# densityHeat

0th

Percentile

##### Kernel Density on a Network using Heat Equation

Computes a kernel density estimate on a linear network using the heat equation.

Keywords
methods, smooth, spatial
##### Usage
densityHeat(x, sigma, …,
at=c("pixels", "points"), leaveoneout=TRUE,
weights = NULL,
dx = NULL, dt = NULL, iterMax = 1e+06, 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) in the same units as the spatial coordinates of x.

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 grid of pixel locations (at="pixels") or only at the 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.

verbose

Logical value specifying whether to print progress reports.

##### Details

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.

Smoothing is performed using 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).

##### Value

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).

##### 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.

density.lpp

• densityHeat
##### Examples
# NOT RUN {
X <- runiflpp(3, simplenet)
D <- densityHeat(X, 0.2)