slrm {spatstat.model}

Spatial Logistic Regression

Description

Fits a spatial logistic regression model to a spatial point pattern.

Usage

slrm(formula, ..., data = NULL, offset = TRUE, link = "logit",
                   dataAtPoints=NULL, splitby=NULL)

Arguments

formula

The model formula. See Details.

...

Optional arguments passed to as.mask determining the pixel resolution for the discretisation of the point pattern.

data

Optional. A list containing data required in the formula. The names of entries in the list should correspond to variable names in the formula. The entries should be point patterns, pixel images or windows.

offset

Logical flag indicating whether the model formula should be augmented by an offset equal to the logarithm of the pixel area.

link

The link function for the regression model. A character string, specifying a link function for binary regression.

dataAtPoints

Optional. Exact values of the covariates at the data points. A data frame, with column names corresponding to variables in the formula, with one row for each point in the point pattern dataset.

splitby

Optional. Character string identifying a window. The window will be used to split pixels into sub-pixels.

Details

This function fits a Spatial Logistic Regression model (Tukey, 1972; Agterberg, 1974) to a spatial point pattern dataset. The logistic function may be replaced by another link function.

The formula specifies the form of the model to be fitted, and the data to which it should be fitted. The formula must be an R formula with a left and right hand side.

The left hand side of the formula is the name of the point pattern dataset, an object of class "ppp".

The right hand side of the formula is an expression, in the usual R formula syntax, representing the functional form of the linear predictor for the model.

Each variable name that appears in the formula may be

• one of the reserved names x and y, referring to the Cartesian coordinates;

• the name of an entry in the list data, if this argument is given;

• the name of an object in the parent environment, that is, in the environment where the call to slrm was issued.

Each object appearing on the right hand side of the formula may be

• a pixel image (object of class "im") containing the values of a covariate;

• a window (object of class "owin"), which will be interpreted as a logical covariate which is TRUE inside the window and FALSE outside it;

• a function in the R language, with arguments x,y, which can be evaluated at any location to obtain the values of a covariate.

See the Examples below.

The fitting algorithm discretises the point pattern onto a pixel grid. The value in each pixel is 1 if there are any points of the point pattern in the pixel, and 0 if there are no points in the pixel. The dimensions of the pixel grid will be determined as follows:

• The pixel grid will be determined by the extra arguments ... if they are specified (for example the argument dimyx can be used to specify the number of pixels).

• Otherwise, if the right hand side of the formula includes the names of any pixel images containing covariate values, these images will determine the pixel grid for the discretisation. The covariate image with the finest grid (the smallest pixels) will be used.

• Otherwise, the default pixel grid size is given by spatstat.options("npixel").

The covariates are evaluated at the centre of each pixel. If dataAtPoints is given, then the covariate values at the corresponding pixels are overwritten by the entries of dataAtPoints (and the spatial coordinates are overwritten by the exact spatial coordinates of the data points).

If link="logit" (the default), the algorithm fits a Spatial Logistic Regression model. This model states that the probability p that a given pixel contains a data point, is related to the covariates through

\log\frac{p}{1-p} = \eta

where \eta is the linear predictor of the model (a linear combination of the covariates, whose form is specified by the formula).

If link="cloglog" then the algorithm fits a model stating that

\log(-\log(1-p)) = \eta

.

If offset=TRUE (the default), the model formula will be augmented by adding an offset term equal to the logarithm of the pixel area. This ensures that the fitted parameters are approximately independent of pixel size. If offset=FALSE, the offset is not included, and the traditional form of Spatial Logistic Regression is fitted.

Value

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

There are many methods for this class, including methods for print, fitted, predict, anova, coef, logLik, terms, update, formula and vcov. Automated stepwise model selection is possible using step. Confidence intervals for the parameters can be computed using confint.

Author(s)

Adrian Baddeley [e-mail] and Rolf Turner [e-mail].

References

Agterberg, F.P. (1974) Automatic contouring of geological maps to detect target areas for mineral exploration. Journal of the International Association for Mathematical Geology 6, 373–395.

Baddeley, A., Berman, M., Fisher, N.I., Hardegen, A., Milne, R.K., Schuhmacher, D., Shah, R. and Turner, R. (2010) Spatial logistic regression and change-of-support for spatial Poisson point processes. Electronic Journal of Statistics 4, 1151–1201. DOI: 10.1214/10-EJS581

Tukey, J.W. (1972) Discussion of paper by F.P. Agterberg and S.C. Robinson. Bulletin of the International Statistical Institute 44 (1) p. 596. Proceedings, 38th Congress, International Statistical Institute.

See Also

anova.slrm, coef.slrm, fitted.slrm, logLik.slrm, plot.slrm, predict.slrm, vcov.slrm

Examples

if(offline <- !interactive()) op <- spatstat.options(npixel=32)

     X <- copper$SouthPoints
     slrm(X ~ 1)
     slrm(X ~ x+y)

     slrm(X ~ x+y, link="cloglog")
     # specify a grid of 2-km-square pixels
     slrm(X ~ 1, eps=2)

     Y <- copper$SouthLines
     Z <- distmap(Y)
     slrm(X ~ Z)
     slrm(X ~ Z, dataAtPoints=list(Z=nncross(X,Y,what="dist")))

     mur <- solapply(murchison, rescale, s=1000, unitname="km")
     mur$dfault <- distfun(mur$faults)
     slrm(gold ~ dfault, data=mur)
     slrm(gold ~ dfault + greenstone, data=mur)
     slrm(gold ~ dfault, data=mur, splitby="greenstone")

if(offline) spatstat.options(op)