Title:	Sequential Outlier Identification for Model-Based Clustering
Version:	0.0.1
Description:	Sequential outlier identification for Gaussian mixture models using the distribution of Mahalanobis distances. The optimal number of outliers is chosen based on the dissimilarity between the theoretical and observed distributions of the scaled squared sample Mahalanobis distances. Also includes an extension for Gaussian linear cluster-weighted models using the distribution of studentized residuals. Doherty, McNicholas, and White (2025) <doi:10.48550/arXiv.2505.11668>.
License:	MIT + file LICENSE
Encoding:	UTF-8
RoxygenNote:	7.3.2
Imports:	ClusterR, dbscan, flexCWM, ggplot2, mixture, mvtnorm, spatstat.univar, stats
Depends:	R (≥ 4.1.0)
LazyData:	true
NeedsCompilation:	no
Packaged:	2025-05-26 15:07:41 UTC; Administrator
Author:	Ultán P. Doherty [aut, cre, cph], Paul D. McNicholas [aut], Arthur White [aut]
Maintainer:	Ultán P. Doherty <dohertyu@tcd.ie>
Repository:	CRAN
Date/Publication:	2025-05-28 15:40:02 UTC

outlierMBC: Sequential Outlier Identification for Model-Based Clustering

Description

Sequential outlier identification for Gaussian mixture models using the distribution of Mahalanobis distances. The optimal number of outliers is chosen based on the dissimilarity between the theoretical and observed distributions of the scaled squared sample Mahalanobis distances. Also includes an extension for Gaussian linear cluster-weighted models using the distribution of studentized residuals. Doherty, McNicholas, and White (2025) doi:10.48550/arXiv.2505.11668.

Author(s)

Maintainer: Ultán P. Doherty dohertyu@tcd.ie (ORCID) [copyright holder]

Authors:

• Paul D. McNicholas paulmc@mcmaster.ca (ORCID)

• Arthur White arwhite@tcd.ie (ORCID)

Move backwards from the minimum to a more conservative solution.

Description

Given a vector of dissimilarity values, each corresponding to a different number of outliers, this function first finds the index and value of the minimum dissimilarity, then moves backwards from right to left to a reasonable solution with a lower index (i.e. lower number of outliers). Limits are placed on the maximum increase in dissimilarity from a single step (max_step_rise) and from all steps (max_total_rise), where both are defined in proportion to the minimum dissimilarity value.

Usage

backtrack(x, max_total_rise = 0.1, max_step_rise = 0.05)

Arguments

Value

backtrack returns a list with two elements, minimum and backtrack:

minimum is a list with the following elements:

ind

Index of the minimum solution.

val

Value of the minimum solution.

backtrack is a list with the following elements:

ind

Index of the backtrack solution.

val

Value of the backtrack solution.

Examples

ombc_gmm_k3n1000o10 <-
  ombc_gmm(gmm_k3n1000o10[, 1:2], comp_num = 3, max_out = 20)

backtrack(ombc_gmm_k3n1000o10$distrib_diff_vec)

Fit a Gaussian mixture model to the backtrack solution.

Description

The backtrack function determines the number of outliers for the backtrack solution and plot_backtrack plots this on a dissimilarity curve. backtrack_gmm fits the mixture model corresponding to the number of outliers selected by the backtrack solution (or any manually specified number of outliers).

Usage

backtrack_gmm(
  x,
  ombc_out,
  max_total_rise = 0.1,
  max_step_rise = 0.05,
  init_model = NULL,
  init_z = NULL,
  manual_outlier_num = NULL,
  verbose = TRUE
)

Arguments

Value

backtrack_gmm returns a list with the following elements:

labels

Vector of mixture component labels with outliers denoted by 0.

outlier_bool

Logical vector indicating if an observation has been classified as an outlier.

outlier_num

Number of observations classified as outliers.

mix

Output from mixture::gpcm fitted to the non-outlier observations.

call

Arguments / parameter values used in this function call.

Examples

ombc_gmm_k3n1000o10 <- ombc_gmm(
  gmm_k3n1000o10[, 1:2],
  comp_num = 3, max_out = 20
)

backtrack_gmm(gmm_k3n1000o10[, 1:2], ombc_gmm_k3n1000o10)

Fit a linear cluster-weighted model to the backtrack solution.

Description

The backtrack function determines the number of outliers for the backtrack solution and plot_backtrack plots this on a dissimilarity curve. backtrack_gmm fits the mixture model corresponding to the number of outliers selected by the backtrack solution (or any manually specified number of outliers).

Usage

backtrack_lcwm(
  xy,
  x,
  ombc_lcwm_out,
  max_total_rise = 0.1,
  max_step_rise = 0.05,
  init_z = NULL,
  manual_outlier_num = NULL,
  verbose = TRUE
)

Arguments

Value

backtrack_gmm returns a list with the following elements:

labels

Vector of component labels with outliers denoted by 0.

outlier_bool

Logical vector indicating if an observation has been classified as an outlier.

outlier_num

Number of observations classified as outliers.

lcwm

Output from flexCWM::cwm fitted to the non-outlier observations.

call

Arguments / parameter values used in this function call.

Examples

gross_lcwm_k3n1000o10 <- find_gross(lcwm_k3n1000o10, 20)

ombc_lcwm_k3n1000o10 <- ombc_lcwm(
  xy = lcwm_k3n1000o10[, c("X1", "Y")],
  x = lcwm_k3n1000o10$X1,
  y_formula = Y ~ X1,
  comp_num = 2,
  max_out = 20,
  mnames = "V",
  gross_outs = gross_lcwm_k3n1000o10$gross_bool
)

backtrack_lcwm_k3n1000o10 <- backtrack_lcwm(
  xy = lcwm_k3n1000o10[, c("X1", "Y")],
  x = lcwm_k3n1000o10$X1,
  ombc_lcwm_out = ombc_lcwm_k3n1000o10
)

Compute the dissimilarity for a Gaussian mixture model and identify the lowest density observation.

Description

At each iteration of ombc_gmm, distrib_diff_gmm computes the dissimilarity value of the current Gaussian mixture model. It also identifies the observation with the lowest mixture density.

Usage

distrib_diff_gmm(x, z, prop, mu, sigma, logdet)

Arguments

Value

distrib_diff_gmm returns a list with the following elements:

distrib_diff

Aggregated dissimilarity across components.

distrib_diff_vec

Vector containing dissimilarity value for each component.

choice_id

Index of observation with lowest mixture density.

removal_dens

Value of the lowest mixture density.

Compute the dissimilarity for a linear cluster-weighted model and identify the lowest density observation.

Description

At each iteration of ombc_lcwm, distrib_diff_lcwm computes the dissimilarity value of the current linear cluster-weighted model. It also identifies the observation with the lowest mixture density.

Usage

distrib_diff_lcwm(x, z, prop, mu, sigma, mod_list, y_sigma, dd_weight = 0.5)

Arguments

Value

distrib_diff_lcwm_lcwm returns a list with the following elements:

distrib_diff

Aggregated dissimilarity across components.

distrib_diff_vec

Vector containing dissimilarity value for each component.

choice_id

Index of observation with lowest mixture density.

removal_dens

Value of the lowest mixture density.

distrib_diff_mat

Two-column matrix containing response and covariate dissimilarities across components.

Compute the dissimilarity for a single component of a Linear CWM.

Description

Computes the covariate dissimilarity value, the response dissimilarity value, and their aggregated dissimilarity value. It also obtains the covariate, response, and joint densities for every observation.

Usage

distrib_diff_lcwm_g(x, z_g, mu_g, sigma_g, mod_g, y_sigma_g, dd_weight = 0.5)

Arguments

Value

distrib_diff_lcwm_lcwm_g returns a list with the following elements:

diff

Aggregated dissimilarity value for this component.

dens

Joint (covariate & response) density of all observations for this component.

diff_x

Covariate dissimilarity value for this component.

diff_y

Response dissimilarity value for this component.

dens_x

Covariate density of all observations for this component.

dens_y

Response density of all observations for this component.

Compute the dissimilarity for a single multivariate Gaussian distribution.

Description

Compute the dissimilarity value and observation densities for a single multivariate Gaussian distribution. This could be a whole component in a Gaussian mixture model or the covariate part of a component in a Linear CWM.

Usage

distrib_diff_mahalanobis(x, z_g, mu_g, sigma_g, logdet_g)

Arguments

Value

distrib_diff_mahalanobis returns a list with the following elements:

diff

Dissimilarity value for this component.

dens

Gaussian density of all observations for this component.

mahalas

Scaled squared sample Mahalanobis distances for all observations with respect to this component.

Compute the response dissimilarity for a single component of a Linear CWM.

Description

Computes the response dissimilarity value and the response density for every observation.

Usage

distrib_diff_residual(x, z_g, mod_g, y_sigma_g)

Arguments

Value

distrib_diff_lcwm_residual returns a list with the following elements:

diff

Response dissimilarity value for this component.

dens

Response density of all observations for this component.

Find gross outliers.

Description

The distance of each observation to its k^{th} nearest neighbour is computed. We assume that the largest max_out kNN distances correspond to potential outliers. We select the next largest kNN distance, outside of the top max_out, as a benchmark value. We multiply this benchmark kNN distance by multiplier to get the minimum threshold for our gross outliers. In other words, a gross outlier must have a kNN distance at least multiplier times greater than all of the observations which we do not consider to be potential outliers.

Usage

find_gross(
  x,
  max_out,
  multiplier = 3,
  k_neighbours = floor(nrow(x)/100),
  manual_threshold = NULL,
  scale = TRUE
)

Arguments

Value

find_gross returns a list with the following elements:

gross_choice

A numeric value indicating the elbow's location.

gross_bool

A logical vector identifying the gross outliers.

gross_curve

ggplot of the highest 2 * max_out kNN distances in decreasing order.

gross_scatter

ggplot of all kNN distances in index order.

Obtain an initial clustering as a component assignment matrix.

Description

Implement the specified initial clustering, either hierarchical clustering or k-means++, and return a binary component assignment matrix.

Usage

get_init_z(
  comp_num,
  dist_mat = NULL,
  x = NULL,
  init_method = c("hc", "kmpp"),
  kmpp_seed = NULL
)

Arguments

Value

A component assignment matrix for initialisation.

Simulated data set consisting of 1000 observations from 3 Gaussian components and 10 outliers.

Description

This data set was simulated using simulate_gmm. There are 500 observations in Component 1, 250 observations in Component 2, and 250 observations in Component 3

Usage

gmm_k3n1000o10

Format

gmm_k3n1000o10

A data frame with 1010 rows and 3 columns:

X1, X2

Continuous variables.

G

Component label: 0 for outliers; 1, 2, or 3 for true points.

Source

For simulation code, see gmm_k3n1000o10.R in data-raw folder at https://github.com/UltanPDoherty/outlierMBC.

Simulated data set consisting of 2000 observations from 3 Gaussian components and 20 outliers.

Description

This data set was simulated using simulate_gmm. There are 1000 observations in Component 1, 500 observations in Component 2, and 500 observations in Component 3.

Usage

gmm_k3n2000o20

Format

gmm_k3n2000o20

A data frame with 2020 rows and 3 columns:

X1, X2

Continuous variables.

G

Component label: 0 for outliers; 1, 2, or 3 for true points.

Source

For simulation code, see gmm_k3n2000o20.R in data-raw folder at https://github.com/UltanPDoherty/outlierMBC.

Simulated data set consisting of 4000 observations from 3 Gaussian components and 40 outliers.

Description

This data set was simulated using simulate_gmm. There are 2000 observations in Component 1, 1000 observations in Component 2, and 1000 observations in Component 3.

Usage

gmm_k3n4000o40

Format

gmm_k3n4000o40

A data frame with 4040 rows and 3 columns:

X1, X2

Continuous variables.

G

Component label: 0 for outliers; 1, 2, or 3 for true points.

Source

For simulation code, see gmm_k3n4000o40.R in data-raw folder at https://github.com/UltanPDoherty/outlierMBC.

Simulated data set consisting of 1000 observations from 3 Gaussian components and 10 outliers.

Description

This data set was simulated using simulate_lcwm. There are 300 observations in Component 1, 300 observations in Component 2, and 400 observations in Component 3

Usage

lcwm_k3n1000o10

Format

lcwm_k3n1000o10

A data frame with 1010 rows and 3 columns:

X1

Continuous explanatory variable.

Y

Continuous response variable.

G

Component label: 0 for outliers; 1, 2, or 3 for true points.

Source

For simulation code, see lcwm_k3n1000o10.R in data-raw folder at https://github.com/UltanPDoherty/outlierMBC.

Simulated data set consisting of 2000 observations from 3 Gaussian components and 20 outliers.

Description

This data set was simulated using simulate_lcwm. There are 600 observations in Component 1, 600 observations in Component 2, and 800 observations in Component 3.

Usage

lcwm_k3n2000o20

Format

lcwm_k3n2000o20

A data frame with 2020 rows and 3 columns:

X1

Continuous explanatory variable.

Y

Continuous response variable.

G

Component label: 0 for outliers; 1, 2, or 3 for true points.

Source

For simulation code, see lcwm_k3n2000o20.R in data-raw folder at https://github.com/UltanPDoherty/outlierMBC.

Simulated data set consisting of 4000 observations from 3 Gaussian components and 40 outliers.

Description

This data set was simulated using simulate_lcwm. There are 1200 observations in Component 1, 1200 observations in Component 2, and 1600 observations in Component 3.

Usage

lcwm_k3n4000o40

Format

lcwm_k3n4000o40

A data frame with 4040 rows and 3 columns:

X1

Continuous explanatory variable.

Y

Continuous response variable.

G

Component label: 0 for outliers; 1, 2, or 3 for true points.

Source

For simulation code, see lcwm_k3n4000o40.R in data-raw folder at https://github.com/UltanPDoherty/outlierMBC.

Constructor for "outliermbc_gmm" S3 class.

Description

Constructor for "outliermbc_gmm" S3 class.

Usage

new_outliermbc_gmm(x = list())

Arguments

Value

"outliermbc_gmm" S3 object.

Constructor for "outliermbc_lcwm" S3 object.

Description

Constructor for "outliermbc_lcwm" S3 object.

Usage

new_outliermbc_lcwm(x = list())

Arguments

Value

"outliermbc_lcwm" S3 object.

Sequentially identify outliers while fitting a Gaussian mixture model.

Description

This function performs model-based clustering and outlier identification. It does so by iteratively fitting a Gaussian mixture model and removing the observation that is least likely under the model. Its procedure is summarised below:

1. Fit a Gaussian mixture model to the data.

2. Compute a dissimilarity between the theoretical and observed distributions of the scaled squared sample Mahalanobis distances for each mixture component.

3. Aggregate across the components to obtain a single dissimilarity value.

4. Remove the observation with the lowest mixture density.

5. Repeat Steps 1-4 until max_out observations have been removed.

6. Identify the number of outliers which minimised the aggregated dissimilarity, remove only those observations, and fit a Gaussian mixture model to the remaining data.

Usage

ombc_gmm(
  x,
  comp_num,
  max_out,
  gross_outs = rep(FALSE, nrow(x)),
  init_scheme = c("update", "reinit", "reuse"),
  mnames = "VVV",
  nmax = 1000,
  atol = 1e-08,
  init_z = NULL,
  init_model = NULL,
  init_method = c("hc", "kmpp"),
  init_scaling = FALSE,
  kmpp_seed = 123,
  fixed_labels = NULL,
  verbose = TRUE
)

Arguments

Value

ombc_gmm returns an object of class "outliermbc_gmm", which is essentially a list with the following elements:

labels

Vector of mixture component labels with outliers denoted by 0.

outlier_bool

Logical vector indicating if an observation has been classified as an outlier.

outlier_num

Number of observations classified as outliers.

outlier_rank

Order in which observations are removed from the data set. Observations which were provisionally removed, including those that were eventually not classified as outliers, are ranked from 1 to max_out. All gross outliers have rank 1. If there are gross_num gross outliers, then the observations removed during the main algorithm itself will be numbered from gross_num + 1 to max_out. Observations that were ever removed have rank 0.

gross_outs

Logical vector identifying the gross outliers. This is identical to the gross_outs vector passed to this function as an argument / input.

mix

Output from mixture::gpcm fitted to the non-outlier observations.

loglike

Vector of log-likelihood values for each iteration.

removal_dens

Vector of mixture densities for the removed observations. These are the lowest mixture densities at each iteration.

distrib_diff_vec

Vector of aggregated cross-component dissimilarity values for each iteration.

distrib_diff_mat

Matrix of component-specific dissimilarity values for each iteration.

call

Arguments / parameter values used in this function call.

version

Version of outlierMBC used in this function call.

conv_status

Logical vector indicating which iterations' mixture models reached convergence during model-fitting.

Examples

ombc_gmm_k3n1000o10 <- ombc_gmm(
  gmm_k3n1000o10[, 1:2],
  comp_num = 3, max_out = 20
)

plot_curve(ombc_gmm_k3n1000o10)

Sequentially identify outliers while fitting a linear cluster-weighted model.

Description

This function performs model-based clustering, clusterwise regression, and outlier identification. It does so by iteratively fitting a linear cluster-weighted model and removing the observation that is least likely under the model. Its procedure is summarised below:

1. Fit a linear cluster-weighted model to the data.

2. Compute a dissimilarity between the theoretical and observed distributions of the scaled squared sample Mahalanobis distances for each mixture component.

3. Compute a dissimilarity between the theoretical and observed distributions of the scaled squared studentised residuals for each mixture component.

4. Aggregate these two dissimilarities to obtain one dissimilarity value for each component.

5. Aggregate across the components to obtain a single dissimilarity value.

6. Remove the observation with the lowest mixture density.

7. Repeat Steps 1-6 until max_out observations have been removed.

8. Identify the number of outliers which minimised the aggregated dissimilarity, remove only those observations, and fit a linear cluster-weighted model to the remaining data.

Usage

ombc_lcwm(
  xy,
  x,
  y_formula,
  comp_num,
  max_out,
  gross_outs = rep(FALSE, nrow(x)),
  init_scheme = c("update", "reinit", "reuse"),
  mnames = "VVV",
  nmax = 1000,
  atol = 1e-08,
  init_z = NULL,
  init_method = c("hc", "kmpp"),
  init_scaling = TRUE,
  kmpp_seed = 123,
  verbose = TRUE,
  dd_weight = 0.5
)

Arguments

Value

ombc_lcwm returns an object of class "outliermbc_lcwm", which is essentially a list with the following elements:

labels

Vector of mixture component labels with outliers denoted by 0.

outlier_bool

Logical vector indicating if an observation has been classified as an outlier.

outlier_num

Number of observations classified as outliers.

outlier_rank

Order in which observations are removed from the data set. Observations which were provisionally removed, including those that were eventually not classified as outliers, are ranked from 1 to max_out. All gross outliers have rank 1. If there are gross_num gross outliers, then the observations removed during the main algorithm itself will be numbered from gross_num + 1 to max_out. Observations that were ever removed have rank 0.

gross_outs

Logical vector identifying the gross outliers. This is identical to the gross_outs vector passed to this function as an argument / input.

lcwm

Output from flexCWM::cwm fitted to the non-outlier observations.

loglike

Vector of log-likelihood values for each iteration.

removal_dens

Vector of mixture densities for the removed observations. These are the lowest mixture densities at each iteration.

distrib_diff_vec

Vector of aggregated cross-component dissimilarity values for each iteration.

distrib_diff_mat

Matrix of component-specific dissimilarity values for each iteration.

distrib_diff_arr

Array of component-specific response and covariate dissimilarity values for each iteration.

call

Arguments / parameter values used in this function call.

version

Version of outlierMBC used in this function call.

conv_status

Logical vector indicating which iterations' mixture models reached convergence during model-fitting.

Examples

gross_lcwm_k3n1000o10 <- find_gross(lcwm_k3n1000o10, 20)

ombc_lcwm_k3n1000o10 <- ombc_lcwm(
  xy = lcwm_k3n1000o10[, c("X1", "Y")],
  x = lcwm_k3n1000o10$X1,
  y_formula = Y ~ X1,
  comp_num = 3,
  max_out = 20,
  mnames = "V",
  gross_outs = gross_lcwm_k3n1000o10$gross_bool
)

plot method for "outliermbc_gmm" S3 class.

Description

plot method for "outliermbc_gmm" S3 class.

Usage

## S3 method for class 'outliermbc_gmm'
plot(x, backtrack = FALSE, ...)

Arguments

Value

A ggplot

plot method for "outliermbc_lcwm" S3 class.

Description

plot method for "outliermbc_lcwm" S3 class.

Usage

## S3 method for class 'outliermbc_lcwm'
plot(x, backtrack = FALSE, ...)

Arguments

Value

A ggplot

Plot the dissimilarity curve showing the backtrack solution.

Description

Plots a rescaled dissimilarity curve where the dissimilarity values (y axis) have been divided by their minimum so that the rescaled minimum is at 1. Vertical lines mark the minimum and backtrack solutions.

Usage

plot_backtrack(ombc_out, max_total_rise = 0.1, max_step_rise = 0.05)

Arguments

Value

plot_backtrack returns a ggplot of the rescaled dissimilarity curve showing the minimum solution and the backtrack solutions.

Plot multiple dissimilarity curves.

Description

Given a range of ombc_gmm outputs, each arising from a different model, this function is designed to produce a graphical aid for selecting the best model. It displays the dissimilarity curves from each of these models on the same plot.

Usage

plot_comparison(ombc_list)

Arguments

Value

plot_comparison returns a ggplot object consisting of multiple dissimilarity curves overlaid on the same plot.

Plot multiple dissimilarity curves.

Description

Given a range of ombc_gmm outputs, each arising from a different model, this function is designed to produce a graphical aid for selecting the best model. It displays the dissimilarity curves from each of these models on the same plot.

Usage

plot_comparison_bic(ombc_list)

Arguments

Value

plot_comparison returns a ggplot object consisting of multiple dissimilarity curves overlaid on the same plot.

Plot the dissimilarity curve.

Description

Given the output from ombc_gmm or ombc_lcwm, this function extracts the dissimilarity value associated with each outlier number and plots them as a curve. It also draws a vertical line at the outlier number which minimised the dissimilarity.

Usage

plot_curve(ombc_out)

Arguments

Value

plot_curve returns a ggplot object showing the dissimilarity values as a curve and marking the minimum solution with a vertical line.

Plot dissimilarity values for multiple solutions.

Description

Given a range of ombc_gmm outputs, each arising from a different model, this function is designed to produce a graphical aid for selecting the best model. It plots the dissimilarity values of the models' minimum and backtrack solutions against their number of components (x_axis = "comp_num"), number of outliers (x_axis = "outlier_num"), or number of parameters (x_axis = "param_num").

Usage

plot_selection(ombc_list, x_axis = c("comp_num", "outlier_num", "param_num"))

Arguments

Value

plot_selection return a ggplot object plotting the minimum dissimilarity and backtrack solutions from a number of outputs from ombc_gmm versus their number of components, outliers, or parameters.

print method for "outliermbc_gmm" S3 class.

Description

print method for "outliermbc_gmm" S3 class.

Usage

## S3 method for class 'outliermbc_gmm'
print(x, backtrack = FALSE, max_total_rise = 0.1, max_step_rise = 0.05, ...)

Arguments

Value

A ggplot

print method for "outliermbc_lcwm" S3 class.

Description

print method for "outliermbc_lcwm" S3 class.

Usage

## S3 method for class 'outliermbc_lcwm'
print(x, backtrack = FALSE, max_total_rise = 0.1, max_step_rise = 0.05, ...)

Arguments

Value

A ggplot

Simulate data from a Gaussian mixture model with outliers.

Description

Simulates data from a Gaussian mixture model, then simulates outliers from a hyper-rectangle, with a rejection step to ensure that the outliers are sufficiently unlikely under the model.

Usage

simulate_gmm(
  n,
  mu,
  sigma,
  outlier_num,
  seed = NULL,
  crit_val = 0.9999,
  range_multiplier = 1.5,
  verbose = TRUE,
  max_rejection = 1e+06
)

Arguments

Details

The simulated outliers are sampled from a Uniform distribution over a hyper-rectangle. For each dimension, the hyper-rectangle is centred at the midpoint between the maximum and minimum values for that variable from all of the Gaussian observations. Its width in that dimension is the distance between the minimum and maximum values for that variable multiplied by the value of range_multiplier. If range_multiplier = 1, then this hyper-rectangle is the axis-aligned minimum bounding box for all of the Gaussian data points in this data set.

The crit_val ensures that it would have been sufficiently unlikely for a simulated outlier to have been sampled from any of the Gaussian components. The Mahalanobis distances of a proposed outlier from each component's mean vector with respect to that component's covariance matrix are computed. If any of these Mahalanobis distances are smaller than the critical value of the appropriate Chi-squared distribution, then the proposed outlier is rejected. In summary, for a Uniform sample to be accepted, it must be sufficiently far from each component in terms of Mahalanobis distance.

Value

simulate_gmm return a data.frame with continuous variables X1, X2, ..., followed by a mixture component label vector G with outliers denoted by 0.

Examples

gmm_k3n1000o10 <- simulate_gmm(
  n = c(500, 250, 250),
  mu = list(c(-1, 0), c(+1, -1), c(+1, +1)),
  sigma = list(diag(c(0.2, 4 * 0.2)), diag(c(0.2, 0.2)), diag(c(0.2, 0.2))),
  outlier_num = 10,
  seed = 123,
  crit_val = 0.9999,
  range_multiplier = 1.5
)

plot(
  gmm_k3n1000o10[, c("X1", "X2")],
  col = gmm_k3n1000o10$G + 1, pch = gmm_k3n1000o10$G + 1
)

Simulate data from a linear cluster-weighted model with outliers.

Description

Simulates data from a linear cluster-weighted model, then simulates outliers from a region around each mixture component, with a rejection step to control how unlikely the outliers are under the model.

Usage

simulate_lcwm(
  n,
  mu,
  sigma,
  beta,
  error_sd,
  outlier_num,
  outlier_type = c("x_and_y", "x_only", "y_only"),
  seed = NULL,
  prob_range = c(1e-08, 1e-06),
  range_multipliers = c(3, 3),
  more_extreme = FALSE
)

Arguments

Details

simulate_lcwm samples a user-defined number of outliers for each component. However, even though an outlier may be associated with one component, it must be outlying with respect to every component.

The covariate values of the simulated outliers for a given component g are sampled from a Uniform distribution over a hyper-rectangle which is specific to that component. For each covariate dimension, the hyper-rectangle is centred at the midpoint between the maximum and minimum values for that variable from all of the Gaussian observations from component g. Its width in that dimension is the distance between the minimum and maximum values for that variable multiplied by the value of range_multiplier[1].

The response values of the simulated outliers for a given component g are obtained by sampling random errors from a Uniform distribution over a univariate interval, simulating covariate values as discussed above, computing the mean response value for those covariate values, then adding this simulated error to the response. The error sampling interval is centred at the midpoint between the maximum and minimum errors for that variable from all of the Gaussian observations from component g. Its width is the distance between the minimum and maximum errors multiplied by the value of range_multiplier[2].

A proposed outlier for component g is rejected if the probability of sampling a more extreme point from any of the components is greater than prob_range[2] or if the probability of sampling a less extreme point from component g is less than prob_range[1]. This can be visualised as a pair of inner and outer envelopes around each component. To be accepted, a proposed outlier must lie inside the outer envelope for its component and outside the inner envelopes of all components. Setting prob_range[1] = 0 will eliminate the outer envelope, while setting prob_range[2] = 0 will eliminate the inner envelope.

By setting outlier_type = "x_only" and giving arbitrary values to error_sd (e.g. a zero vector) and beta (e.g. a list of zero vectors), then ignoring the simulated Y variable, simulate_lcwm can be used to simulate a Gaussian mixture model. Since simulate_lcwm simulates component-specific outliers from sampling regions around each component, rather than a single sampling region around all of the components, this will not be equivalent to simulate_gmm. simulate_lcwm also allows the user to set an upper bound on how unlikely an outlier is, as well as a lower bound, whereas simulate_gmm only sets a lower bound.

Value

simulate_lcwm returns a data.frame with continuous variables X1, X2, ..., followed by a continuous response variable, Y, and a mixture component label vector G with outliers denoted by 0. The optional variable more_extreme may be included, if specified by the corresponding argument.

Examples

lcwm_k3n1000o10 <- simulate_lcwm(
  n = c(300, 300, 400),
  mu = list(c(3), c(6), c(3)),
  sigma = list(as.matrix(1), as.matrix(0.1), as.matrix(1)),
  beta = list(c(0, 0), c(-75, 15), c(0, 5)),
  error_sd = c(1, 1, 1),
  outlier_num = c(3, 3, 4),
  outlier_type = "x_and_y",
  seed = 123,
  prob_range = c(1e-8, 1e-6),
  range_multipliers = c(1, 2)
)

plot(
  lcwm_k3n1000o10[, c("X1", "Y")],
  col = lcwm_k3n1000o10$G + 1,
  pch = lcwm_k3n1000o10$G + 1
)

Check if a new sample satisfies the outlier criteria.

Description

This function checks whether a given sample is an acceptable outlier with respect to prob_range and also computes the probability of sampling a more extreme point from component g.

Usage

test_outlier_ombc(
  outlier_type,
  mu,
  sigma,
  beta,
  error_sd,
  x_sample,
  y_sample,
  prob_range,
  g
)

Arguments

Value

test_outlier_ombc returns a vector consisting of a logical value indicating whether the new sample satisfies the outlier checks, and a numeric value giving the probability of sampling a more extreme point from component g.

Run mixture::gpcm and try alternative covariance structures or initialisations if necessary.

Description

If mixture::gpcm returns an error, this function first tries the other covariance structures, and then tries a k-means initialisation.

Usage

try_mixture_gpcm(x, comp_num, mnames, z, nmax, atol, fixed_labels)

Arguments

Value

Object of class "gpcm" outputted by mixture::gpcm.

Produce a single sample that passes the outlier checks.

Description

This function calls uniform_sample_lcwm to sample a proposed outlier and then calls test_outlier_ombc to check if it satisfies the required criteria.

Usage

uniform_outlier_ombc(
  outlier_type,
  mu,
  sigma,
  beta,
  error_sd,
  g,
  uniform_spans,
  prob_range
)

Arguments

Value

uniform_outlier_ombc returns a simulated outlier as a vector containing its covariate values, response value, and its component label 0. This vector's final element is the probability of sampling a more extreme Gaussian point from this outlier's associated component.

Sample a potential outlier.

Description

If outlier_type = "x_and_y", then both the covariate values and response error of the outlier proposed by this function will be Uniformly distributed. If outlier_type = "x_only", then the covariate values will be Uniformly distributed but the response error will be Normally distributed. If outlier_type = "y_only", then the response error will be Uniformly distributed but the covariate values will be Normally distributed.

Usage

uniform_sample_lcwm(
  outlier_type,
  mu_g,
  sigma_g,
  beta_g,
  error_sd_g,
  uniform_spans_g
)

Arguments

Value

uniform_sample_lcwm returns a list with the following elements:

x

Vector of covariate values.

y

Response value.

Obtain the span of the observations for each component.

Description

Determine the minimum and maximum values for each covariate / explanatory variable and for the response errors from all Gaussian observations.

Usage

uniform_spans_lcwm(range_multipliers, covariates_g, errors_g)

Arguments

Value

uniform_spans_lcwm returns a 2-column matrix. The final row contains the minimum and maximum values of the response errors, while the previous rows contain the minimum and maximum values for each covariate.

Validator for "outliermbc_gmm" S3 class.

Description

Validator for "outliermbc_gmm" S3 class.

Usage

validate_outliermbc_gmm(x)

Arguments

Validator for "outliermbc_lcwm" S3 class.

Description

Validator for "outliermbc_lcwm" S3 class.

Usage

validate_outliermbc_lcwm(x)

Arguments