ICS_outlier: Outlier Detection Using ICS

Description

In a multivariate framework outlier(s) are detected using ICS. The function performs ICS() and decides automatically about the number of invariant components to use to search for the outliers and the number of outliers detected on these components. Currently the function is restricted to the case of searching outliers only on the first components.

Usage

ICS_outlier(
  X,
  S1 = ICS_cov,
  S2 = ICS_cov4,
  S1_args = list(),
  S2_args = list(),
  ICS_algorithm = c("whiten", "standard", "QR"),
  method = "norm_test",
  test = "agostino.test",
  n_eig = 10000,
  level_test = 0.05,
  adjust = TRUE,
  level_dist = 0.025,
  n_dist = 10000,
  type = "smallprop",
  n_cores = NULL,
  iseed = NULL,
  pkg = "ICSOutlier",
  q_type = 7,
  ...
)

Value

An object of S3-class 'ICS_Out' which contains:

outliers: A vector containing ones for outliers and zeros for non outliers.
ics_distances: A numeric vector containing the squared ICS distances.
ics_dist_cutoff: The cut-off for the distances to decide if an observation is outlying or not.
level_dist: The level for deciding upon the cut-off value for the ICS distances.
level_test: The initial level for selecting the invariant coordinates.
method: Name of the method used to decide upon the number of ICS components.
index: Vector giving the indices of the ICS components selected.
test: The name of the normality test as specified in the function call.
criterion: Vector giving the marginal levels for the components selection.
adjust: Wether the initial level used to decide upon the number of components has been adjusted for multiple testing or not.
type: Currently always the string "smallprop".
n_dist: Number of simulations performed to decide upon the cut-off for the ICS distances.
n_eig: Number of simulations performed for selecting the ICS components based on simulations.
S1_label: Name of S1.
S2_label : Name of S2.

Arguments

X: a numeric matrix or data frame containing the data to be transformed.
S1: an object of class "ICS_scatter" or a function that contains the location vector and scatter matrix as location and scatter components.
S2: an object of class "ICS_scatter" or a function that contains the location vector and scatter matrix as location and scatter components.
S1_args: a list containing additional arguments for S1.
S2_args: a list containing additional arguments for S2.
ICS_algorithm: a character string specifying with which algorithm the invariant coordinate system is computed. Possible values are "whiten", "standard" or "QR".
method: name of the method used to select the ICS components involved to compute ICS distances. Options are "norm_test" and "simulation". Depending on the method either comp_norm_test or comp_simu_test are used.
test: name of the marginal normality test to use if method = "norm_test". Possibilities are "jarque.test", "anscombe.test", "bonett.test", "agostino.test", "shapiro.test".Default is "agostino.test".
n_eig: number of simulations performed to derive the cut-off values for selecting the ICS components. Only if method = "simulation". See comp_simu_test for details.
level_test: for the comp_norm_test or comp_simu_test functions. The initial level for selecting the invariant coordinates.
adjust: logical. For selecting the invariant coordinates, the level of the test can be adjusted for each component to deal with multiple testing. See comp_norm_test and comp_simu_test for details. Default is TRUE.
level_dist: level for the dist_simu_test function. The (1-level)th quantile used to determine the cut-off value for the ICS distances.
n_dist: number of simulations performed to derive the cut-off value for the ICS distances. See dist_simu_test for details.
type: currently the only option is "smallprop" which means that only the first ICS components can be selected. See comp_norm_test or comp_simu_test for details.
n_cores: number of cores to be used in dist_simu_test and comp_simu_test. If NULL or 1, no parallel computing is used. Otherwise makeCluster with type = "PSOCK" is used.
iseed: If parallel computation is used the seed passed on to clusterSetRNGStream. Default is NULL which means no fixed seed is used.
pkg: When using parallel computing, a character vector listing all the packages which need to be loaded on the different cores via require. Must be at least "ICSOutlier" and must contain the packages needed to compute the scatter matrices.
q_type: specifies the quantile algorithm used in quantile.
...: passed on to other methods.

Author

Aurore Archimbaud and Klaus Nordhausen

Details

The ICS method has attractive properties for outlier detection in the case of a small proportion of outliers. As for PCA three steps have to be performed:(i) select the components most useful for the detection, (ii) compute distances as outlierness measures for all observation and finally (iii) label outliers using some cut-off value.

This function performs these three steps automatically:

For choosing the components of interest two methods are proposed: "norm_test" based on some marginal normality tests (see details in comp_norm_test) or "simulation" based on a parallel analysis (see details in comp_simu_test). These two approaches lie on the intrinsic property of ICS in case of a small proportion of outliers with the choice of S1 "more robust" than S2, which ensures to find outliers on the first components. Indeed when using S1 = ICS_cov and S2 = ICS_cov4, the Invariant Coordinates are ordered according to their classical Pearson kurtosis values in decreasing order. The information to find the outliers should be then contained in the first k non-normal directions.
Then the ICS distances are computed as the Euclidean distances on the selected k centered components \(Z_k\).
Finally the outliers are identified based on a cut-off derived from simulations. If the distance of an observation exceeds the expectation under the normal model, this observation is labeled as outlier (see details in dist_simu_test).

As a rule of thumb, the percentage of contamination should be limited to 10% in case of a mixture of gaussian distributions and using the default combination of locations and scatters for ICS.

References

Archimbaud, A., Nordhausen, K. and Ruiz-Gazen, A. (2018), ICS for multivariate outlier detection with application to quality control. Computational Statistics & Data Analysis, 128:184-199. ISSN 0167-9473. tools:::Rd_expr_doi("10.1016/j.csda.2018.06.011").

Examples

Run this code

# ReliabilityData example: the observations 414 and 512 are suspected to be outliers  
library(REPPlab)
data(ReliabilityData)
# For demo purpose only small mDist value, but as extreme quantiles
# are of interest mDist should be much larger. Also number of cores used
# should be larger if available
icsOutlierDA <- ICS_outlier(ReliabilityData, S1 = ICS_tM, S2 = ICS_cov, 
level_dist = 0.01, n_dist = 50, n_cores = 1)
icsOutlierDA
summary(icsOutlierDA)
plot(icsOutlierDA)

if (FALSE) {
  # For using several cores and for using a scatter function from a different package
  # Using the parallel package to detect automatically the number of cores
  library(parallel)
  # ICS with MCD estimates and the usual estimates
  # Need to create a wrapper for the CovMcd function to return first the location estimate
  # and the scatter estimate secondly.
  data(HTP)
 library(ICSClust)
  # For demo purpose only small m value, should select the first seven components
  icsOutlier <- ICS_outlier(HTP, S1 = ICS_mcd_rwt, S2 = ICS_cov,
                            S1_args = list(location = TRUE, alpha = 0.75),
                            n_eig = 50, level_test = 0.05, adjust = TRUE,
                            level_dist = 0.025, n_dist = 50,
                            n_cores =  detectCores()-1, iseed = 123,
                            pkg = c("ICSOutlier", "ICSClust"))
  icsOutlier
}

# Exemple of no direction and hence also no outlier
set.seed(123)
X = rmvnorm(500, rep(0, 2), diag(rep(0.1,2)))
icsOutlierJB <- ICS_outlier(X, test = "jarque.test", level_dist = 0.01,
                            level_test = 0.01, n_dist = 100, n_cores = 1)
summary(icsOutlierJB)
plot(icsOutlierJB)
rm(.Random.seed)

# Example of no outlier
set.seed(123)
X = matrix(rweibull(1000, 4, 4), 500, 2)
X = apply(X,2, function(x){ifelse(x<5 & x>2, x, runif(sum(!(x<5 & x>2)), 5, 5.5))})
icsOutlierAG <- ICS_outlier(X, test = "anscombe.test", level_dist = 0.01,
                            level_test = 0.05, n_dist = 100, n_cores = 1)
summary(icsOutlierAG)
plot(icsOutlierAG)
rm(.Random.seed)

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