# densityHeat

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

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

##### 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.

##### See Also

##### Examples

```
# NOT RUN {
X <- runiflpp(3, simplenet)
D <- densityHeat(X, 0.2)
plot(D, style="w", main="", adjust=2)
densityHeat(X, 0.2, at="points")
Dw <- densityHeat(X, 0.2, weights=c(1,2,-1))
# }
```

*Documentation reproduced from package spatstat, version 1.64-1, License: GPL (>= 2)*