GET (version 1.0-5)

deviation_test: Deviation test

Description

Crop the curve set to the interval of distances [r_min, r_max], calculate residuals, scale the residuals and perform a deviation test with a chosen deviation measure. The deviation tests are well known in spatial statistics; in GET they are provided for comparative purposes. Some (maximum type) of the deviation test have their corresponding envelope tests available, see Myllymäki et al., 2017 (and 'unscaled', 'st' and 'qdir' in global_envelope_test).

Usage

deviation_test(
  curve_set,
  r_min = NULL,
  r_max = NULL,
  use_theo = TRUE,
  scaling = "qdir",
  measure = "max",
  alternative = c("two.sided", "less", "greater"),
  savedevs = FALSE
)

Value

If 'savedevs=FALSE' (default), the p-value is returned. If 'savedevs=TRUE', then a list containing the p-value and calculated deviation measures \(u_i\), \(i=1,...,\text{nsim}+1\) (where \(u_1\) corresponds to the data pattern) is returned.

Arguments

curve_set

A residual curve_set object. Can be obtained by using residual().

r_min

The minimum radius to include.

r_max

The maximum radius to include.

use_theo

Whether to use the theoretical summary function or the mean of the functions in the curve_set.

scaling

The name of the scaling to use. Options include 'none', 'q', 'qdir' and 'st'. 'qdir' is default.

measure

The deviation measure to use. Default is 'max'. Must be one of the following: 'max', 'int' or 'int2'.

alternative

A character string specifying the alternative hypothesis when measure = 'max'; otherwise ignored. Must be one of the following: "two.sided" (default), "less" or "greater".

savedevs

Logical. Should the global rank values k_i, i=1,...,nsim+1 be returned? Default: FALSE.

Details

The deviation test is based on a test function \(T(r)\) and it works as follows:

1) The test function estimated for the data, \(T_1(r)\), and for nsim simulations from the null model, \(T_2(r), ...., T_{nsim+1}(r)\), must be saved in 'curve_set' and given to the deviation_test function.

2) The deviation_test function then

• Crops the functions to the chosen range of distances \([r_{\min}, r_{\max}]\).

• If the curve_set does not consist of residuals (see residual), then the residuals \(d_i(r) = T_i(r) - T_0(r)\) are calculated, where \(T_0(r)\) is the expectation of \(T(r)\) under the null hypothesis. If use_theo = TRUE, the theoretical value given in the curve_set$theo is used for as \(T_0(r)\), if it is given. Otherwise, \(T_0(r)\) is estimated by the mean of \(T_j(r)\), \(j=2,...,nsim+1\).

• Scales the residuals. Options are

  • 'none' No scaling. Nothing done.

  • 'q' Quantile scaling.

  • 'qdir' Directional quantile scaling.

  • 'st' Studentised scaling.

  See for details Myllymäki et al. (2013).

• Calculates the global deviation measure \(u_i\), \(i=1,...,nsim+1\), see options for 'measure'.

  • 'max' is the maximum deviation measure

    $$u_i = \max_{r \in [r_{\min}, r_{\max}]} | w(r)(T_i(r) - T_0(r))|$$

    If alternative = "greater", then instead

    $$u_i = \max_{r \in [r_{\min}, r_{\max}]} [ w(r)(T_i(r) - T_0(r)) ] $$

    i.e. the largest values will have the largest \(u_i\).

    If alternative = "less", then instead

    $$u_i = \max_{r \in [r_{\min}, r_{\max}]} [- w(r)(T_i(r) - T_0(r)) ] $$

    i.e. the smallest values will have the largest \(u_i\).

  • 'int2' is the integral deviation measure

    $$u_i = \int_{r_{\min}}^{r_{\max}} ( w(r)(T_i(r) - T_0(r)) )^2 dr$$

  • 'int' is the 'absolute' integral deviation measure

    $$u_i = \int_{r_{\min}}^{r_{\max}} |w(r)(T_i(r) - T_0(r))| dr$$

• Calculates the p-value.

References

Myllymäki, M., Grabarnik, P., Seijo, H. and Stoyan. D. (2015). Deviation test construction and power comparison for marked spatial point patterns. Spatial Statistics 11: 19-34. doi: 10.1016/j.spasta.2014.11.004

Myllymäki, M., Mrkvička, T., Grabarnik, P., Seijo, H. and Hahn, U. (2017). Global envelope tests for spatial point patterns. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79: 381–404. doi: 10.1111/rssb.12172

Examples

## Testing complete spatial randomness (CSR)
#-------------------------------------------
if(require("spatstat.explore", quietly=TRUE)) {
  pp <- unmark(spruces)
  nsim <- 999
  nsim <- 19
  # Generate nsim simulations under CSR, calculate L-function for the data and simulations
  env <- envelope(pp, fun="Lest", nsim=nsim, savefuns=TRUE, correction="translate")
  # The deviation test using the integral deviation measure
  res <- deviation_test(env, measure='int')
  res
  # or
  res <- deviation_test(env, r_min=0, r_max=7, measure='int2')
}