Uses cross-validation to select a smoothing bandwidth for the estimation of relative risk.

```
bw.relrisk(X, method = "likelihood", nh = spatstat.options("n.bandwidth"),
hmin=NULL, hmax=NULL, warn=TRUE)
```

X

A multitype point pattern (object of class `"ppp"`

which has factor valued marks).

method

Character string determining the cross-validation method.
Current options are `"likelihood"`

,
`"leastsquares"`

or
`"weightedleastsquares"`

.

nh

Number of trial values of smoothing bandwith `sigma`

to consider. The default is 32.

hmin, hmax

Optional. Numeric values.
Range of trial values of smoothing bandwith `sigma`

to consider. There is a sensible default.

warn

Logical. If `TRUE`

, issue a warning if the minimum of
the cross-validation criterion occurs at one of the ends of the
search interval.

A numerical value giving the selected bandwidth.
The result also belongs to the class `"bw.optim"`

which can be plotted.

This function selects an appropriate bandwidth for the nonparametric
estimation of relative risk using `relrisk`

.

Consider the indicators \(y_{ij}\) which equal \(1\) when data point \(x_i\) belongs to type \(j\), and equal \(0\) otherwise. For a particular value of smoothing bandwidth, let \(\hat p_j(u)\) be the estimated probabilities that a point at location \(u\) will belong to type \(j\). Then the bandwidth is chosen to minimise either the likelihood, the squared error, or the approximately standardised squared error, of the indicators \(y_{ij}\) relative to the fitted values \(\hat p_j(x_i)\). See Diggle (2003).

The result is a numerical value giving the selected bandwidth `sigma`

.
The result also belongs to the class `"bw.optim"`

allowing it to be printed and plotted. The plot shows the cross-validation
criterion as a function of bandwidth.

The range of values for the smoothing bandwidth `sigma`

is set by the arguments `hmin, hmax`

. There is a sensible default,
based on multiples of Stoyan's rule of thumb `bw.stoyan`

.

If the optimal bandwidth is achieved at an endpoint of the
interval `[hmin, hmax]`

, the algorithm will issue a warning
(unless `warn=FALSE`

). If this occurs, then it is probably advisable
to expand the interval by changing the arguments `hmin, hmax`

.

Computation time depends on the number `nh`

of trial values
considered, and also on the range `[hmin, hmax]`

of values
considered, because larger values of `sigma`

require
calculations involving more pairs of data points.

Diggle, P.J. (2003)
*Statistical analysis of spatial point patterns*,
Second edition. Arnold.

Kelsall, J.E. and Diggle, P.J. (1995)
Kernel estimation of relative risk.
*Bernoulli* **1**, 3--16.

```
# NOT RUN {
data(urkiola)
# }
# NOT RUN {
b <- bw.relrisk(urkiola)
b
plot(b)
b <- bw.relrisk(urkiola, hmax=20)
plot(b)
# }
```

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