Given a point pattern and two spatial covariates \(Z_1\) and \(Z_2\), construct a smooth estimate of the relative risk of the pair \((Z_1,Z_2)\).

`rho2hat(object, cov1, cov2, ..., method=c("ratio", "reweight"))`

A pixel image (object of class `"im"`

). Also
belongs to the special class `"rho2hat"`

which has a plot method.

- object
A point pattern (object of class

`"ppp"`

), a quadrature scheme (object of class`"quad"`

) or a fitted point process model (object of class`"ppm"`

).- cov1,cov2
The two covariates. Each argument is either a

`function(x,y)`

or a pixel image (object of class`"im"`

) providing the values of the covariate at any location, or one of the strings`"x"`

or`"y"`

signifying the Cartesian coordinates.- ...
Additional arguments passed to

`density.ppp`

to smooth the scatterplots.- method
Character string determining the smoothing method. See Details.

Adrian Baddeley Adrian.Baddeley@curtin.edu.au

This is a bivariate version of `rhohat`

.

If `object`

is a point pattern, this command
produces a smoothed version of the scatterplot of
the values of the covariates `cov1`

and `cov2`

observed at the points of the point pattern.

The covariates `cov1,cov2`

must have continuous values.

If `object`

is a fitted point process model, suppose `X`

is
the original data point pattern to which the model was fitted. Then
this command assumes `X`

is a realisation of a Poisson point
process with intensity function of the form
$$
\lambda(u) = \rho(Z_1(u), Z_2(u)) \kappa(u)
$$
where \(\kappa(u)\) is the intensity of the fitted model
`object`

, and \(\rho(z_1,z_2)\) is a function
to be estimated. The algorithm computes a smooth estimate of the
function \(\rho\).

The `method`

determines how the density estimates will be
combined to obtain an estimate of \(\rho(z_1, z_2)\):

If

`method="ratio"`

, then \(\rho(z_1, z_2)\) is estimated by the ratio of two density estimates. The numerator is a (rescaled) density estimate obtained by smoothing the points \((Z_1(y_i), Z_2(y_i))\) obtained by evaluating the two covariate \(Z_1, Z_2\) at the data points \(y_i\). The denominator is a density estimate of the reference distribution of \((Z_1,Z_2)\).If

`method="reweight"`

, then \(\rho(z_1, z_2)\) is estimated by applying density estimation to the points \((Z_1(y_i), Z_2(y_i))\) obtained by evaluating the two covariate \(Z_1, Z_2\) at the data points \(y_i\), with weights inversely proportional to the reference density of \((Z_1,Z_2)\).

Baddeley, A., Chang, Y.-M., Song, Y. and Turner, R. (2012)
Nonparametric estimation of the dependence of a point
process on spatial covariates.
*Statistics and Its Interface* **5** (2), 221--236.

`rhohat`

,
`methods.rho2hat`

```
attach(bei.extra)
plot(rho2hat(bei, elev, grad))
if(require("spatstat.model")) {
fit <- ppm(bei ~elev, covariates=bei.extra)
# \donttest{
plot(rho2hat(fit, elev, grad))
# }
plot(rho2hat(fit, elev, grad, method="reweight"))
}
```

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