lurking

Lurking variable plot

Plot spatial point process residuals against a covariate

Usage

lurking(object, covariate, type="eem",
                    cumulative=TRUE,
                    clipwindow=default.clipwindow(object),
                    rv,
                    plot.sd, plot.it=TRUE,
                    typename,
                    covname, ...)

Details

This function generates a `lurking variable' plot for a fitted point process model. Residuals from the model represented by object are plotted against the covariate specified by covariate. This plot can be used to reveal departures from the fitted model, in particular, to reveal that the point pattern depends on the covariate.

First the residuals from the fitted model (Baddeley et al, 2004) are computed at each quadrature point, or alternatively the `exponential energy marks' (Stoyan and Grabarnik, 1991) are computed at each data point. The argument type selects the type of residual or weight. See diagnose.ppm for options and explanation.

A lurking variable plot for point processes (Baddeley et al, 2004) displays either the cumulative sum of residuals/weights (if cumulative = TRUE) or a kernel-weighted average of the residuals/weights (if cumulative = FALSE) plotted against the covariate. The empirical plot (solid lines) is shown together with its expected value assuming the model is true (dashed lines) and optionally also the pointwise two-standard-deviation limits (dotted lines). To be more precise, let $Z(u)$ denote the value of the covariate at a spatial location $u$.

• Ifcumulative=TRUEthen we plot$H(z)$against$z$, where$H(z)$is the sum of the residuals over all quadrature points where the covariate takes a value less than or equal to$z$, or the sum of the exponential energy weights over all data points where the covariate takes a value less than or equal to$z$.
• Ifcumulative=FALSEthen we plot$h(z)$against$z$, where$h(z)$is the derivative of$H(z)$, computed approximately by spline smoothing.

If the empirical and theoretical curves deviate substantially from one another, the interpretation is that the fitted model does not correctly account for dependence on the covariate. The correct form (of the spatial trend part of the model) may be suggested by the shape of the plot. If plot.sd = TRUE, then superimposed on the lurking variable plot are the pointwise two-standard-deviation error limits for $H(x)$ calculated for the inhomogeneous Poisson process. The default is plot.sd = TRUE for Poisson models and plot.sd = FALSE for non-Poisson models.

The argument object must be a fitted point process model (object of class "ppm") typically produced by the maximum pseudolikelihood fitting algorithm ppm).

The argument covariate is either a numeric vector, a pixel image, or an R language expression. If it is a numeric vector, it is assumed to contain the values of the covariate for each of the quadrature points in the fitted model. The quadrature points can be extracted by quad.ppm(object).

If covariate is a pixel image, it is assumed to contain the values of the covariate at each location in the window. The values of this image at the quadrature points will be extracted.

Alternatively, if covariate is an expression, it will be evaluated in the same environment as the model formula used in fitting the model object. It must yield a vector of the same length as the number of quadrature points. The expression may contain the terms x and y representing the cartesian coordinates, and may also contain other variables that were available when the model was fitted. Certain variable names are reserved words; see ppm.

Note that lurking variable plots for the $x$ and $y$ coordinates are also generated by diagnose.ppm, amongst other types of diagnostic plots. This function is more general in that it enables the user to plot the residuals against any chosen covariate that may have been present.

References

Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2004) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617--666.

Stoyan, D. and Grabarnik, P. (1991) Second-order characteristics for stochastic structures connected with Gibbs point processes. Mathematische Nachrichten, 151:95--100.

See Also

residuals.ppm, diagnose.ppm, residuals.ppm, qqplot.ppm, eem, ppm

Aliases

• lurking

Examples

data(nztrees)
  fit <- ppm(nztrees, ~x, Poisson())
  lurking(fit, expression(x))
  lurking(fit, expression(x), cumulative=FALSE)