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By Nechaev A. A.

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1 Covariates in a List of Images 41500 42000 42500 43000 The format (a), in which covariates is a list of images, would typically be used when the covariate values are computed from other data. + + + +++++ + ++ ++ + +++ ++ + + + + + ++ ++ + ++ + ++ + + + ++ + ++ + +++ ++ +++ +++ + +++ ++++ +++++ + +++ + ++ + + + + ++ + + +++++++++ + + + + ++ +++++++++ + ++ ++++ + + + ++ + + + + + + + + + + + + + + + +++ +++++ ++++ +++ + +++ +++ + ++ + + ++ ++ ++ +++ + + +++++ + + + + ++ ++ + + ++++ ++ ++ + + ++++ + + + + + +++ ++++++ + ++ + +++ + + +++++ ++++++ ++ + + ++ ++ + + +++ + ++ + + ++ + + +++ + + + + + ++ + + ++ + + ++ + ++++ ++ +++ ++++ + + + + + + +++ + + +++ ++ ++ ++ + ++ ++ ++ +++++ + ++ +++ + + ++ +++ +++ + +++ +++++ + + ++ ++ + ++ +++++++ ++ +++ +++ + ++ ++ ++++ + ++++ ++ +++ ++ ++++ + +++ ++ + + + + + + + ++ + + + + ++ + + + + + + + + + ++ + + + ++ ++ + + ++ + + + ++ ++ ++ + + +++ +++ ++ ++ ++++ + ++++ +++ + + + + ++ + + ++ + + ++++++ + + + ++ +++ ++++++ + ++ + + + + + + + + ++ +++++ +++ +++++ + ++ + + +++ + ++ ++ +++ +++ ++ ++++ +++++ +++ ++ + +++ ++++ ++ +++++ ++ + + ++ ++++ + +++ + ++ + + + ++++++ + +++ ++++ + + + + + ++ + + + + + + + + ++++ + + + + + + ++++ ++ +++ + + + + ++ ++ +++++++ + ++ + + ++ ++ + + + + + +++ + + ++ 35000 35500 + 36000 Fig.

Then the model formula should invoke the factor command on this image. For example if fim is an image with integer values which represent levels of a factor, then ppm(X, ˜factor(f), Poisson(), covariates=list(f=fim)) ﬁts the nonstationary Poisson process with an intensity that depends on the levels of this factor. Care must be taken to ensure the correct interpretation of the factor levels [72, p. ]. 2 Covariates in a Data Frame Typically we would use the data frame format (b) if the values of the spatial covariates can only be observed at certain locations.

3. Examples of marked point patterns. (a) continuous marks. Mark values (tree diameters) represented as radii of circles. The Longleaf Pines dataset, available as longleaf. (b) categorical marks. Mark values (cell types) represented as different graphical symbols. ppp, read data from a ﬁle using scanpp, manipulate existing point pattern objects using a variety of tools, or generate a random pattern using one of the simulation routines. Suppose, for example, that we have data for a point pattern observed in the rectangle [0, 10] × [0, 4].