R: Smoothed Relative Density of Pairs of Covariate Values

rho2hat {spatstat.core} | R Documentation

## Smoothed Relative Density of Pairs of Covariate Values

### Description

Given a point pattern and two spatial covariates Z1 and Z2, construct a smooth estimate of the relative risk of the pair (Z1, Z2).

### Usage

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

### Arguments

`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.

### Details

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(Z1(u), Z2(u)) * kappa(u)

where kappa(u) is the intensity of the fitted model `object`, and rho(z1, z2) 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(z1, z2):

- If `method="ratio"`, then rho(z1,z2) is estimated by the ratio of two density estimates. The numerator is a (rescaled) density estimate obtained by smoothing the points (Z1(y[i]), Z2(y[i])) obtained by evaluating the two covariate Z1, Z2 at the data points y[i]. The denominator is a density estimate of the reference distribution of (Z1, Z2).

- If `method="reweight"`, then rho(z1,z2) is estimated by applying density estimation to the points (Z1(y[i]), Z2(y[i])) obtained by evaluating the two covariate Z1, Z2 at the data points y[i], with weights inversely proportional to the reference density of (Z1, Z2).

### Value

A pixel image (object of class `"im"`). Also belongs to the special class `"rho2hat"` which has a plot method.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au

### References

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.

### See Also

`rhohat`, `methods.rho2hat`

### Examples

      data(bei)
      attach(bei.extra)
      plot(rho2hat(bei, elev, grad))
      fit <- ppm(bei, ~elev, covariates=bei.extra)
      # plot(rho2hat(fit, elev, grad))
      plot(rho2hat(fit, elev, grad, method="reweight"))

[Package spatstat.core version 2.0-0 Index]