Kernel Estimate of Intensity on a Linear Network

Estimates the intensity of a point process on a linear network by applying kernel smoothing to the point pattern data.

methods, smooth, spatial
# S3 method for lpp
density(x, sigma=NULL, …,
        distance=c("path", "euclidean"),

# S3 method for splitppx density(x, sigma=NULL, …)


Point pattern on a linear network (object of class "lpp") to be smoothed.


Smoothing bandwidth (standard deviation of the kernel). A single numerical value in the same units as the spatial coordinates of x.

Additional arguments controlling the algorithm and the spatial resolution of the result. These arguments are passed either to densityQuick.lpp, densityHeat or densityEqualSplit depending on the algorithm chosen.


Optional. Numeric vector of weights associated with the points of x. Weights may be positive, negative or zero.


Character string (partially matched) specifying whether to use a kernel based on paths in the network (distance="path", the default) or a two-dimensional kernel (distance="euclidean").


Character string specifying the smoothing kernel. See dkernel for possible options.


Logical value indicating whether to compute the “equal-split continuous” smoother (continuous=TRUE, the default) or the “equal-split discontinuous” smoother (continuous=FALSE). Applies only when distance="path".


Kernel smoothing is applied to the points of x using either a kernel based on path distances in the network, or a two-dimensional kernel. The result is a pixel image on the linear network (class "linim") which can be plotted.

  • If distance="path" (the default) then the smoothing is performed using a kernel based on path distances in the network, as described in described in Okabe and Sugihara (2012) and McSwiggan et al (2016).

    • If continuous=TRUE (the default), smoothing is performed using the “equal-split continuous” rule described in Section 9.2.3 of Okabe and Sugihara (2012). The resulting function is continuous on the linear network.

    • If continuous=FALSE, smoothing is performed using the “equal-split discontinuous” rule described in Section 9.2.2 of Okabe and Sugihara (2012). The resulting function is continuous except at the network vertices.

    • In the default case (where distance="path" and continuous=TRUE and kernel="gaussian", computation is performed rapidly by solving the classical heat equation on the network, as described in McSwiggan et al (2016). The arguments are passed to densityHeat which performs the computation. Computational time is short, but increases quadratically with sigma.

    • In all other cases, computation is performed by path-tracing as described in Okabe and Sugihara (2012); the arguments are passed to densityEqualSplit which performs the computation. Computation time can be extremely long, and increases exponentially with sigma.

  • If distance="euclidean", the smoothing is performed using a two-dimensional kernel. The arguments are passed to densityQuick.lpp to perform the computation. Computation time is very short. See the help for densityQuick.lpp for further details.

There is also a method for split point patterns on a linear network (class "splitppx") which will return a list of pixel images.


A pixel image on the linear network (object of class "linim").

Infinite bandwidth

If sigma=Inf, the resulting density estimate is constant over all locations, and is equal to the average density of points per unit length. (If the network is not connected, then this rule is applied separately to each connected component of the network).


McSwiggan, G., Baddeley, A. and Nair, G. (2016) Kernel density estimation on a linear network. Scandinavian Journal of Statistics 44, 324--345.

Okabe, A. and Sugihara, K. (2012) Spatial analysis along networks. Wiley.

See Also

lpp, linim, densityQuick.lpp

  • density.lpp
  • density.splitppx
  X <- runiflpp(3, simplenet)
  D <- density(X, 0.2, verbose=FALSE)
  plot(D, style="w", main="", adjust=2)
  Dq <- density(X, 0.2, distance="euclidean")
  plot(Dq, style="w", main="", adjust=2)
  Dw <- density(X, 0.2, weights=c(1,2,-1), verbose=FALSE)
  De <- density(X, 0.2, kernel="epanechnikov", verbose=FALSE)
  Ded <- density(X, 0.2, kernel="epanechnikov", continuous=FALSE, verbose=FALSE)
# }
Documentation reproduced from package spatstat, version 1.64-1, License: GPL (>= 2)

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