Description Usage Arguments Details Value Note Author(s) References See Also Examples

View source: R/plot.etasclass.R

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.

1 2 3 |

`x` |
an |

`pdf` |
If |

`file` |
name of the pdf file |

`ngrid` |
number of points for each direction ( |

`nclass` |
number of class for each direction ( |

`tfixed` |
If a positive value is given, then the triggered intensity at time |

`flag.3D` |
If |

`flag.log` |
If |

`...` |
other arguments. |

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{ν}_{\ell j}}{√{\hat{ν}_{\ell j}}}
\qquad (\ell =1,2,...,nclass; \ j=1,2,...,nclass)
*

For each cell *lj* we have observed (*n_{\ell j}*) and
theoretical frequency (*\hat{ν}_{\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{ν}_{\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{ν}_{\ell j}}{√{{}_b \hat{ν}_{\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{λ}(t)=
\int \int_{Ω_{(x,y)}}\,
\hat{λ}(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:

*τ_i =
\int_{t_0}^{t_{i}}
\hat{λ}(t) \ d t
*

Then, a plot of *τ_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.

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` |
x grid values. |

`y.grid` |
y grid values. |

`back.grid` |
background intensity estimated on a |

`trig.grid` |
triggered intensities estimated on a grid of |

`tot.grid` |
total intensities estimated on a grid of |

`tfixed` |
the fixed time for which intensity is estimated and visualized. |

`totfixed.grid` |
total intensities estimated on a grid of |

`back.grid` |
background space intensity estimated for observed points. |

`trig.grid` |
triggered space intensities estimated for observed points. |

`tot.grid` |
total space intensities estimated for observed points. |

`teo1` |
matrix of |

`teo2` |
matrix of |

`emp1` |
matrix of |

`emp2` |
matrix of |

`t.trasf` |
vector of transformed times. |

In this first version the x-y space region, where the point process is defined, by default is a rectangle embedding the catalog values.

Marcello Chiodi, Giada Adelfio

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`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 | ```
## Not run:
data("italycatalog")
# load a sample catalog of the italian seismicity
class(italycatalog)<-"eqcat"
etas.flp<-etasclass(italycatalog,description="etas flp",magn.threshold=3.1,thinning=FALSE,flp=TRUE,
is.backconstant=FALSE,magn.threshold.back=3.5,sectoday=TRUE,
onlytime=FALSE,declustering=TRUE,epsmax=0.00001,
params.ind=c(1,1,1,1,1,1,1,1),k0=0.005,c=0.005,p=1.01,a=1.05,gamma=0.6,q=1.52,d=1.1,
compsqm=TRUE,usenlm=TRUE,ndeclust=15)
# execution of etasclass for events with minimum magnitude of 3.1.
# The events with magnitude at least 3.5 are used to build a first approximation
# for the background intensity function
# (magn.threshold.back=3.5)
# plot method
plot(etas.flp)
## End(Not run)
``` |

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