densityHeat.lpp: Kernel Density on a Network using Heat Equation

In spatstat.linnet: Linear Networks Functionality of the 'spatstat' Family

Kernel Density on a Network using Heat Equation

Description

Given a point pattern on a linear network, compute a kernel estimate of intensity, by solving the heat equation.

Usage

  ## S3 method for class 'lpp'
densityHeat(x, sigma=NULL, ...,
              at=c("pixels", "points"), leaveoneout=TRUE,
              weights = NULL,
              dx = NULL, dt = NULL, iterMax = 1e+06,
              finespacing = TRUE, verbose=FALSE)

Arguments

x	Point pattern on a linear network (object of class "lpp") to be smoothed.
sigma	Smoothing bandwidth (standard deviation of the kernel). A numeric value in the same units as the spatial coordinates of x. Alternatively sigma may be a function which selects a bandwidth when applied to X, for example, bw.scott.iso or bw.lppl. There is a sensible default.
...	Arguments passed to as.mask determining the resolution of the result. (Any other arguments are ignored.)
at	String specifying whether to compute the intensity values at a fine grid of pixel locations on the network (at="pixels", the default) or only at the data points of x (at="points").
leaveoneout	Logical value indicating whether to compute a leave-one-out estimator. Applicable only when at="points".
weights	Optional. Numeric vector of weights associated with the points of x. Weights may be positive, negative or zero.
dx	Optional. Spacing of the sampling points along the network. A single number giving a distance value in the same units as x.
dt	Optional. Time step in the heat equation solver. A single number.
iterMax	Maximum number of iterations.
finespacing	Logical value specifying whether the discrete approximation is required to be accurate along every segment of the network, no matter how short the segment is. See the section on Discretisation.
verbose	Logical value specifying whether to print progress reports.

Details

The function densityHeat is generic. This is the method for the class "lpp" of points on a linear network.

Kernel smoothing is applied to the points of x using a kernel based on path distances in the network. If at="pixels" (the default), the result is a pixel image on the linear network (class "linim") which can be plotted. If at="points" the result is a numeric vector giving the density estimates at the data points of x.

The smoothing operation is equivalent to the “equal-split continuous” rule described in Section 9.2.3 of Okabe and Sugihara (2012). However, the actual computation is performed rapidly, by solving the classical time-dependent heat equation on the network, as described in McSwiggan et al (2016). Computational time is short, but increases quadratically with sigma.

If at="points" and leaveoneout=TRUE, a leave-one-out estimate is computed at each data point (that is, the estimate at each data point x[i] is based on all of the points except x[i]) using the truncated series approximation of McSwiggan et al (2019).

The argument sigma specifies the smoothing bandwidth. If sigma is missing or NULL, the default is one-eighth of the length of the shortest side of the bounding box of x. If sigma is a function in the R language, it is assumed to be a bandwidth selection rule, and it will be applied to x to compute the bandwidth value.

Value

If at="pixels" (the default), a pixel image on the linear network (object of class "linim").

If at="points", a numeric vector with one entry for each point of x.

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).

Discretisation and Error Messages

The arguments dx, dt and iterMax determine the discretisation of the network, according to a set of rules. The argument finespacing determines which rule will be applied.

The arguments dx, dt, iterMax are connected by several constraints; specifying one of these arguments will affect the default values of the other two arguments.

The argument finespacing specifies whether a very fine spacing of sample points is required, in order to attain hihg accuracy.

• If finespacing=TRUE (the default), then the sample point spacing dx must not exceed one-third of the length of the shortest segment of the network. This ensures that the discrete approximation is accurate along every segment, no matter how short the segment is. However, this may not be feasible if it implies a very large number of sample points, or a large number of iterations: in such cases, the code may terminate with an error about illegal values of dx, dt or iterMax.

• If finespacing=FALSE, then the sample point spacing dx will be about one-half the width of a pixel in the default pixellation of the window of x. This is usually a much coarser resolution than the one selected by finespacing=TRUE. If it is too coarse, the pixel resolution can be refined using the arguments dimyx, eps or xy passed to as.mask. For example, dimyx=512 would specify a 512 x 512 pixel grid. The default pixel resolution can be changed for the remainder of the R session by spatstat.options('npixel').

Author(s)

\adrian

and Greg McSwiggan.

References

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

McSwiggan, G., Baddeley, A. and Nair, G. (2019) Estimation of relative risk for events on a linear network. Statistics and Computing 30, 469–484.

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

See Also

density.lpp

Examples

  X <- runiflpp(3, simplenet)
  D <- densityHeat(X, 0.2)
  plot(D, style="w", main="", adjust=2)
  densityHeat.lpp(X, 0.2, at="points")
  Dw <- densityHeat(X, 0.2, weights=c(1,2,-1))

spatstat.linnet documentation built on Sept. 20, 2024, 5:06 p.m.