Package 'spatstat.local' reference manual

Title:	Extension to 'spatstat' for Local Composite Likelihood
Description:	Extension to the 'spatstat' package, enabling the user to fit point process models to point pattern data by local composite likelihood ('geographically weighted regression').
Authors:	Adrian Baddeley [aut, cre]
Maintainer:	Adrian Baddeley <[email protected]>
License:	GPL (>= 2)
Version:	5.1-0
Built:	2024-09-14 05:38:01 UTC
Source:	https://github.com/baddstats/spatstat.local

Local Composite Likelihood

Description

Extension of the spatstat package, for fitting spatial point process models by local composite likelihood.

Details

The main functions are

`locppm`	Local likelihood fit of Poisson model
	Local pseudolikelihood fit of Gibbs model
`locmincon`	Local minimum contrast fit
	of Neyman-Scott or Cox model
`loccit`	Local composite likelihood fit
	of Neyman-Scott or Cox model

Author(s)

Adrian Baddeley [email protected].

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Cross Validated Bandwidth Selection for Locally Fitted Point Process Model

Description

Uses cross-validation to select a smoothing bandwidth for locally fitting a Cox or cluster point process model.

Usage

bw.loccit(..., use.fft=TRUE, 
               srange = NULL, ns = 9, sigma = NULL,
               fftopt=list(), 
               verbose = TRUE)
bw.loccit(..., use.fft=TRUE, 
               srange = NULL, ns = 9, sigma = NULL,
               fftopt=list(), 
               verbose = TRUE)

Arguments

`...`	Arguments passed to `kppm` to fit the homogeneous version of the model.
`use.fft`	Logical value indicating whether to use a quick-and-dirty approximation based on a first order Taylor expansion.
`srange`	Range of values of the smoothing parameter `sigma` to be searched. A numeric vector of length 2 giving the minimum and maximum values of `sigma`.
`ns`	Number of values of the smoothing parameter `sigma` in the range `srange` to be searched. A positive integer.
`sigma`	Vector of values of the smoothing parameter to be searched.
`fftopt`	Developer use only.
`verbose`	Logical value indicating whether to display progress reports.

Details

This function determines the optimal value of the smoothing parameter sigma to be used in a call to loccit.

The function loccit fits a Cox or cluster point process model to point pattern data by local composite likelihood. The degree of local smoothing is controlled by a smoothing parameter sigma which is an argument to loccit.

For each value of sigma in a search interval, the function bw.loccit fits the model locally and evaluates a cross-validation criterion. The optimal value of sigma is returned.

Value

A numerical value giving the selected bandwidth. The result also belongs to the class "bw.optim" which can be plotted.

Author(s)

Adrian Baddeley [email protected].

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

  X <- redwood[owin(c(0,1), c(-1,-1/2))]
  Ns <- if(interactive()) 16 else 2
  b <- bw.loccit(X, ~1, "Thomas", srange=c(0.07, 0.14), ns=Ns)
  b
  plot(b)
X <- redwood[owin(c(0,1), c(-1,-1/2))]
  Ns <- if(interactive()) 16 else 2
  b <- bw.loccit(X, ~1, "Thomas", srange=c(0.07, 0.14), ns=Ns)
  b
  plot(b)

Cross Validated Bandwidth Selection for Locally Fitted Point Process Model

Description

Uses cross-validation to select a smoothing bandwidth for locally fitting a Poisson or Gibbs point process model.

Usage

bw.locppm(...,
          method = c("fft", "exact", "taylor"), 
          srange = NULL, ns = 9, sigma = NULL,
          additive = TRUE, 
          verbose = TRUE)
bw.locppm(...,
          method = c("fft", "exact", "taylor"), 
          srange = NULL, ns = 9, sigma = NULL,
          additive = TRUE, 
          verbose = TRUE)

Arguments

`...`	Arguments passed to `ppm` to fit the homogeneous version of the model.
`method`	Method of calculation. The default `method="fft"` is much faster than the other choices.
`srange`	Range of values of the smoothing parameter `sigma` to be searched. A numeric vector of length 2 giving the minimum and maximum values of `sigma`.
`ns`	Number of values of the smoothing parameter `sigma` in the range `srange` to be searched. A positive integer.
`sigma`	Vector of values of the smoothing parameter to be searched. Overrides the values of `ns` and `srange`.
`additive`	Logical value indicating whether to calculate the leverage approximation on the scale of the intensity (`additive=TRUE`) or the log intensity (`additive=FALSE`). Applies only when `method = "taylor"`.
`verbose`	Logical value indicating whether to display progress reports.

Details

This function determines the optimal value of the smoothing parameter sigma to be used in a call to locppm.

The function locppm fits a Poisson or Gibbs point process model to point pattern data by local composite likelihood. The degree of local smoothing is controlled by a smoothing parameter sigma which is an argument to locppm.

This function bw.locppm determines the optimal value of sigma by cross-validation. For each value of sigma in a search interval, the function bw.locppm fits the model locally with smoothing bandwidth sigma, and evaluates the composite likelihood cross-validation criterion LCV(sigma) defined in Baddeley (2016), section 3.2. The value of sigma which maximises LCV(sigma) is returned.

Value

A numerical value giving the selected bandwidth. The result also belongs to the class "bw.optim" which can be plotted.

Author(s)

Adrian Baddeley [email protected].

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

  Ns <- if(interactive()) 16 else 2
  b <- bw.locppm(swedishpines, ~1, srange=c(2.5,4.5), ns=Ns)
  b
  plot(b)
Ns <- if(interactive()) 16 else 2
  b <- bw.locppm(swedishpines, ~1, srange=c(2.5,4.5), ns=Ns)
  b
  plot(b)

Homogeneity Test for Local Poisson or Gibbs Model

Description

Conducts a Monte Carlo test of homogeneity for a locally-fitted Poisson or Gibbs point process model.

Usage

   homtest(X, ..., nsim = 19,
           test = c("residuals", "score", "taylor", "likelihood"),
           locations = c("coarse", "fine", "split"),
           ladjust = NULL,
           use.fft = NULL,
           simul = NULL,
           verbose = TRUE, Xname = NULL)
homtest(X, ..., nsim = 19,
           test = c("residuals", "score", "taylor", "likelihood"),
           locations = c("coarse", "fine", "split"),
           ladjust = NULL,
           use.fft = NULL,
           simul = NULL,
           verbose = TRUE, Xname = NULL)

Arguments

`X`	A point pattern (object of class `"ppp"`).
`...`	Additional arguments passed to `locppm` to determine the locally fitted model, and passed to `ppm` to determine the homogeneous model.
`nsim`	Number of simulations for the Monte Carlo test.
`test`	The local test statistic to be used: either `"likelihood"` for the local likelihood ratio test statistic, `"taylor"` for the Taylor approximation to the local likelihood ratio test statistic, `"score"` for the local score test statistic, or `"residuals"` for the squared local residuals.
`locations`, `use.fft`	Arguments passed to `locppm` to control the calculation of variances (if `method="local"`). See Details.
`ladjust`	Optional argument passed to `homtestmap` specifying a data-dependent adjustment of the test statistic.
`simul`	Optional information that determines how to simulate the realisations from the null hypothesis. An expression in the R language (that will be evaluated `nsim` times to obtain the simulated patterns), or a list that contains the simulated point patterns.
`verbose`	Logical value indicating whether to print progress reports.
`Xname`	Optional character string name for the dataset `X`, to be printed in the test output.

Details

This function performs a Monte Carlo test of the null hypothesis of homogeneity (i.e.\ constant parameter values) for the locally-fitted Poisson point process or Gibbs point process specified by the arguments.

The type of test is controlled by the argument test.

test="likelihood": the locally fitted model is computed as locppm(X, ...). The local composite likelihood ratio test statistic of this model is computed at each location, and the mean of this statistic over the window is computed.
test="taylor": the locally fitted model is computed as locppm(X, ...). The Taylor approximation to the local composite likelihood ratio test statistic of this model is computed at each location, and the mean of this statistic over the window is computed.
test="score": the locally fitted model is computed as locppm(X, ...). The local score test statistic of this model is computed at each location, and the mean of this statistic over the window is computed.
method="residuals": the homogeneous model is fitted as ppm(X, ...). The smoothed score residuals of this model are computed at each location, and the mean of the squared norm over the window is computed.

The test statistic is computed for the data pattern X and for each of nsim simulated realisations from the homogeneous model. The Monte Carlo $p$ -value is computed.

Value

An object of class "htest" containing the test outcome.

Author(s)

Adrian Baddeley [email protected].

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

   ## Not run: 
   homtest(swedishpines)
   
## End(Not run)
   
## Not run: 
   homtest(swedishpines)
   
## End(Not run)

Test Statistic for Homogeneity Test

Description

Compute the test statistic for the test of homogeneity of a locally-fitted Poisson or Gibbs point process model.

Usage

homteststat(object, ..., verbose = FALSE)

homtestmap(object, ...,
           what=c("components", "statistic", "pvalue"),
           test = c("score", "taylor", "likelihood"),
           ladjust=c("none", "moment", "PSS"),
           calibrate=c("chisq", "Satterthwaite", "firstmoment"),
           simple = !is.null(theta0),
           theta0 = NULL,
           poolmoments=NULL,
           sigma = NULL, 
           saveall = FALSE, 
           use.fft = TRUE,
           verbose = TRUE)

## S3 method for class 'homtestmap'
update(object, ..., 
           what=NULL, test=NULL, ladjust=NULL,
           calibrate=NULL, saveall=FALSE, poolmoments=NULL)
homteststat(object, ..., verbose = FALSE)

homtestmap(object, ...,
           what=c("components", "statistic", "pvalue"),
           test = c("score", "taylor", "likelihood"),
           ladjust=c("none", "moment", "PSS"),
           calibrate=c("chisq", "Satterthwaite", "firstmoment"),
           simple = !is.null(theta0),
           theta0 = NULL,
           poolmoments=NULL,
           sigma = NULL, 
           saveall = FALSE, 
           use.fft = TRUE,
           verbose = TRUE)

## S3 method for class 'homtestmap'
update(object, ..., 
           what=NULL, test=NULL, ladjust=NULL,
           calibrate=NULL, saveall=FALSE, poolmoments=NULL)

Arguments

`object`	Locally-fitted point process (object of class `"locppm"`) or an object previously computed by `homtestmap`.
`...`	For `homteststat`, arguments passed to `homtestmap`. For `homtestmap`, additional unmatched arguments are ignored.
`what`	Character string (partially matched) indicating whether to return the vector components of the local test statistic, or the value of the local test statistic, or the local $p$ -values.
`test`	Character string (partially matched) indicating whether to perform the local score test (`test="score"`), or the local composite likelihood ratio test approximately (`test="taylor"`) or exactly (`test="likelihood"`).
`ladjust`	Character string (partially matched) specifying an adjustment to the composite likelihood ratio test statistic.
`calibrate`	Character string (partially matched) specifying how to calculate $p$ -values from the test statistic.
`simple`	Logical value indicating whether to treat the fitted model as a simple null hypothesis (`simple=TRUE`) or as an estimate of the parameters in a composite null hypothesis (`simple=FALSE`, the default).
`theta0`	Coefficient vector specifying a simple null hypothesis.
`poolmoments`	Logical value indicating how to calculate the reference distribution for the likelihood ratio test statistic (and thus how to calculate $p$ -values). See Details.
`sigma`	Smoothing bandwidth.
`saveall`	Logical value indicating whether to compute a complete set of sufficient statistics and save them as an attribute of the result. See Details.
`use.fft`	For software testing purposes only. Logical value indicating whether to use data computed by the Fast Fourier Transform.
`verbose`	Logical value indicating whether to print progress reports.

Details

These functions are used by homtest to perform a Monte Carlo test of the null hypothesis of homogeneity (i.e.\ constant parameter values) for the locally-fitted Poisson point process or Gibbs point process object.

The function homtestmap computes either the local likelihood ratio test statistic or the local score test statistic. If what="statistic", then the result is a scalar-valued function giving the local values of the test statistic. If what="pvalue", the result is a scalar-valued function $p(v)$ giving the local $p$ -value at each location $v$ . If what="components", the result is a vector-valued function $T(v)$ containing the components of the quadratic form; the squared norm of $T(v)$ is equal to the desired test statistic at each location $v$ .

If saveall=TRUE, then a complete set of sufficient statistics is calculated and stored as an attribute of the result. This makes it possible to compute all of the statistics and $p$ values described above.

The function update.homtestmap, a method for the generic function update, converts an object of class "homtestmap" from one of these formats to another, where possible. Except in trivial cases, this requires that the "homtestmap" object was computed with saveall=TRUE.

The function homteststat computes the mean of the local test statistic or the mean of the local $p$ -values over the observation window.

To compute the $p$ -values when test="likelihood" or test="taylor", the values of the local likelihood ratio test statistic are referred to a gamma distribution whose first two moments are estimated from the data. If poolmoments=FALSE, the local estimates of the moments are used; if poolmoments=TRUE, the spatial average of these estimates is used. The default is to use pooling whenever it is theoretically justified, namely when the template model is a stationary point process.

Finer control over the computation is possible using the arguments ... passed to locppm.

Value

For homteststat, a numeric value giving the test statistic.

For homtestmap and update.homtestmap, a spatially-sampled function object (class "ssf"; see ssf). This object also belongs to the special class "homtestmap" which has a print method.

Author(s)

Adrian Baddeley [email protected].

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

   example(locppm)
   plot(H <- homtestmap(fit))
   H
example(locppm)
   plot(H <- homtestmap(fit))
   H

Locally Fitted Cluster or Cox Point Process Model

Description

Fits a Neyman-Scott cluster process or Cox point process model using a locally-weighted composite likelihood.

Usage

loccit(X, trend = ~1,
       clusters = c("Thomas", "MatClust", "Cauchy", "VarGamma", "LGCP"),
       covariates = NULL,
       ...,
       diagnostics = FALSE,
       taylor = FALSE,
       sigma = NULL, f = 1/4,
       clustargs = list(), control = list(), 
       rmax,
       covfunargs=NULL, use.gam=FALSE, nd=NULL, eps=NULL,
       niter=3,
       fftopt = list(),
       verbose = TRUE)
loccit(X, trend = ~1,
       clusters = c("Thomas", "MatClust", "Cauchy", "VarGamma", "LGCP"),
       covariates = NULL,
       ...,
       diagnostics = FALSE,
       taylor = FALSE,
       sigma = NULL, f = 1/4,
       clustargs = list(), control = list(), 
       rmax,
       covfunargs=NULL, use.gam=FALSE, nd=NULL, eps=NULL,
       niter=3,
       fftopt = list(),
       verbose = TRUE)

Arguments

`X`	Point pattern.
`trend`	Formula (without a left hand side) specifying the form of the logarithm of the intensity.
`clusters`	Character string determining the cluster model. Partially matched.
`covariates`	The values of any spatial covariates (other than the Cartesian coordinates) required by the model. A named list of pixel images, functions, windows or numeric constants.
`diagnostics`	Whether to perform auxiliary calculations in addition to the local estimates of the model parameters.
`...`	Additional arguments passed to `as.mask` to control the spatial resolution in the Fast Fourier Transform.
`taylor`	Logical value indicating whether to fit the model exactly at each spatial location (`taylor=FALSE`, the default) or to compute a first-order Taylor approximation to the fitted parameters (`taylor=TRUE`). The Taylor approximation is much faster.
`sigma`	Standard deviation of Gaussian kernel for local likelihood.
`f`	Argument passed to `bw.frac` to compute a value for `sigma` if it is missing or `NULL`.
`clustargs`	List of additional parameters for the cluster model, passed to the function `RFcov` in the `RandomFields` package.
`control`	List of control arguments passed to the generic optimisation function `optim`.
`rmax`	Maximum distance between pairs of points that will contribute to the composite likelihood.
`covfunargs`, `use.gam`, `nd`, `eps`	Arguments passed to `ppm` to control the intensity model and intensity fitting.
`niter`	Number of iterations in algorithm if `taylor=FALSE`.
`fftopt`	Developer use only.
`verbose`	Logical. If `TRUE`, print progress reports.

Details

This function fits a Cox or cluster process model to point pattern data locally, using the local Palm likelihood technique (Baddeley, 2016, section 8).

It can be used in the same way as kppm and effectively performs local fitting of the same model.

Value

An object of class "loccit".

Author(s)

Adrian Baddeley [email protected].

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

   X <- redwood[owin(c(0,1), c(-1,-1/2))]
   fit <- loccit(X, ~1, "Thomas", nd=5, control=list(maxit=20))
   fit
X <- redwood[owin(c(0,1), c(-1,-1/2))]
   fit <- loccit(X, ~1, "Thomas", nd=5, control=list(maxit=20))
   fit

Locally Fitted Cluster or Cox Point Process Model

Description

Fits a Neyman-Scott cluster process or Cox point process model using local minimum contrast.

Usage

locmincon(..., sigma = NULL, f = 1/4, verbose = TRUE,
                 localstatargs = list(), LocalStats = NULL,
                 tau = NULL)
locmincon(..., sigma = NULL, f = 1/4, verbose = TRUE,
                 localstatargs = list(), LocalStats = NULL,
                 tau = NULL)

Arguments

`...`	Arguments passed to `kppm` to determine the template homogeneous model.
`sigma`	Standard deviation of Gaussian kernel for local likelihood.
`f`	Argument passed to `bw.frac` to compute a value for `sigma` if it is missing or `NULL`.
`verbose`	Logical. If `TRUE`, print progress reports.
`localstatargs`	Optional. List of arguments to be passed to the local statistic `localK`, `localKinhom`, `localpcf` or `localpcfinhom`.
`LocalStats`	Optional. Values of the local statistics, if they have already been computed.
`tau`	Optional. Bandwidth for smoothing the fitted cluster parameters.

Details

The template or homogeneous model is first fitted by kppm. The statistic used to fit the template model is determined (as explained in the help for kppm) by the arguments statistic and trend.

The local version of this statistic is then computed. If statistic="K" and trend=~1 for example, the template model is fitted using the $K$ function Kest, and the local version is the local $K$ function localK. The possibilities are:

`statistic`	stationary?	template	local
`"K"`	yes	`Kest`	`localK`
`"K"`	no	`Kinhom`	`localKinhom`
`"pcf"`	yes	`pcf`	`localpcf`
`"pcf"`	no	`pcfinhom`	`localpcfinhom`

These local functions, one for each data point, are then spatially averaged, using a Gaussian kernel with standard deviation sigma. Finally the model is fitted to each of the averaged local functions to obtain a local fit at each data point.

Value

Object of class "locmincon".

Author(s)

Adrian Baddeley [email protected].

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

   X <- redwood[owin(c(0,1), c(-1,-1/2))]
   fit <- locmincon(X, ~1, "Thomas", sigma=0.07)
   fit
X <- redwood[owin(c(0,1), c(-1,-1/2))]
   fit <- locmincon(X, ~1, "Thomas", sigma=0.07)
   fit

Locally Fitted Poisson or Gibbs Point Process Model

Description

Fits Poisson or Gibbs point process model using local likelihood or pseudolikelihood.

Usage

locppm(..., sigma = NULL, f = 1/4,
       vcalc = c("none", "t", "hessian", "hom", "lik", "full"),
       locations=c("split", "fine", "coarse"),
       ngrid = NULL, grideps = NULL, verbose = TRUE,
       use.fft=FALSE, fft.algorithm="closepairs")
locppm(..., sigma = NULL, f = 1/4,
       vcalc = c("none", "t", "hessian", "hom", "lik", "full"),
       locations=c("split", "fine", "coarse"),
       ngrid = NULL, grideps = NULL, verbose = TRUE,
       use.fft=FALSE, fft.algorithm="closepairs")

Arguments

`...`	Arguments passed to `ppm` to fit the homogeneous model.
`sigma`	Standard deviation of Gaussian kernel for local likelihood.
`f`	Argument passed to `bw.frac` to compute a value for `sigma` if it is missing or `NULL`.
`vcalc`	Type of variance calculation to be performed. See Details.
`locations`	Spatial locations for local calculations. See Details.
`ngrid`	Dimensions of coarse grid, if used. See Details. Incompatible with `grideps`.
`grideps`	Grid spacing of coarse grid, if used. See Details. Incompatible with `ngrid`.
`verbose`	Logical. If `TRUE`, print progress reports.
`use.fft`	Logical value indicating whether to perform computations using the Fast Fourier Transform. With `use.fft = TRUE` the code runs much faster but some quantities are not computed exactly. See Details.
`fft.algorithm`	Developer use only.

Details

This function fits a Poisson or Gibbs point process model to point pattern data by local likelihood or local pseudolikelihood respectively.

This command should be used in the same way as ppm. The point pattern data and the specification of the model are given in the leading arguments ... which are passed directly to ppm.

In all cases, the local estimates of the coefficients are computed. However, because the variance calculations are time-consuming, the default is not to perform them. This is controlled by the argument vcalc.

vcalc = "none":: no variance calculations are performed.
vcalc = "t":: the $t$ statistic for each parameter is computed for the local model.
vcalc = "hessian":: the local Hessian matrix is computed, and its negative inverse is used as a surrogate for the local variance.
vcalc = "hom":: No local fitting is performed. Calculations are performed only for the homogeneous (template) model. The variance of the local parameter estimates under the homogeneous model is computed.
vcalc = "lik":: In addition to the calculations for vcalc="hom" described above, if use.fft=FALSE the algorithm also computes the local composite likelihood ratio test statistic for the test of homogeneity. If use.fft=TRUE then vcalc="lik" is equivalent to vcalc="hom".
vcalc = "full":: all variance calculations are performed for the local model.

The spatial locations, where the model fits and variance calculations are performed, are determined by the argument locations.

locations = "fine":: The calculations are performed at every quadrature point of the model. This can take a very long time.
locations = "coarse":: The calculations are performed at the points of a coarse grid with dimensions specified by ngrid or grideps.
locations = "split":: The fitted coefficients are computed at every quadrature point of the model, but the variance calculations (if any) are performed at a coarse grid of locations, specified by ngrid or grideps. If neither ngrid nor grideps is specified, the default is ngrid=10.

If use.fft=FALSE (the default), all desired quantities are computed exactly, by an iterative algorithm that fits a separate model at each spatial location. This can be quite slow.

If use.fft=TRUE, we only compute quantities that can be obtained using the Fast Fourier Transform, resulting in much faster calculations (sometimes 3 orders of magnitude faster) when locations="fine". Properties of the homogeneous model are computed accurately. Properties of the locally-fitted model are approximated by a first order Taylor expansion.

Value

An object of class "locppm" representing the fitted model.

Author(s)

Adrian Baddeley [email protected].

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

   fit <- locppm(swedishpines, ~1, sigma=9, nd=20)
   fit
fit <- locppm(swedishpines, ~1, sigma=9, nd=20)
   fit

Methods for Local Cluster or Cox Models

Description

Methods for various generic functions, for the class "locmincon" of locally fitted cluster or Cox point process models.

Usage

  ## S3 method for class 'locmincon'
as.ppp(X, ...)

  ## S3 method for class 'locmincon'
print(x, ...)
## S3 method for class 'locmincon'
as.ppp(X, ...)

  ## S3 method for class 'locmincon'
print(x, ...)

Arguments

`x`, `X`	A locally-fitted Cox or cluster point process model (object of class `"locmincon"`).
`...`	Additional arguments

Details

Objects of class "locmincon" represent locally fitted cluster or Cox point process models.

The functions documented here provided methods for this class, for the generic functions as.ppp and print.

Value

as.ppp returns an object of class "ppp".

print returns NULL.

Author(s)

Adrian Baddeley

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

  example(locmincon)
  fit
  as.ppp(fit)  
example(locmincon)
  fit
  as.ppp(fit)

Methods for Local Gibbs Models

Description

Methods for various generic functions, for the class "locppm" of locally fitted Gibbs point process models.

Usage

  ## S3 method for class 'locppm'
as.interact(object)

  ## S3 method for class 'locppm'
as.ppm(object)

  ## S3 method for class 'locppm'
coef(object, ..., which = c("local", "homogeneous"))

  ## S3 method for class 'locppm'
confint(object, parm, level = 0.95, ..., which = c("local", "homogeneous"))

  ## S3 method for class 'locppm'
is.poisson(x)

  ## S3 method for class 'locppm'
print(x, ...)
## S3 method for class 'locppm'
as.interact(object)

  ## S3 method for class 'locppm'
as.ppm(object)

  ## S3 method for class 'locppm'
coef(object, ..., which = c("local", "homogeneous"))

  ## S3 method for class 'locppm'
confint(object, parm, level = 0.95, ..., which = c("local", "homogeneous"))

  ## S3 method for class 'locppm'
is.poisson(x)

  ## S3 method for class 'locppm'
print(x, ...)

Arguments

`object`, `x`	A locally-fitted Gibbs point process model (object of class `"locppm"`).
`...`	Additional arguments passed to the default method (for `confint.locppm`) or ignored (by `coef.locppm`).
`which`	Character string determining whether to perform calculations for the local Gibbs model (`which="local"`, the default) or the corresponding homogeneous Gibbs model (`which="homogeneous"`).
`parm`	The parameter or parameters for which a confidence interval is desired. A character string or character vector matching the names of `coef(object)`, or an index or index vector that can be applied to `coef(object)`.
`level`	Confidence level: a number between 0 and 1.

Details

Objects of class "locppm" represent locally fitted Gibbs point process models.

The functions documented here provided methods for this class, for the generic functions as.interact, as.ppm, coef, confint, is.poisson and print.

For the coef and confint methods, the calculations can be performed either on the locally fitted model or on its homogeneous equivalent, by changing the argument which.

Value

as.interact returns an interaction structure (object of class "interact").

as.ppm returns a fitted Gibbs model (object of class "ppm").

coef and confint return a numeric vector if which="homogeneous" and an object of class "ssf" if which="local".

is.poisson returns a logical value.

print returns NULL.

Author(s)

Adrian Baddeley

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

   fit <- locppm(swedishpines, ~1, sigma=9, nd=20,
                 vcalc="full", locations="coarse")
   fit
   is.poisson(fit)
   coef(fit)
   coef(fit, which="homogeneous")
   confint(fit)
   confint(fit, which="homogeneous")
   as.ppm(fit)
   as.interact(fit)
fit <- locppm(swedishpines, ~1, sigma=9, nd=20,
                 vcalc="full", locations="coarse")
   fit
   is.poisson(fit)
   coef(fit)
   coef(fit, which="homogeneous")
   confint(fit)
   confint(fit, which="homogeneous")
   as.ppm(fit)
   as.interact(fit)

Plot a Locally Fitted Cluster or Cox Point Process Model

Description

Plot an object of class "loccit" representing a locally-fitted cluster or Cox point process model.

Usage

## S3 method for class 'loccit'
plot(x, ...,
               what = c("modelpar", "coefs", "lambda"),
               how = c("smoothed", "exact"), which = NULL,
               pre=NULL, post=NULL)
## S3 method for class 'loccit'
plot(x, ...,
               what = c("modelpar", "coefs", "lambda"),
               how = c("smoothed", "exact"), which = NULL,
               pre=NULL, post=NULL)

Arguments

`x`	The model to be plotted. A locally-fitted cluster or Cox point process model (object of class `"loccit"`).
`...`	Arguments passed to `plot.ppp` or `plot.im` to control the plot.
`what`	Character string determining which quantities to display: `"modelpar"` for the cluster model parameters, `"coefs"` for the trend coefficients, or `"lambda"` for the fitted intensity.
`how`	Character string determining whether to display the fitted parameter values at the data points (`how="exact"`) or the smoothed fitted parameters as pixel images (`how="smoothed"`).
`which`	Optional. Which component(s) of the vector-valued quantity to display. An index or index vector. Default is to plot all components.
`pre`, `post`	Transformations to apply before and after smoothing.

Details

This is a method for the generic command plot for the class "loccit".

The argument which, if present, specifies which fitted parameters are displayed. It may be any kind of index for a numeric vector.

The quantities are computed at irregularly-placed points. If how="exact" the exact computed values will be displayed as circles centred at the locations where they were computed. If how="smoothed" these values will be kernel-smoothed using Smooth.ppp and displayed as a pixel image.

Value

NULL.

Author(s)

Adrian Baddeley [email protected].

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

   X <- redwood[owin(c(0,1), c(-1,-1/2))]
   fitc <- loccit(X, ~1, "Thomas", nd=5, control=list(maxit=20))
   plot(fitc, how="exact")  
   plot(fitc, how="smoothed")  
X <- redwood[owin(c(0,1), c(-1,-1/2))]
   fitc <- loccit(X, ~1, "Thomas", nd=5, control=list(maxit=20))
   plot(fitc, how="exact")  
   plot(fitc, how="smoothed")

Plot a Locally Fitted Cluster or Cox Point Process Model

Description

Plot an object of class "locmincon" representing a locally-fitted cluster or Cox point process model.

Usage

## S3 method for class 'locmincon'
plot(x, ...,
               how = c("exact", "smoothed"),
               which = NULL, sigma = NULL, do.points = TRUE)
## S3 method for class 'locmincon'
plot(x, ...,
               how = c("exact", "smoothed"),
               which = NULL, sigma = NULL, do.points = TRUE)

Arguments

`x`	The model to be plotted. A locally-fitted cluster or Cox point process model (object of class `"locmincon"` or `"loccit"`).
`...`	Arguments passed to `plot.ppp` or `plot.im` to control the plot.
`how`	Character string determining whether to display the fitted parameter values at the data points (`how="exact"`) or the smoothed fitted parameters as pixel images (`how="smoothed"`).
`which`	Optional. Which component(s) of the vector-valued quantity to display. An index or index vector. Default is to plot all components.
`sigma`	Numeric. Smoothing bandwidth to be used if `how="smoothed"`.
`do.points`	Logical. Whether to display the original point data as well.

Details

This is a method for the generic command plot for the class "locmincon".

The argument which, if present, specifies which fitted parameters are displayed. It may be any kind of index for a numeric vector.

Value

NULL.

Author(s)

Adrian Baddeley [email protected].

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

   X <- redwood[owin(c(0,1), c(-1,-1/2))]
   fitm <- locmincon(X, ~1, "Thomas", sigma=0.07)
   plot(fitm, how="smoothed")  
   plot(fitm, how="exact")  
X <- redwood[owin(c(0,1), c(-1,-1/2))]
   fitm <- locmincon(X, ~1, "Thomas", sigma=0.07)
   plot(fitm, how="smoothed")  
   plot(fitm, how="exact")

Plot a Locally Fitted Poisson or Gibbs Model

Description

Plot an object of class "locppm" representing a locally-fitted Poisson or Gibbs point process model.

Usage

## S3 method for class 'locppm'
plot(x, ..., what = "cg", which = NULL)

## S3 method for class 'locppm'
contour(x, ..., what = "cg", which = NULL)
## S3 method for class 'locppm'
plot(x, ..., what = "cg", which = NULL)

## S3 method for class 'locppm'
contour(x, ..., what = "cg", which = NULL)

Arguments

`x`	A locally-fitted Poisson or Gibbs point process model (object of class `"locppm"`).
`...`	Arguments passed to `plot.ssf` to control the plot.
`what`	What quantity to display. A character string. The default is to display the fitted coefficient vectors.
`which`	Which component(s) of the vector-valued quantity to display. An index or index vector.

Details

These are methods for the generic commands plot and contour, for the class "locppm".

The argument what specifies what quantity will be displayed:

`"cg"`	Fitted coefficients of local model
`"vg"`	Local variance matrix for Gibbs model
`"vh"`	Local variance matrix for homogeneous model
`"tg"`	$t$ -statistics based on `"coefs"` and `"vg"`

Typically these quantities are vector-valued (matrices are converted to vectors). The argument which, if present, specifies which elements of the vector are displayed. It may be any kind of index for a numeric vector.

The plotting is performed by plot.ssf.

Value

NULL.

Author(s)

Adrian Baddeley [email protected].

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

   fit <- locppm(swedishpines, ~1, sigma=9, nd=20,
             vcalc="hessian", locations="coarse")
   plot(fit)  
   plot(fit, what="Vg")  
fit <- locppm(swedishpines, ~1, sigma=9, nd=20,
             vcalc="hessian", locations="coarse")
   plot(fit)  
   plot(fit, what="Vg")

Prediction for Locally-Fitted Cox or Cluster Model

Description

Computes the fitted intensity of a locally-fitted Cox process or cluster process model.

Usage

  ## S3 method for class 'loccit'
predict(object, ...)

  ## S3 method for class 'loccit'
fitted(object, ..., new.coef=NULL)
## S3 method for class 'loccit'
predict(object, ...)

  ## S3 method for class 'loccit'
fitted(object, ..., new.coef=NULL)

Arguments

`object`	Locally fitted point process model (object of class `"loccit"` fitted by `loccit`).
`...`	Arguments passed to `predict.locppm`.
`new.coef`	New values for the fitted coefficients. A matrix in which each row gives the fitted coefficients at one of the quadrature points of the model.

Details

The fitted intensity is computed.

Value

An object of class "ssf" as described in ssf.

Author(s)

Adrian Baddeley [email protected].

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

  X <- redwood[owin(c(0,1), c(-1,-1/2))]
  fit <- loccit(X, ~1, "Thomas", nd=5, control=list(maxit=20))
  lam <- predict(fit)
X <- redwood[owin(c(0,1), c(-1,-1/2))]
  fit <- loccit(X, ~1, "Thomas", nd=5, control=list(maxit=20))
  lam <- predict(fit)

Prediction of a Locally Fitted Poisson or Gibbs Point Process Model

Description

Computes the fitted intensity of a locally-fitted Poisson point process model, or the fitted intensity, trend or conditional intensity of a locally-fitted Gibbs point process model.

Usage

## S3 method for class 'locppm'
fitted(object, ...,
                        type = c("cif", "trend", "intensity"),
                        new.coef=NULL)

## S3 method for class 'locppm'
predict(object, ...,
                         type = c("cif", "trend", "intensity"),
                         locations=NULL, new.coef=NULL)
## S3 method for class 'locppm'
fitted(object, ...,
                        type = c("cif", "trend", "intensity"),
                        new.coef=NULL)

## S3 method for class 'locppm'
predict(object, ...,
                         type = c("cif", "trend", "intensity"),
                         locations=NULL, new.coef=NULL)

Arguments

`object`	A locally-fitted Poisson or Gibbs point process model (object of class `"locppm"`).
`...`	Currently ignored.
`new.coef`	New vector or matrix of values for the model coefficients.
`locations`	Point pattern of locations where prediction should be computed.
`type`	Character string (partially matched) specifying the type of predicted value: the conditional intensity `"cif"` (the default), or the first order trend, or the intensity. For Poisson models all three options are equivalent.

Details

These are methods for the generic functions fitted and predict for the class "locppm" of locally-fitted Gibbs point process models.

The fitted method computes, for each quadrature point v (or in general, at each point v where a local model was fitted), the intensity of the locally-fitted model at v. The result is a numeric vector.

The predict computes the fitted intensity at any specified set of locations, and returns the result as an ssf object.

Value

For fitted.locppm, a numeric vector.

For predict.locppm, an object of class "ssf" as described in ssf.

Author(s)

Adrian Baddeley [email protected].

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

  fit <- locppm(cells, sigma=0.1, use.fft=TRUE)
  lam <- predict(fit)
fit <- locppm(cells, sigma=0.1, use.fft=TRUE)
  lam <- predict(fit)

Sibling Probability of Locally Fitted Cluster Point Process

Description

Computes the sibling probability of a locally fitted cluster point process model.

Usage

  ## S3 method for class 'loccit'
psib(object)

  ## S3 method for class 'locmincon'
psib(object)
## S3 method for class 'loccit'
psib(object)

  ## S3 method for class 'locmincon'
psib(object)

Arguments

object

Fitted cluster point process model (object of class "loccit" or "locmincon").

Details

In a Poisson cluster process, two points are called siblings if they belong to the same cluster, that is, if they had the same parent point. If two points of the process are separated by a distance $r$ , the probability that they are siblings is $p(r) = 1 - 1/g(r)$ where $g$ is the pair correlation function of the process.

The value $p(0) = 1 - 1/g(0)$ is the probability that, if two points of the process are situated very close to each other, they came from the same cluster. This probability is an index of the strength of clustering, with high values suggesting strong clustering.

This concept was proposed in Baddeley, Rubak and Turner (2015, page 479) and Baddeley (2016).

The function psib is generic, with methods for "kppm", "loccit" and "locmincon".

The functions described here are the methods for locally-fitted cluster models of class "loccit" and "locmincon". They compute the spatially-varying sibling probability of the locally-fitted model.

Value

A spatially sampled function (object of class "ssf") giving the spatially-varying sibling probability.

Author(s)

Adrian Baddeley [email protected].

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

   ## Not run: 
   fit <- loccit(redwood, ~1, "Thomas")
   
## End(Not run)
   
   fit
   plot(psib(fit))
## Not run: 
   fit <- loccit(redwood, ~1, "Thomas")
   
## End(Not run)
   
   fit
   plot(psib(fit))

Smooth a Locally Fitted Cluster or Cox Point Process Model

Description

Applies kernel smoothing to the fitted cluster parameters of a locally-fitted cluster or Cox point process model.

Usage

## S3 method for class 'locmincon'
Smooth(X, tau = NULL, ...)
## S3 method for class 'locmincon'
Smooth(X, tau = NULL, ...)

Arguments

`X`	Object of class `"locmincon"`.
`tau`	Smoothing bandwidth.
`...`	Additional arguments passed to `Smooth.ppp` controlling the smoothing and the pixel resolution.

Details

An object of class "locmincon" represents a locally-fitted Cox or cluster point process model. It provides estimates of the cluster parameters at each of the data points of the original point pattern dataset.

The parameter estimates will be smoothed using a Gaussian kernel with standard deviation tau.

Value

A pixel image or a list of pixel images.

Author(s)

Adrian Baddeley [email protected].

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

   fit <- locmincon(redwood)
   Smooth(fit, tau=0.1)
fit <- locmincon(redwood)
   Smooth(fit, tau=0.1)

Smooth a locally fitted Gibbs model

Description

Applies kernel smoothing to one of the components of a locally-fitted Gibbs point process model.

Usage

## S3 method for class 'locppm'
Smooth(X, ..., what = "cg")
## S3 method for class 'locppm'
Smooth(X, ..., what = "cg")

Arguments

`X`	A locally-fitted Gibbs point process model (object of class `"locppm"`).
`...`	Arguments passed to `Smooth.ppp` to control the smoothing.
`what`	Component to be smoothed. A character string. The default is to smooth the fitted coefficient vectors.

Details

This function extracts the selected quantity from the fitted object and spatially smooths it using Smooth.ppp. The result is a pixel image or a list of pixel images.

Value

A pixel image or a list of pixel images.

Author(s)

Adrian Baddeley [email protected].

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

  fit <- locppm(cells, sigma=0.1, use.fft=TRUE)
  plot(Smooth(fit))
fit <- locppm(cells, sigma=0.1, use.fft=TRUE)
  plot(Smooth(fit))

Test of Effect in Locally Fitted Point Process Model

Description

Perform a local $t$ -test for the presence of a covariate effect in a locally fitted Poisson or Gibbs point process model.

Usage

ttestmap(object, term, ...,
         method = c("exact", "hessian", "taylor"),
         grid = FALSE, 
         ngrid = NULL, grideps = NULL,
         verbose = TRUE)
ttestmap(object, term, ...,
         method = c("exact", "hessian", "taylor"),
         grid = FALSE, 
         ngrid = NULL, grideps = NULL,
         verbose = TRUE)

Arguments

`object`	Locally fitted Poisson or Gibbs point process model (object of class `"locppm"`).
`term`	Term to be dropped from the model. A character string matching a term in the model formula
`...`	Ignored.
`method`	Choice of method to be used to evaluate the $t$ statistic. See Details.
`grid`	Logical. If `FALSE`, compute the test statistic at all quadrature points of the model. If `TRUE`, compute at a coarse grid of locations.
`ngrid`	Number of grid points (in each axis direction) for the coarse grid. Incompatible with `grideps`.
`grideps`	Spacing (horizontal and vertical) between grid points for the coarse grid. Incompatible with `ngrid`.
`verbose`	Logical value indicating whether to print progress reports.

Details

The argument object should be a locally-fitted Poisson or Gibbs point process model (object of class "locppm" created by locppm).

This function computes the local $t$ test statistic for the test that a particular covariate effect in the model is zero. This is described in Baddeley (2016, sections 3 and 5).

Value

Object of class "ssf".

Author(s)

Adrian Baddeley [email protected].

References

Baddeley, A. (2017) Local composite likelihood for spatial point patterns. Spatial Statistics 22, 261–295. DOI: 10.1016/j.spasta.2017.03.001

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Examples

 fit <- with(copper,
        locppm(Points, ~D, covariates=list(D=distfun(Lines)), nd=c(7,15)))
 plot(ttestmap(fit, "D"))
fit <- with(copper,
        locppm(Points, ~D, covariates=list(D=distfun(Lines)), nd=c(7,15)))
 plot(ttestmap(fit, "D"))

Evaluate an Expression for a Locally Fitted Model

Description

Given a locally-fitted Cox or cluster point process model, evaluate an expression involving the fitted cluster parameters.

Usage

  ## S3 method for class 'locmincon'
with(data, ...)

  ## S3 method for class 'loccit'
with(data, ...)
## S3 method for class 'locmincon'
with(data, ...)

  ## S3 method for class 'loccit'
with(data, ...)

Arguments

`data`	An object of class `"locmincon"` or `"loccit"` representing a locally-fitted Cox or cluster point process model.
`...`	Arguments passed to `with.default` specifying the expression to be evaluated.

Details

These are method for the generic function with for the classes "locmincon" and "loccit".

An object of class "locmincon" or "loccit" represents a locally-fitted Cox or cluster point process model. It contains a data frame which provides estimates of the cluster parameters at each of the data points of the original point pattern dataset.

The expression specified by ... will be evaluated in this dataframe. If the result of evaluation is a data frame with one row for each data point, or a numeric vector with one entry for each data point, then the result will be an object of class "ssf" containing this information. Otherwise, the result will be a numeric vector.

Value

An object of class "ssf" or a numeric vector.

Author(s)

Adrian Baddeley [email protected].

Examples

  example(locmincon)
  with(fit, kappa * sigma2)
  example(locmincon)
  with(fit, kappa * sigma2)
example(locmincon)
  with(fit, kappa * sigma2)
  example(locmincon)
  with(fit, kappa * sigma2)

Package 'spatstat.local'

Help Index

Local Composite Likelihood

Description

Details

Author(s)

References

Cross Validated Bandwidth Selection for Locally Fitted Point Process Model

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Cross Validated Bandwidth Selection for Locally Fitted Point Process Model

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Homogeneity Test for Local Poisson or Gibbs Model

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Test Statistic for Homogeneity Test

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Locally Fitted Cluster or Cox Point Process Model

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Locally Fitted Cluster or Cox Point Process Model

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Locally Fitted Poisson or Gibbs Point Process Model

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Methods for Local Cluster or Cox Models

Description

Usage