Different plots of the output of an object of class etasclass.By default the space-time region is the same used for the estimation of the ETAS model.
Background, triggered and total space intensities
are also computed and plotted for a grid of values.
If a positive value is given for tfixed, then the triggered intensity at time tfixed is estimated and visualized.
A tipical use can be with tfixed a day after a big earthquake.
Starting with the package version 1.2.0 different kind of residual analysis are computed and visualized, separately for the space and time dimensions. (8 plot on three windows for the space and 2 plots on one window for the time)
For space dimension,
Space residuals are computed dividing the observed rectangular space area in
a equally spaced grid of nclass intervals for each dimension, so to
divide the observed space area in nclass x nclass rectangular cells.
We obtain the classical comparison between observed and
theoretical frequencies. All frequencies are related to the whole time interval (and thus theoretical frequencies are obtained integrating estimated intensities with respect to time).
Fifth graph (image plot)
We define nclass x nclass standardized
residuals:
$$z_{\ell j} \ = \frac{n_{\ell j}-\hat{\nu}_{\ell j}}{\sqrt{\hat{\nu}_{\ell j}}}
\qquad (\ell =1,2,...,nclass; \ j=1,2,...,nclass)$$
For each cell $\ell j$ we have observed ($n_{\ell j}$) and
theoretical frequency ($\hat{\nu}_{\ell j}$).
Sixth graph (image plot)
We used a similar technique to compute residuals for the
background seismicity only, to check if at least the estimation of
the background component is appropriate. To this purpose the
observed background frequencies (${}_b n_{ \ell j}$) are now
computed by the sum of the estimated weights rho.weights and the theoretical background frequency ${}_b \hat{\nu}_{\ell j}$ by the estimated
marginal space background intensity in each cell.
From these quantities we obtain
nclass x nclass standardized residuals for the background intensity
only:
$${}_b z_{\ell j} \ = \frac{{}_b n_{\ell j} \ - \ {}_b
\hat{\nu}_{\ell j}}{\sqrt{{}_b \hat{\nu}_{\ell j}}} \qquad (\ell =1,2,...,nclass; \
j=1,2,...,nclass)$$
seventh plot: (space intensities (integrated over time))
A 3x2 plot: first column for observed vs.theoretical , second column for standardized residuals vs theoretical values. First row for total intensity, second row for background intensity, and third row for their difference, the triggered intensities
eight-th graph:
To check departure of the model for the time dimension, we first
integrated the estimated intensity function with respect to the
observed space region, so to obtain an estimated time process (a
one dimensional ETAS model):
$$\hat{\lambda}(t)=
\int \int_{\Omega_{(x,y)}}\,
\hat{\lambda}(x,y,t)\,d x \, d y$$
As known, a non-homogeneous time
process can be transformed to a homogeneous one through the
integral transformation:
$$\tau_i =
\int_{t_0}^{t_{i}}
\hat{\lambda}(t) \ d t$$
Then, a plot of $\tau_i$ versus
$i$ can give information about the departures of the models in the
time dimension. In particular, this plot, together with a plot of
the estimated time intensities, drawn on the same graphic winodw, can inform on the time at which
departures are more evident
If pdf=TRUE all graphs are printed on a pdf file, as spcified by file; otherwise
default screen device is used.