ETAS (version 0.7.2)

resid.etas: Residuals Analysis and Diagnostics Plots

Description

A function to compute and plot spatial and temporal residuals as well as transformed times for a fitted ETAS model.

Usage

resid.etas(fit, type="raw", n.temp=1000, dimyx=NULL)

Value

The function produces plots of temporal and smoothed spatial residuals, transformed times \(\tau_j\) against \(j\) and Q-Q plot of \(U_j\).

It also returns a list with components

• tau the transformed times

• U related quantities with the transformed times

• tres the temporal residuals

• sres the smoothed spatial residuals

Arguments

fit

A fitted ETAS model. An object of class "etas".

type

A character string specifying the type residuals to be computed. Options are "raw" (the default case), "reciprocal" and "pearson".

n.temp

An integer specifying the number of partition points for temporal residuals.

dimyx

Dimensions of the discretization for the smoothed spatial residuals. A numeric vector of length 2.

Author

Abdollah Jalilian

Details

The function computes the temporal residuals $$R^{temp}(I_j, h) = \sum_{i=1}^{N} \delta_i 1[t_i \in I_j] h(t_i) \lambda^{temp}(t_i|H_{t_i}) - \int_{I_j} h(t)\lambda^{temp}(t|H_t) d t$$ for \(I_j=((j-1)T/n.temp, jT/n.temp]\), \(j=1,...,n.temp\), and the (smoothed version of) spatial residuals $$R^{spat}(B_j, h) = h(\tilde{x}_i, \tilde{y}_i) \lambda^{spat}(\tilde{x}_i, \tilde{y}_i)(\tilde{\delta}_i - \tilde{w}_i)$$ for a Berman-Turner quadrature scheme with quadrature points \((\tilde{x}_i, \tilde{y}_i)\) and quadrature weights \(\tilde{w}_i\), \(i=1,...,n.spat\). Raw, reciprocal and Pearson residuals obtain with \(h=1\), \(h=1/\lambda\) and \(h=1/\sqrt{\lambda}\), respectively.

In addition, the function computes transformed times $$\tau_j=\int_{0}^{t_j} \lambda^{temp}(t|H_t) d t$$ and $$U_j = 1 - \exp(-(t_j - t_{j-1}))$$

References

Baddeley A, Rubak E, Turner R (2015). Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press, London. https://www.routledge.com/Spatial-Point-Patterns-Methodology-and-Applications-with-R/Baddeley-Rubak-Turner/p/book/9781482210200.

Baddeley A, Turner R (2000). Practical Maximum Pseudolikelihood for Spatial Point Patterns. Australian & New Zealand Journal of Statistics, 42(3), 283--322. tools:::Rd_expr_doi("10.1111/1467-842X.00128").

Baddeley A, Turner R, Moller J, Hazelton M (2005). Residual Analysis for Spatial Point Processes. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67(5), 617--666. tools:::Rd_expr_doi("10.1111/j.1467-9868.2005.00519.x").

Ogata Y (1988). Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes. Journal of the American Statistical Association, 83(401), 9--27. tools:::Rd_expr_doi("10.2307/2288914").

Zhuang J (2006). Second-order Residual Analysis of Spatiotemporal Point Processes and Applications in Model Evaluation Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68(4), 635--653. tools:::Rd_expr_doi("10.1111/j.1467-9868.2006.00559.x").

See Also

etas

Examples

  iran.cat <- catalog(iran.quakes, time.begin="1973/01/01",
     study.start="1986/01/01", study.end="2016/01/01",
     lat.range=c(26, 40), long.range=c(44, 63), mag.threshold=5)
  print(iran.cat)
  if (FALSE) {
  plot(iran.cat)}

  # setting initial parameter values
  param0 <- c(0.46, 0.23, 0.022, 2.8, 1.12, 0.012, 2.4, 0.35)

  # fitting the model
  if (FALSE) {
  iran.fit <- etas(iran.cat, param0=param0)

  # diagnostic plots
  resid.etas(iran.fit)}