rpca: Randomized principal component analysis

In rsvd: Randomized Singular Value Decomposition

Randomized principal component analysis

Description

Compute principal components of an optionally centered and scaled matrix using a randomized or deterministic singular value decomposition (SVD). Loadings describe the components, scores locate observations in those coordinates, and variances measure the contribution of each component.

Usage

rpca(
  A,
  k = NULL,
  center = TRUE,
  scale = TRUE,
  retx = TRUE,
  p = 10,
  q = 2,
  rand = TRUE
)

## S3 method for class 'rpca'
print(x, ...)

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

## S3 method for class 'rpca'
predict(object, newdata, ...)

Arguments

A	A numeric matrix or data frame with observations in rows and variables in columns. Dense real or complex data and real sparse matrices from Matrix are supported. Rows containing missing values are omitted with a warning; all remaining values must be finite. At least two rows must remain. In the dimensions below, m is the number of retained observations and n is the number of variables.
k	A positive integer giving the number of principal components to return. A fractional value of at least one is truncated toward zero. Values greater than min(m, n) are reduced to that limit. The default, NULL, returns min(m, n) components using the deterministic SVD. The randomized method is most useful when k is much smaller than both matrix dimensions.
center	Logical; subtract the training column means before computing the SVD (default TRUE). FALSE leaves the column means unchanged.
scale	Logical; divide each processed column by its scale factor (default TRUE). This gives unit sample variance for centered columns whose scale factors are at least 1e-8; see Details.
retx	Logical; return the fitted component scores in x (default TRUE). With FALSE, x is NULL; loadings and new-data prediction remain available.
p	A nonnegative integer oversampling parameter for rsvd (default 10). Ignored for the deterministic SVD.
q	A nonnegative integer number of power iterations for rsvd (default 2). Ignored for the deterministic SVD.
rand	Logical; use the randomized SVD when TRUE and k is supplied. With FALSE, or k = NULL, use the deterministic SVD. Requesting fewer components does not make the deterministic backend a partial SVD algorithm.
...	Additional arguments; currently ignored by the print, summary, and predict methods.
object, x	An object returned by rpca.
newdata	A numeric matrix, data frame, or real sparse Matrix containing new observations. Supply the same variables in the same column order as the training data, preferably with no missing values; see Details.

Details

Principal components describe data along orthogonal directions in the space of the variables. Each loading vector weights the original variables to specify one direction. Retaining k components gives a low-rank approximation to the processed data. For centered data, the leading components describe directions of greatest sample variance.

With rand = TRUE and a supplied k, rpca uses rsvd to approximate the dominant singular subspace. Finding a small subspace first reduces the size of the subsequent SVD, with the greatest savings for large matrices and small requested ranks. Increasing p enlarges the sampled subspace, and increasing q spends more work refining it to improve accuracy. Use set.seed before fitting for reproducible random draws.

Centering and scaling

Centering makes the components describe deviations from the training column means. Scaling adjusts the relative influence of variables with different units or magnitudes. Both operations are enabled by default. As in prcomp, center = FALSE leaves the column means unchanged and accepts either already-centered or uncentered data. Scaling remains enabled unless scale = FALSE; prcomp disables scaling by default.

With scale = TRUE, each column is divided by the square root of its sum of squared magnitudes divided by m-1, after any requested centering. For centered data this is the sample standard deviation. Scale factors below 1e-8 are replaced by one for compatibility, so constant and nearly constant columns are retained. Complex columns use squared magnitudes and therefore have real scale factors.

Interpreting variance

The total variance var includes every direction in the processed data, including those omitted from the approximation. It is computed as the squared Frobenius norm of the processed matrix divided by m-1. For uncentered data this quantity measures squared energy divided by m-1 and includes the contribution of the column means. The proportion attributed to a retained component is its eigvals entry divided by var, so the denominator stays the same as the retained rank changes. At least two complete observations are required; variance proportions are undefined when the processed matrix has zero total variance.

Using a fit

Fitted scores are constructed from the SVD as U_k D_k. Multiplying them by the conjugate transpose of the loadings reconstructs the approximation in processed variable coordinates. Undo any scaling and centering to express it in the original input coordinates.

predict(object, newdata) applies the stored training centering and scaling, then multiplies the processed observations by object$rotation. Reusing the training means and scale factors places new observations in the same coordinate system. Supply complete rows in the training column order; the method does not reorder columns by name or omit incomplete new rows automatically. Missing values generally propagate to the scores. For compatibility, NaN entries produced or retained during scaling are replaced by zero. Prediction also works when the fit was created with retx = FALSE. For a randomized fit, projecting the training data with predict can differ from the fitted scores in object$x because the randomized subspace omits part of the data, represented by the range-finder residual.

The print and summary methods describe a fit. plot.rpca draws the available scree and factor plots; the individual plotting helpers can also be called directly.

Sparse input

Subtracting nonzero column means would fill in many entries of a sparse matrix. With rand = TRUE and a supplied k, rpca instead incorporates centering and scaling into its matrix products for real sparse inputs. This avoids forming the entire centered matrix. Storage includes the sparse entries and dense working factors of width up to k + p, capped by the matrix dimensions. Loadings, fitted scores, and predicted scores are dense. The computation estimates ordinary PCA loadings and imposes no sparsity constraint on them.

With rand = FALSE or k = NULL, a sparse input is converted to a dense matrix for the deterministic SVD.

Value

rpca returns a list of class "rpca" with these components:

rotation

A dense n by k matrix of orthonormal loadings (exact or approximate right singular vectors of the processed data).

eigvals

A numeric vector of length k containing the retained squared singular values divided by m-1. These estimate the leading eigenvalues of the processed cross-product matrix divided by m-1, which is the sample covariance matrix for centered data.

sdev

A numeric vector of length k containing the square roots of eigvals.

var

The squared Frobenius norm of the processed matrix divided by m-1, including directions not returned. For centered data this is the total sample variance.

x

A dense m by k matrix of fitted scores, or NULL when retx = FALSE. Rows correspond to complete observations retained from the input.

center, scale

Numeric vectors containing the training column means and scale factors, respectively, or FALSE when the corresponding operation was disabled. Centers can be complex; scale factors are real.

The predict method returns a dense matrix with one row per new observation and one column per retained component. The summary method returns a numeric matrix with rows for explained variance, standard deviation, proportion of variance, and cumulative proportion. Its entries are rounded to three decimal places. The print method displays standard deviations, eigenvalues, and loadings, and invisibly returns the displayed loading matrix.

Note

Singular vectors are defined only up to sign (or a constant of modulus one for complex data). With repeated singular values, any orthonormal basis of the corresponding subspace is valid. Individual loading vectors can therefore differ between runs or implementations without changing the represented subspace.

Author(s)

N. Benjamin Erichson

References

• N. B. Erichson, S. Voronin, S. L. Brunton and J. N. Kutz. 2019. Randomized Matrix Decompositions Using R. Journal of Statistical Software, 89(11), 1–48. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v089.i11")}.

• N. Halko, P. G. Martinsson and J. A. Tropp. 2011. Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions. SIAM Review, 53(2), 217–288. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1137/090771806")}.

See Also

rsvd, prcomp, plot.rpca, ggscreeplot, ggindplot, ggcorplot, rrpca for a low-rank plus sparse decomposition.

Examples

# Fit two components and project observations held out from the fit.
training <- log(as.matrix(iris[1:145, 1:4]))
new_data <- log(as.matrix(iris[146:150, 1:4]))
set.seed(42)
fit <- rpca(training, k = 2)
summary(fit)
predict(fit, new_data)

# Reconstruct in input coordinates (here, log measurements).
reconstructed <- fit$x %*% Conj(t(fit$rotation))
reconstructed <- sweep(reconstructed, 2, fit$scale, "*")
reconstructed <- sweep(reconstructed, 2, fit$center, "+")

# k = NULL uses the deterministic SVD and returns all components.
full <- rpca(training)
length(full$sdev)

# Sparse input: implicit preprocessing, with training scores omitted.
A <- Matrix::bdiag(outer(1:12, 1:6), outer(1:12, 6:1))
set.seed(42)
sparse_fit <- rpca(A, k = 2, retx = FALSE)
predict(sparse_fit, A[1:2, , drop = FALSE])

if (requireNamespace("ggplot2", quietly = TRUE)) {
  plot(fit, type = "ratio", groups = iris$Species[1:145],
       ind_labels = FALSE, ellipse = FALSE)
}

rsvd documentation built on Nov. 2, 2026, 5:06 p.m.