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Estimate principal component directions

Source: R/pc_dir.R
dp_pc_dir.Rd

This function returns the principal component directions used by the plotting and estimation functions in this package. By default, it computes the usual non-private directions from the sample covariance matrix. If g_dppca = TRUE, it computes private directions using the spherical-transformation version of g-DPPCA proposed by Kim and Jung (2025) : it adds Gaussian noise to the spherical Kendall matrix and then extracts its leading eigenvectors.

Usage

dp_pc_dir(
  X,
  k,
  center = TRUE,
  standardize = FALSE,
  g_dppca = FALSE,
  eps = NULL,
  delta = NULL,
  cpp.option = FALSE
)

Arguments

X

A numeric matrix or data frame. Rows correspond to observations and columns correspond to variables.

k

Number of principal component directions to return. Must be an integer between 1 and the number of columns in X.

center

A logical value indicating whether to center the columns of X before computing principal component directions. The default is TRUE.

standardize

A logical value indicating whether to scale the columns of X by their sample standard deviations after optional centering. The default is FALSE.

g_dppca

Whether to compute private principal component directions using the spherical Kendall mechanism based on the g-DPPCA method. If FALSE, the usual non-private directions are computed from the sample covariance matrix. The default is FALSE.

eps

Positive number defining the epsilon privacy parameter for private principal component directions. Required when g_dppca = TRUE.

delta

Number in (0, 1) defining the delta privacy parameter for private principal component directions. Required when g_dppca = TRUE.

cpp.option

A logical value reserved for a future C++ implementation of the spherical Kendall matrix. Currently only FALSE is supported.

Value

A numeric matrix with ncol(X) rows and k columns. The columns are orthonormal principal component directions.

Details

The non-private option computes leading eigenvectors of the sample covariance matrix of the preprocessed data. The private option is based on the spherical Kendall mechanism of Kim and Jung (2025) : it first forms the spherical Kendall matrix from pairwise normalized differences, adds symmetric Gaussian noise, and then computes leading eigenvectors. The final eigenvector matrix is re-orthonormalized by QR decomposition. For a detailed procedure and mathematical formulations, refer https://yejinjo0220.github.io/dppca/articles/pc_direction.

References

Kim M, Jung S (2025). “Robust and Differentially Private Principal Component Analysis.” Statistical Analysis and Data Mining: An ASA Data Science Journal, 18(6), e70053. doi:10.1002/sam.70053 .

Examples

data(gau, package = "dppca")

# Use a small subset to keep the example fast.
X <- gau[1:200, ]

# Non-private principal component directions
V <- dp_pc_dir(X, k = 2)
head(V)
#>              [,1]        [,2]
#> [1,] -0.276390432 -0.05512405
#> [2,] -0.008155916 -0.03392589
#> [3,] -0.433823614 -0.26807870
#> [4,] -0.007115806  0.43519507
#> [5,]  0.406349436  0.12107397
#> [6,]  0.108264920  0.07128666


# Private principal component directions
set.seed(123)
V_private <- dp_pc_dir(
  X,
  k = 2,
  g_dppca = TRUE,
  eps = 2,
  delta = 1e-3
)
head(V_private)
#>              [,1]       [,2]
#> [1,] -0.223077596  0.3476042
#> [2,] -0.050923111 -0.2384854
#> [3,] -0.018308885  0.1395241
#> [4,]  0.091084919  0.1563828
#> [5,]  0.009964607 -0.2978056
#> [6,]  0.161973207  0.1422014