global_envelope_test: Global envelope test

View source: R/envelopes.r

global_envelope_testR Documentation

Global envelope test

Description

Global envelope test, global envelopes and p-values

Usage

global_envelope_test(
  curve_sets,
  type = "erl",
  alpha = 0.05,
  alternative = c("two.sided", "less", "greater"),
  ties = "erl",
  probs = c(0.025, 0.975),
  quantile.type = 7,
  central = "mean",
  nstep = 2,
  ...
)

Arguments

curve_sets

A curve_set (see create_curve_set) or an envelope object of spatstat containing a data function and simulated functions. If an envelope object is given, it must contain the summary functions from the simulated patterns which can be achieved by setting savefuns = TRUE when calling the envelope function. Alternatively, a list of curve_set or envelope objects can be given.

type

The type of the global envelope with current options for 'rank', 'erl', 'cont', 'area', 'qdir', 'st' and 'unscaled'. See details.

alpha

The significance level. The 100(1-alpha)% global envelope will be calculated. If a vector of values is provided, the global envelopes are calculated for each value.

alternative

A character string specifying the alternative hypothesis. Must be one of the following: "two.sided" (default), "less" or "greater". The last two options only available for types 'rank', 'erl', 'cont' and 'area'.

ties

The method to obtain a unique p-value when type = 'rank'. Possible values are 'midrank', 'random', 'conservative', 'liberal' and 'erl'. For 'conservative' the resulting p-value will be the highest possible. For 'liberal' the p-value will be the lowest possible. For 'random' the rank of the obs within the tied values is uniformly sampled so that the resulting p-value is at most the conservative option and at least the liberal option. For 'midrank' the mid-rank within the tied values is taken. For 'erl' the extreme rank length p-value is calculated. The default is 'erl'.

probs

A two-element vector containing the lower and upper quantiles for the measure 'q' or 'qdir', in that order and on the interval [0, 1]. The default values are 0.025 and 0.975, suggested by Myllymäki et al. (2015, 2017).

quantile.type

As type argument of quantile, how to calculate quantiles for 'q' or 'qdir'.

central

Either "mean" or "median". If the curve sets do not contain the component theo for the theoretical central function, then the central function (used for plotting only) is calculated either as the mean or median of functions provided in the curve sets. For 'qdir', 'st' and 'unscaled' only the mean is allowed as an option, due to their definition.

nstep

1 or 2 for how to contruct a combined global envelope if list of curve sets is provided. 2 (default) for a two-step combining procedure, 1 for one-step.

...

Additional parameters to be passed to central_region.

Details

Given a curve_set (see create_curve_set for how to create such an object) or an envelope object of spatstat, which contains both the data curve (or function or vector) T_1(r) (in the component obs) and the simulated curves T_2(r),...,T_(s+1)(r) (in the component sim_m), the function global_envelope_test performs a global envelope test. The functionality of the function is rather similar to the function central_region, but in addition to ordering the functions from the most extreme one to the least extreme one using different measures and providing the global envelopes with intrinsic graphical interpretation, p-values are calculated for the test. Thus, while central_region can be used to construct global envelopes in a general setting, the function global_envelope_test is devoted to testing as its name suggests.

The function global_envelope_test is the main function for global envelope tests (for simple hypotheses). Different type of global envelope tests can be performed. We use such ordering of the functions for which we are able to construct global envelopes with intrinsic graphical interpretation.

  • 'rank': the completely non-parametric rank envelope test (Myllymäki et al., 2017) based on minimum of pointwise ranks

  • 'erl': the completely non-parametric rank envelope test based on extreme rank lengths (Myllymäki et al., 2017; Mrkvička et al., 2018) based on number of minimal pointwise ranks

  • 'cont': the completely non-parametric rank envelope test based on continuous rank (Hahn, 2015; Mrkvička et al., 2019) based on minimum of continuous pointwise ranks

  • 'area': the completely non-parametric rank envelope test based on area rank (Mrkvička et al., 2019) based on area between continuous pointwise ranks and minimum pointwise ranks for those argument (r) values for which pointwise ranks achieve the minimum (it is a combination of erl and cont)

  • "qdir", the directional quantile envelope test, protected against unequal variance and asymmetry of T(r) for different distances r (Myllymäki et al., 2015, 2017)

  • "st", the studentised envelope test, protected against unequal variance of T(r) for different distances r (Myllymäki et al., 2015, 2017)

  • "unscaled", the unscaled envelope (providing a baseline) that has a contant width and that corresponds to the classical maximum deviation test (Ripley, 1981).

The first four types are global rank envelopes. The 'rank' envelope test is a completely non-parametric test, which provides the 100(1-alpha) T(r) on the chosen interval of distances and associated p-values. The other three are modifications of 'rank' to treat the ties in the extreme rank ordering on which the 'rank' test is based on. The last three envelopes are global scaled maximum absolute difference (MAD) envelope tests. The unscaled envelope test leads to envelopes with constant width over the distances r. Thus, it suffers from unequal variance of T(r) over the distances r and from the asymmetry of distribution of T(r). We recommend to use the other global envelope tests available. The unscaled envelope is provided as a reference.

See Myllymäki and Mrkvička (2020, Section 2.), i.e. vignette("GET"), for more detailed description of the measures and the corresponding envelopes.

See vignette("pointpatterns") for examples of point pattern analyses.

Value

Either an object of class "global_envelope" or "combined_global_envelope", similarly as the objects returned by central_region.

The global_envelope is essentially a data frame containing columns

  • the values of the argument r at which the test was made, copied from the argument curve_sets with the corresponding names

  • obs = values of the data function, copied from the argument curve_sets (unlike for central regions, obs always exists for a global envelope test)

  • lo = the lower envelope; in case of a vector of alpha values, several 'lo' exist with names paste0("lo.", 100*(1-alpha))

  • hi = the upper envelope; in case of a vector of alpha values, several 'lo' exist with names paste0("hi.", 100*(1-alpha))

  • central = a central curve as specified in the argument central.

Moreover, the returned object has the same attributes as the global_envelope object returned by central_region and in addition

  • p = A point estimate for the p-value (default is the mid-rank p-value).

and in the case that type = 'rank' also

  • p_interval = The p-value interval [p_liberal, p_conservative].

  • ties = As the argument ties.

The combined_global_envelope is a list of global_envelope objects containing the above mentioned columns and which all together form the global envelope. It has the same attributes as described in central_region, and in addition also the p-value p. The 2d classes are attached as described in central_region.

Procedure

1) First the curves are ranked from the most extreme one to the least extreme one by a measure that is specified by the argument type. The options are

  • 'rank': extreme ranks (Myllymäki et al., 2017)

  • 'erl': extreme rank lengths (Myllymäki et al., 2017; Mrkvička et al., 2018)

  • 'cont': continuous ranks (Hahn, 2015; Mrkvička et al., 2019)

  • 'area': area ranks (Mrkvička et al., 2019)

  • 'qdir': the directional quantile maximum absolute deviation (MAD) measure (Myllymäki et al., 2015, 2017)

  • 'st': the studentized MAD measure (Myllymäki et al., 2015, 2017)

  • 'unscaled': the unscaled MAD measure (Ripley, 1981)

2) Based on the measures used to rank the functions, the 100(1-alpha)% global envelope is provided. It corresponds to the 100*coverage% central region.

3) P-values: In the case type="rank", based on the extreme ranks k_i, i=1, ..., s+1, the p-interval is calculated. Because the extreme ranks contain ties, there is not just one p-value. The p-interval is given by the most liberal and the most conservative p-value estimate. Also a single p-value is calculated. By default this single p-value is the extreme rank length p-value ("erl") as specified by the argument ties. If the case of other measures, a (single) p-value based on the given ordering of the functions is calculated and returned in the attribute p.

Number of simulations

For the global "rank" envelope test, Myllymäki et al. (2017) recommended to use at least 2500 simulations for testing at the significance level alpha = 0.05 for single function tests, based on experiments with summary functions for point processes evaluated approximately at 500 argument values. In this case, the width of the p-interval associated with the extreme rank measure tended to be smaller than 0.01. The tests 'erl', 'cont' and 'area', similarly as the MAD deviation/envelope tests 'qdir', 'st' and 'unscaled', allow in principle a lower number of simulations to be used than the test based on extreme ranks ('rank'), because no ties occur for these measures. If affordable, we recommend in any case some thousands of simulations for all the measures to achieve a good power and repeatability of the test. If the dimension of the test functions is higher, also the number of simulations should preferably be higher.

Tests based on several functions

If a list of (suitable) objects are provided in the argument curve_sets, then by default (nstep = 2) the two-step combining procedure is used to perform the combined global test as described in Myllymäki and Mrkvička (2020). If nstep = 1 and the lengths of the multivariate vectors in each component of the list are equal, then the one-step combining procedure is used where the functions are concatenated together into a one long vector.

References

Mrkvička, T., Myllymäki, M. and Hahn, U. (2017). Multiple Monte Carlo testing, with applications in spatial point processes. Statistics & Computing 27(5), 1239-1255. doi: 10.1007/s11222-016-9683-9

Mrkvička, T., Myllymäki, M., Jilek, M. and Hahn, U. (2020) A one-way ANOVA test for functional data with graphical interpretation. Kybernetika 56(3), 432-458. doi: 10.14736/kyb-2020-3-0432

Mrkvička, T., Myllymäki, M., Kuronen, M. and Narisetty, N. N. (2022) New methods for multiple testing in permutation inference for the general linear model. Statistics in Medicine 41(2), 276-297. doi: 10.1002/sim.9236

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

Myllymäki, M. and Mrkvička, T. (2020). GET: Global envelopes in R. arXiv:1911.06583 [stat.ME]

Ripley, B.D. (1981). Spatial statistics. Wiley, New Jersey.

See Also

plot.global_envelope, central_region, GET.composite

Examples

# Goodness-of-fit testing for simple hypothesis
if(require("spatstat.explore", quietly=TRUE)) {
  # Testing complete spatial randomness (CSR)
  #==========================================
  X <- unmark(spruces)

  nsim <- 1999 # Number of simulations
  

  # Illustration of general workflow for simple hypotheses
  #=======================================================
  # First illustrate the general workflow for the test by this example
  # of CSR test for a point pattern X using the empirical L-function.
  # Define the argument values at which the functions are evaluated
  obs.L <- Lest(X, correction="translate")
  r <- obs.L[['r']]
  # The test function for the data
  obs <- obs.L[['trans']] - r
  # Prepare simulations and calculate test functions for them at same r as 'obs'
  sim <- matrix(nrow=length(r), ncol=nsim)
  for(i in 1:nsim) {
    sim.X <- runifpoint(ex=X) # simulation under CSR
    sim[, i] <- Lest(sim.X, correction="translate", r=r)[['trans']] - r
  }
  # Create a curve_set containing argument values, observed and simulated functions
  cset <- create_curve_set(list(r=r, obs=obs, sim_m=sim))
  # Perform the test
  res <- global_envelope_test(cset, type="erl")
  plot(res) + ggplot2::ylab(expression(italic(hat(L)(r)-r)))

  # Simple hypothesis for a point pattern utilizing the spatstat package
  #=====================================================================
  # Generate nsim simulations under CSR, calculate L-function for the data and simulations
  env <- envelope(X, fun="Lest", nsim=nsim,
                  savefuns=TRUE, # save the functions
                  correction="translate", # edge correction for L
                  transform=expression(.-r), # centering
                  simulate=expression(runifpoint(ex=X))) # Simulate CSR
  # The rank envelope test (ERL)
  res <- global_envelope_test(env, type="erl")
  # Plot the result
  plot(res)

  ## Advanced use:
  # Choose the interval of distances [r_min, r_max] (at the same time create a curve_set from 'env')
  cset <- crop_curves(env, r_min=1, r_max=7)
  # Do the rank envelope test (erl)
  res <- global_envelope_test(cset, type="erl")
  plot(res) + ggplot2::ylab(expression(italic(L(r)-r)))

  # A combined global envelope test
  #================================
  # As an example test CSR of the saplings point pattern by means of
  # L, F, G and J functions.
  data(saplings)
  X <- as.ppp(saplings, W=square(75))

  nsim <- 499 # Number of simulations
  
  # Specify distances for different test functions
  n <- 500 # the number of r-values
  rmin <- 0; rmax <- 20; rstep <- (rmax-rmin)/n
  rminJ <- 0; rmaxJ <- 8; rstepJ <- (rmaxJ-rminJ)/n
  r <- seq(0, rmax, by=rstep)    # r-distances for Lest
  rJ <- seq(0, rmaxJ, by=rstepJ) # r-distances for Fest, Gest, Jest
  

  # Perform simulations of CSR and calculate the L-functions
  env_L <- envelope(X, nsim=nsim,
   simulate=expression(runifpoint(ex=X)),
   fun="Lest", correction="translate",
   transform=expression(.-r), # Take the L(r)-r function instead of L(r)
   r=r,                       # Specify the distance vector
   savefuns=TRUE,             # Save the estimated functions
   savepatterns=TRUE)         # Save the simulated patterns
  # Take the simulations from the returned object
  simulations <- attr(env_L, "simpatterns")
  # Then calculate the other test functions F, G, J for each simulated pattern
  env_F <- envelope(X, nsim=nsim, simulate=simulations,
                    fun="Fest", correction="Kaplan", r=rJ,
                    savefuns=TRUE)
  env_G <- envelope(X, nsim=nsim, simulate=simulations,
                    fun="Gest", correction="km", r=rJ,
                    savefuns=TRUE)
  env_J <- envelope(X, nsim=nsim, simulate=simulations,
                    fun="Jest", correction="none", r=rJ,
                    savefuns=TRUE)

  # Crop the curves to the desired r-interval I
  curve_set_L <- crop_curves(env_L, r_min=rmin, r_max=rmax)
  curve_set_F <- crop_curves(env_F, r_min=rminJ, r_max=rmaxJ)
  curve_set_G <- crop_curves(env_G, r_min=rminJ, r_max=rmaxJ)
  curve_set_J <- crop_curves(env_J, r_min=rminJ, r_max=rmaxJ)

  res <- global_envelope_test(curve_sets=list(L=curve_set_L, F=curve_set_F,
                                              G=curve_set_G, J=curve_set_J))
  plot(res)
  plot(res, labels=c("L(r)-r", "F(r)", "G(r)", "J(r)"))
}

# A test based on a low dimensional random vector
#================================================
# Let us generate some example data.
X <- matrix(c(-1.6,1.6),1,2) # data pattern X=(X_1,X_2)
if(requireNamespace("mvtnorm", quietly=TRUE)) {
  Y <- mvtnorm::rmvnorm(200,c(0,0),matrix(c(1,0.5,0.5,1),2,2)) # simulations
  plot(Y, xlim=c(min(X[,1],Y[,1]), max(X[,1],Y[,1])), ylim=c(min(X[,2],Y[,2]), max(X[,2],Y[,2])))
  points(X, col=2)

  # Test the null hypothesis is that X is from the distribution of Y's (or if it is an outlier).

  # Case 1. The test vector is (X_1, X_2)
  cset1 <- create_curve_set(list(r=1:2, obs=as.vector(X), sim_m=t(Y)))
  res1 <- global_envelope_test(cset1)
  plot(res1)

  # Case 2. The test vector is (X_1, X_2, (X_1-mean(Y_1))*(X_2-mean(Y_2))).
  t3 <- function(x, y) { (x[,1]-mean(y[,1]))*(x[,2]-mean(y[,2])) }
  cset2 <- create_curve_set(list(r=1:3, obs=c(X[,1],X[,2],t3(X,Y)), sim_m=rbind(t(Y), t3(Y,Y))))
  res2 <- global_envelope_test(cset2)
  plot(res2)
}

GET documentation built on Nov. 16, 2022, 5:09 p.m.