This is the main method to visualize graphically 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 plotted for a grid of values.
# S3 method for etasclass
plot(x,pdf=FALSE,file ="etasplot", ngrid=201,nclass=10,tfixed=0,flag.log=FALSE,...)This
plot method computes, among others, back.grid, trig.grid,
with coordinates x.grid and y.grid used to obtain image plots of background, triggered and total spatial estimated intensities
(see etasclass to see the details of the mixed estimation method used).
x grid values.
y grid values.
background intensity estimated on a ngrid x ngrid grid.
triggered intensities estimated on a grid of ngrid x ngrid points.
total intensities estimated on a grid of ngrid x ngrid points.
the fixed time for which intensity is estimated and visualized.
total intensities estimated on a grid of ngrid x ngrid points at time tfixed.
background space intensity estimated for observed points.
triggered space intensities estimated for observed points.
total space intensities estimated for observed points.
matrix of nclass*nclass cells with theoretical total space intensities.
matrix of nclass*nclass cells with theoretical background space intensities.
matrix of nclass*nclass cells with empirical total space intensities.
matrix of nclass*nclass cells with empirical background space intensities.
vector of transformed times.
an etaclass object.
If TRUE, then 2D plots are sent to a pdf file
name of the pdf file
number of points for each direction (x, y) of a ngrid*ngrid grid where estimated intensities must be evaluated. Default value= 201.
number of class for each direction (x, y) of a grid of nclass*nclass cells where estimated intensities must be evaluated.
Must divide ngrid-1. Default value= 10.
If a positive value is given, then the triggered intensity at time tfixed is estimated and visualized.
If TRUE then a log scale is used to plot intensities.
other arguments.
Marcello Chiodi, Giada Adelfio
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.
For space dimension, four plot are drawn with triggered, observed, total intensity, and total intensity with points.
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)
Then two plots are printed for space residuals for total and background intensities
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.
Adelfio G, Chiodi M (2009).Second-Order Diagnostics for Space-Time Point Processes with Application to Seismic Events. Environmetrics, 20(8), 895-911. doi:10.1002/env.961.
Adelfio G, Chiodi M (2015). FLP Estimation of Semi-Parametric Models for Space-Time Point Processes and Diagnostic Tools. Spatial Statistics, 14(B), 119-132. doi:10.1016/j.spasta.2015.06.004.
Adelfio G, Schoenberg FP (2009). Point Process Diagnostics Based on Weighted Second- Order Statistics and Their Asymptotic Properties. The Annals of the Institute of Statistical Mathematics, 61(4), 929-948. doi:10.1007/s10463-008-0177-1.
Chiodi, M. and Adelfio, G., (2017) Mixed Non-Parametric and Parametric Estimation Techniques in R Package etasFLP for Earthquakes' Description. Journal of Statistical Software, vol. 76 (3), pp. 1-28. DOI: 10.18637/jss.v076.i03.
etasclass, eqcat, profile.etasclass