# rhohat

##### Smoothing Estimate of Covariate Transformation

Computes a smoothing estimate of the intensity of a point process, as a function of a spatial covariate.

##### Usage

```
rhohat(object, covariate, ...,
method=c("ratio", "reweight", "transform"),
smoother=c("kernel", "local"),
dimyx=NULL, eps=NULL,
n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL,
bwref=bw,
covname, confidence=0.95)
```

##### 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"`

). - covariate
- Either a
`function(x,y)`

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

) providing the values of the covariate at any location. Alternatively one of the strings`"x"`

or`"y"`

signifying the Cartesian - method
- Character string determining the smoothing method. See Details.
- smoother
- Character string determining the smoothing algorithm. See Details.
- dimyx,eps
- Arguments passed to
`as.mask`

to control the pixel resolution at which the covariate will be evaluated. - bw
- Smoothing bandwidth or bandwidth rule
(passed to
`density.default`

). - adjust
- Smoothing bandwidth adjustment factor
(passed to
`density.default`

). - n, from, to
- Arguments passed to
`density.default`

to control the number and range of values at which the function will be estimated. - bwref
- Optional. An alternative value of
`bw`

to use when smoothing the reference density (the density of the covariate values observed at all locations in the window). - ...
- Additional arguments passed to
`density.default`

or`locfit`

. - covname
- Optional. Character string to use as the name of the covariate.
- confidence
- Confidence level for confidence intervals. A number between 0 and 1.

##### Details

If `object`

is a point pattern, this command assumes that
`object`

is a realisation of a Poisson point process with
intensity function $\lambda(u)$ of the form
$$\lambda(u) = \rho(Z(u))$$ where
$Z$ is the spatial covariate function given by `covariate`

,
and $\rho(z)$ is a function to be estimated.
This command computes estimators of $\rho(z)$
proposed by Baddeley and Turner (2005)
and Baddeley et al (2012).

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
$$\lambda(u) = \rho(Z(u)) \kappa(u)$$
where $\kappa(u)$ is the intensity of the fitted model
`object`

. A smoothing estimator of $\rho(z)$ is computed.

The estimation procedure is determined by the character strings
`method`

and `smoother`

.
The estimation procedure involves computing several density estimates
and combining them.
The algorithm used to compute density estimates is
determined by `smoother`

:

- If
`smoother="kernel"`

, each the smoothing procedure is based on fixed-bandwidth kernel density estimation, performed by`density.default`

. - If
`smoother="local"`

, the smoothing procedure is based on local likelihood density estimation, performed by`locfit`

.

`method`

determines how the density estimates will be
combined to obtain an estimate of $\rho(z)$:
- If
`method="ratio"`

, then$\rho(z)$is estimated by the ratio of two density estimates. The numerator is a (rescaled) density estimate obtained by smoothing the values$Z(y_i)$of the covariate$Z$observed at the data points$y_i$. The denominator is a density estimate of the reference distribution of$Z$. - If
`method="reweight"`

, then$\rho(z)$is estimated by applying density estimation to the values$Z(y_i)$of the covariate$Z$observed at the data points$y_i$, with weights inversely proportional to the reference density of$Z$. - If
`method="transform"`

, the smoothing method is variable-bandwidth kernel smoothing, implemented by applying the Probability Integral Transform to the covariate values, yielding values in the range 0 to 1, then applying edge-corrected density estimation on the interval$[0,1]$, and back-transforming.

##### Value

- A function value table (object of class
`"fv"`

) containing the estimated values of $\rho$ for a sequence of values of $Z$. Also belongs to the class`"rhohat"`

which has special methods for`print`

,`plot`

and`predict`

.

##### 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*, in press.
Baddeley, A. and Turner, R. (2005)
Modelling spatial point patterns in R.
In: A. Baddeley, P. Gregori, J. Mateu, R. Stoica, and D. Stoyan,
editors, *Case Studies in Spatial Point Pattern Modelling*,
Lecture Notes in Statistics number 185. Pages 23--74.
Springer-Verlag, New York, 2006.
ISBN: 0-387-28311-0.

##### See Also

##### Examples

```
X <- rpoispp(function(x,y){exp(3+3*x)})
rho <- rhohat(X, "x")
rho <- rhohat(X, function(x,y){x})
plot(rho)
curve(exp(3+3*x), lty=3, col=2, add=TRUE)
rhoB <- rhohat(X, "x", method="reweight")
rhoC <- rhohat(X, "x", method="transform")
fit <- ppm(X, ~x)
rr <- rhohat(fit, "y")
```

*Documentation reproduced from package spatstat, version 1.27-0, License: GPL (>= 2)*