This function computes several kinds of residuals for the fit of
a point process model to a spatial point pattern on a linear network.
It is an extension of the method of Baddeley et al (2005)
to point process models on a network.
Use plot.msr to plot the residuals directly.
The argument object must be a fitted point process model
on a network (object of class "lppm").
Such objects are produced by the model-fitting algorithm lppm.
This fitted model object contains complete
information about the original data pattern.
Residuals are attached both to the data points and to some
other points in the window of observation (namely, to the dummy
points of the quadrature scheme used to fit the model).
If the fitted model is correct, then the sum of the
residuals over all (data and dummy) points in a spatial region \(B\)
has mean zero. For further explanation, see Baddeley et al (2005).
The type of residual
is chosen by the argument type. Current options are
"raw":the raw residuals
$$
r_j = z_j - w_j \lambda_j
$$
at the quadrature points \(u_j\),
where \(z_j\) is the indicator equal to 1 if \(u_j\)
is a data point and 0 if \(u_j\) is a dummy point;
\(w_j\) is the quadrature weight attached to
\(u_j\); and
$$\lambda_j = \hat\lambda(u_j,x)$$
is the conditional intensity of the fitted model at \(u_j\).
These are the spatial analogue of the martingale residuals
of a one-dimensional counting process.
"inverse":the `inverse-lambda' residuals (Baddeley et al, 2005)
$$
r^{(I)}_j = \frac{r_j}{\lambda_j}
= \frac{z_j}{\lambda_j} - w_j
$$
obtained by dividing the raw residuals by
the fitted conditional intensity. These are
a counterpart of the exponential energy marks
(see eem).
"pearson":the Pearson residuals (Baddeley et al, 2005)
$$
r^{(P)}_j = \frac{r_j}{\sqrt{\lambda_j}}
= \frac{z_j}{\sqrt{\lambda_j}}
- w_j \sqrt{\lambda_j}
$$
obtained by dividing the raw residuals by the
square root of the fitted conditional intensity.
The Pearson residuals are standardised, in the sense
that if the model (true and fitted) is Poisson,
then the sum of the Pearson residuals in a spatial region \(B\)
has variance equal to the area of \(B\).
"score":the score residuals (Baddeley et al, 2005)
$$
r_j = (z_j - w_j \lambda_j) x_j
$$
obtained by multiplying the raw residuals \(r_j\)
by the covariates \(x_j\) for quadrature point \(j\).
The score residuals always sum to zero.
The result of residuals.ppm is a measure
(object of class "msr").
Use plot.msr to plot the residuals directly.
Use integral.msr to compute the total residual.