Align Loadings of Principal Components

Description

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

Usage

align_PC(PC, N, p, type = "largest", vec = NULL)

Arguments

Details

For input type possible values are "largest", "maxvar","nonzero","mean","scalar". For option "maxvar" the variable with the highest absolute value in the loading is scaled to be positive (per neighborhood, per loading). For option "nonzero" the variable with largest distance to zero in the entries is scaled to be positive (per neighborhood, per loading). For option "scalar" the variable is scaled in a way, that the scalar product between the loading and the respective part of vec is positive (per neighborhood, per loading). If vec is of size p times k, the same vector is used for all neighborhoods. Option "mean" is option "scalar" with vec being the mean of the loadings per variable across neighborhoods. Option "largest" scales the largest absolute value to be positive per neighborhood and per PC. Option "none" does nothing and returns PC.

Value

Returns a matrix of loadings of size Np times k.

Examples

x = matrix(c(1, 0, 0, 0, sqrt(0.5), -sqrt(0.5), 0, 0,
             0, sqrt(1/3), -sqrt(1/3), sqrt(1/3), sqrt(0.5), sqrt(0.5), 0, 0),
           ncol = 2)
align_PC(PC = x, N = 2, p = 4, type = "largest")
align_PC(PC = x, N = 2, p = 4, type = "mean")

Biplot for PCAloc

Description

Biplot for PCAloc

Usage

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

Arguments

Details

Additional parameters that can be given to the function are:

Value

Returns version of biplot for PCAloc object.

Examples

# set seed
set.seed(236)

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

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

# sparse PCA
pca = sparsePCAloc(eta = 0.3, gamma = 0.7, cor = FALSE, COVS = covs$MRCDcov,
             n_max = 1000, increase_rho = list(TRUE, 50, 1), trace = FALSE)

# plot biplot
biplot(pca, alpha = 0.4, shape = 16, size = 2, color = "variable")

Contamination Through Swapping

Description

This function swaps observations completely random in order to introduce contamination in the data. Used in parameter_tuning.

Usage

contamination_random(cont, data)

Arguments

Value

A matrix with switched observations.

Examples

# set seed
set.seed(1)

# get data
data(weatherAUT2021)

# switch 5% of observations
contamination_random(cont = 0.05, data = weatherAUT2021[,1:6])

Objective function value for local sparse PCA

Description

Objective function value for local sparse PCA

Usage

eval_objective(PC, eta, gamma, COVS)

Arguments

Value

Returns value of the objective function for given v.

Examples

S1 = matrix(c(1, 0.9, 0.8, 0.5,
              0.9, 1.1, 0.7, 0.4,
              0.8, 0.7, 1.5, 0.2,
              0.5, 0.4, 0.2, 1), ncol = 4)
S2 = t(S1)%*% S1
S2 = S2/2

eval_objective(PC = c(1,0,0,0,sqrt(2),0,0,-sqrt(2)),
               eta = 1, gamma = 0.5,
               COVS = list(S1, S2))

Explained Variance summarized over Groups

Description

Explained Variance summarized over Groups

Usage

explained_var(COVS, PC, k, type = "scaled", cor = FALSE, gamma = 0.5)

Arguments

Value

Returns scalar

Examples

S1 = matrix(c(1, 0.9, 0.8, 0.5,
              0.9, 1.1, 0.7, 0.4,
              0.8, 0.7, 1.5, 0.2,
              0.5, 0.4, 0.2, 1), ncol = 4)
S2 = t(S1)%*% S1
S2 = S2/2

explained_var(COVS = list(S1, S2),
              PC = c(1,0,0,0,sqrt(2),0,0,-sqrt(2)),
              k = 1,
              cor = FALSE,
              gamma = 0.5)

explained_var(COVS = list(cov2cor(S1), cov2cor(S2)),
              PC = c(1,0,0,0,sqrt(2),0,0,-sqrt(2)),
              k = 1,
              cor = TRUE,
              gamma = 0.5)

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

rescale_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 Technique based on ssMRCD

Description

This function applies the local outlier detection method based on the spatially smoothed MRCD estimator developed in Puchhammer and Filzmoser (2023).

Usage

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

Arguments

Value

Returns an object of class "locOuts" with following components:

References

Puchhammer P. and Filzmoser P. (2023): Spatially smoothed robust covariance estimation for local outlier detection. doi:10.48550/arXiv.2305.05371

See Also

See also functions ssMRCD, plot.locOuts, summary.locOuts.

Examples

# data construction
data = matrix(rnorm(2000), ncol = 4)
coords = matrix(rnorm(1000), ncol = 2)
groups = sample(1:10, 500, replace = TRUE)
lambda = 0.3

# apply function
outs = local_outliers_ssMRCD(data = data,
                             coords = coords,
                             groups = groups,
                             lambda = lambda,
                             k = 10)
outs

Calculation of Objective Function

Description

Calculation of the value of the objective function for the ssMRCD for a given list of matrices, lambda and a weighting matrix according to formula (3) in Puchhammer and Filzmoser (2023).

Usage

objective_matrix(matrix_list, lambda, weights)

Arguments

Value

Returns the value of the objective function using matrices K_i.

References

Puchhammer P. and Filzmoser P. (2023): Spatially smoothed robust covariance estimation for local outlier detection. doi:10.48550/arXiv.2305.05371

Examples

# construct matrices
k1 = matrix(c(1,2,3,4), nrow = 2)
k2 = matrix(c(1,3,5,7), nrow = 2)

# construct weighting matrix
W = matrix(c(0, 1, 1, 0), nrow = 2)

objective_matrix(list(k1, k2), 0.5, W)

Optimal Smoothing Parameter for ssMRCD based on Local Outliers

Description

This function provides insight into the effects of different parameter settings.

Usage

parameter_tuning(
  data,
  coords,
  groups,
  lambda = c(0, 0.25, 0.5, 0.75, 0.9),
  weights = NULL,
  k = NULL,
  dist = NULL,
  cont = 0.05,
  repetitions = 5
)

Arguments

Value

Returns a matrix of average false-negative rate (FNR) values and the total number of outliers found by the method as aproxy for the false-positive rate. 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. Also a plot is returned representing these average. The best parameter selection depends on the goal of the analysis.

Examples

# get data set
data("weatherAUT2021")

# make neighborhood assignments
cut_lon = c(9:16, 18)
cut_lat = c(46, 47, 47.5, 48, 49)
N = ssMRCD::groups_gridbased(weatherAUT2021$lon, weatherAUT2021$lat, cut_lon, cut_lat)
table(N)
N[N == 2] = 1
N[N == 3] = 4
N[N == 5] = 4
N[N == 6] = 7
N[N == 11] = 15
N = as.numeric(as.factor(N))

# tune parameters
set.seed(123)
parameter_tuning(data = weatherAUT2021[, 1:6 ],
                 coords = weatherAUT2021[, c("lon", "lat")],
                 groups = N,
                 lambda = c(0.5, 0.75),
                 k = c(10),
                 repetitions = 1)

Plotting method PCAloc object

Description

Plotting method PCAloc object

Usage

## S3 method for class 'PCAloc'
plot(
  x,
  type = c("loadings", "screeplot", "scores", "score_distances", "biplot"),
  ...
)

Arguments

Value

Returns plots in ggplot2.

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 = sparsePCAloc(eta = 0.3, gamma = 0.7, cor = FALSE, COVS = covs$MRCDcov,
             n_max = 1000, increase_rho = list(TRUE, 50, 1), trace = FALSE)

# align loadings
pca$PC = align_PC(PC = pca$PC, N = pca$N, p = pca$p, type = "mean")

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

Diagnostic Plots for Local Outlier Detection

Description

This function plots different diagnostic plots for local outlier detection. It can be applied to an object of class "locOuts" which is the output of the function local_outliers_ssMRCD.

Usage

## S3 method for class 'locOuts'
plot(
  x,
  type = c("hist", "spatial", "lines", "3D"),
  colour = "all",
  focus = NULL,
  pos = NULL,
  alpha = 0.3,
  data = NULL,
  add_map = TRUE,
  ...
)

Arguments

Details

Regarding the parameter type the value "hist" corresponds to a plot of the histogram of the next distances together with the used cutoff-value. When using "spatial" the coordinates of each observation are plotted and colorized according to the color setting. The "lines" plot is used with the index focus of one observation whose out/inlyingness to its neighborhood should by plotted. The whole data set is scaled to the range [0,1] and the scaled value of the selected observation and its neighbors are plotted. Outliers are plotted in orange. The "3D" setting leads to a 3D-plot using the colour setting as height. The view can be adapted using the parameters theta and phi.

For the colour setting possible values are "all" (all next distances are used and colored in an orange palette), "onlyOuts" (only outliers are plotted in orange, inliers are plotted in grey) and "outScore" (the next distance divided by the cutoff value is used to colourize the points; inliers are colorized in blue, outliers in orange).

Value

Returns plots regarding next distances and spatial context.

See Also

local_outliers_ssMRCD

Examples

# set seed
set.seed(1)

# make locOuts object
data = matrix(rnorm(2000), ncol = 4)
coords = matrix(rnorm(1000), ncol = 2)
groups = sample(1:10, 500, replace = TRUE)
lambda = 0.3

# local outlier detection
outs = local_outliers_ssMRCD(data = data,
                             coords = coords,
                             groups = groups,
                             lambda = lambda,
                             k = 10)

# plot results
plot(outs, type = "hist")
plot(outs, type = "spatial", colour = "outScore")
plot(outs, type = "3D", colour = "outScore", theta = 0)
plot(outs, type ="lines", focus = outs$outliers[1])

Plot Method for ssMRCD Object

Description

Plots diagnostics for function output of ssMRCD regarding convergence behavior and the resulting covariances matrices.

Usage

## S3 method for class 'ssMRCD'
plot(
  x,
  type = c("convergence", "ellipses"),
  centersN = NULL,
  colour_scheme = "none",
  xlim_upper = 9,
  manual_rescale = 1,
  legend = TRUE,
  xlim = NULL,
  ylim = NULL,
  ...
)

Arguments

Details

For type = "convergence" a plot is produced displaying the convergence behaviour. Each line represents a different initial value used for the c-step iteration. On the x-axis the iteration step is plotted with the corresponding value of the objective function. Not monotonically lines are plotted in red.

For type = "ellipses" and more than a 2-dimensional data setting plotting the exact tolerance ellipse is not possible anymore. Instead the two eigenvectors with highest eigenvalue from the MCD used on the full data set without neighborhood assignments are taken and used as axis for the tolerance ellipses of the ssMRCD covariance estimators. The tolerance ellipse for the global MCD covariance is plotted in grey in the upper left corner. It is possible to set the colour scheme to "trace" to see the overall amount of variabilty and compare the plotted covariance and the real trace to see how much variance is not plotted. For "regularity" the regularization of each covariance is shown.

Value

Returns plots of the ssMRCD methodology and results.

See Also

ssMRCD, summary.ssMRCD, local_outliers_ssMRCD, plot.locOuts

Examples

# set seed
set.seed(1)

# create data set
data = matrix(rnorm(2000), ncol = 4)
coords = matrix(rnorm(1000), ncol = 2)
groups = sample(1:10, 500, replace = TRUE)
lambda = 0.3

# calculate ssMRCD by using the local outlier detection method
outs = local_outliers_ssMRCD(data = data,
                             coords = coords,
                             groups = groups,
                             lambda = lambda,
                             k = 10)

# plot ssMRCD object included in outs
plot(x = outs$ssMRCD,
     centersN = outs$centersN,
     colour_scheme = "trace",
     legend = FALSE)

Plots of loadings of PCAloc object

Description

Plots of loadings of PCAloc object

Usage

plot_loadings(object, ...)

Arguments

Details

Additional parameters that can be given to the function are:

Value

Returns loading heatmap for component k.

Examples

# set seed
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))

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

# sparse PCA
pca = sparsePCAloc(eta = 0.3, gamma = 0.7, cor = FALSE, COVS = covs$MRCDcov,
             n_max = 1000, increase_rho = list(TRUE, 50, 1), trace = FALSE)

# plot score distances
plot_loadings(object = pca,
            k = 1,
            size = 2)

Distance-distance plot of scores of PCA

Description

Distance-distance plot of scores of PCA

Usage

plot_score_distances(X, PC, groups, ssMRCD, k, ...)

Arguments

Details

Additional parameters that can be given to the function are:

Value

Returns distance-distance plot of orthogonal and score distance.

Examples

# set seed
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))

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

# sparse PCA
pca = sparsePCAloc(eta = 0.3, gamma = 0.7, cor = FALSE, COVS = covs$MRCDcov,
             n_max = 1000, increase_rho = list(TRUE, 50, 1), trace = FALSE)

# plot score distances
plot_score_distances(PC = pca$PC,
                     groups = groups,
                     X = data,
                     ssMRCD = covs,
                     k = 2,
                     alpha = 0.4,
                     shape = 16,
                     size = 2)

Plots of score distribution

Description

Plots of score distribution

Usage

plot_scores(X, PC, groups, ssMRCD, ...)

Arguments

Details

Additional parameters that can be given to the function are:

Value

Returns histograms of scores for component k.

Examples

# set seed
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))

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

# sparse PCA
pca = sparsePCAloc(eta = 0.3, gamma = 0.7, cor = FALSE, COVS = covs$MRCDcov,
             n_max = 1000, increase_rho = list(TRUE, 50, 1), trace = FALSE)

# plot score distances
plot_scores(PC = pca$PC,
            groups = groups,
            X = data,
            ssMRCD = covs,
            k = 1,
            alpha = 0.4,
            shape = 16,
            size = 2)

Rescale Weight Matrix

Description

Given a matrix with values for neighborhood influences the function rescales the matrix in order to get an appropriate weight matrix used for the function ssMRCD.

Usage

rescale_weights(W)

Arguments

Value

An appropriately scaled weight matrix.

See Also

ssMRCD, local_outliers_ssMRCD, geo_weights

Examples

W = matrix(c(0, 1, 2,
             1, 0, 1,
             2, 1, 0), nrow = 3)
rescale_weights(W)

Extracting Residuals from Local Fit

Description

Extracting Residuals from Local Fit

Usage

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

Arguments

Details

Other input variables are:

If X and groups are provided, alpha is set to one and all residuals are used. If remove_outliers is TRUE, alpha is set to 1 automatically.

Value

Returns either all residuals or the mean of the residual norms lower than the alpha- Quantile.

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
localCovs = ssMRCD(x, weights = W, lambda = 0.5)

# residuals of model
residuals(localCovs, remove_outliers = TRUE, mean = FALSE)

# residuals of new data
residuals(localCovs,
      X = matrix(rnorm(20), ncol = 2, nrow = 10),
      groups = rep(2, 10),
      mean =TRUE)

Restructure Data Matrix as List

Description

This function restructures neighborhood information given by a data matrix containing all information and one neighborhood assignment vector. It returns a list of data matrices used in ssMRCD.

Usage

restructure_as_list(data, groups)

Arguments

Value

Returns a list containing the observations per neighborhood assignment. The list is sorted according to the order of the first appearance in the groups vector.

Examples

# data matrix
data = matrix(rnorm(n = 3000), ncol = 3)
N_assign = sample(x = 1:10, size = 1000, replace = TRUE)

restructure_as_list(data, N_assign)

Scale Data Locally

Description

Scale Data Locally

Usage

scale_ssMRCD(
  ssMRCD,
  X = NULL,
  groups = NULL,
  multivariate = FALSE,
  center_only = FALSE
)

Arguments

Value

Returns matrix of observations. If X = NULL X from the ssMRCD object is used and sorted according to group numbering.

See Also

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
localCovs = ssMRCD(x, weights = W, lambda = 0.5)

# scale used data
scale_ssMRCD(localCovs,
      multivariate = TRUE)

# scale new data
scale_ssMRCD(localCovs,
      X = matrix(rnorm(20), ncol = 2, nrow = 10),
      groups = rep(2, 10),
      multivariate =TRUE)

Calculate Scores for local sparse PCA

Description

Calculate Scores for local sparse PCA

Usage

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

Arguments

Value

Returns a list with scores and univariately and locally centered and scaled observations.

See Also

ssMRCD, scale_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
loccovs = ssMRCD(x, weights = W, lambda = 0.5)

# calculate PCA
pca = sparsePCAloc(eta = 1, gamma = 0.5, cor = FALSE,
                   COVS = loccovs$MRCDcov,
                   increase_rho = list(FALSE, 20, 1))

# calculate scores
scores(X = rbind(x1, x2), PC = pca$PC,
       groups = rep(c(1,2), each = 100), ssMRCD = loccovs)

Orthogonal Distances for PCAloc

Description

Orthogonal Distances for PCAloc

Usage

scores.OD(X, PC, groups, ssMRCD)

Arguments

Value

Returns vector of orthogonal distances of observations.

See Also

scores, scores.SD, sparsePCAloc, scale_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
loccovs = ssMRCD(x, weights = W, lambda = 0.5)

# calculate PCA
pca = sparsePCAloc(eta = 1, gamma = 0.5, cor = FALSE,
                   COVS = loccovs$MRCDcov,
                   increase_rho = list(FALSE, 20, 1))

# calculate scores
scores.OD(X = rbind(x1, x2), PC = pca$PC,
          groups = rep(c(1,2), each = 100), ssMRCD = loccovs)

Score Distances for PCAloc

Description

Score Distances for PCAloc

Usage

scores.SD(X, PC, groups, ssMRCD)

Arguments

Value

Returns vector of score distances of observations.

See Also

scores, scores.OD, sparsePCAloc, scale_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
loccovs = ssMRCD(x, weights = W, lambda = 0.5)

# calculate PCA
pca = sparsePCAloc(eta = 1, gamma = 0.5, cor = FALSE,
                   COVS = loccovs$MRCDcov,
                   increase_rho = list(FALSE, 20, 1))

# calculate scores
scores.SD(X = rbind(x1, x2), PC = pca$PC,
          groups = rep(c(1,2), each = 100), ssMRCD = loccovs)

Screeplot for PCAloc

Description

Screeplot for PCAloc

Usage

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

Arguments

Details

Additional parameters that can be given to the function are:

Value

Returns version of scree plot and cumulative explained variance per group for PCAloc object.

Examples

# set seed
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))

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

# sparse PCA
pca = sparsePCAloc(eta = 0.3, gamma = 0.7, cor = FALSE, COVS = covs$MRCDcov,
             n_max = 1000, increase_rho = list(TRUE, 50, 1), trace = FALSE)

# plot biplot
screeplot(pca, text = TRUE, cutoff = 0.8, size = 2)

Optimal Smoothing Parameter for ssMRCD based on Residuals

Description

The optimal smoothing value for the ssMRCD estimator is based on the residuals and the trimmed mean of the norm.

Usage

select_smoothing(
  X,
  groups,
  weights,
  lambda = seq(0, 1, 0.1),
  TM = NULL,
  alpha = 0.75,
  seed = 123436,
  return_all = TRUE,
  cores = 1
)

Arguments

Value

Examples

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

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

select_smoothing (X = rbind(x1, x2),
                 groups = rep(c(1,2), each = 100),
                 weights = W,
                 lambda = seq(0, 1, 0.1),
                 return_all = TRUE,
                 cores = 1)

Optimal Sparsity Parameter Selection for PCA

Description

Optimal Sparsity Parameter Selection for PCA

Usage

select_sparsity(
  COVS,
  k = 1,
  rho = NULL,
  cor = FALSE,
  eta = seq(0, 5, by = 0.2),
  gamma = seq(0, 1, 0.05),
  eps_threshold = 0.001,
  eps_root = 0.1,
  eps_ADMM = 1e-04,
  n_max = 300,
  adjust_eta = FALSE,
  cores = 1,
  increase_rho = list(TRUE, 100, 1),
  convergence_plot = FALSE,
  trace = FALSE,
  stop.sparse = TRUE
)

Arguments

Details

The input increase_rho consists of a logical indicating if rho should be adjusted if algorithm did not converged within the given maximal number of iterations. Two integers specify the maximal rho that is allowed and the step size.

Value

Returns list with

Examples

C1 = matrix(c(1,0,0,0.9), ncol = 2)
C2 = matrix(c(1.1, 0.1, 0.1, 1), ncol = 2)
C3 = matrix(c(1.2, 0.2, 0.2, 1), ncol = 2)

select_sparsity(COVS = list(C1, C2, C3),
                k = 1,
                rho = 5,
                eta = c(0, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5 ,0.75,  1),
                gamma =c(0, 0.25, 0.5, 0.75, 1),
                eps_threshold = 0.005,
                increase_rho = list(FALSE, 20, 5))

Calculate Sparse Principle Components

Description

Calculate Sparse Principle Components

Usage

sparsePCAloc(
  eta,
  gamma,
  COVS,
  cor = FALSE,
  rho = NULL,
  k = NULL,
  eps_threshold = NULL,
  eps_ADMM = 1e-04,
  n_max = 200,
  eps_root = 0.1,
  maxiter_root = 50,
  increase_rho = list(TRUE, 100, 1),
  convergence_plot = TRUE,
  starting_value = NULL,
  adjust_eta = TRUE,
  trace = TRUE
)

Arguments

Details

The input increase_rho consists of a logical indicating if rho should be adjusted if algorithm did not converged within the given maximal number of iterations. Two integers specify the maximal rho that is allowed and the step size.

Value

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

Examples

C1 = diag(c(1.1, 0.9, 0.6))
C2 = matrix(c(1.1, 0.1, -0.1,
              0.1, 1.0, -0.2,
             -0.1, -0.2, 0.7), ncol = 3)
C3 = (C1 + C2)/2

sparsePCAloc(eta = 1, gamma = 0.5, cor = FALSE, COVS = list(C1, C2, C3),
             n_max = 100, increase_rho = list(FALSE, 100, 1))

Entry-wise Sparsity in the Loadings

Description

Entry-wise Sparsity in the Loadings

Usage

sparsity_entries(PC, N, p, tolerance = 0, k = 1, scaled = TRUE)

Arguments

Value

Returns either a percentage (scaled = TRUE) or the amount of zero-values entries (scaled = FALSE).

Examples

PC = matrix(c(1,0,2,3,0,7,0,1,0,1,0.001,0), ncol = 2)
sparsity_entries(PC, N = 2, p = 3, tolerance = 0, k = 1, scaled = FALSE)
sparsity_entries(PC, N = 2, p = 3, tolerance = 0.001, k = 2, scaled = TRUE)

Group-wise Sparsity in the Loadings

Description

Group-wise Sparsity in the Loadings

Usage

sparsity_group(PC, N, p, tolerance = 0, k = 1, scaled = TRUE)

Arguments

Value

Returns either a matrix of percentages (scaled = TRUE) or the amounts of zero-values entries (scaled = FALSE) for each group/neighborhood.

Examples

PC = matrix(c(1,0,2,3,0,7,0,1,0,1,0.001,0), ncol = 2)
sparsity_group(PC, N = 2, p = 3, tolerance = 0, k = 1, scaled = FALSE)
sparsity_group(PC, N = 2, p = 3, tolerance = 0.001, k = 2, scaled = TRUE)

Mixed Sparsity of the Loadings

Description

Mixed Sparsity of the Loadings

Usage

sparsity_mixed(PC, p, N, k = 1, tolerance = 0.001, mean = "arithmetic")

Arguments

Value

Returns the geometric mean of the percentage of entry-wise and group-wise sparsity.

Examples

PC = matrix(c(1,0,2,3,0,7,0,1,0,1,0.001,0), ncol = 2)
sparsity_mixed(PC, N = 2, p = 3, tolerance = 0, k = 1)
sparsity_mixed(PC, N = 2, p = 3, tolerance = 0.001, k = 2, mean = "geometric")

Entry-wise Sparsity in the Loadings per Group

Description

Entry-wise Sparsity in the Loadings per Group

Usage

sparsity_summary(PC, N, p, tolerance = 0, k = 1, scaled = FALSE)

Arguments

Value

Returns either a matrix of percentages (scaled = TRUE) or the amounts of zero-values entries (scaled = FALSE) for each group/neighborhood.

Examples

PC = matrix(c(1,0,2,3,0,7,0,1,0,1,0.001,0), ncol = 2)
sparsity_summary(PC, N = 2, p = 3, tolerance = 0, k = 1, scaled = FALSE)
sparsity_summary(PC, N = 2, p = 3, tolerance = 0.001, k = 2, scaled = TRUE)

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,
  TM = NULL,
  alpha = 0.75,
  maxcond = 50,
  maxcsteps = 200,
  n_initialhsets = NULL
)

Arguments

Value

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

References

Puchhammer P. and Filzmoser P. (2023): Spatially smoothed robust covariance estimation for local outlier detection. doi:10.48550/arXiv.2305.05371

See Also

plot.ssMRCD, summary.ssMRCD, restructure_as_list

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, weights = W, lambda = 0.5)

Summary method for PCAloc

Description

Summary method for PCAloc

Usage

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

Arguments

Value

Summary for PCAloc

See Also

sparsePCAloc

Examples

#'
C1 = diag(c(1.1, 0.9, 0.6))
C2 = matrix(c(1.1, 0.1, -0.1,
              0.1, 1.0, -0.2,
             -0.1, -0.2, 0.7), ncol = 3)
C3 = (C1 + C2)/2

pca = sparsePCAloc(eta = 1, gamma = 0.5, cor = FALSE, COVS = list(C1, C2, C3),
             n_max = 100, increase_rho = list(FALSE, 100, 1), trace = FALSE)

summary(pca)

Summary of Local Outlier Detection

Description

Prints a summary of the locOuts object obtained by the function local_outliers_ssMRCD.

Usage

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

Arguments

Value

Prints a summary of the locOuts object.

See Also

plot.locOuts

Examples

# set seed
set.seed(1)

# make locOuts object
data = matrix(rnorm(2000), ncol = 4)
coords = matrix(rnorm(1000), ncol = 2)
groups = sample(1:10, 500, replace = TRUE)
lambda = 0.3

# local outlier detection
outs = local_outliers_ssMRCD(data = data,
                             coords = coords,
                             groups = groups,
                             lambda = lambda,
                             k = 10)

# summary method
summary(outs)

Summary Method for ssMRCD Object

Description

Summarises most important information of output ssMRCD.

Usage

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

Arguments

Value

Prints a summary of the ssMRCD object.

See Also

See also ssMRCD, plot.ssMRCD.

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, rescale_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)