Regularized Covariance Matrix Calculation

Description

Computes a regularized covariance matrix using a convex combination of a target matrix and the sample covariance matrix.

Arguments

Details

The function calculates the sample covariance matrix of the centered data XX and combines it with the target covariance matrix mT, scaled by the factor scfac. The regularization is controlled by the parameter rho, where rho = 1 results in using only the target matrix, and rho = 0 uses only the sample covariance.

Value

A named list containing:

Align Loadings of Principal Components

Description

Aligns loadings per neighborhood for better visualization and comparison. Different options are available.

Usage

align(PC, type = "largest", vec = NULL)

Arguments

Value

Returns an array of loadings of size p times k times N.

Examples

set.seed(1)

# Note, that in this example the vectors are not feasible loadings.
COVS = list("a"= matrix(runif(16, -1, 1), 4), "b" = matrix(runif(16, -1, 1), 4))
COVS = lapply(COVS, function(x) x %*% t(x))

pca = msPCA(eta = 1, gamma = 0.5, COVS = COVS, k = 2)
x = pca$PC

align(PC = x, type = "largest")
align(PC = x, type = "maxvar")
align(PC = x, type = "mean")
align(PC = x, type = "nonzero")
align(PC = x, type = "scalar", vec = c(1,1,1,1))

Biplot Visualization for msPCA Objects

Description

Creates a biplot of the first two principal components from an msPCA object, displaying variables or groups in the component space.

Usage

## S3 method for class 'msPCA'
biplot(x, ...)

Arguments

Value

A ggplot2 object representing the biplot of the first two principal components.

Examples

set.seed(236)
data <- matrix(rnorm(2000), ncol = 4)
groups <- sample(1:10, 500, replace = TRUE)
W <- time_weights(N = 10, c(3,2,1))
covs <- ssMRCD(data, groups = groups, weights = W, lambda = 0.3)
pca <- msPCA(eta = 0.3, gamma = 0.5,COVS = covs$MRCDcov, k = 2)
pca$PC = align(PC = pca$PC, type = "largest")
biplot(pca, alpha = 1, shape = 16, size = 5, color = "variable")

Calculation of the cellwise robust multi-group Gaussian mixture model

Description

Performs robust estimation of multivariate location and scatter within predefined groups using an iiterative EM-based algorithm.

Usage

cellMGGMM(
  X,
  groups,
  alpha = 0.5,
  hperc = 0.75,
  nsteps = 100,
  crit = 1e-04,
  silent = TRUE,
  maxcond = 100
)

Arguments

Value

A list containing:

X

The original data matrix.

Ximp

The imputed and/or scaled data matrix.

groups

Vector specifying group assignments from the input.

class

Vector indicating the most likely group membership for each observation, as inferred by the model.

mu

A list of estimated location (mean) vectors for each group.

Sigma

A list of estimated covariance matrices for each group.

Sigmai

A list of estimated inverse covariance matrices for each group.

probs

A matrix of class probabilities for each observation (rows = observations, columns = groups).

pi_groups

A matrix of estimated mixture probabilities, where rows correspond to groups and columns to distributions.

W

A binary matrix indicating outlying cells (0 = outlier, 1 = no outlier).

Q

A matrix of penalty weights.

Sigma_reg

A list of estimated target (regularization) matrices.

rho

A vector of regularization factors used in the estimation.

alpha

Flexibility parameter, as provided in the function input.

hperc

A matrix or vector indicating the percentage of outlying cells per variable and group, based on input.

nsteps

The number of iteration steps taken until convergence.

objvals

The values of the objective function across the iteration steps.

References

Puchhammer, P., Wilms, I., & Filzmoser, P. (2025). A smooth multi-group Gaussian Mixture Model for cellwise robust covariance estimation. ArXiv preprint doi:10.48550/arXiv.2504.02547.

See Also

residuals_mggmm

Examples

data("weatherAUT2021")
cut_lon = c(min(weatherAUT2021$lon)-0.2, 12, 16, max(weatherAUT2021$lon) + 0.2)
cut_lat = c(min(weatherAUT2021$lat)-0.2, 48, max(weatherAUT2021$lat) + 0.2)
groups = groups_gridbased(weatherAUT2021$lon, weatherAUT2021$lat, cut_lon, cut_lat)
N = length(unique(groups))
model = cellMGGMM(X = weatherAUT2021[, c("p", "s", "vv", "t", "rsum", "rel")],
                 groups = groups,
                 alpha = 0.5)

Perform Concentration Step

Description

This function performs concentration steps by iteratively updating the neighborhoods of items based on Mahalanobis distances. The function computes the covariance matrix and updates the neighborhood list until convergence or a maximum number of iterations is reached.

Usage

cstep(init, maxcsteps, which_indices, lambda, weights, mT)

Arguments

Details

The function updates the neighborhoods of each item based on Mahalanobis distances, recalculates means and covariances, and checks for convergence. If the neighborhoods do not change between iterations, the optimization stops early.

Value

A list containing:

• numit An integer representing the number of iterations performed.

• out The updated list of items with updated neighborhoods and additional information.

• obj_value A numeric vector of objective values at each iteration, including the initial value.

Computes Mahalanobis Distances for a Given Set of H-Subsets

Description

This function calculates the Mahalanobis distances for a set of observations by centering them with the mean vector and using a covariance matrix computed as a weighted combination of the covariance matrix of the current item and the covariance matrices of its neighbors.

Usage

dist_cstep(init, i, lambda, weights)

Arguments

Details

The Mahalanobis distances are computed using the covariance matrix, which is a weighted combination of the current item's covariance matrix and those of its neighbors. The covariance matrix is smoothed using the parameter lambda and the distances are computed as (x_centered^T * Cov_matrix_chol_inv * x_centered) for each observation.

Value

A numeric vector of distances for each observation in the centered matrix.

Inverse Geographic Weight Matrix

Description

Calculates a inverse-distance based weight matrix for the function ssMRCD (see details).

Usage

geo_weights(coordinates, groups)

Arguments

Details

First, the centers (means of the coordinates given) c_i of each neighborhood is calculated. Then, the Euclidean distance between the centers is calculated and the weight is based on the inverse distance between two neighborhoods,

w_{ij} = \frac{1}{dist(c_i, c_j)}.

It is scaled according to a weight matrix.

Value

Returns a weighting matrix W and the coordinates of the centers per neighborhood centersN.

See Also

time_weights

Examples

coordinates = matrix(rnorm(1000), ncol = 2, nrow = 500)
groups = sample(1:5, 500, replace = TRUE)

geo_weights(coordinates, groups)

Creates Grid-Based Neighborhood Structure

Description

This function creates a grid-based neighborhood structure for the ssMRCD function using cut-off values for two coordinate axis.

Usage

groups_gridbased(x, y, cutx, cuty)

Arguments

Value

Returns a neighborhood assignment vector for the coordinates x and y.

Examples

# get data
data(weatherAUT2021)

# set cut-off values
cut_lon = c(9:16, 18)
cut_lat = c(46, 47, 47.5, 48, 49)

# create neighborhood assignments
groups_gridbased(weatherAUT2021$lon,
                      weatherAUT2021$lat,
                      cut_lon,
                      cut_lat)

Local Outlier Detection using Spatially Smoothed MRCD

Description

Identifies local multivariate outliers using spatially smoothed robust covariance estimation (ssMRCD) as proposed by Puchhammer and Filzmoser (2023). For each observation, the Mahalanobis distance to its nearest neighbor is computed, and an adjusted boxplot is used to detect outliers.

Usage

locOuts(data, coords, groups, lambda, weights = NULL, k = NULL, dist = NULL)

Arguments

Value

An object of class "locOuts" containing:

outliers

Indices of detected outliers.

next_distance

Vector of Mahalanobis next distances (min distance to neighbors).

cutoff

Upper fence of the adjusted boxplot used as outlier threshold.

coords

Matrix of observation coordinates.

data

Original data matrix.

groups

Group assignments.

k, dist

Neighborhood comparison parameters used.

centersN

Centers of neighborhoods.

matneighbor

Binary matrix indicating which neighbors were used for each observation.

ssMRCD

The fitted ssMRCD object.

References

Puchhammer, P. and Filzmoser, P. (2023). Spatially Smoothed Robust Covariance Estimation for Local Outlier Detection. *Journal of Computational and Graphical Statistics*, 33(3), 928–940. doi:10.1080/10618600.2023.2277875

See Also

ssMRCD, plot.locOuts

Examples

data <- matrix(rnorm(2000), ncol = 4)
coords <- matrix(runif(1000), ncol = 2)
groups <- sample(1:10, 500, replace = TRUE)
result <- locOuts(data, coords, groups, lambda = 0.3, k = 10)

Compute Sparse Multi-Source Principal Components

Description

Estimates sparse principal components from multiple covariance or correlation matrices using an ADMM-based optimization routine (see Puchhammer, Wilms and Filzmoser, 2024).

Usage

msPCA(
  eta,
  gamma,
  COVS,
  k = ncol(COVS[[1]]),
  adjust_eta = TRUE,
  convergence_plot = FALSE,
  n_max = 200,
  rho = list(NA, TRUE, 100, 1),
  eps = c(1e-05, 1e-04, 0.1, 50)
)

Arguments

Value

An object of class "msPCA" containing the following elements:

References

Puchhammer, P., Wilms, I., & Filzmoser, P. (2024). Sparse outlier-robust PCA for multi-source data. *ArXiv Preprint*. doi:10.48550/arXiv.2407.16299

See Also

ssMRCD, plot.msPCA, summary.msPCA, biplot.msPCA, screeplot.msPCA, scores, align

Examples

C1 = diag(c(1.1, 0.9, 0.6, 0.5, 2))
C2 = matrix(runif(25, -1, 1), 5, 5)
C2 = t(C2) %*% C2
C3 = matrix(runif(25, -1, 1), 5, 5)
C3 = t(C3) %*% C3

pca1 = msPCA(eta = 1, gamma = 0.5, COVS = list(C1, C2, C3), k = 3,
             n_max = 100, rho = list(NA, TRUE, 100, 1))
summary(pca1)

pca2 = msPCA(eta = seq(0, 3, 0.25), gamma = 1, COVS = list(C1, C2, C3), k = 3,
             n_max = 100, rho = list(NA, TRUE, 100, 1))
summary(pca2)

Objective Function for Init Object

Description

Calculates the objective function based on the matrices stored in the init object.

Usage

objective_init(init_object, lambda, weights)

Arguments

Value

Returns the value of the objective function.

Calculation of Objective Function

Description

Calculation of the value of the objective function for a given list of matrices, lambda, and a weighting matrix.

Usage

objective_matrix(matrix_list, lambda, weights)

Arguments

Value

Returns the value of the objective function.

Diagnostic Plots for Local Outlier Detection ('locOuts')

Description

Produces diagnostic plots for local outlier detection results returned by locOuts. Available visualizations include a histogram of next distances, spatial distribution of next distances, and a parallel coordinate plot (PCP) for a selected observation and their neighborhood.

Usage

## S3 method for class 'locOuts'
plot(
  x,
  type = c("hist", "spatial", "pcp"),
  scale = c("none", "minmax", "zscore"),
  bins = 30,
  observation = 1,
  ...
)

Arguments

Details

The function visualizes outlier behavior in different ways:

• Histogram: Shows the distribution of next distances across observations. The cutoff is shown as a dashed line.

• Spatial Plot: 2D plot of observation coordinates. Color encodes the ratio of next distance to cutoff.

• Parallel Coordinate Plot (PCP): Shows scaled values across all variables for a selected observation (in red) and its neighbors (in blue or grey). The type of scaling can be controlled via the scale parameter.

Value

A named list with elements:

p_hist

ggplot object of the histogram (or NULL if not requested).

p_spatial

ggplot object of the spatial plot (or NULL).

p_pcp

ggplot object of the parallel coordinate plot (or NULL).

See Also

locOuts

Examples

set.seed(1)
data <- matrix(rnorm(2000), ncol = 4)
coords <- matrix(rnorm(1000), ncol = 2)
groups <- sample(1:10, 500, replace = TRUE)
outs <- locOuts(data = data,
                coords = coords,
                groups = groups,
                lambda = 0.3,
                k = 10)

# Generate all plots
plots <- plot(outs,
              type = c("hist", "spatial", "pcp"),
              observation = outs$outliers[1],
              scale = "minmax")
plots$p_hist
plots$p_spatial
plots$p_pcp

Plot Method for msPCA Objects

Description

Generic plotting interface for objects of class "msPCA". Depending on the type argument, this function visualizes loadings, scores, screeplots, biplots, or score distances.

Usage

## S3 method for class 'msPCA'
plot(x, type = c("loadings"), ...)

Arguments

Details

If type = "loadings" a heatmap of loadings across groups is displayed. Optional input arguments are:

If type = "screeplot" boxplots of the explained variance per component and cumulative variance per group are plotted. Optional input arguments are:

If type = "biplot" the loadings are visualized in the first and second component over all groups. Optional input arguments are:

If type = "scores" a histogram of the k-th scores per group are shown. Optional input arguments are:

If type = "score_distances" a distance-distance plot of score and orthogonal distances is shown for each group. of the k-th scores per group are shown. Optional input arguments are:

Value

A ggplot2 object or list of plots, depending on the type.

See Also

msPCA, align, screeplot.msPCA, , biplot.msPCA, scores,

Examples

# set seed
set.seed(236)

# create data and setup
data = matrix(rnorm(2000), ncol = 4)
groups = sample(1:10, 500, replace = TRUE)
W = time_weights(N = 10, c(3,2,1))

# calculate covariances
covs = ssMRCD(data, groups = groups, weights = W, lambda = 0.3)

# calculate sparse PCA
pca = msPCA(eta = 1.3, gamma = 0.7, COVS = covs$MRCDcov)

# plot screeplot
plot(x = pca, type = "screeplot")

# align and plot loadings
pca$PC = align(PC = pca$PC, type = "mean")
plot(x = pca, type = "loadings", k = 1)
pca$PC = align(PC = pca$PC, type = "maxvar")
plot(x = pca, type = "loadings", k = 1)
pca$PC = align(PC = pca$PC, type = "largest")
plot(x = pca, type = "loadings", k = 1)

# plot different PCA plots
plot(x = pca, type = "score_distances", k = 2,
groups = groups, X = data, mu = covs$MRCDmu, Sigma = covs$MRCDcov)
plot(x = pca, type = "biplot", color = "variable")
plot(x = pca, type = "scores", ssMRCD = covs, k = 1)
plot(x = pca, type = "loadings", k = 1)

Plot Method for ssMRCD Object

Description

Produces diagnostic plots for an object of class "ssMRCD" including convergence behavior and visualizations of the covariance matrices.

Usage

## S3 method for class 'ssMRCD'
plot(
  x,
  type = c("convergence", "ellipses", "ellipses_geo"),
  variables = NULL,
  geo_centers = NULL,
  manual_rescale = 1,
  tolerance = 0.95,
  ...
)

Arguments

Details

type = "convergence": Displays the convergence behavior of the objective function across C-step iterations for each initialization. Red lines indicate non-monotonic convergence.

type = "ellipses": Shows Mahalanobis tolerance ellipses for each group based on their estimated covariance matrix. Only the two variables specified in variables are visualized. The global MCD ellipse may be shown for comparison (if included elsewhere).

type = "ellipses_geo": Projects group-wise covariance ellipses onto a geographical coordinate system (e.g., spatial map), using the positions given in geo_centers. The ellipses are scaled using manual_rescale and drawn using the same two variables as for type = "ellipses".

Value

A named list of ggplot2 plot objects:

See Also

ssMRCD, locOuts, plot.locOuts

Examples

set.seed(1)
data <- matrix(rnorm(2000), ncol = 4)
colnames(data) <- paste0("V", 1:4)
coords <- matrix(rnorm(1000), ncol = 2)
groups <- sample(1:10, 500, replace = TRUE)
lambda <- 0.3

outs <- locOuts(data = data,
                              coords = coords,
                              groups = groups,
                              lambda = lambda,
                              k = 10)

# Plot convergence
plot(x = outs$ssMRCD, type = "convergence")

# Plot ellipses in variable space
plot(x = outs$ssMRCD, type = "ellipses", variables = c("V1", "V2"))

# Plot ellipses in geographical space
centers <- matrix(rnorm(20), ncol = 2)  # example centers for 10 groups
plot(x = outs$ssMRCD, type = "ellipses_geo", geo_centers = centers, variables = c("V1", "V2"))

Residual Method from an ssMRCD Object

Description

Computes group-wise Mahalanobis residuals (standardized distances) using the robust local covariance and location estimates from an ssMRCD object. Residuals can be computed for the fitted data or for new data, and optionally summarized as a trimmed mean.

Usage

## S3 method for class 'ssMRCD'
residuals(object, ...)

Arguments

Details

The function supports several modes of use, controlled by the type argument in ...:

type

"residuals" (default), "trimmed_mean", or "additional_data".

X

A numeric matrix of new observations to compute residuals for. Required if type = "additional_data".

groups

A vector of group assignments for the new data in X. Required if type = "additional_data".

alpha

A numeric value (default taken from the ssMRCD object if missing) indicating the quantile for trimmed mean calculation. Only used if type = "trimmed_mean".

Notes:

• If type = "residuals", residuals are computed for the original data stored in the ssMRCD object.

• If type = "additional_data", both X and groups must be provided. All residuals of X are returned (i.e., alpha = 1 is used internally).

• If type = "trimmed_mean", the mean of the alpha proportion of smallest residual norms is returned. This is also used for parameter tuning.

Value

Depending on the type:

"residuals" or "additional_data"

A numeric matrix of residuals.

"trimmed_mean"

A single numeric value: the trimmed mean of residual norms.

See Also

ssMRCD

Examples

# Create data
x1 <- matrix(runif(200), ncol = 2)
x2 <- matrix(rnorm(200), ncol = 2)
x <- list(x1, x2)

# Define neighborhood weights
W <- matrix(c(0, 1, 1, 0), ncol = 2)

# Compute ssMRCD
localCovs <- ssMRCD(x, weights = W, lambda = 0.5)

# Residuals for original data (all)
residuals(localCovs, type = "residuals")

# Trimmed mean of residual norms
residuals(localCovs, type = "trimmed_mean", alpha = 0.8)

# Residuals for new data
newX <- matrix(rnorm(20), ncol = 2, nrow = 10)
newGroups <- rep(2, 10)
residuals(localCovs, type = "additional_data", X = newX, groups = newGroups)

Calculation of Residuals for the Multi-Group GMM

Description

This function calculates the cell-wise residuals for each observation based on the fitted parameters of a multi-group Gaussian Mixture Model (GMM) and the cellwise outlyingness pattern in matrix 'W'.

Usage

residuals_mggmm(X, groups, Sigma, mu, probs, W, set_to_zero = TRUE)

Arguments

Details

Positive values of residuals mean that the observed value of the outlying variable is higher than would have been expected based on the other observed variables, negative values mean that the observed value is lower than expected. For non-outlying cells (i.e. where 'W[i, j] == 1'), the residual is set to zero.

Value

A numeric matrix of residuals of the same dimension as 'X', where each cell represents the standardized deviation from the model-based conditional expectation, or zero if the cell was not flagged as outlying in 'W'.

References

Puchhammer, P., Wilms, I., & Filzmoser, P. (2025). A smooth multi-group Gaussian Mixture Model for cellwise robust covariance estimation. ArXiv preprint doi:10.48550/arXiv.2504.02547.

See Also

cellMGGMM

Examples

data("weatherAUT2021")
cut_lon = c(min(weatherAUT2021$lon)-0.2, 12, 16, max(weatherAUT2021$lon) + 0.2)
cut_lat = c(min(weatherAUT2021$lat)-0.2, 48, max(weatherAUT2021$lat) + 0.2)
groups = ssMRCD::groups_gridbased(weatherAUT2021$lon, weatherAUT2021$lat, cut_lon, cut_lat)
N = length(unique(groups))
model = cellMGGMM(X = weatherAUT2021[, c("p", "s", "vv", "t", "rsum", "rel")],
                 groups = groups,
                 alpha = 0.5)
res = residuals_mggmm(X =  weatherAUT2021[, c("p", "s", "vv", "t", "rsum", "rel")],
                groups = groups,
                Sigma = model$Sigma,
                mu = model$mu,
                probs = model$probs,
                W = model$W)

Locally Center and/or Scale or Data Using an ssMRCD Object

Description

Applies local standardization (scaling and/or centering) of either the original data from an ssMRCD object or new data provided via the X argument, using group-wise robust means and variances from the ssMRCD estimation.

Usage

## S3 method for class 'ssMRCD'
scale(x, ...)

Arguments

Details

For each group, the function applies scaling (or just centering) using the robust location and scale (square root of the diagonal of the covariance) estimates obtained during ssMRCD estimation.

Value

A numeric matrix of the same dimension as X, where each observation has been standardized (or centered) using the corresponding group-wise robust mean and (if applicable) variance from the ssMRCD model. If X = NULL, the original data from the ssMRCD object is returned in scaled form, sorted according to group labels.

See Also

ssMRCD

Examples

# Simulated example
x1 <- matrix(runif(200), ncol = 2)
x2 <- matrix(rnorm(200), ncol = 2)
x <- list(x1, x2)

W <- matrix(c(0, 1, 1, 0), ncol = 2)
localCovs <- ssMRCD(x, weights = W, lambda = 0.5)

# Scale original data
sc = scale(localCovs)

# Scale new observations
sc = scale(localCovs,
           list(X = matrix(rnorm(20), ncol = 2, nrow = 10),
           groups = rep(2, 10)))

# Center only
sc = scale(localCovs,
           list(X = matrix(rnorm(20), ncol = 2, nrow = 10),
           groups = rep(2, 10),
           center_only = TRUE))

Calculate Scores and Distances for Multi-Source PCA

Description

Computes principal component scores, score distances (SD), and orthogonal distances (OD) for observations grouped into multiple sources using the multi-source PCA model. The function supports either an 'ssMRCD' object for robust local centering and scaling or manually provided group-wise means ('mu') and covariances ('Sigma').

Usage

scores(PC, ssMRCD = NULL, X = NULL, groups = NULL, mu = NULL, Sigma = NULL)

Arguments

Value

A list with the following components:

scores

Matrix of principal component scores for each observation.

SD

Numeric vector of score distances, i.e., Mahalanobis distances in the PCA space.

OD

Numeric vector of orthogonal distances (reconstruction error orthogonal to PCA space).

X_centered

Locally centered input data.

See Also

ssMRCD, scale.ssMRCD, msPCA

Examples

# create data set
x1 = matrix(runif(200), ncol = 2)
x2 = matrix(rnorm(200), ncol = 2)
x = list(x1, x2)

# create weighting matrix
W = matrix(c(0, 1, 1, 0), ncol = 2)

# calculate ssMRCD
loccovs = ssMRCD(x, weights = W, lambda = 0.5)

# calculate PCA
pca = msPCA(eta = 1, gamma = 0.5,COVS = loccovs$MRCDcov)

# calculate scores
scores(PC = pca$PC, ssMRCD = loccovs)

scores(PC = pca$PC,
       X = rbind(x1, x2),
       groups = rep(c(1,2), each = 100),
       mu = loccovs$MRCDmu,
       Sigma = loccovs$MRCDcov)

Scree Plot and Cumulative Explained Variance for msPCA Objects

Description

Generates a scree plot displaying the explained variance of principal components and a heatmap of cumulative explained variance per group for an object of class msPCA.

Usage

## S3 method for class 'msPCA'
screeplot(x, ...)

Arguments

Value

A list containing two ggplot2 plot objects:

• A boxplot-based scree plot showing explained variance per principal component across groups.

• A tile plot showing cumulative explained variance per group and principal component.

Examples

set.seed(236)
data <- matrix(rnorm(1500), ncol = 5)
groups <- sample(1:5, 300, replace = TRUE)
W <- time_weights(N = 5, c(3,2,1))
covs <- ssMRCD(data, groups = groups, weights = W, lambda = 0.3)
pca <- msPCA(eta = 0.3, gamma = 0.7, COVS = covs$MRCDcov)
screeplot(pca, text = TRUE, cutoff = 0.8, size = 4, textrotate = 0)
screeplot(pca, text = FALSE, cutoff = 0.6)

Spatially Smoothed MRCD Estimator

Description

The ssMRCD function calculates the spatially smoothed MRCD estimator from Puchhammer and Filzmoser (2023).

Usage

ssMRCD(
  X,
  groups = NULL,
  weights,
  lambda = 0.5,
  tuning = list(method = NULL, plot = FALSE, k = 10, repetitions = 5, cont = 0.05),
  TM = NULL,
  alpha = 0.75,
  maxcond = 50,
  maxcsteps = 200,
  n_initialhsets = NULL
)

Arguments

Details

The necessary list elements for the parameter tuning depend on the method specified. For both tuning approaches (residual-based or contamination-based) the element method needs to be specified to "residuals" and "local contamination", respectively. The boolean list element plot is available for both methods and specifies if a plot should be constructed after tuning.

For tuning$method = "local contamination", additional information needs to be passed. The number of nearest neighbors tuning$k used for the local outlier detection method is 10 by default. The percentage of exchanged/contaminated observations is specified by tuning$cont and is set to 0.05 by default. Also the coordinates must be given in tuning$coords and the number of repetitions for the switching procedure, tuning$repetitions.

For tuning$method = "local contamination" no optimal value is returned but the choice has to be made by the user. Be aware that the FNR does not take into account that there are also natural outliers included in the data set that might or might not be found. The best parameter selection depends on the goal of the analysis and whether false negatives should be avoided or whether the number of flagged outliers should be low.

Value

The output depends on whether parameters are tuned. If there is no tuning the output is an object of class "ssMRCD" containing the following elements:

If parameters are tuned, the output consists of:

References

Puchhammer P. and Filzmoser P. (2023). Spatially Smoothed Robust Covariance Estimation for Local Outlier Detection. Journal of Computational and Graphical Statistics, 33(3), 928–940. doi:10.1080/10618600.2023.2277875

See Also

plot.ssMRCD

Examples

# create data set
x1 = matrix(runif(200), ncol = 2)
x2 = matrix(rnorm(200), ncol = 2)
x = list(x1, x2)

# create weighting matrix
W = matrix(c(0, 1, 1, 0), ncol = 2)

# calculate ssMRCD
ssMRCD(X = x, weights = W, lambda = 0.5)

Summary Method for msPCA Objects

Description

Provides a summary of a sparse multi-source PCA ('msPCA') result, including group-wise sparsity and explained variance.

Usage

## S3 method for class 'msPCA'
summary(object, ...)

Arguments

Value

Prints a summary to the console. No return value.

Band weight matrix for time series groupings

Description

Band weight matrix for time series groupings

Usage

time_weights(N, off_diag)

Arguments

Value

Returns weight matrix for time series groups appropriate for ssMRCD.

See Also

geo_weights

Examples

time_weights(N = 10, off_diag = c(2,1))

Austrian Weather Data 2021

Description

This data is a subset of the GeoSphere Austria monthly weather data of 2021 averaged using the median. Stations with missing values are removed.

Usage

weatherAUT2021

Format

A data frame with 183 rows and 10 columns:

name

Unique name of the weather station in German.

lon, lat

Longitude and latitude of the weather station.

alt

Altitude of the weather station (meter).

p

Average air pressure (hPa).

s

Monthly sum of sunshine duration (hours).

vv

Wind velocity (meter/second).

t

Air temperature in 2 meters above the ground in (°C).

rsum

Average daily sum of precipitation (mm).

rel

Relative air humidity (percent).

Source

The original data was downloaded here (December 2022): https://data.hub.geosphere.at/dataset/klima-v1-1m.

References

Data Source: GeoSphere Austria - https://data.hub.geosphere.at.

Examples

data(weatherAUT2021)
summary(weatherAUT2021)

Vienna Weather Time Series (1960-2023)

Description

This data is a subset of the GeoSphere Austria daily weather data of the time 1960-2023 for the weather station Hohe Warte in Vienna.

Usage

weatherHoheWarte

Format

A data frame with 23372 rows and 18 columns including 13 weather measurements:

time

Time of measurement in date format.

cloud_cover

Daily mean of cloud coverage, values between 1 and 100.

global_radiation

Daily sum of global radiation (J/cm^2).

vapor_pressure

Daily mean of vapour pressuer (hPa).

max_wind_speed

Maximal wind speed (m/s).

air_pressure

Daily mean of air pressure (hPa).

relative_humidity

Daily mean of relative humidity (percent).

precipitation

Daily sum of precepitation (mm).

sight

Sight distance at 1pm (m).

sunshine_duration

Daily sum of sunshine duration (h).

temperature_max

Daily maximum of temperature at 2m air height (°C).

temperature_min

Daily minimum of temperature at 2m air height (°C).

temperature_mean

Daily mean of temperature at 2m air height (°C).

wind_velocity

Daily mean of wind speed (m/s).

year

Year of measurement.

month

Month of measurement.

day

Day of the year of measurement.

season

Season of measuremen (1 = winter, 2 = spring, 3 = summer, 4 = fall).

Source

The original data was downloaded here (April 2024): https://data.hub.geosphere.at/dataset/klima-v2-1d.

References

Data Source: GeoSphere Austria - https://data.hub.geosphere.at.

Examples

data(weatherHoheWarte)
summary(weatherHoheWarte)