric smoothing method which does not assume a particular form for the relationship.

If object is a point pattern, and baseline is missing or null, this command assumes that object is a realisation of a Poisson point process with intensity function \(\lambda(u)\) of the form $$\lambda(u) = \rho(Z(u))$$ where \(Z\) is the spatial covariate function given by covariate, and \(\rho(z)\) is a function to be estimated. This command computes estimators of \(\rho(z)\) proposed by Baddeley and Turner (2005) and Baddeley et al (2012).

The covariate \(Z\) must have continuous values.

If object is a point pattern, and baseline is given, then the intensity function is assumed to be $$\lambda(u) = \rho(Z(u)) B(u)$$ where \(B(u)\) is the baseline intensity at location \(u\). A smoothing estimator of the relative intensity \(\rho(z)\) is computed.

If object is a fitted point process model, suppose X is the original data point pattern to which the model was fitted. Then this command assumes X is a realisation of a Poisson point process with intensity function of the form $$ \lambda(u) = \rho(Z(u)) \kappa(u) $$ where \(\kappa(u)\) is the intensity of the fitted model object. A smoothing estimator of \(\rho(z)\) is computed.

The estimation procedure is determined by the character strings method and smoother and the argument horvitz. The estimation procedure involves computing several density estimates and combining them. The algorithm used to compute density estimates is determined by smoother:

  • If smoother="kernel", each the smoothing procedure is based on fixed-bandwidth kernel density estimation, performed by density.default.

  • If smoother="local", the smoothing procedure is based on local likelihood density estimation, performed by locfit.

The method determines how the density estimates will be combined to obtain an estimate of \(\rho(z)\):

  • If method="ratio", then \(\rho(z)\) is estimated by the ratio of two density estimates. The numerator is a (rescaled) density estimate obtained by smoothing the values \(Z(y_i)\) of the covariate \(Z\) observed at the data points \(y_i\). The denominator is a density estimate of the reference distribution of \(Z\).

  • If method="reweight", then \(\rho(z)\) is estimated by applying density estimation to the values \(Z(y_i)\) of the covariate \(Z\) observed at the data points \(y_i\), with weights inversely proportional to the reference density of \(Z\).

  • If method="transform", the smoothing method is variable-bandwidth kernel smoothing, implemented by applying the Probability Integral Transform to the covariate values, yielding values in the range 0 to 1, then applying edge-corrected density estimation on the interval \([0,1]\), and back-transforming.

If horvitz=TRUE, then the calculations described above are modified by using Horvitz-Thompson weighting. The contribution to the numerator from each data point is weighted by the reciprocal of the baseline value or fitted intensity value at that data point; and a corresponding adjustment is made to the denominator.

The covariate will be evaluated on a fine grid of locations, with spatial resolution controlled by the arguments dimyx,eps,nd,random. In two dimensions (i.e. if object is of class "ppp", "ppm" or "quad") the arguments dimyx, eps are passed to as.mask to control the pixel resolution. On a linear network (i.e. if object is of class "lpp") the argument nd specifies the total number of test locations on the linear network, eps specifies the linear separation between test locations, and random specifies whether the test locations have a randomised starting position.

If the argument weights is present, then the contribution from each data point X[i] to the estimate of \(\rho\) is multiplied by weights[i].

If the argument subset is present, then the calculations are performed using only the data inside this spatial region.

Value

A function value table (object of class "fv") containing the estimated values of \(\rho\) for a sequence of values of \(Z\). Also belongs to the class "rhohat" which has special methods for print, plot and predict.

Categorical and discrete covariates

This technique assumes that the covariate has continuous values. It is not applicable to covariates with categorical (factor) values or discrete values such as small integers. For a categorical covariate, use intensity.quadratcount applied to the result of quadratcount(X, tess=covariate).

References

Baddeley, A., Chang, Y.-M., Song, Y. and Turner, R. (2012) Nonparametric estimation of the dependence of a point process on spatial covariates. Statistics and Its Interface 5 (2), 221--236.

Baddeley, A. and Turner, R. (2005) Modelling spatial point patterns in R. In: A. Baddeley, P. Gregori, J. Mateu, R. Stoica, and D. Stoyan, editors, Case Studies in Spatial Point Pattern Modelling, Lecture Notes in Statistics number 185. Pages 23--74. Springer-Verlag, New York, 2006. ISBN: 0-387-28311-0.

See Also

rho2hat, methods.rhohat, parres.

See ppm for a parametric method for the same problem.

Aliases
  • rhohat
  • rhohat.ppp
  • rhohat.quad
  • rhohat.ppm
  • rhohat.lpp
  • rhohat.lppm
Examples
# NOT RUN {
  X <-  rpoispp(function(x,y){exp(3+3*x)})
  rho <- rhohat(X, "x")
  rho <- rhohat(X, function(x,y){x})
  plot(rho)
  curve(exp(3+3*x), lty=3, col=2, add=TRUE)

  rhoB <- rhohat(X, "x", method="reweight")
  rhoC <- rhohat(X, "x", method="transform")

  
# }
# NOT RUN {
  fit <- ppm(X, ~x)
  rr <- rhohat(fit, "y")

# linear network
  Y <- runiflpp(30, simplenet)
  rhoY <- rhohat(Y, "y")
# }
Documentation reproduced from package spatstat, version 1.55-1, License: GPL (>= 2)

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