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DP Loading Plots

Source: vignettes/tutorials-loading-plots.Rmd
tutorials-loading-plots.Rmd

This tutorial explains how to estimate private principal component (PC) directions with dp_loading(), inspect the fitted results, and create vector, point, and heatmap plots with plot(). It covers R arguments, examples, and visualization, including optional ordinary PCA comparisons.

Here, loading means a coefficient of a unit-length PC direction vector. Rows of the direction matrix represent variables and columns represent PCs and loading plots visualize variables. For the spherical Kendall construction, Gaussian mechanism, and interpretation of the coefficients, see DP Directions Methodology. For installation, input requirements, example datasets, and general budget allocation, see Getting Started.

1. Load the example data

We use gau, a synthetic dataset with 1,000 observations and 20 numeric variables. For more details, see the documentation for gau.

library(dppca)
data("gau", package = "dppca")

X <- gau
dim(X)
#> [1] 1000   20

2. Set the privacy budget and estimate PC directions

For dp_loading(), the entire privacy budget is allocated to private direction estimation, so no split argument is needed.

loading_privacy <- privacy_control(
  eps = 3,
  delta = 1e-4
)

Now, estimate the private directions. We set non_private = TRUE to include ordinary PCA directions for comparison.

set.seed(123)
loading_fit <- dp_loading(
  X,
  privacy = loading_privacy,
  non_private = TRUE
)

loading_fit
#> Differentially Private PC Directions
#> Data: 1000 observations x 20 variables
#> Components: 20 PC directions
#> Total privacy budget: eps = 3 ; delta = 1e-04 
#> Non-private reference: included

The main arguments of dp_loading() are:

Argument Description
X Numeric input matrix or data frame; rows are observations and columns are variables.
privacy The loading budget from privacy_control(); when only the total eps and delta are given, the full budget is used by dp_loading().
cpp.option Use the compiled direction estimator (TRUE, the default), or the R implementation (FALSE).
non_private Compute ordinary PCA directions for comparison when TRUE (default: FALSE).

The optional non-private reference is computed from the sample covariance matrix and is not a differentially private output.

3. Inspect the result

The fitted object stores private directions at the top level, an optional ordinary PCA reference under nonprivate, and input dimensions and privacy settings under metadata:

Result Description
loading_fit$directions Private PC direction coefficients (a p by p matrix).
loading_fit$nonprivate Ordinary PCA direction coefficients, or NULL if not requested.
loading_fit$metadata$n, $p Number of observations and numeric variables.
loading_fit$metadata$privacy Privacy parameters used for private direction estimation.

For this dataset, each direction matrix has 20 rows (variables) and 20 columns (PCs). Each coefficient indicates the weight of a variable in a particular PC direction.

dim(loading_fit$directions)
#> [1] 20 20
round(loading_fit$directions[1:6, 1:3, drop = FALSE], 3)
#>       PC1    PC2    PC3
#> V1 -0.420 -0.142 -0.055
#> V2 -0.448  0.075 -0.155
#> V3 -0.476 -0.009  0.102
#> V4 -0.432 -0.023  0.212
#> V5 -0.450  0.191 -0.063
#> V6  0.027  0.583  0.371

The summary() function reports the estimated directions and the privacy settings without rerunning the estimator. Its returned summary object retains the direction matrices under directions and nonprivate:

summary(loading_fit)
#> DP PC Directions Summary
#> Data: 1000 observations x 20 variables
#> Components: 20 PC directions
#> 
#> Private PC directions  (first 6 variables x first 4 PCs):
#>        PC1      PC2     PC3     PC4
#> V1 -0.4200 -0.14198 -0.0549  0.2245
#> V2 -0.4475  0.07532 -0.1552 -0.2335
#> V3 -0.4757 -0.00875  0.1022 -0.0361
#> V4 -0.4321 -0.02322  0.2124  0.0591
#> V5 -0.4500  0.19129 -0.0628 -0.0116
#> V6  0.0267  0.58292  0.3706  0.2886
#> 
#> Non-private PC directions  (first 6 variables x first 4 PCs):
#>        PC1      PC2     PC3     PC4
#> V1 -0.4403 -0.00758 -0.0684  0.2466
#> V2 -0.4554  0.04056  0.1534 -0.1527
#> V3 -0.4443  0.02665  0.0218 -0.3505
#> V4 -0.4367  0.01992 -0.1261  0.0381
#> V5 -0.4521  0.08331  0.0298  0.2080
#> V6  0.0408  0.49751 -0.4353  0.0211
#> 
#> Total privacy budget: eps = 3 ; delta = 1e-04 
#> Non-private reference: included 
#> Full direction matrices are available in $directions and $nonprivate.

4. Plot the result

The plot() method for a dp_loading object provides three display types: "vector", "point", and "heatmap".

Plot type What it shows
"vector" Each variable as an arrow from the origin to its coefficients on two selected PCs.
"point" The same pair of PC coefficients shown as a point instead of an arrow.
"heatmap" Signed direction coefficients for selected variables across one or more PCs.

4.1. Vector plot

A vector plot displays each variable using its coefficients on the two selected PC axes. For example, a variable with coefficients (v_{j1}, v_{j2}) appears as an arrow from (0,0) to (v_{j1}, v_{j2}). The arrows represent variables with their lengths reflecting the magnitudes of the selected direction coefficients.

vector_plot <- plot(
  loading_fit,
  type = "vector"
)

Comparison of non-private and differentially private PCA direction coefficients shown as variable arrows.

By default, the plot displays PC1 and PC2, selecting the ten variables with the largest coefficient norms. Since non_private = TRUE was used when fitting, the non-private reference appears on the left and the private result on the right.

4.2. Point and heatmap plots

A point plot displays the same coefficient pairs without arrows. A heatmap shows the sign and magnitude of the coefficients across selected PCs.

To display all three plot types together, simply omit type:

loading_plots <- plot(loading_fit)

Non-private and private PCA direction coefficients shown in vector, point, and heatmap plots.

By default, all three plot types display PC1 and PC2. When multiple types are selected, the non-private plots appear in the top row and the private plots in the bottom row.

In the heatmaps, negative coefficients are shown in blue and positive coefficients in red by default. Each private and non-private heatmap uses its own symmetric color scale, so comparisons should be based on the displayed values or colorbars rather than on color intensity alone.

The displayed PCs and variables can be customized using axes, heatmap_components, and variables, as described in the next section.

5. Select PCs and variables

The plot() method provides three arguments for selecting which PCs and variables to display:

Argument Description
axes Selects the two PCs displayed in vector and point plots (default: c(1, 2)).
heatmap_components Selects the PCs displayed in heatmaps (default: NULL, which uses the PCs specified by axes).
variables Selects the variables displayed in each plot (default: 10).

For example, the following code displays PC1 and PC3 in the vector plot and PC1 through PC4 in the heatmap, using the first eight variables:

plot(
  loading_fit,
  type = c("vector", "heatmap"),
  axes = c(1, 3),
  heatmap_components = 1:4,
  variables = 1:8  # Alternatively, select by name using paste0("V", 1:8)
)

The variables argument supports several ways to select variables:

Example Selection
variables = 6 Selects the six variables with the largest coefficient norms across the displayed PCs (default: variables = 10).
variables = "all" All variables.
variables = c("V1", "V3", "V5") Variables selected by name.
variables = c(1, 3, 5) Variables selected by column position.

When a single number is specified (e.g., variables = 6), variables are selected based on the magnitudes of their coefficients across the PCs displayed in each plot. Thus, vector and point plots select the same variables, while heatmaps may select different variables depending on heatmap_components. Variable selection is performed separately for private and non-private results. To display the same variables across all plots, specify their names or column positions explicitly.

6. Customize the appearance

Use loading_plot_control() to change colors, text sizes, and heatmap appearance. Specify plot type, PC axes, and variable selection directly in plot().

custom_plot <- plot(
  loading_fit,
  type = "point",
  axes = c(1, 2),
  variables = 1:8,
  plot_control = loading_plot_control(
    nonprivate_color = "black",
    private_color = "navy",
    point_size = 3,
    label_size = 3.5,
    base_size = 12,
    title_size = 13,
    private_title = "Private directions"
  )
)

Customized visualization of differentially private PCA direction coefficients.

Option Purpose
nonprivate_color, private_color Colors used for ordinary and private vectors or points.
arrow_size, point_size, label_size Sizes of arrows, points, and variable labels.
heatmap_values, heatmap_digits, heatmap_text_size Display and formatting of coefficient labels inside heatmap cells.
heatmap_colors Negative and positive heatmap colors, supplied as c(low = ..., high = ...).
base_size, title_size General text and title sizes.
nonprivate_title, private_title Titles of the ordinary and private panels.
xlab, ylab Axis labels for vector and point plots.

For complete argument descriptions, see ?dp_loading, ?plot.dp_loading, and ?loading_plot_control.

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